DT2H2X---1.9.5.THM-Ref.s

%------------------------------------------------------------------------------
% File     : DT2H2X---1.9.5
% Problem  : ITP223^3 : 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 : n027.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 09:06:16 AM UTC 2026

% Result   : Theorem 37.37s 9.71s
% Output   : Refutation 37.37s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   11
%            Number of leaves      :   10
% Syntax   : Number of formulae    :   46 (  43 unt;   0 typ;   0 def)
%            Number of atoms       :   51 (  46 equ;   0 cnn)
%            Maximal formula atoms :    2 (   1 avg)
%            Number of connectives :  457 (   7   ~;   0   |;   3   &; 447   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    5 (   2 avg)
%            Number of types       :   84 (  83 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1254 (1251 usr;  60 con; 0-8 aty)
%            Number of variables   :   62 (  32   ^;  30   !;   0   ?;  62   :)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(type_def_30,type,
    set_Pr5085853215250843933omplex: $tType ).

thf(type_def_31,type,
    set_Pr6218003697084177305l_real: $tType ).

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

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

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

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

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

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

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

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

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

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

thf(type_def_42,type,
    set_Pr958786334691620121nt_int: $tType ).

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

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

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

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

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

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

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

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

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

thf(type_def_52,type,
    list_P8526636022914148096eger_o: $tType ).

thf(type_def_53,type,
    list_P7333126701944960589_nat_o: $tType ).

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

thf(type_def_55,type,
    list_P3521021558325789923at_int: $tType ).

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

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

thf(type_def_58,type,
    list_P1726324292696863441at_num: $tType ).

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

thf(type_def_60,type,
    list_P5647936690300460905T_VEBT: $tType ).

thf(type_def_61,type,
    produc8025551001238799321T_VEBT: $tType ).

thf(type_def_62,type,
    list_P3744719386663036955um_num: $tType ).

thf(type_def_63,type,
    list_P3126845725202233233VEBT_o: $tType ).

thf(type_def_64,type,
    produc334124729049499915VEBT_o: $tType ).

thf(type_def_65,type,
    list_P7037539587688870467BT_nat: $tType ).

thf(type_def_66,type,
    produc9072475918466114483BT_nat: $tType ).

thf(type_def_67,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(type_def_68,type,
    produc8243902056947475879T_VEBT: $tType ).

thf(type_def_69,type,
    list_P4002435161011370285od_o_o: $tType ).

thf(type_def_70,type,
    list_P3795440434834930179_o_int: $tType ).

thf(type_def_71,type,
    list_P7495141550334521929T_VEBT: $tType ).

thf(type_def_72,type,
    list_P5087981734274514673_int_o: $tType ).

thf(type_def_73,type,
    list_P5707943133018811711nt_int: $tType ).

thf(type_def_74,type,
    list_P7524865323317820941T_VEBT: $tType ).

thf(type_def_75,type,
    list_P4547456442757143711BT_int: $tType ).

thf(type_def_76,type,
    option4927543243414619207at_nat: $tType ).

thf(type_def_77,type,
    produc4411394909380815293omplex: $tType ).

thf(type_def_78,type,
    produc2422161461964618553l_real: $tType ).

thf(type_def_79,type,
    set_Pr1261947904930325089at_nat: $tType ).

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

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

thf(type_def_82,type,
    set_op4508134149509766951at_nat: $tType ).

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

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

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

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

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

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

thf(func_def_0,type,
    set_fi4554929511873752355omplex: $tType ).

thf(func_def_1,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(func_def_2,type,
    set_fi7789364187291644575l_real: $tType ).

thf(func_def_3,type,
    set_op4508134149509766951at_nat: $tType ).

thf(func_def_4,type,
    filter6041513312241820739omplex: $tType ).

thf(func_def_5,type,
    list_P7037539587688870467BT_nat: $tType ).

thf(func_def_6,type,
    list_P4547456442757143711BT_int: $tType ).

thf(func_def_7,type,
    list_P5647936690300460905T_VEBT: $tType ).

thf(func_def_8,type,
    list_P7524865323317820941T_VEBT: $tType ).

thf(func_def_9,type,
    produc8243902056947475879T_VEBT: $tType ).

thf(func_def_10,type,
    set_Pr5085853215250843933omplex: $tType ).

thf(func_def_11,type,
    produc8923325533196201883nteger: $tType ).

thf(func_def_12,type,
    list_P3126845725202233233VEBT_o: $tType ).

thf(func_def_13,type,
    list_P7495141550334521929T_VEBT: $tType ).

thf(func_def_14,type,
    filter2146258269922977983l_real: $tType ).

thf(func_def_15,type,
    list_P8526636022914148096eger_o: $tType ).

thf(func_def_16,type,
    option4927543243414619207at_nat: $tType ).

thf(func_def_17,type,
    set_Pr6218003697084177305l_real: $tType ).

thf(func_def_18,type,
    list_P3744719386663036955um_num: $tType ).

thf(func_def_19,type,
    list_P1726324292696863441at_num: $tType ).

thf(func_def_20,type,
    list_P6011104703257516679at_nat: $tType ).

thf(func_def_21,type,
    list_P3521021558325789923at_int: $tType ).

thf(func_def_22,type,
    list_P5707943133018811711nt_int: $tType ).

thf(func_def_23,type,
    produc9072475918466114483BT_nat: $tType ).

thf(func_def_24,type,
    produc8025551001238799321T_VEBT: $tType ).

thf(func_def_25,type,
    set_Pr1261947904930325089at_nat: $tType ).

thf(func_def_26,type,
    set_Pr958786334691620121nt_int: $tType ).

thf(func_def_27,type,
    produc4411394909380815293omplex: $tType ).

thf(func_def_28,type,
    list_P7333126701944960589_nat_o: $tType ).

thf(func_def_29,type,
    list_P5087981734274514673_int_o: $tType ).

thf(func_def_30,type,
    list_P3795440434834930179_o_int: $tType ).

thf(func_def_31,type,
    produc334124729049499915VEBT_o: $tType ).

thf(func_def_32,type,
    produc6271795597528267376eger_o: $tType ).

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

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

thf(func_def_35,type,
    produc2422161461964618553l_real: $tType ).

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

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

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

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

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

thf(func_def_41,type,
    list_P4002435161011370285od_o_o: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(func_def_83,type,
    archim2889992004027027881ng_rat: rat > int ).

thf(func_def_84,type,
    archim7802044766580827645g_real: real > int ).

thf(func_def_85,type,
    archim3151403230148437115or_rat: rat > int ).

thf(func_def_86,type,
    archim6058952711729229775r_real: real > int ).

thf(func_def_87,type,
    archimedean_frac_rat: rat > rat ).

thf(func_def_88,type,
    archim2898591450579166408c_real: real > real ).

thf(func_def_89,type,
    archim7778729529865785530nd_rat: rat > int ).

thf(func_def_90,type,
    archim8280529875227126926d_real: real > int ).

thf(func_def_91,type,
    binomial: nat > nat > nat ).

thf(func_def_92,type,
    gbinomial_complex: complex > nat > complex ).

thf(func_def_93,type,
    gbinomial_int: int > nat > int ).

thf(func_def_94,type,
    gbinomial_nat: nat > nat > nat ).

thf(func_def_95,type,
    gbinomial_rat: rat > nat > rat ).

thf(func_def_96,type,
    gbinomial_real: real > nat > real ).

thf(func_def_97,type,
    bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(func_def_98,type,
    bit_and_not_num: num > num > option_num ).

thf(func_def_99,type,
    bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_100,type,
    bit_concat_bit: nat > int > int > int ).

thf(func_def_101,type,
    bit_or_not_num_neg: num > num > num ).

thf(func_def_102,type,
    bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_103,type,
    bit_ri7632146776885996613nteger: code_integer > code_integer ).

thf(func_def_104,type,
    bit_ri7919022796975470100ot_int: int > int ).

thf(func_def_105,type,
    bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).

thf(func_def_106,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(func_def_107,type,
    bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).

thf(func_def_108,type,
    bit_se725231765392027082nd_int: int > int > int ).

thf(func_def_109,type,
    bit_se727722235901077358nd_nat: nat > nat > nat ).

thf(func_def_110,type,
    bit_se8568078237143864401it_int: nat > int > int ).

thf(func_def_111,type,
    bit_se8570568707652914677it_nat: nat > nat > nat ).

thf(func_def_112,type,
    bit_se1345352211410354436nteger: nat > code_integer > code_integer ).

thf(func_def_113,type,
    bit_se2159334234014336723it_int: nat > int > int ).

thf(func_def_114,type,
    bit_se2161824704523386999it_nat: nat > nat > nat ).

thf(func_def_115,type,
    bit_se2119862282449309892nteger: nat > code_integer ).

thf(func_def_116,type,
    bit_se2000444600071755411sk_int: nat > int ).

thf(func_def_117,type,
    bit_se2002935070580805687sk_nat: nat > nat ).

thf(func_def_118,type,
    bit_se1409905431419307370or_int: int > int > int ).

thf(func_def_119,type,
    bit_se1412395901928357646or_nat: nat > nat > nat ).

thf(func_def_120,type,
    bit_se7788150548672797655nteger: nat > code_integer > code_integer ).

thf(func_def_121,type,
    bit_se545348938243370406it_int: nat > int > int ).

thf(func_def_122,type,
    bit_se547839408752420682it_nat: nat > nat > nat ).

thf(func_def_123,type,
    bit_se2793503036327961859nteger: nat > code_integer > code_integer ).

thf(func_def_124,type,
    bit_se7879613467334960850it_int: nat > int > int ).

thf(func_def_125,type,
    bit_se7882103937844011126it_nat: nat > nat > nat ).

thf(func_def_126,type,
    bit_se1745604003318907178nteger: nat > code_integer > code_integer ).

thf(func_def_127,type,
    bit_se2923211474154528505it_int: nat > int > int ).

thf(func_def_128,type,
    bit_se2925701944663578781it_nat: nat > nat > nat ).

thf(func_def_129,type,
    bit_se8260200283734997820nteger: nat > code_integer > code_integer ).

thf(func_def_130,type,
    bit_se4203085406695923979it_int: nat > int > int ).

thf(func_def_131,type,
    bit_se4205575877204974255it_nat: nat > nat > nat ).

thf(func_def_132,type,
    bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).

thf(func_def_133,type,
    bit_se6526347334894502574or_int: int > int > int ).

thf(func_def_134,type,
    bit_se6528837805403552850or_nat: nat > nat > nat ).

thf(func_def_135,type,
    bit_se9216721137139052372nteger: code_integer > nat > $o ).

thf(func_def_136,type,
    bit_se1146084159140164899it_int: int > nat > $o ).

thf(func_def_137,type,
    bit_se1148574629649215175it_nat: nat > nat > $o ).

thf(func_def_138,type,
    bit_take_bit_num: nat > num > option_num ).

thf(func_def_139,type,
    bit_un7362597486090784418nd_num: num > num > option_num ).

thf(func_def_140,type,
    bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_141,type,
    bit_un2480387367778600638or_num: num > num > option_num ).

thf(func_def_142,type,
    bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_143,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

thf(func_def_144,type,
    code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_145,type,
    code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_146,type,
    code_int_of_integer: code_integer > int ).

thf(func_def_147,type,
    code_integer_of_int: int > code_integer ).

thf(func_def_148,type,
    code_integer_of_num: num > code_integer ).

thf(func_def_149,type,
    code_nat_of_integer: code_integer > nat ).

thf(func_def_150,type,
    code_negative: num > code_integer ).

thf(func_def_151,type,
    code_positive: num > code_integer ).

thf(func_def_152,type,
    code_Target_negative: num > int ).

thf(func_def_153,type,
    code_Target_positive: num > int ).

thf(func_def_154,type,
    code_T6385005292777649522of_nat: nat > int ).

thf(func_def_155,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

thf(func_def_156,type,
    comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).

thf(func_def_157,type,
    comple4887499456419720421f_real: set_real > real ).

thf(func_def_158,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

thf(func_def_159,type,
    complete_Sup_Sup_int: set_int > int ).

thf(func_def_160,type,
    complete_Sup_Sup_nat: set_nat > nat ).

thf(func_def_161,type,
    comple1385675409528146559p_real: set_real > real ).

thf(func_def_162,type,
    arg: complex > real ).

thf(func_def_163,type,
    cis: real > complex ).

thf(func_def_164,type,
    cnj: complex > complex ).

thf(func_def_165,type,
    complex2: real > real > complex ).

thf(func_def_166,type,
    im: complex > real ).

thf(func_def_167,type,
    re: complex > real ).

thf(func_def_168,type,
    csqrt: complex > complex ).

thf(func_def_169,type,
    imaginary_unit: complex ).

thf(func_def_170,type,
    differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).

thf(func_def_171,type,
    has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).

thf(func_def_172,type,
    has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).

thf(func_def_173,type,
    adjust_div: product_prod_int_int > int ).

thf(func_def_174,type,
    adjust_mod: int > int > int ).

thf(func_def_175,type,
    divmod_nat: nat > nat > product_prod_nat_nat ).

thf(func_def_176,type,
    eucl_rel_int: int > int > product_prod_int_int > $o ).

thf(func_def_177,type,
    unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).

thf(func_def_178,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

thf(func_def_179,type,
    unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).

thf(func_def_180,type,
    unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).

thf(func_def_181,type,
    unique5052692396658037445od_int: num > num > product_prod_int_int ).

thf(func_def_182,type,
    unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).

thf(func_def_183,type,
    unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_184,type,
    unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).

thf(func_def_185,type,
    unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_186,type,
    comm_s8582702949713902594nteger: code_integer > nat > code_integer ).

thf(func_def_187,type,
    comm_s2602460028002588243omplex: complex > nat > complex ).

thf(func_def_188,type,
    comm_s4660882817536571857er_int: int > nat > int ).

thf(func_def_189,type,
    comm_s4663373288045622133er_nat: nat > nat > nat ).

thf(func_def_190,type,
    comm_s4028243227959126397er_rat: rat > nat > rat ).

thf(func_def_191,type,
    comm_s7457072308508201937r_real: real > nat > real ).

thf(func_def_192,type,
    semiri3624122377584611663nteger: nat > code_integer ).

thf(func_def_193,type,
    semiri5044797733671781792omplex: nat > complex ).

thf(func_def_194,type,
    semiri1406184849735516958ct_int: nat > int ).

thf(func_def_195,type,
    semiri1408675320244567234ct_nat: nat > nat ).

thf(func_def_196,type,
    semiri773545260158071498ct_rat: nat > rat ).

thf(func_def_197,type,
    semiri2265585572941072030t_real: nat > real ).

thf(func_def_198,type,
    invers8013647133539491842omplex: complex > complex ).

thf(func_def_199,type,
    inverse_inverse_rat: rat > rat ).

thf(func_def_200,type,
    inverse_inverse_real: real > real ).

thf(func_def_201,type,
    at_bot_real: filter_real ).

thf(func_def_202,type,
    at_top_int: filter_int ).

thf(func_def_203,type,
    at_top_nat: filter_nat ).

thf(func_def_204,type,
    at_top_real: filter_real ).

thf(func_def_205,type,
    eventually_nat: ( nat > $o ) > filter_nat > $o ).

thf(func_def_206,type,
    eventually_real: ( real > $o ) > filter_real > $o ).

thf(func_def_207,type,
    filterlim_int_real: ( int > real ) > filter_real > filter_int > $o ).

thf(func_def_208,type,
    filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).

thf(func_def_209,type,
    filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).

thf(func_def_210,type,
    filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).

thf(func_def_211,type,
    filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).

thf(func_def_212,type,
    princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).

thf(func_def_213,type,
    princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).

thf(func_def_214,type,
    finite_card_o: set_o > nat ).

thf(func_def_215,type,
    finite_card_complex: set_complex > nat ).

thf(func_def_216,type,
    finite_card_int: set_int > nat ).

thf(func_def_217,type,
    finite_card_nat: set_nat > nat ).

thf(func_def_218,type,
    finite410649719033368117t_unit: set_Product_unit > nat ).

thf(func_def_219,type,
    finite_card_set_nat: set_set_nat > nat ).

thf(func_def_220,type,
    finite_card_char: set_char > nat ).

thf(func_def_221,type,
    finite3207457112153483333omplex: set_complex > $o ).

thf(func_def_222,type,
    finite_finite_int: set_int > $o ).

thf(func_def_223,type,
    finite_finite_nat: set_nat > $o ).

thf(func_def_224,type,
    bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).

thf(func_def_225,type,
    bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).

thf(func_def_226,type,
    bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).

thf(func_def_227,type,
    comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_228,type,
    comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_229,type,
    comp_C3531382070062128313er_num: ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ).

thf(func_def_230,type,
    comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).

thf(func_def_231,type,
    comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).

thf(func_def_232,type,
    comp_int_real_real: ( int > real ) > ( real > int ) > real > real ).

thf(func_def_233,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(func_def_234,type,
    comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).

thf(func_def_235,type,
    id_o: $o > $o ).

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

thf(func_def_237,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(func_def_238,type,
    inj_on_nat_char: ( nat > char ) > set_nat > $o ).

thf(func_def_239,type,
    inj_on_real_real: ( real > real ) > set_real > $o ).

thf(func_def_240,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_241,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_242,type,
    map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).

thf(func_def_243,type,
    map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).

thf(func_def_244,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_245,type,
    strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).

thf(func_def_246,type,
    the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).

thf(func_def_247,type,
    fun_is_measure_int: ( int > nat ) > $o ).

thf(func_def_248,type,
    gcd_Gcd_int: set_int > int ).

thf(func_def_249,type,
    gcd_Gcd_nat: set_nat > nat ).

thf(func_def_250,type,
    bezw: nat > nat > product_prod_int_int ).

thf(func_def_251,type,
    bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_252,type,
    gcd_gcd_Code_integer: code_integer > code_integer > code_integer ).

thf(func_def_253,type,
    gcd_gcd_int: int > int > int ).

thf(func_def_254,type,
    gcd_gcd_nat: nat > nat > nat ).

thf(func_def_255,type,
    gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_256,type,
    abs_abs_Code_integer: code_integer > code_integer ).

thf(func_def_257,type,
    abs_abs_complex: complex > complex ).

thf(func_def_258,type,
    abs_abs_int: int > int ).

thf(func_def_259,type,
    abs_abs_rat: rat > rat ).

thf(func_def_260,type,
    abs_abs_real: real > real ).

thf(func_def_261,type,
    minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).

thf(func_def_262,type,
    minus_minus_complex: complex > complex > complex ).

thf(func_def_263,type,
    minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_264,type,
    minus_minus_int: int > int > int ).

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

thf(func_def_266,type,
    minus_minus_rat: rat > rat > rat ).

thf(func_def_267,type,
    minus_minus_real: real > real > real ).

thf(func_def_268,type,
    minus_minus_set_o: set_o > set_o > set_o ).

thf(func_def_269,type,
    minus_811609699411566653omplex: set_complex > set_complex > set_complex ).

thf(func_def_270,type,
    minus_minus_set_int: set_int > set_int > set_int ).

thf(func_def_271,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(func_def_272,type,
    minus_minus_set_rat: set_rat > set_rat > set_rat ).

thf(func_def_273,type,
    minus_minus_set_real: set_real > set_real > set_real ).

thf(func_def_274,type,
    minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(func_def_275,type,
    minus_5127226145743854075T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_276,type,
    one_one_Code_integer: code_integer ).

thf(func_def_277,type,
    one_one_complex: complex ).

thf(func_def_278,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(func_def_279,type,
    one_one_int: int ).

thf(func_def_280,type,
    one_one_nat: nat ).

thf(func_def_281,type,
    one_one_rat: rat ).

thf(func_def_282,type,
    one_one_real: real ).

thf(func_def_283,type,
    plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).

thf(func_def_284,type,
    plus_plus_complex: complex > complex > complex ).

thf(func_def_285,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_286,type,
    plus_plus_int: int > int > int ).

thf(func_def_287,type,
    plus_plus_nat: nat > nat > nat ).

thf(func_def_288,type,
    plus_plus_num: num > num > num ).

thf(func_def_289,type,
    plus_plus_rat: rat > rat > rat ).

thf(func_def_290,type,
    plus_plus_real: real > real > real ).

thf(func_def_291,type,
    sgn_sgn_Code_integer: code_integer > code_integer ).

thf(func_def_292,type,
    sgn_sgn_complex: complex > complex ).

thf(func_def_293,type,
    sgn_sgn_int: int > int ).

thf(func_def_294,type,
    sgn_sgn_rat: rat > rat ).

thf(func_def_295,type,
    sgn_sgn_real: real > real ).

thf(func_def_296,type,
    times_3573771949741848930nteger: code_integer > code_integer > code_integer ).

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

thf(func_def_298,type,
    times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_299,type,
    times_times_int: int > int > int ).

thf(func_def_300,type,
    times_times_nat: nat > nat > nat ).

thf(func_def_301,type,
    times_times_num: num > num > num ).

thf(func_def_302,type,
    times_times_rat: rat > rat > rat ).

thf(func_def_303,type,
    times_times_real: real > real > real ).

thf(func_def_304,type,
    uminus1351360451143612070nteger: code_integer > code_integer ).

thf(func_def_305,type,
    uminus1482373934393186551omplex: complex > complex ).

thf(func_def_306,type,
    uminus_uminus_int: int > int ).

thf(func_def_307,type,
    uminus_uminus_rat: rat > rat ).

thf(func_def_308,type,
    uminus_uminus_real: real > real ).

thf(func_def_309,type,
    uminus_uminus_set_o: set_o > set_o ).

thf(func_def_310,type,
    uminus8566677241136511917omplex: set_complex > set_complex ).

thf(func_def_311,type,
    uminus1532241313380277803et_int: set_int > set_int ).

thf(func_def_312,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(func_def_313,type,
    uminus2201863774496077783et_rat: set_rat > set_rat ).

thf(func_def_314,type,
    uminus612125837232591019t_real: set_real > set_real ).

thf(func_def_315,type,
    uminus613421341184616069et_nat: set_set_nat > set_set_nat ).

thf(func_def_316,type,
    uminus8041839845116263051T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_317,type,
    zero_z3403309356797280102nteger: code_integer ).

thf(func_def_318,type,
    zero_zero_complex: complex ).

thf(func_def_319,type,
    zero_z5237406670263579293d_enat: extended_enat ).

thf(func_def_320,type,
    zero_zero_int: int ).

thf(func_def_321,type,
    zero_zero_nat: nat ).

thf(func_def_322,type,
    zero_zero_rat: rat ).

thf(func_def_323,type,
    zero_zero_real: real ).

thf(func_def_324,type,
    groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).

thf(func_def_325,type,
    groups4538972089207619220nt_int: ( int > int ) > set_int > int ).

thf(func_def_326,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(func_def_327,type,
    groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).

thf(func_def_328,type,
    groups1705073143266064639nt_int: ( int > int ) > set_int > int ).

thf(func_def_329,type,
    groups705719431365010083at_int: ( nat > int ) > set_nat > int ).

thf(func_def_330,type,
    groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).

thf(func_def_331,type,
    groups3417619833198082522nteger: ( $o > code_integer ) > code_integer > list_o > code_integer ).

thf(func_def_332,type,
    groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).

thf(func_def_333,type,
    groups9119017779487936845_o_nat: ( $o > nat ) > nat > list_o > nat ).

thf(func_def_334,type,
    the_int: ( int > $o ) > int ).

thf(func_def_335,type,
    the_real: ( real > $o ) > real ).

thf(func_def_336,type,
    if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).

thf(func_def_337,type,
    if_Code_integer: $o > code_integer > code_integer > code_integer ).

thf(func_def_338,type,
    if_complex: $o > complex > complex > complex ).

thf(func_def_339,type,
    if_int: $o > int > int > int ).

thf(func_def_340,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(func_def_341,type,
    if_nat: $o > nat > nat > nat ).

thf(func_def_342,type,
    if_option_num: $o > option_num > option_num > option_num ).

thf(func_def_343,type,
    if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).

thf(func_def_344,type,
    if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_345,type,
    if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).

thf(func_def_346,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_347,type,
    if_rat: $o > rat > rat > rat ).

thf(func_def_348,type,
    if_real: $o > real > real > real ).

thf(func_def_349,type,
    if_set_int: $o > set_int > set_int > set_int ).

thf(func_def_350,type,
    abs_Integ: product_prod_nat_nat > int ).

thf(func_def_351,type,
    rep_Integ: int > product_prod_nat_nat ).

thf(func_def_352,type,
    int_ge_less_than: int > set_Pr958786334691620121nt_int ).

thf(func_def_353,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

thf(func_def_354,type,
    nat2: int > nat ).

thf(func_def_355,type,
    power_int_real: real > int > real ).

thf(func_def_356,type,
    ring_11222124179247155820nteger: set_Code_integer ).

thf(func_def_357,type,
    ring_1_Ints_complex: set_complex ).

thf(func_def_358,type,
    ring_1_Ints_int: set_int ).

thf(func_def_359,type,
    ring_1_Ints_rat: set_rat ).

thf(func_def_360,type,
    ring_1_Ints_real: set_real ).

thf(func_def_361,type,
    ring_18347121197199848620nteger: int > code_integer ).

thf(func_def_362,type,
    ring_17405671764205052669omplex: int > complex ).

thf(func_def_363,type,
    ring_1_of_int_int: int > int ).

thf(func_def_364,type,
    ring_1_of_int_rat: int > rat ).

thf(func_def_365,type,
    ring_1_of_int_real: int > real ).

thf(func_def_366,type,
    semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(func_def_367,type,
    sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).

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

thf(func_def_369,type,
    sup_sup_nat: nat > nat > nat ).

thf(func_def_370,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(func_def_371,type,
    lattic1921953407002678535_Max_o: set_o > $o ).

thf(func_def_372,type,
    lattic8263393255366662781ax_int: set_int > int ).

thf(func_def_373,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(func_def_374,type,
    lattic4275903605611617917x_real: set_real > real ).

thf(func_def_375,type,
    bfun_nat_real: ( nat > real ) > filter_nat > $o ).

thf(func_def_376,type,
    at_infinity_real: filter_real ).

thf(func_def_377,type,
    append_int: list_int > list_int > list_int ).

thf(func_def_378,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(func_def_379,type,
    distinct_int: list_int > $o ).

thf(func_def_380,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

thf(func_def_381,type,
    cons_int: int > list_int > list_int ).

thf(func_def_382,type,
    cons_nat: nat > list_nat > list_nat ).

thf(func_def_383,type,
    nil_int: list_int ).

thf(func_def_384,type,
    nil_nat: list_nat ).

thf(func_def_385,type,
    set_o2: list_o > set_o ).

thf(func_def_386,type,
    set_complex2: list_complex > set_complex ).

thf(func_def_387,type,
    set_int2: list_int > set_int ).

thf(func_def_388,type,
    set_nat2: list_nat > set_nat ).

thf(func_def_389,type,
    set_real2: list_real > set_real ).

thf(func_def_390,type,
    set_set_nat2: list_set_nat > set_set_nat ).

thf(func_def_391,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_392,type,
    size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).

thf(func_def_393,type,
    nth_o: list_o > nat > $o ).

thf(func_def_394,type,
    nth_Code_integer: list_Code_integer > nat > code_integer ).

thf(func_def_395,type,
    nth_complex: list_complex > nat > complex ).

thf(func_def_396,type,
    nth_int: list_int > nat > int ).

thf(func_def_397,type,
    nth_nat: list_nat > nat > nat ).

thf(func_def_398,type,
    nth_num: list_num > nat > num ).

thf(func_def_399,type,
    nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).

thf(func_def_400,type,
    nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).

thf(func_def_401,type,
    nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).

thf(func_def_402,type,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).

thf(func_def_403,type,
    nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).

thf(func_def_404,type,
    nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).

thf(func_def_405,type,
    nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).

thf(func_def_406,type,
    nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).

thf(func_def_407,type,
    nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).

thf(func_def_408,type,
    nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).

thf(func_def_409,type,
    nth_real: list_real > nat > real ).

thf(func_def_410,type,
    nth_set_nat: list_set_nat > nat > set_nat ).

thf(func_def_411,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_412,type,
    product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

thf(func_def_413,type,
    product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(func_def_414,type,
    product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(func_def_415,type,
    produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).

thf(func_def_416,type,
    product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).

thf(func_def_417,type,
    product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(func_def_418,type,
    produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

thf(func_def_419,type,
    product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(func_def_420,type,
    product_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).

thf(func_def_421,type,
    product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(func_def_422,type,
    product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).

thf(func_def_423,type,
    produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(func_def_424,type,
    product_num_num: list_num > list_num > list_P3744719386663036955um_num ).

thf(func_def_425,type,
    product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(func_def_426,type,
    produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

thf(func_def_427,type,
    produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(func_def_428,type,
    produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(func_def_429,type,
    replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_430,type,
    upto: int > int > list_int ).

thf(func_def_431,type,
    upto_aux: int > int > list_int > list_int ).

thf(func_def_432,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(func_def_433,type,
    suc: nat > nat ).

thf(func_def_434,type,
    compow_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(func_def_435,type,
    case_nat_o: $o > ( nat > $o ) > nat > $o ).

thf(func_def_436,type,
    case_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(func_def_437,type,
    case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).

thf(func_def_438,type,
    pred: nat > nat ).

thf(func_def_439,type,
    semiri3842193898606819883omplex: set_complex ).

thf(func_def_440,type,
    semiring_1_Nats_int: set_int ).

thf(func_def_441,type,
    semiri4939895301339042750nteger: nat > code_integer ).

thf(func_def_442,type,
    semiri8010041392384452111omplex: nat > complex ).

thf(func_def_443,type,
    semiri1314217659103216013at_int: nat > int ).

thf(func_def_444,type,
    semiri1316708129612266289at_nat: nat > nat ).

thf(func_def_445,type,
    semiri681578069525770553at_rat: nat > rat ).

thf(func_def_446,type,
    semiri5074537144036343181t_real: nat > real ).

thf(func_def_447,type,
    size_size_list_o: list_o > nat ).

thf(func_def_448,type,
    size_s3445333598471063425nteger: list_Code_integer > nat ).

thf(func_def_449,type,
    size_s3451745648224563538omplex: list_complex > nat ).

thf(func_def_450,type,
    size_size_list_int: list_int > nat ).

thf(func_def_451,type,
    size_size_list_nat: list_nat > nat ).

thf(func_def_452,type,
    size_size_list_num: list_num > nat ).

thf(func_def_453,type,
    size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).

thf(func_def_454,type,
    size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).

thf(func_def_455,type,
    size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).

thf(func_def_456,type,
    size_s4246224855604898693_int_o: list_P5087981734274514673_int_o > nat ).

thf(func_def_457,type,
    size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).

thf(func_def_458,type,
    size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > nat ).

thf(func_def_459,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).

thf(func_def_460,type,
    size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).

thf(func_def_461,type,
    size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).

thf(func_def_462,type,
    size_size_list_real: list_real > nat ).

thf(func_def_463,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

thf(func_def_464,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

thf(func_def_465,type,
    size_size_num: num > nat ).

thf(func_def_466,type,
    size_size_option_num: option_num > nat ).

thf(func_def_467,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

thf(func_def_468,type,
    size_size_char: char > nat ).

thf(func_def_469,type,
    size_size_VEBT_VEBT: vEBT_VEBT > nat ).

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

thf(func_def_471,type,
    nat_list_encode_rel: list_nat > list_nat > $o ).

thf(func_def_472,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

thf(func_def_473,type,
    nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_474,type,
    nat_prod_encode: product_prod_nat_nat > nat ).

thf(func_def_475,type,
    nat_set_decode: nat > set_nat ).

thf(func_def_476,type,
    nat_set_encode: set_nat > nat ).

thf(func_def_477,type,
    nat_triangle: nat > nat ).

thf(func_def_478,type,
    root: nat > real > real ).

thf(func_def_479,type,
    sqrt: real > real ).

thf(func_def_480,type,
    bitM: num > num ).

thf(func_def_481,type,
    inc: num > num ).

thf(func_def_482,type,
    neg_nu8804712462038260780nteger: code_integer > code_integer ).

thf(func_def_483,type,
    neg_nu7009210354673126013omplex: complex > complex ).

thf(func_def_484,type,
    neg_numeral_dbl_int: int > int ).

thf(func_def_485,type,
    neg_numeral_dbl_rat: rat > rat ).

thf(func_def_486,type,
    neg_numeral_dbl_real: real > real ).

thf(func_def_487,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

thf(func_def_488,type,
    neg_nu6511756317524482435omplex: complex > complex ).

thf(func_def_489,type,
    neg_nu3811975205180677377ec_int: int > int ).

thf(func_def_490,type,
    neg_nu3179335615603231917ec_rat: rat > rat ).

thf(func_def_491,type,
    neg_nu6075765906172075777c_real: real > real ).

thf(func_def_492,type,
    neg_nu5831290666863070958nteger: code_integer > code_integer ).

thf(func_def_493,type,
    neg_nu8557863876264182079omplex: complex > complex ).

thf(func_def_494,type,
    neg_nu5851722552734809277nc_int: int > int ).

thf(func_def_495,type,
    neg_nu5219082963157363817nc_rat: rat > rat ).

thf(func_def_496,type,
    neg_nu8295874005876285629c_real: real > real ).

thf(func_def_497,type,
    neg_numeral_sub_int: num > num > int ).

thf(func_def_498,type,
    bit0: num > num ).

thf(func_def_499,type,
    bit1: num > num ).

thf(func_def_500,type,
    one: num ).

thf(func_def_501,type,
    case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).

thf(func_def_502,type,
    size_num: num > nat ).

thf(func_def_503,type,
    num_of_nat: nat > num ).

thf(func_def_504,type,
    numera6620942414471956472nteger: num > code_integer ).

thf(func_def_505,type,
    numera6690914467698888265omplex: num > complex ).

thf(func_def_506,type,
    numera1916890842035813515d_enat: num > extended_enat ).

thf(func_def_507,type,
    numeral_numeral_int: num > int ).

thf(func_def_508,type,
    numeral_numeral_nat: num > nat ).

thf(func_def_509,type,
    numeral_numeral_rat: num > rat ).

thf(func_def_510,type,
    numeral_numeral_real: num > real ).

thf(func_def_511,type,
    pow: num > num > num ).

thf(func_def_512,type,
    pred_numeral: num > nat ).

thf(func_def_513,type,
    sqr: num > num ).

thf(func_def_514,type,
    none_num: option_num ).

thf(func_def_515,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(func_def_516,type,
    some_num: num > option_num ).

thf(func_def_517,type,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

thf(func_def_518,type,
    case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).

thf(func_def_519,type,
    case_option_int_num: int > ( num > int ) > option_num > int ).

thf(func_def_520,type,
    case_option_num_num: num > ( num > num ) > option_num > num ).

thf(func_def_521,type,
    case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).

thf(func_def_522,type,
    map_option_num_num: ( num > num ) > option_num > option_num ).

thf(func_def_523,type,
    size_option_num: ( num > nat ) > option_num > nat ).

thf(func_def_524,type,
    size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).

thf(func_def_525,type,
    bot_bot_o_o: $o > $o ).

thf(func_def_526,type,
    bot_bot_complex_o: complex > $o ).

thf(func_def_527,type,
    bot_bot_int_o: int > $o ).

thf(func_def_528,type,
    bot_bot_nat_o: nat > $o ).

thf(func_def_529,type,
    bot_bot_real_o: real > $o ).

thf(func_def_530,type,
    bot_bot_set_nat_o: set_nat > $o ).

thf(func_def_531,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(func_def_532,type,
    bot_bot_nat: nat ).

thf(func_def_533,type,
    bot_bot_set_o: set_o ).

thf(func_def_534,type,
    bot_bot_set_complex: set_complex ).

thf(func_def_535,type,
    bot_bot_set_int: set_int ).

thf(func_def_536,type,
    bot_bot_set_nat: set_nat ).

thf(func_def_537,type,
    bot_bot_set_real: set_real ).

thf(func_def_538,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(func_def_539,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

thf(func_def_540,type,
    ord_le6747313008572928689nteger: code_integer > code_integer > $o ).

thf(func_def_541,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

thf(func_def_542,type,
    ord_less_int: int > int > $o ).

thf(func_def_543,type,
    ord_less_nat: nat > nat > $o ).

thf(func_def_544,type,
    ord_less_num: num > num > $o ).

thf(func_def_545,type,
    ord_less_rat: rat > rat > $o ).

thf(func_def_546,type,
    ord_less_real: real > real > $o ).

thf(func_def_547,type,
    ord_less_set_o: set_o > set_o > $o ).

thf(func_def_548,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

thf(func_def_549,type,
    ord_less_set_int: set_int > set_int > $o ).

thf(func_def_550,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(func_def_551,type,
    ord_less_set_real: set_real > set_real > $o ).

thf(func_def_552,type,
    ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).

thf(func_def_553,type,
    ord_le3480810397992357184T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(func_def_554,type,
    ord_le3102999989581377725nteger: code_integer > code_integer > $o ).

thf(func_def_555,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(func_def_556,type,
    ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).

thf(func_def_557,type,
    ord_le4104064031414453916r_real: filter_real > filter_real > $o ).

thf(func_def_558,type,
    ord_less_eq_int: int > int > $o ).

thf(func_def_559,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(func_def_560,type,
    ord_less_eq_num: num > num > $o ).

thf(func_def_561,type,
    ord_less_eq_rat: rat > rat > $o ).

thf(func_def_562,type,
    ord_less_eq_real: real > real > $o ).

thf(func_def_563,type,
    ord_less_eq_set_o: set_o > set_o > $o ).

thf(func_def_564,type,
    ord_le211207098394363844omplex: set_complex > set_complex > $o ).

thf(func_def_565,type,
    ord_less_eq_set_int: set_int > set_int > $o ).

thf(func_def_566,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(func_def_567,type,
    ord_less_eq_set_real: set_real > set_real > $o ).

thf(func_def_568,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(func_def_569,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(func_def_570,type,
    ord_max_Code_integer: code_integer > code_integer > code_integer ).

thf(func_def_571,type,
    ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_572,type,
    ord_max_int: int > int > int ).

thf(func_def_573,type,
    ord_max_nat: nat > nat > nat ).

thf(func_def_574,type,
    order_Greatest_nat: ( nat > $o ) > nat ).

thf(func_def_575,type,
    order_9091379641038594480t_real: ( nat > real ) > $o ).

thf(func_def_576,type,
    order_mono_nat_nat: ( nat > nat ) > $o ).

thf(func_def_577,type,
    order_mono_nat_real: ( nat > real ) > $o ).

thf(func_def_578,type,
    order_mono_real_real: ( real > real ) > $o ).

thf(func_def_579,type,
    order_5726023648592871131at_nat: ( nat > nat ) > $o ).

thf(func_def_580,type,
    order_7092887310737990675l_real: ( real > real ) > $o ).

thf(func_def_581,type,
    top_top_set_o: set_o ).

thf(func_def_582,type,
    top_top_set_int: set_int ).

thf(func_def_583,type,
    top_top_set_nat: set_nat ).

thf(func_def_584,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(func_def_585,type,
    top_top_set_real: set_real ).

thf(func_def_586,type,
    top_top_set_char: set_char ).

thf(func_def_587,type,
    power_8256067586552552935nteger: code_integer > nat > code_integer ).

thf(func_def_588,type,
    power_power_complex: complex > nat > complex ).

thf(func_def_589,type,
    power_power_int: int > nat > int ).

thf(func_def_590,type,
    power_power_nat: nat > nat > nat ).

thf(func_def_591,type,
    power_power_rat: rat > nat > rat ).

thf(func_def_592,type,
    power_power_real: real > nat > real ).

thf(func_def_593,type,
    produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).

thf(func_def_594,type,
    produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_595,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(func_def_596,type,
    product_Pair_nat_o: nat > $o > product_prod_nat_o ).

thf(func_def_597,type,
    product_Pair_nat_int: nat > int > product_prod_nat_int ).

thf(func_def_598,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(func_def_599,type,
    product_Pair_nat_num: nat > num > product_prod_nat_num ).

thf(func_def_600,type,
    produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).

thf(func_def_601,type,
    product_Pair_num_num: num > num > product_prod_num_num ).

thf(func_def_602,type,
    produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).

thf(func_def_603,type,
    produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).

thf(func_def_604,type,
    produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).

thf(func_def_605,type,
    produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_606,type,
    produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).

thf(func_def_607,type,
    produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).

thf(func_def_608,type,
    produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).

thf(func_def_609,type,
    produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_610,type,
    produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).

thf(func_def_611,type,
    produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(func_def_612,type,
    produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

thf(func_def_613,type,
    produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_614,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_615,type,
    produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_616,type,
    produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).

thf(func_def_617,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_618,type,
    produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).

thf(func_def_619,type,
    produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).

thf(func_def_620,type,
    produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).

thf(func_def_621,type,
    product_fst_int_int: product_prod_int_int > int ).

thf(func_def_622,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(func_def_623,type,
    product_snd_int_int: product_prod_int_int > int ).

thf(func_def_624,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(func_def_625,type,
    fract: int > int > rat ).

thf(func_def_626,type,
    frct: product_prod_int_int > rat ).

thf(func_def_627,type,
    field_5140801741446780682s_real: set_real ).

thf(func_def_628,type,
    normalize: product_prod_int_int > product_prod_int_int ).

thf(func_def_629,type,
    of_int: int > rat ).

thf(func_def_630,type,
    quotient_of: rat > product_prod_int_int ).

thf(func_def_631,type,
    real_V2521375963428798218omplex: set_complex ).

thf(func_def_632,type,
    real_V3694042436643373181omplex: complex > complex > real ).

thf(func_def_633,type,
    real_V975177566351809787t_real: real > real > real ).

thf(func_def_634,type,
    real_V1022390504157884413omplex: complex > real ).

thf(func_def_635,type,
    real_V7735802525324610683m_real: real > real ).

thf(func_def_636,type,
    real_V4546457046886955230omplex: real > complex ).

thf(func_def_637,type,
    real_V1803761363581548252l_real: real > real ).

thf(func_def_638,type,
    divide6298287555418463151nteger: code_integer > code_integer > code_integer ).

thf(func_def_639,type,
    divide1717551699836669952omplex: complex > complex > complex ).

thf(func_def_640,type,
    divide_divide_int: int > int > int ).

thf(func_def_641,type,
    divide_divide_nat: nat > nat > nat ).

thf(func_def_642,type,
    divide_divide_rat: rat > rat > rat ).

thf(func_def_643,type,
    divide_divide_real: real > real > real ).

thf(func_def_644,type,
    dvd_dvd_Code_integer: code_integer > code_integer > $o ).

thf(func_def_645,type,
    dvd_dvd_complex: complex > complex > $o ).

thf(func_def_646,type,
    dvd_dvd_int: int > int > $o ).

thf(func_def_647,type,
    dvd_dvd_nat: nat > nat > $o ).

thf(func_def_648,type,
    dvd_dvd_rat: rat > rat > $o ).

thf(func_def_649,type,
    dvd_dvd_real: real > real > $o ).

thf(func_def_650,type,
    modulo364778990260209775nteger: code_integer > code_integer > code_integer ).

thf(func_def_651,type,
    modulo_modulo_int: int > int > int ).

thf(func_def_652,type,
    modulo_modulo_nat: nat > nat > nat ).

thf(func_def_653,type,
    zero_n356916108424825756nteger: $o > code_integer ).

thf(func_def_654,type,
    zero_n1201886186963655149omplex: $o > complex ).

thf(func_def_655,type,
    zero_n2684676970156552555ol_int: $o > int ).

thf(func_def_656,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

thf(func_def_657,type,
    zero_n2052037380579107095ol_rat: $o > rat ).

thf(func_def_658,type,
    zero_n3304061248610475627l_real: $o > real ).

thf(func_def_659,type,
    suminf_real: ( nat > real ) > real ).

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

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

thf(func_def_662,type,
    collect_o: ( $o > $o ) > set_o ).

thf(func_def_663,type,
    collect_complex: ( complex > $o ) > set_complex ).

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

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

thf(func_def_666,type,
    collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).

thf(func_def_667,type,
    collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).

thf(func_def_668,type,
    collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).

thf(func_def_669,type,
    collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).

thf(func_def_670,type,
    collect_real: ( real > $o ) > set_real ).

thf(func_def_671,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

thf(func_def_672,type,
    pow_nat: set_nat > set_set_nat ).

thf(func_def_673,type,
    image_o_o: ( $o > $o ) > set_o > set_o ).

thf(func_def_674,type,
    image_o_int: ( $o > int ) > set_o > set_int ).

thf(func_def_675,type,
    image_o_nat: ( $o > nat ) > set_o > set_nat ).

thf(func_def_676,type,
    image_o_real: ( $o > real ) > set_o > set_real ).

thf(func_def_677,type,
    image_complex_o: ( complex > $o ) > set_complex > set_o ).

thf(func_def_678,type,
    image_1468599708987790691omplex: ( complex > complex ) > set_complex > set_complex ).

thf(func_def_679,type,
    image_complex_int: ( complex > int ) > set_complex > set_int ).

thf(func_def_680,type,
    image_complex_nat: ( complex > nat ) > set_complex > set_nat ).

thf(func_def_681,type,
    image_complex_real: ( complex > real ) > set_complex > set_real ).

thf(func_def_682,type,
    image_932796090930683071T_VEBT: ( complex > vEBT_VEBT ) > set_complex > set_VEBT_VEBT ).

thf(func_def_683,type,
    image_int_o: ( int > $o ) > set_int > set_o ).

thf(func_def_684,type,
    image_int_complex: ( int > complex ) > set_int > set_complex ).

thf(func_def_685,type,
    image_int_int: ( int > int ) > set_int > set_int ).

thf(func_def_686,type,
    image_int_nat: ( int > nat ) > set_int > set_nat ).

thf(func_def_687,type,
    image_int_real: ( int > real ) > set_int > set_real ).

thf(func_def_688,type,
    image_nat_o: ( nat > $o ) > set_nat > set_o ).

thf(func_def_689,type,
    image_nat_complex: ( nat > complex ) > set_nat > set_complex ).

thf(func_def_690,type,
    image_nat_int: ( nat > int ) > set_nat > set_int ).

thf(func_def_691,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(func_def_692,type,
    image_nat_real: ( nat > real ) > set_nat > set_real ).

thf(func_def_693,type,
    image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).

thf(func_def_694,type,
    image_nat_char: ( nat > char ) > set_nat > set_char ).

thf(func_def_695,type,
    image_nat_VEBT_VEBT: ( nat > vEBT_VEBT ) > set_nat > set_VEBT_VEBT ).

thf(func_def_696,type,
    image_num_option_num: ( num > option_num ) > set_num > set_option_num ).

thf(func_def_697,type,
    image_4198897800814241419at_nat: ( product_prod_nat_nat > option4927543243414619207at_nat ) > set_Pr1261947904930325089at_nat > set_op4508134149509766951at_nat ).

thf(func_def_698,type,
    image_rat_rat: ( rat > rat ) > set_rat > set_rat ).

thf(func_def_699,type,
    image_real_o: ( real > $o ) > set_real > set_o ).

thf(func_def_700,type,
    image_real_complex: ( real > complex ) > set_real > set_complex ).

thf(func_def_701,type,
    image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).

thf(func_def_702,type,
    image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).

thf(func_def_703,type,
    image_real_int: ( real > int ) > set_real > set_int ).

thf(func_def_704,type,
    image_real_nat: ( real > nat ) > set_real > set_nat ).

thf(func_def_705,type,
    image_real_real: ( real > real ) > set_real > set_real ).

thf(func_def_706,type,
    image_real_VEBT_VEBT: ( real > vEBT_VEBT ) > set_real > set_VEBT_VEBT ).

thf(func_def_707,type,
    image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).

thf(func_def_708,type,
    image_7998606247489673935omplex: ( set_complex > set_complex ) > set_set_complex > set_set_complex ).

thf(func_def_709,type,
    image_524474410958335435et_int: ( set_int > set_int ) > set_set_int > set_set_int ).

thf(func_def_710,type,
    image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).

thf(func_def_711,type,
    image_2436557299294012491t_real: ( set_real > set_real ) > set_set_real > set_set_real ).

thf(func_def_712,type,
    image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat ).

thf(func_def_713,type,
    image_1661326939266726661T_VEBT: ( set_VEBT_VEBT > set_VEBT_VEBT ) > set_set_VEBT_VEBT > set_set_VEBT_VEBT ).

thf(func_def_714,type,
    image_char_nat: ( char > nat ) > set_char > set_nat ).

thf(func_def_715,type,
    image_VEBT_VEBT_o: ( vEBT_VEBT > $o ) > set_VEBT_VEBT > set_o ).

thf(func_def_716,type,
    image_VEBT_VEBT_int: ( vEBT_VEBT > int ) > set_VEBT_VEBT > set_int ).

thf(func_def_717,type,
    image_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > set_nat ).

thf(func_def_718,type,
    image_VEBT_VEBT_real: ( vEBT_VEBT > real ) > set_VEBT_VEBT > set_real ).

thf(func_def_719,type,
    image_3375948659692109573T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_720,type,
    insert_o: $o > set_o > set_o ).

thf(func_def_721,type,
    insert_complex: complex > set_complex > set_complex ).

thf(func_def_722,type,
    insert_int: int > set_int > set_int ).

thf(func_def_723,type,
    insert_nat: nat > set_nat > set_nat ).

thf(func_def_724,type,
    insert_real: real > set_real > set_real ).

thf(func_def_725,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(func_def_726,type,
    insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_727,type,
    is_singleton_o: set_o > $o ).

thf(func_def_728,type,
    is_singleton_int: set_int > $o ).

thf(func_def_729,type,
    is_singleton_nat: set_nat > $o ).

thf(func_def_730,type,
    is_singleton_real: set_real > $o ).

thf(func_def_731,type,
    is_sin24926331636114728T_VEBT: set_VEBT_VEBT > $o ).

thf(func_def_732,type,
    the_elem_o: set_o > $o ).

thf(func_def_733,type,
    the_elem_int: set_int > int ).

thf(func_def_734,type,
    the_elem_nat: set_nat > nat ).

thf(func_def_735,type,
    the_elem_real: set_real > real ).

thf(func_def_736,type,
    the_elem_set_nat: set_set_nat > set_nat ).

thf(func_def_737,type,
    the_elem_VEBT_VEBT: set_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_738,type,
    set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).

thf(func_def_739,type,
    set_or1266510415728281911st_int: int > int > set_int ).

thf(func_def_740,type,
    set_or1269000886237332187st_nat: nat > nat > set_nat ).

thf(func_def_741,type,
    set_or1222579329274155063t_real: real > real > set_real ).

thf(func_def_742,type,
    set_or4662586982721622107an_int: int > int > set_int ).

thf(func_def_743,type,
    set_or4665077453230672383an_nat: nat > nat > set_nat ).

thf(func_def_744,type,
    set_ord_atLeast_nat: nat > set_nat ).

thf(func_def_745,type,
    set_ord_atLeast_real: real > set_real ).

thf(func_def_746,type,
    set_ord_atMost_int: int > set_int ).

thf(func_def_747,type,
    set_ord_atMost_nat: nat > set_nat ).

thf(func_def_748,type,
    set_or6656581121297822940st_int: int > int > set_int ).

thf(func_def_749,type,
    set_or6659071591806873216st_nat: nat > nat > set_nat ).

thf(func_def_750,type,
    set_or5832277885323065728an_int: int > int > set_int ).

thf(func_def_751,type,
    set_or5834768355832116004an_nat: nat > nat > set_nat ).

thf(func_def_752,type,
    set_or1633881224788618240n_real: real > real > set_real ).

thf(func_def_753,type,
    set_or1210151606488870762an_nat: nat > set_nat ).

thf(func_def_754,type,
    set_or5849166863359141190n_real: real > set_real ).

thf(func_def_755,type,
    set_ord_lessThan_int: int > set_int ).

thf(func_def_756,type,
    set_ord_lessThan_nat: nat > set_nat ).

thf(func_def_757,type,
    set_or5984915006950818249n_real: real > set_real ).

thf(func_def_758,type,
    ascii_of: char > char ).

thf(func_def_759,type,
    char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).

thf(func_def_760,type,
    size_char: char > nat ).

thf(func_def_761,type,
    comm_s629917340098488124ar_nat: char > nat ).

thf(func_def_762,type,
    integer_of_char: char > code_integer ).

thf(func_def_763,type,
    unique3096191561947761185of_nat: nat > char ).

thf(func_def_764,type,
    topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).

thf(func_def_765,type,
    topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).

thf(func_def_766,type,
    topolo6980174941875973593q_real: ( nat > real ) > $o ).

thf(func_def_767,type,
    topolo2177554685111907308n_real: real > set_real > filter_real ).

thf(func_def_768,type,
    topolo2815343760600316023s_real: real > filter_real ).

thf(func_def_769,type,
    topolo4055970368930404560y_real: ( nat > real ) > $o ).

thf(func_def_770,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

thf(func_def_771,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

thf(func_def_772,type,
    arccos: real > real ).

thf(func_def_773,type,
    arcosh_real: real > real ).

thf(func_def_774,type,
    arcsin: real > real ).

thf(func_def_775,type,
    arctan: real > real ).

thf(func_def_776,type,
    arsinh_real: real > real ).

thf(func_def_777,type,
    artanh_real: real > real ).

thf(func_def_778,type,
    cos_complex: complex > complex ).

thf(func_def_779,type,
    cos_real: real > real ).

thf(func_def_780,type,
    cos_coeff: nat > real ).

thf(func_def_781,type,
    cosh_complex: complex > complex ).

thf(func_def_782,type,
    cosh_real: real > real ).

thf(func_def_783,type,
    cot_complex: complex > complex ).

thf(func_def_784,type,
    cot_real: real > real ).

thf(func_def_785,type,
    diffs_real: ( nat > real ) > nat > real ).

thf(func_def_786,type,
    exp_complex: complex > complex ).

thf(func_def_787,type,
    exp_real: real > real ).

thf(func_def_788,type,
    ln_ln_real: real > real ).

thf(func_def_789,type,
    log: real > real > real ).

thf(func_def_790,type,
    pi: real ).

thf(func_def_791,type,
    powr_real: real > real > real ).

thf(func_def_792,type,
    sin_complex: complex > complex ).

thf(func_def_793,type,
    sin_real: real > real ).

thf(func_def_794,type,
    sin_coeff: nat > real ).

thf(func_def_795,type,
    sinh_complex: complex > complex ).

thf(func_def_796,type,
    sinh_real: real > real ).

thf(func_def_797,type,
    tan_complex: complex > complex ).

thf(func_def_798,type,
    tan_real: real > real ).

thf(func_def_799,type,
    tanh_complex: complex > complex ).

thf(func_def_800,type,
    tanh_real: real > real ).

thf(func_def_801,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_802,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_803,type,
    vEBT_Leaf: $o > $o > vEBT_VEBT ).

thf(func_def_804,type,
    vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_805,type,
    vEBT_size_VEBT: vEBT_VEBT > nat ).

thf(func_def_806,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).

thf(func_def_807,type,
    vEBT_VEBT_high: nat > nat > nat ).

thf(func_def_808,type,
    vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).

thf(func_def_809,type,
    vEBT_VEBT_low: nat > nat > nat ).

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

thf(func_def_811,type,
    vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_812,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).

thf(func_def_813,type,
    vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_814,type,
    vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).

thf(func_def_815,type,
    vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_816,type,
    vEBT_invar_vebt: vEBT_VEBT > nat > $o ).

thf(func_def_817,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(func_def_818,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

thf(func_def_819,type,
    vEBT_v4011308405150292612up_rel: nat > nat > $o ).

thf(func_def_820,type,
    vEBT_VEBT_height: vEBT_VEBT > nat ).

thf(func_def_821,type,
    vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_822,type,
    accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).

thf(func_def_823,type,
    accp_nat: ( nat > nat > $o ) > nat > $o ).

thf(func_def_824,type,
    accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(func_def_825,type,
    accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_826,type,
    accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).

thf(func_def_827,type,
    accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).

thf(func_def_828,type,
    accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

thf(func_def_829,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(func_def_830,type,
    fChoice_real: ( real > $o ) > real ).

thf(func_def_831,type,
    member_o: $o > set_o > $o ).

thf(func_def_832,type,
    member_Code_integer: code_integer > set_Code_integer > $o ).

thf(func_def_833,type,
    member_complex: complex > set_complex > $o ).

thf(func_def_834,type,
    member_int: int > set_int > $o ).

thf(func_def_835,type,
    member_nat: nat > set_nat > $o ).

thf(func_def_836,type,
    member_option_num: option_num > set_option_num > $o ).

thf(func_def_837,type,
    member3954567711264315760at_nat: option4927543243414619207at_nat > set_op4508134149509766951at_nat > $o ).

thf(func_def_838,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(func_def_839,type,
    member_rat: rat > set_rat > $o ).

thf(func_def_840,type,
    member_real: real > set_real > $o ).

thf(func_def_841,type,
    member_set_o: set_o > set_set_o > $o ).

thf(func_def_842,type,
    member_set_complex: set_complex > set_set_complex > $o ).

thf(func_def_843,type,
    member_set_int: set_int > set_set_int > $o ).

thf(func_def_844,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(func_def_845,type,
    member_set_real: set_real > set_set_real > $o ).

thf(func_def_846,type,
    member_set_set_nat: set_set_nat > set_set_set_nat > $o ).

thf(func_def_847,type,
    member_set_VEBT_VEBT: set_VEBT_VEBT > set_set_VEBT_VEBT > $o ).

thf(func_def_848,type,
    member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).

thf(func_def_849,type,
    deg: nat ).

thf(func_def_850,type,
    m: nat ).

thf(func_def_851,type,
    na: nat ).

thf(func_def_852,type,
    summary: vEBT_VEBT ).

thf(func_def_853,type,
    treeList: list_VEBT_VEBT ).

thf(func_def_872,type,
    sP0: real > real > real > $o ).

thf(func_def_873,type,
    sP1: num > real > real > $o ).

thf(func_def_874,type,
    sP2: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_875,type,
    sP3: real > real > real > $o ).

thf(func_def_876,type,
    sP4: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_877,type,
    sP5: real > real > real > $o ).

thf(func_def_878,type,
    sP6: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_879,type,
    sP7: real > num > real > $o ).

thf(func_def_880,type,
    sP8: ( int > $o ) > int > $o ).

thf(func_def_881,type,
    sP9: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_882,type,
    sP10: list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_883,type,
    sP11: num > real > real > $o ).

thf(func_def_884,type,
    sP12: set_VEBT_VEBT > set_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_885,type,
    sP13: set_VEBT_VEBT > set_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_886,type,
    sP14: ( int > $o ) > int > int > $o ).

thf(func_def_887,type,
    sP15: ( int > $o ) > int > int > $o ).

thf(func_def_888,type,
    sP16: nat > nat > nat > $o ).

thf(func_def_889,type,
    sP17: nat > nat > nat > $o ).

thf(func_def_890,type,
    sP18: nat > nat > nat > $o ).

thf(func_def_891,type,
    sP19: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_892,type,
    sP20: int > int > int > $o ).

thf(func_def_893,type,
    sP21: int > int > int > $o ).

thf(func_def_894,type,
    sP22: int > int > int > $o ).

thf(func_def_895,type,
    sP23: real > num > real > $o ).

thf(func_def_896,type,
    sP24: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_897,type,
    sP25: ( int > $o ) > $o ).

thf(func_def_898,type,
    sP26: real > real > real > $o ).

thf(func_def_899,type,
    sP27: real > real > real > $o ).

thf(func_def_900,type,
    sP28: real > num > real > $o ).

thf(func_def_901,type,
    sP29: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_902,type,
    sP30: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > $o ).

thf(func_def_903,type,
    sP31: real > real > real > $o ).

thf(func_def_904,type,
    sP32: nat > nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_905,type,
    sP33: list_VEBT_VEBT > vEBT_VEBT > nat > $o ).

thf(func_def_906,type,
    sP34: nat > nat > nat > list_VEBT_VEBT > nat > $o ).

thf(func_def_907,type,
    sP35: list_VEBT_VEBT > vEBT_VEBT > nat > $o ).

thf(func_def_908,type,
    sP36: nat > vEBT_VEBT > $o ).

thf(func_def_909,type,
    sP37: nat > vEBT_VEBT > $o ).

thf(func_def_910,type,
    sP38: vEBT_VEBT > nat > $o ).

thf(func_def_911,type,
    sP39: vEBT_VEBT > nat > $o ).

thf(func_def_912,type,
    sP40: real > num > real > $o ).

thf(func_def_913,type,
    sP41: real > real > ( real > real > $o ) > $o ).

thf(func_def_914,type,
    sP42: real > real > real > $o ).

thf(func_def_915,type,
    sP43: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > $o ).

thf(func_def_916,type,
    sP44: nat > ( nat > $o ) > nat > $o ).

thf(func_def_917,type,
    sP45: real > num > real > $o ).

thf(func_def_918,type,
    sP46: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_919,type,
    sP47: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > $o ).

thf(func_def_920,type,
    sP48: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > $o ).

thf(func_def_921,type,
    sP49: real > real > real > $o ).

thf(func_def_922,type,
    sP50: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_923,type,
    sP51: nat > vEBT_VEBT > list_VEBT_VEBT > $o ).

thf(func_def_924,type,
    sP52: num > real > real > $o ).

thf(func_def_925,type,
    sP53: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_926,type,
    sK54: ( real > real ) > real ).

thf(func_def_927,type,
    sK55: ( real > real ) > real ).

thf(func_def_928,type,
    sK56: nat > nat > nat ).

thf(func_def_929,type,
    sK57: real > ( int > $o ) > int ).

thf(func_def_930,type,
    sK58: nat > nat > nat ).

thf(func_def_931,type,
    sK59: nat > nat > nat ).

thf(func_def_932,type,
    sK60: nat > real > real ).

thf(func_def_933,type,
    sK61: real > real > real ).

thf(func_def_934,type,
    sK62: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_935,type,
    sK63: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_936,type,
    sK64: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_937,type,
    sK65: real > int ).

thf(func_def_938,type,
    sK66: nat > list_VEBT_VEBT ).

thf(func_def_939,type,
    sK67: real > nat ).

thf(func_def_940,type,
    sK68: nat > nat > nat ).

thf(func_def_941,type,
    sK69: vEBT_VEBT > list_VEBT_VEBT > nat ).

thf(func_def_942,type,
    sK70: nat > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_943,type,
    sK71: real > real ).

thf(func_def_944,type,
    sK72: int > int > int ).

thf(func_def_945,type,
    sK73: nat > nat ).

thf(func_def_946,type,
    sK74: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_947,type,
    sK75: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_948,type,
    sK76: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_949,type,
    sK77: real > real > nat ).

thf(func_def_950,type,
    sK78: ( real > real ) > real ).

thf(func_def_951,type,
    sK79: ( real > real ) > real ).

thf(func_def_952,type,
    sK80: list_VEBT_VEBT > ( vEBT_VEBT > $o ) > nat ).

thf(func_def_953,type,
    sK81: list_VEBT_VEBT > ( vEBT_VEBT > $o ) > vEBT_VEBT ).

thf(func_def_954,type,
    sK82: int > nat ).

thf(func_def_955,type,
    sK83: int > nat ).

thf(func_def_956,type,
    sK84: nat > nat > nat ).

thf(func_def_957,type,
    sK85: int > int ).

thf(func_def_958,type,
    sK86: int > ( int > $o ) > int ).

thf(func_def_959,type,
    sK87: int > ( int > $o ) > int ).

thf(func_def_960,type,
    sK88: ( nat > $o ) > nat ).

thf(func_def_961,type,
    sK89: ( nat > $o ) > nat ).

thf(func_def_962,type,
    sK90: real > real > real ).

thf(func_def_963,type,
    sK91: nat > nat > nat ).

thf(func_def_964,type,
    sK92: ( nat > $o ) > nat > nat ).

thf(func_def_965,type,
    sK93: ( nat > $o ) > nat > nat ).

thf(func_def_966,type,
    sK94: ( nat > $o ) > nat ).

thf(func_def_967,type,
    sK95: ( nat > nat > $o ) > nat ).

thf(func_def_968,type,
    sK96: ( nat > nat > $o ) > nat ).

thf(func_def_969,type,
    sK97: ( nat > nat > $o ) > nat ).

thf(func_def_970,type,
    sK98: ( nat > nat > $o ) > nat ).

thf(func_def_971,type,
    sK99: ( nat > nat > $o ) > nat ).

thf(func_def_972,type,
    sK100: ( int > $o ) > int > int ).

thf(func_def_973,type,
    sK101: nat > nat ).

thf(func_def_974,type,
    sK102: int > nat ).

thf(func_def_975,type,
    sK103: ( nat > $o ) > nat > nat ).

thf(func_def_976,type,
    sK104: ( nat > $o ) > nat > nat ).

thf(func_def_977,type,
    sK105: int > nat ).

thf(func_def_978,type,
    sK106: real > ( int > $o ) > int ).

thf(func_def_979,type,
    sK107: ( nat > nat > $o ) > nat ).

thf(func_def_980,type,
    sK108: ( nat > nat > $o ) > nat ).

thf(func_def_981,type,
    sK109: ( nat > nat > $o ) > nat ).

thf(func_def_982,type,
    sK110: ( nat > nat > $o ) > nat ).

thf(func_def_983,type,
    sK111: ( nat > nat > $o ) > nat ).

thf(func_def_984,type,
    sK112: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_985,type,
    sK113: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_986,type,
    sK114: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_987,type,
    sK115: real > real ).

thf(func_def_988,type,
    sK116: real > nat ).

thf(func_def_989,type,
    sK117: nat > ( nat > $o ) > nat > nat ).

thf(func_def_990,type,
    sK118: ( nat > nat ) > nat ).

thf(func_def_991,type,
    sK119: int > nat ).

thf(func_def_992,type,
    sK120: ( nat > $o ) > nat > nat ).

thf(func_def_993,type,
    sK121: nat > nat ).

thf(func_def_994,type,
    sK122: int > ( int > $o ) > int ).

thf(func_def_995,type,
    sK123: ( real > real ) > real ).

thf(func_def_996,type,
    sK124: ( real > real ) > real ).

thf(func_def_997,type,
    sK125: nat > nat ).

thf(func_def_998,type,
    sK126: int > int > int ).

thf(func_def_999,type,
    sK127: ( nat > nat > $o ) > nat ).

thf(func_def_1000,type,
    sK128: ( nat > nat > $o ) > nat ).

thf(func_def_1001,type,
    sK129: ( nat > nat > $o ) > nat ).

thf(func_def_1002,type,
    sK130: ( nat > nat > $o ) > nat ).

thf(func_def_1003,type,
    sK131: int > int ).

thf(func_def_1004,type,
    sK132: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1005,type,
    sK133: int > int ).

thf(func_def_1006,type,
    sK134: ( real > real > $o ) > real ).

thf(func_def_1007,type,
    sK135: ( real > real > $o ) > real ).

thf(func_def_1008,type,
    sK136: ( real > real > $o ) > real ).

thf(func_def_1009,type,
    sK137: ( real > real > $o ) > real ).

thf(func_def_1010,type,
    sK138: ( real > real > $o ) > real ).

thf(func_def_1011,type,
    sK139: ( int > $o ) > int > int ).

thf(func_def_1012,type,
    sK140: real > nat ).

thf(func_def_1013,type,
    sK141: real > real ).

thf(func_def_1014,type,
    sK142: nat > nat > nat ).

thf(func_def_1015,type,
    sK143: nat > nat > nat ).

thf(func_def_1016,type,
    sK144: nat > nat > nat ).

thf(func_def_1017,type,
    sK145: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1018,type,
    sK146: int > ( int > $o ) > int ).

thf(func_def_1019,type,
    sK147: int > ( int > $o ) > int ).

thf(func_def_1020,type,
    sK148: ( int > $o ) > int ).

thf(func_def_1021,type,
    sK149: ( int > $o ) > int > int ).

thf(func_def_1022,type,
    sK150: int > ( int > $o ) > int ).

thf(func_def_1023,type,
    sK151: nat > nat ).

thf(func_def_1024,type,
    sK152: nat > nat > nat ).

thf(func_def_1025,type,
    sK153: int > int ).

thf(func_def_1026,type,
    sK154: int > int ).

thf(func_def_1027,type,
    sK155: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1028,type,
    sK156: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1029,type,
    sK157: list_VEBT_VEBT > nat ).

thf(func_def_1030,type,
    sK158: nat > list_VEBT_VEBT > nat > nat > nat > nat ).

thf(func_def_1031,type,
    sK159: nat > list_VEBT_VEBT > nat > nat > nat > nat ).

thf(func_def_1032,type,
    sK160: nat > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1033,type,
    sK161: vEBT_VEBT > list_VEBT_VEBT > nat > nat ).

thf(func_def_1034,type,
    sK162: vEBT_VEBT > list_VEBT_VEBT > nat > nat ).

thf(func_def_1035,type,
    sK163: nat > nat ).

thf(func_def_1036,type,
    sK164: nat > nat > nat ).

thf(func_def_1037,type,
    sK165: ( nat > $o ) > nat ).

thf(func_def_1038,type,
    sK166: int > nat ).

thf(func_def_1039,type,
    sK167: ( list_VEBT_VEBT > $o ) > list_VEBT_VEBT ).

thf(func_def_1040,type,
    sK168: set_VEBT_VEBT > set_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_1041,type,
    sK169: real > real ).

thf(func_def_1042,type,
    sK170: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1043,type,
    sK171: ( int > $o ) > int > int > int ).

thf(func_def_1044,type,
    sK172: ( int > $o ) > int > int > int ).

thf(func_def_1045,type,
    sK173: ( int > $o ) > int > int > int ).

thf(func_def_1046,type,
    sK174: ( int > $o ) > int > int > int ).

thf(func_def_1047,type,
    sK175: ( nat > real ) > nat ).

thf(func_def_1048,type,
    sK176: ( nat > real ) > nat ).

thf(func_def_1049,type,
    sK177: nat > nat > nat ).

thf(func_def_1050,type,
    sK178: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1051,type,
    sK179: nat > ( nat > $o ) > nat ).

thf(func_def_1052,type,
    sK180: ( nat > $o ) > nat ).

thf(func_def_1053,type,
    sK181: ( int > int > $o ) > int ).

thf(func_def_1054,type,
    sK182: ( int > int > $o ) > int ).

thf(func_def_1055,type,
    sK183: ( int > int > $o ) > int ).

thf(func_def_1056,type,
    sK184: ( int > int > $o ) > int ).

thf(func_def_1057,type,
    sK185: nat > ( nat > $o ) > nat ).

thf(func_def_1058,type,
    sK186: nat > ( nat > $o ) > nat ).

thf(func_def_1059,type,
    sK187: ( nat > real ) > nat ).

thf(func_def_1060,type,
    sK188: ( nat > real ) > nat ).

thf(func_def_1061,type,
    sK189: nat > nat ).

thf(func_def_1062,type,
    sK190: ( nat > real ) > nat ).

thf(func_def_1063,type,
    sK191: ( nat > real ) > nat ).

thf(func_def_1064,type,
    sK192: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1065,type,
    sK193: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1066,type,
    sK194: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1067,type,
    sK195: int > int ).

thf(func_def_1068,type,
    sK196: ( nat > nat ) > nat ).

thf(func_def_1069,type,
    sK197: ( nat > nat ) > nat ).

thf(func_def_1070,type,
    sK198: real > int ).

thf(func_def_1071,type,
    sK199: real > nat > real ).

thf(func_def_1072,type,
    sK200: nat > nat > nat ).

thf(func_def_1073,type,
    sK201: nat > nat > nat ).

thf(func_def_1074,type,
    sK202: nat > nat > nat ).

thf(func_def_1075,type,
    sK203: nat > nat ).

thf(func_def_1076,type,
    sK204: ( nat > nat ) > nat ).

thf(func_def_1077,type,
    sK205: ( nat > nat ) > nat ).

thf(func_def_1078,type,
    sK206: set_VEBT_VEBT > vEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_1079,type,
    sK207: ( int > real ) > int ).

thf(func_def_1080,type,
    sK208: ( int > real ) > int ).

thf(func_def_1081,type,
    sK209: real > real ).

thf(func_def_1082,type,
    sK210: ( nat > $o ) > nat ).

thf(func_def_1083,type,
    sK211: real > real > real > real ).

thf(func_def_1084,type,
    sK212: ( nat > nat ) > nat ).

thf(func_def_1085,type,
    sK213: ( nat > $o ) > nat > nat ).

thf(func_def_1086,type,
    sK214: ( nat > $o ) > nat > nat ).

thf(func_def_1087,type,
    sK215: nat > nat > nat ).

thf(func_def_1088,type,
    sK216: nat > nat ).

thf(func_def_1089,type,
    sK217: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1090,type,
    sK218: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1091,type,
    sK219: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1092,type,
    sK220: int > ( int > $o ) > int ).

thf(func_def_1093,type,
    sK221: ( int > $o ) > int ).

thf(func_def_1094,type,
    sK222: ( int > $o ) > int ).

thf(func_def_1095,type,
    sK223: real > real ).

thf(func_def_1096,type,
    sK224: nat > nat ).

thf(func_def_1097,type,
    sK225: ( nat > nat > $o ) > nat ).

thf(func_def_1098,type,
    sK226: ( nat > nat > $o ) > nat ).

thf(func_def_1099,type,
    sK227: ( nat > nat > $o ) > nat ).

thf(func_def_1100,type,
    sK228: ( nat > nat > $o ) > nat ).

thf(func_def_1101,type,
    sK229: ( nat > $o ) > nat ).

thf(func_def_1102,type,
    sK230: nat > nat > nat ).

thf(func_def_1103,type,
    sK231: ( nat > int ) > nat ).

thf(func_def_1104,type,
    sK232: real > real ).

thf(func_def_1105,type,
    sK233: int > int ).

thf(func_def_1106,type,
    sK234: int > int > int ).

thf(func_def_1107,type,
    sK235: nat > nat ).

thf(func_def_1108,type,
    sK236: nat > nat ).

thf(func_def_1109,type,
    sK237: int > int ).

thf(func_def_1110,type,
    sK238: nat > nat > nat > nat ).

thf(func_def_1111,type,
    sK239: nat > nat > nat > nat ).

thf(func_def_1112,type,
    sK240: nat > nat ).

thf(func_def_1113,type,
    sK241: ( real > $o ) > real ).

thf(func_def_1114,type,
    sK242: ( real > $o ) > real ).

thf(func_def_1115,type,
    sK243: ( real > $o ) > nat ).

thf(func_def_1116,type,
    sK244: ( nat > real ) > nat ).

thf(func_def_1117,type,
    sK245: real > real > nat ).

thf(func_def_1118,type,
    sK246: real > real ).

thf(func_def_1119,type,
    sK247: nat > nat > nat ).

thf(func_def_1120,type,
    sK248: ( real > $o ) > nat ).

thf(func_def_1121,type,
    sK249: ( real > $o ) > real ).

thf(func_def_1122,type,
    sK250: ( real > $o ) > real ).

thf(func_def_1123,type,
    sK251: real > int ).

thf(func_def_1124,type,
    sK252: ( nat > $o ) > nat ).

thf(func_def_1125,type,
    sK253: nat > nat ).

thf(func_def_1126,type,
    sK254: ( nat > $o ) > int > nat ).

thf(func_def_1127,type,
    sK255: real > real > real > real ).

thf(func_def_1128,type,
    sK256: nat > nat ).

thf(func_def_1129,type,
    sK257: nat > nat > nat ).

thf(func_def_1130,type,
    sK258: nat > ( nat > $o ) > nat ).

thf(func_def_1131,type,
    sK259: nat > ( nat > $o ) > nat ).

thf(func_def_1132,type,
    sK260: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1133,type,
    sK261: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1134,type,
    sK262: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1135,type,
    sK263: int > int ).

thf(func_def_1136,type,
    sK264: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1137,type,
    sK265: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1138,type,
    sK266: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1139,type,
    sK267: int > int > int ).

thf(func_def_1140,type,
    sK268: nat > nat ).

thf(func_def_1141,type,
    sK269: real > real ).

thf(func_def_1142,type,
    sK270: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1143,type,
    sK271: vEBT_VEBT > nat ).

thf(func_def_1144,type,
    sK272: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1145,type,
    sK273: list_VEBT_VEBT > nat ).

thf(func_def_1146,type,
    sK274: int > ( int > $o ) > int ).

thf(func_def_1147,type,
    sK275: real > nat ).

thf(func_def_1148,type,
    sK276: ( int > real ) > int ).

thf(func_def_1149,type,
    sK277: ( int > real ) > int ).

thf(func_def_1150,type,
    sK278: real > real ).

thf(func_def_1151,type,
    sK279: int > int ).

thf(func_def_1152,type,
    sK280: vEBT_VEBT > option4927543243414619207at_nat ).

thf(func_def_1153,type,
    sK281: vEBT_VEBT > nat ).

thf(func_def_1154,type,
    sK282: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1155,type,
    sK283: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1156,type,
    sK284: vEBT_VEBT > $o ).

thf(func_def_1157,type,
    sK285: vEBT_VEBT > $o ).

thf(func_def_1158,type,
    sK286: nat > nat > nat ).

thf(func_def_1159,type,
    sK287: nat > nat ).

thf(func_def_1160,type,
    sK288: nat > nat ).

thf(func_def_1161,type,
    sK289: int > int ).

thf(func_def_1162,type,
    sK290: int > ( int > $o ) > int ).

thf(func_def_1163,type,
    sK291: ( real > int ) > real ).

thf(func_def_1164,type,
    sK292: ( real > int ) > real ).

thf(func_def_1165,type,
    sK293: int > int > int ).

thf(func_def_1166,type,
    sK294: nat > nat ).

thf(func_def_1167,type,
    sK295: list_VEBT_VEBT > ( vEBT_VEBT > $o ) > vEBT_VEBT ).

thf(func_def_1168,type,
    sK296: real > real > real ).

thf(func_def_1169,type,
    sK297: int > nat ).

thf(func_def_1170,type,
    sK298: int > nat ).

thf(func_def_1171,type,
    sK299: real > real ).

thf(func_def_1172,type,
    sK300: ( nat > nat > $o ) > nat ).

thf(func_def_1173,type,
    sK301: ( nat > nat > $o ) > nat ).

thf(func_def_1174,type,
    sK302: ( nat > nat > $o ) > nat ).

thf(func_def_1175,type,
    sK303: ( nat > nat > $o ) > nat ).

thf(func_def_1176,type,
    sK304: real > nat > real ).

thf(func_def_1177,type,
    sK305: nat > nat ).

thf(func_def_1178,type,
    sK306: ( real > real ) > real ).

thf(func_def_1179,type,
    sK307: ( real > real ) > real ).

thf(func_def_1180,type,
    sK308: real > real > nat ).

thf(func_def_1181,type,
    sK309: real > nat ).

thf(func_def_1182,type,
    sK310: real > real > real ).

thf(func_def_1183,type,
    sK311: real > nat ).

thf(func_def_1184,type,
    sK312: vEBT_VEBT > $o ).

thf(func_def_1185,type,
    sK313: vEBT_VEBT > $o ).

thf(func_def_1186,type,
    sK314: nat > vEBT_VEBT > option4927543243414619207at_nat ).

thf(func_def_1187,type,
    sK315: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1188,type,
    sK316: nat > vEBT_VEBT > nat ).

thf(func_def_1189,type,
    sK317: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1190,type,
    sK318: real > real ).

thf(func_def_1191,type,
    sK319: nat > nat > nat ).

thf(func_def_1192,type,
    sK320: vEBT_VEBT > nat > nat ).

thf(func_def_1193,type,
    sK321: vEBT_VEBT > nat > nat ).

thf(func_def_1194,type,
    sK322: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1195,type,
    sK323: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1196,type,
    sK324: vEBT_VEBT > nat > nat ).

thf(func_def_1197,type,
    sK325: vEBT_VEBT > nat > nat ).

thf(func_def_1198,type,
    sK326: vEBT_VEBT > nat > nat ).

thf(func_def_1199,type,
    sK327: vEBT_VEBT > nat > nat ).

thf(func_def_1200,type,
    sK328: vEBT_VEBT > nat > nat ).

thf(func_def_1201,type,
    sK329: vEBT_VEBT > nat > nat ).

thf(func_def_1202,type,
    sK330: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1203,type,
    sK331: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1204,type,
    sK332: nat > vEBT_VEBT > nat ).

thf(func_def_1205,type,
    sK333: nat > vEBT_VEBT > nat ).

thf(func_def_1206,type,
    sK334: nat > vEBT_VEBT > nat ).

thf(func_def_1207,type,
    sK335: nat > vEBT_VEBT > nat ).

thf(func_def_1208,type,
    sK336: nat > vEBT_VEBT > nat ).

thf(func_def_1209,type,
    sK337: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1210,type,
    sK338: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1211,type,
    sK339: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1212,type,
    sK340: nat > vEBT_VEBT > nat ).

thf(func_def_1213,type,
    sK341: nat > vEBT_VEBT > nat ).

thf(func_def_1214,type,
    sK342: nat > vEBT_VEBT > nat ).

thf(func_def_1215,type,
    sK343: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1216,type,
    sK344: nat > list_VEBT_VEBT > nat ).

thf(func_def_1217,type,
    sK345: nat > list_VEBT_VEBT > nat ).

thf(func_def_1218,type,
    sK346: vEBT_VEBT > $o ).

thf(func_def_1219,type,
    sK347: vEBT_VEBT > $o ).

thf(func_def_1220,type,
    sK348: nat > nat > nat ).

thf(func_def_1221,type,
    sK349: ( int > $o ) > nat ).

thf(func_def_1222,type,
    sK350: ( int > $o ) > nat ).

thf(func_def_1223,type,
    sK351: ( nat > $o ) > nat ).

thf(func_def_1224,type,
    sK352: real > int ).

thf(func_def_1225,type,
    sK353: int > int ).

thf(func_def_1226,type,
    sK354: real > real > real ).

thf(func_def_1227,type,
    sK355: ( nat > int ) > nat ).

thf(func_def_1228,type,
    sK356: ( nat > $o ) > nat > nat ).

thf(func_def_1229,type,
    sK357: ( nat > $o ) > nat > nat ).

thf(func_def_1230,type,
    sK358: ( real > int ) > real ).

thf(func_def_1231,type,
    sK359: ( real > int ) > real ).

thf(func_def_1232,type,
    sK360: ( real > real ) > real ).

thf(func_def_1233,type,
    sK361: ( real > real ) > real ).

thf(func_def_1234,type,
    sK362: ( nat > $o ) > nat > nat > nat ).

thf(func_def_1235,type,
    sK363: ( nat > $o ) > nat ).

thf(func_def_1236,type,
    sK364: nat > nat > nat ).

thf(func_def_1237,type,
    sK365: int > int ).

thf(func_def_1238,type,
    sK366: ( nat > $o ) > nat ).

thf(func_def_1239,type,
    sK367: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1240,type,
    sK368: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1241,type,
    sK369: ( real > nat ) > real ).

thf(func_def_1242,type,
    sK370: ( real > nat ) > real ).

thf(func_def_1243,type,
    sK371: ( nat > nat ) > nat ).

thf(func_def_1244,type,
    sK372: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1245,type,
    sK373: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1246,type,
    sK374: list_VEBT_VEBT > nat ).

thf(func_def_1247,type,
    sK375: vEBT_VEBT > nat ).

thf(func_def_1248,type,
    sK376: ( nat > int ) > nat ).

thf(func_def_1249,type,
    sK377: real > int ).

thf(func_def_1250,type,
    sK378: nat > nat ).

thf(func_def_1251,type,
    sK379: nat > nat ).

thf(func_def_1252,type,
    sK380: int > nat ).

thf(func_def_1253,type,
    sK381: int > nat ).

thf(func_def_1254,type,
    sK382: real > real > ( real > real > $o ) > real ).

thf(func_def_1255,type,
    sK383: real > real > real > ( real > real > $o ) > real ).

thf(func_def_1256,type,
    sK384: real > real > real > ( real > real > $o ) > real ).

thf(func_def_1257,type,
    sK385: ( real > real > $o ) > real ).

thf(func_def_1258,type,
    sK386: ( real > real > $o ) > real ).

thf(func_def_1259,type,
    sK387: ( real > real > $o ) > real ).

thf(func_def_1260,type,
    sK388: real > real ).

thf(func_def_1261,type,
    sK389: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1262,type,
    sK390: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1263,type,
    sK391: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1264,type,
    sK392: int > nat ).

thf(func_def_1265,type,
    sK393: nat > nat ).

thf(func_def_1266,type,
    sK394: nat > ( nat > $o ) > nat ).

thf(func_def_1267,type,
    sK395: ( nat > real ) > nat ).

thf(func_def_1268,type,
    sK396: int > int ).

thf(func_def_1269,type,
    sK397: real > real > nat ).

thf(func_def_1270,type,
    sK398: real > real ).

thf(func_def_1271,type,
    sK399: nat > ( nat > $o ) > nat > nat ).

thf(func_def_1272,type,
    sK400: nat > ( nat > $o ) > nat > nat ).

thf(func_def_1273,type,
    sK401: real > real > real ).

thf(func_def_1274,type,
    sK402: ( real > nat ) > real ).

thf(func_def_1275,type,
    sK403: ( real > nat ) > real ).

thf(func_def_1276,type,
    sK404: ( nat > nat > $o ) > nat ).

thf(func_def_1277,type,
    sK405: ( nat > nat > $o ) > nat ).

thf(func_def_1278,type,
    sK406: ( nat > nat > $o ) > nat ).

thf(func_def_1279,type,
    sK407: ( nat > nat > $o ) > nat ).

thf(func_def_1280,type,
    sK408: ( nat > nat > $o ) > nat ).

thf(func_def_1281,type,
    sK409: vEBT_VEBT > $o ).

thf(func_def_1282,type,
    sK410: vEBT_VEBT > $o ).

thf(func_def_1283,type,
    sK411: vEBT_VEBT > option4927543243414619207at_nat ).

thf(func_def_1284,type,
    sK412: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1285,type,
    sK413: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1286,type,
    sK414: vEBT_VEBT > nat ).

thf(func_def_1287,type,
    sK415: int > int > int ).

thf(func_def_1288,type,
    sK416: real > int ).

thf(func_def_1289,type,
    sK417: ( nat > $o ) > nat ).

thf(func_def_1290,type,
    sK418: int > nat ).

thf(func_def_1291,type,
    sK419: ( int > int > $o ) > int ).

thf(func_def_1292,type,
    sK420: ( int > int > $o ) > int ).

thf(func_def_1293,type,
    sK421: ( int > int > $o ) > int ).

thf(func_def_1294,type,
    sK422: ( int > int > $o ) > int ).

thf(func_def_1295,type,
    sK423: ( int > int > $o ) > int ).

thf(func_def_1296,type,
    sK424: ( real > nat ) > real ).

thf(func_def_1297,type,
    sK425: ( real > nat ) > real ).

thf(func_def_1298,type,
    sK426: ( real > int ) > real ).

thf(func_def_1299,type,
    sK427: ( real > int ) > real ).

thf(func_def_1300,type,
    sK428: real > int ).

thf(func_def_1301,type,
    sK429: int > nat ).

thf(func_def_1302,type,
    sK430: real > real ).

thf(func_def_1303,type,
    sK431: nat > ( nat > $o ) > nat ).

thf(func_def_1304,type,
    sK432: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1305,type,
    sK433: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1306,type,
    sK434: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1307,type,
    sK435: list_VEBT_VEBT > ( vEBT_VEBT > $o ) > nat ).

thf(func_def_1308,type,
    sK436: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1309,type,
    sK437: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1310,type,
    sK438: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1311,type,
    sK439: ( nat > int ) > nat ).

thf(func_def_1312,type,
    sK440: ( nat > $o ) > nat ).

thf(func_def_1313,type,
    sK441: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1314,type,
    sK442: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1315,type,
    sK443: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1316,type,
    sK444: ( nat > nat ) > nat ).

thf(func_def_1317,type,
    sK445: nat > nat ).

thf(func_def_1318,type,
    sK446: ( int > $o ) > int > int ).

thf(func_def_1319,type,
    sK447: ( int > $o ) > int > int ).

thf(func_def_1320,type,
    sK448: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1321,type,
    sK449: ( int > $o ) > int ).

thf(func_def_1322,type,
    sK450: ( nat > $o ) > nat ).

thf(func_def_1323,type,
    sK451: int > int ).

thf(func_def_1324,type,
    sK452: real > real > nat ).

thf(func_def_1325,type,
    sK453: real > real ).

thf(func_def_1326,type,
    sK454: ( nat > $o ) > nat > nat ).

thf(func_def_1327,type,
    sK455: ( nat > $o ) > nat ).

thf(func_def_1328,type,
    sK456: nat > nat > nat ).

thf(func_def_1329,type,
    sK457: int > nat ).

thf(func_def_1330,type,
    sK458: int > nat ).

thf(func_def_1331,type,
    sK459: ( real > real ) > real ).

thf(func_def_1332,type,
    sK460: ( real > real ) > real ).

thf(func_def_1333,type,
    sK461: ( int > real ) > int ).

thf(func_def_1334,type,
    sK462: ( int > real ) > int ).

thf(func_def_1335,type,
    sK463: nat > vEBT_VEBT > list_VEBT_VEBT > nat ).

thf(func_def_1336,type,
    sK464: nat > vEBT_VEBT > list_VEBT_VEBT > nat ).

thf(func_def_1337,type,
    sK465: nat > list_VEBT_VEBT > nat > nat > nat > nat ).

thf(func_def_1338,type,
    sK466: nat > list_VEBT_VEBT > nat > nat > nat > nat ).

thf(func_def_1339,type,
    sK467: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1340,type,
    sK468: list_VEBT_VEBT > nat ).

thf(func_def_1341,type,
    sK469: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1342,type,
    sK470: int > nat ).

thf(func_def_1343,type,
    sK471: int > nat ).

thf(func_def_1344,type,
    sK472: nat > ( nat > $o ) > nat ).

thf(func_def_1345,type,
    sK473: nat > ( nat > $o ) > nat ).

thf(func_def_1346,type,
    sK474: int > int ).

thf(func_def_1347,type,
    sK475: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_1348,type,
    sK476: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1349,type,
    sK477: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1350,type,
    sK478: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1352,type,
    ph480: 
      !>[X0: $tType] : X0 ).

thf(f40037,plain,
    $false,
    inference(subsumption_resolution,[status(thm)],[f40036,f39951]) ).

thf(f39951,plain,
    ! [X0: nat,X1: nat] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ X0 ) @ ( semiri1314217659103216013at_int @ X1 ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ X0 ) @ ( semiri1314217659103216013at_int @ X1 ) ) ) ) ),
    inference(beta_eta_normalization,[status(thm)],[f33249]) ).

thf(f33249,plain,
    ! [X0: nat,X1: nat] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ X0 ) @ ( semiri1314217659103216013at_int @ X1 ) )
      = ( semiri1314217659103216013at_int
        @ ( ^ [Y0: nat,Y1: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ Y1 ) ) )
          @ X0
          @ X1 ) ) ),
    inference(definition_unfolding,[status(thm)],[f28634,f32230]) ).

thf(f32230,plain,
    ( plus_plus_nat
    = ( ^ [Y0: nat,Y1: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ Y1 ) ) ) ) ),
    inference(cnf_transformation,[status(thm)],[f11142]) ).

thf(f11142,plain,
    ( plus_plus_nat
    = ( ^ [Y0: nat,Y1: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ Y1 ) ) ) ) ),
    inference(fool_elimination,[status(thm)],[f11141]) ).

thf(f11141,plain,
    ( ( ^ [X0: nat,X1: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ X0 ) @ ( semiri1314217659103216013at_int @ X1 ) ) ) )
    = plus_plus_nat ),
    inference(rectify,[status(thm)],[f9157]) ).

thf(f9157,axiom,
    ( ( ^ [X39: nat,X40: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ X39 ) @ ( semiri1314217659103216013at_int @ X40 ) ) ) )
    = plus_plus_nat ),
    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_9156_nat__plus__as__int) ).

thf(f28634,plain,
    ! [X0: nat,X1: nat] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ X0 ) @ ( semiri1314217659103216013at_int @ X1 ) )
      = ( semiri1314217659103216013at_int @ ( plus_plus_nat @ X0 @ X1 ) ) ),
    inference(cnf_transformation,[status(thm)],[f22808]) ).

thf(f22808,plain,
    ! [X0: nat,X1: nat] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ X0 ) @ ( semiri1314217659103216013at_int @ X1 ) )
      = ( semiri1314217659103216013at_int @ ( plus_plus_nat @ X0 @ X1 ) ) ),
    inference(rectify,[status(thm)],[f155]) ).

thf(f155,axiom,
    ! [X12: nat,X13: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ X12 @ X13 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ X12 ) @ ( semiri1314217659103216013at_int @ X13 ) ) ),
    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_154_int__ops_I5_J) ).

thf(f40036,plain,
    ( ( semiri1314217659103216013at_int @ ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ one_one_nat ) @ ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ summary @ ( set_VEBT_VEBT2 @ treeList ) ) ) ) ) ) ) )
   != ( plus_plus_int @ ( semiri1314217659103216013at_int @ one_one_nat ) @ ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ summary @ ( set_VEBT_VEBT2 @ treeList ) ) ) ) ) ) ),
    inference(forward_demodulation,[status(thm)],[f40035,f39818]) ).

thf(f39818,plain,
    ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ summary @ ( set_VEBT_VEBT2 @ treeList ) ) ) ) )
    = ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ na ) @ ( semiri1314217659103216013at_int @ one_one_nat ) ) ) ) ) ) ) ) ),
    inference(beta_eta_normalization,[status(thm)],[f36599]) ).

thf(f36599,plain,
    ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ summary @ ( set_VEBT_VEBT2 @ treeList ) ) ) ) )
    = ( ^ [Y0: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ Y0 ) ) )
      @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( semiri5074537144036343181t_real
          @ ( ^ [Y0: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ one_one_nat ) ) )
            @ na ) ) ) ) ),
    inference(definition_unfolding,[status(thm)],[f32612,f32511,f32629]) ).

thf(f32629,plain,
    ( m
    = ( ^ [Y0: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ one_one_nat ) ) )
      @ na ) ),
    inference(definition_unfolding,[status(thm)],[f30171,f32625]) ).

thf(f32625,plain,
    ( suc
    = ( ^ [Y0: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ one_one_nat ) ) ) ) ),
    inference(definition_unfolding,[status(thm)],[f29899,f28641]) ).

thf(f28641,plain,
    ( one_one_int
    = ( semiri1314217659103216013at_int @ one_one_nat ) ),
    inference(cnf_transformation,[status(thm)],[f137]) ).

thf(f137,axiom,
    ( one_one_int
    = ( semiri1314217659103216013at_int @ one_one_nat ) ),
    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_136_int__ops_I2_J) ).

thf(f29899,plain,
    ( suc
    = ( ^ [Y0: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ one_one_int ) ) ) ),
    inference(cnf_transformation,[status(thm)],[f19014]) ).

thf(f19014,plain,
    ( suc
    = ( ^ [Y0: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ one_one_int ) ) ) ),
    inference(fool_elimination,[status(thm)],[f19013]) ).

thf(f19013,plain,
    ( ( ^ [X0: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ X0 ) @ one_one_int ) ) )
    = suc ),
    inference(rectify,[status(thm)],[f7503]) ).

thf(f7503,axiom,
    ( ( ^ [X39: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ X39 ) @ one_one_int ) ) )
    = suc ),
    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_7502_Suc__as__int) ).

thf(f30171,plain,
    ( m
    = ( suc @ na ) ),
    inference(cnf_transformation,[status(thm)],[f19921]) ).

thf(f19921,plain,
    ( ( m
      = ( suc @ na ) )
    & ( ( ord_less_eq_nat @ one_one_nat @ na )
      = $true ) ),
    inference(fool_elimination,[status(thm)],[f19920]) ).

thf(f19920,plain,
    ( ( ord_less_eq_nat @ one_one_nat @ na )
    & ( m
      = ( suc @ na ) ) ),
    inference(rectify,[status(thm)],[f320]) ).

thf(f320,axiom,
    ( ( ord_less_eq_nat @ one_one_nat @ na )
    & ( m
      = ( suc @ na ) ) ),
    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_319__C00_C) ).

thf(f32511,plain,
    ( archim7802044766580827645g_real
    = ( ^ [Y0: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ Y0 ) ) ) ) ),
    inference(cnf_transformation,[status(thm)],[f19665]) ).

thf(f19665,plain,
    ( archim7802044766580827645g_real
    = ( ^ [Y0: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ Y0 ) ) ) ) ),
    inference(fool_elimination,[status(thm)],[f19664]) ).

thf(f19664,plain,
    ( archim7802044766580827645g_real
    = ( ^ [X0: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X0 ) ) ) ) ),
    inference(rectify,[status(thm)],[f5036]) ).

thf(f5036,axiom,
    ( archim7802044766580827645g_real
    = ( ^ [X22: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X22 ) ) ) ) ),
    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_5035_ceiling__def) ).

thf(f32612,plain,
    ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) )
    = ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ summary @ ( set_VEBT_VEBT2 @ treeList ) ) ) ) ) ),
    inference(cnf_transformation,[status(thm)],[f3]) ).

thf(f3,axiom,
    ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) )
    = ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ summary @ ( set_VEBT_VEBT2 @ treeList ) ) ) ) ) ),
    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_2__C3_C) ).

thf(f40035,plain,
    ( ( semiri1314217659103216013at_int @ ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ one_one_nat ) @ ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ summary @ ( set_VEBT_VEBT2 @ treeList ) ) ) ) ) ) ) )
   != ( plus_plus_int @ ( semiri1314217659103216013at_int @ one_one_nat ) @ ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ na ) @ ( semiri1314217659103216013at_int @ one_one_nat ) ) ) ) ) ) ) ) ) ),
    inference(forward_demodulation,[status(thm)],[f40034,f38868]) ).

thf(f38868,plain,
    ! [X2: nat,X3: list_VEBT_VEBT,X0: vEBT_VEBT,X1: option4927543243414619207at_nat] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ X1 @ X2 @ X3 @ X0 ) )
      = ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ one_one_nat ) @ ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ X3 ) ) ) ) ) ) ) ),
    inference(beta_eta_normalization,[status(thm)],[f36003]) ).

thf(f36003,plain,
    ! [X2: nat,X3: list_VEBT_VEBT,X0: vEBT_VEBT,X1: option4927543243414619207at_nat] :
      ( ( ^ [Y0: nat,Y1: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ Y1 ) ) )
        @ one_one_nat
        @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ X3 ) ) ) ) )
      = ( vEBT_VEBT_height @ ( vEBT_Node @ X1 @ X2 @ X3 @ X0 ) ) ),
    inference(definition_unfolding,[status(thm)],[f31903,f32230]) ).

thf(f31903,plain,
    ! [X2: nat,X3: list_VEBT_VEBT,X0: vEBT_VEBT,X1: option4927543243414619207at_nat] :
      ( ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ X3 ) ) ) ) )
      = ( vEBT_VEBT_height @ ( vEBT_Node @ X1 @ X2 @ X3 @ X0 ) ) ),
    inference(cnf_transformation,[status(thm)],[f23897]) ).

thf(f23897,plain,
    ! [X0: vEBT_VEBT,X1: option4927543243414619207at_nat,X2: nat,X3: list_VEBT_VEBT] :
      ( ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ X3 ) ) ) ) )
      = ( vEBT_VEBT_height @ ( vEBT_Node @ X1 @ X2 @ X3 @ X0 ) ) ),
    inference(rectify,[status(thm)],[f91]) ).

thf(f91,axiom,
    ! [X11: vEBT_VEBT,X8: option4927543243414619207at_nat,X9: nat,X10: list_VEBT_VEBT] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ X8 @ X9 @ X10 @ X11 ) )
      = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ X11 @ ( set_VEBT_VEBT2 @ X10 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_90_VEBT__internal_Oheight_Osimps_I2_J) ).

thf(f40034,plain,
    ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ one_one_nat ) @ ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ na ) @ ( semiri1314217659103216013at_int @ one_one_nat ) ) ) ) ) ) ) ) )
   != ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ na ) @ ( semiri1314217659103216013at_int @ ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ na ) @ ( semiri1314217659103216013at_int @ one_one_nat ) ) ) ) ) ) @ treeList @ summary ) ) ) ),
    inference(beta_eta_normalization,[status(thm)],[f36014]) ).

thf(f36014,plain,
    ( ( semiri1314217659103216013at_int
      @ ( vEBT_VEBT_height
        @ ( vEBT_Node @ none_P5556105721700978146at_nat
          @ ( ^ [Y0: nat,Y1: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ Y1 ) ) )
            @ na
            @ ( ^ [Y0: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ one_one_nat ) ) )
              @ na ) )
          @ treeList
          @ summary ) ) )
   != ( plus_plus_int @ ( semiri1314217659103216013at_int @ one_one_nat )
      @ ( ^ [Y0: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ Y0 ) ) )
        @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( semiri5074537144036343181t_real
            @ ( ^ [Y0: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ one_one_nat ) ) )
              @ na ) ) ) ) ) ),
    inference(definition_unfolding,[status(thm)],[f31918,f32630,f28641,f32511,f32629]) ).

thf(f32630,plain,
    ( deg
    = ( ^ [Y0: nat,Y1: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ Y1 ) ) )
      @ na
      @ ( ^ [Y0: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ Y0 ) @ ( semiri1314217659103216013at_int @ one_one_nat ) ) )
        @ na ) ) ),
    inference(definition_unfolding,[status(thm)],[f31916,f32230,f32629]) ).

thf(f31916,plain,
    ( deg
    = ( plus_plus_nat @ na @ m ) ),
    inference(cnf_transformation,[status(thm)],[f86]) ).

thf(f86,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ),
    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_85__C3_Ohyps_C_I4_J) ).

thf(f31918,plain,
    ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ ( vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary ) ) )
   != ( plus_plus_int @ one_one_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ) ),
    inference(cnf_transformation,[status(thm)],[f22710]) ).

thf(f22710,plain,
    ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ ( vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary ) ) )
   != ( plus_plus_int @ one_one_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ) ),
    inference(flattening,[status(thm)],[f10243]) ).

thf(f10243,negated_conjecture,
    ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ ( vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary ) ) )
   != ( plus_plus_int @ one_one_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ) ),
    inference(negated_conjecture,[status(cth)],[f10242]) ).

thf(f10242,conjecture,
    ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ ( vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary ) ) )
    = ( plus_plus_int @ one_one_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ) ),
    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',conj_0) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13  % Problem  : ITP223^3 : 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 : n027.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 08:42:52 EST 2026
% 0.16/0.36  % CPUTime  : 
% 0.16/0.36  Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.24/0.48  ---- Original DTF file ---
% 0.24/0.48  thf(spec,logic,$$dhol).
% 0.24/0.48  %------------------------------------------------------------------------------
% 0.24/0.48  % File     : ITP223^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.24/0.48  % Domain   : Interactive Theorem Proving
% 0.24/0.48  % Problem  : Sledgehammer problem VEBT_Height 00096_003879
% 0.24/0.48  % Version  : [Des22] axioms.
% 0.24/0.48  % English  :
% 0.24/0.48  
% 0.24/0.48  % Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.24/0.48  %          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.24/0.48  % Source   : [Des22]
% 0.24/0.48  % Names    : 0064_VEBT_Height_00096_003879 [Des22]
% 0.24/0.48  
% 0.24/0.48  % Status   : Theorem
% 0.24/0.48  % Rating   : ? v9.2.1
% 0.24/0.48  % Syntax   : Number of formulae    : 11096 (6303 unt; 854 typ;   0 def)
% 0.24/0.48  %            Number of atoms       : 25787 (11916 equ;   0 cnn)
% 0.24/0.48  %            Maximal formula atoms :   71 (   2 avg)
% 0.24/0.48  %            Number of connectives : 97747 (2534   ~; 484   |;1438   &;84971   @)
% 0.24/0.48  %                                         (   0 <=>;8320  =>;   0  <=;   0 <~>)
% 0.24/0.48  %            Maximal formula depth :   39 (   5 avg)
% 0.24/0.48  %            Number of types       :   84 (  83 usr)
% 0.24/0.48  %            Number of type conns  : 2308 (2308   >;   0   *;   0   +;   0  <<)
% 0.24/0.48  %            Number of symbols     :  774 ( 771 usr;  60 con; 0-8 aty)
% 0.24/0.48  %            Number of variables   : 22069 (1269   ^;20210   !; 590   ?;22069   :)
% 0.24/0.48  % SPC      : TH0_THM_EQU_NAR
% 0.24/0.48  
% 0.24/0.48  % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.24/0.48  %            from the van Emde Boas Trees session in the Archive of Formal
% 0.24/0.48  %            proofs - 
% 0.24/0.48  %            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.24/0.48  %            2022-02-17 18:14:03.703
% 0.24/0.48  % Bugfixes : Added ()s around negated sides of equations.
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% 0.24/0.48  thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
% 0.24/0.48      sup_sup_int: int > int > int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 0.24/0.48      sup_sup_nat: nat > nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.24/0.48      sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001_Eo,type,
% 0.24/0.48      lattic1921953407002678535_Max_o: set_o > $o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
% 0.24/0.48      lattic8263393255366662781ax_int: set_int > int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 0.24/0.48      lattic8265883725875713057ax_nat: set_nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Real__Oreal,type,
% 0.24/0.48      lattic4275903605611617917x_real: set_real > real ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.24/0.48      bfun_nat_real: ( nat > real ) > filter_nat > $o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
% 0.24/0.48      at_infinity_real: filter_real ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 0.24/0.48      append_int: list_int > list_int > list_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 0.24/0.48      append_nat: list_nat > list_nat > list_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Odistinct_001t__Int__Oint,type,
% 0.24/0.48      distinct_int: list_int > $o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 0.24/0.48      linord2614967742042102400et_nat: set_nat > list_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 0.24/0.48      cons_int: int > list_int > list_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 0.24/0.48      cons_nat: nat > list_nat > list_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 0.24/0.48      nil_int: list_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 0.24/0.48      nil_nat: list_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_Oset_001_Eo,type,
% 0.24/0.48      set_o2: list_o > set_o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 0.24/0.48      set_complex2: list_complex > set_complex ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 0.24/0.48      set_int2: list_int > set_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 0.24/0.48      set_nat2: list_nat > set_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 0.24/0.48      set_real2: list_real > set_real ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.24/0.48      set_set_nat2: list_set_nat > set_set_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 0.24/0.48      set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 0.24/0.48      size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001_Eo,type,
% 0.24/0.48      nth_o: list_o > nat > $o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
% 0.24/0.48      nth_Code_integer: list_Code_integer > nat > code_integer ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 0.24/0.48      nth_complex: list_complex > nat > complex ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Int__Oint,type,
% 0.24/0.48      nth_int: list_int > nat > int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.24/0.48      nth_nat: list_nat > nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Num__Onum,type,
% 0.24/0.48      nth_num: list_num > nat > num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.24/0.48      nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
% 0.24/0.48      nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
% 0.24/0.48      nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.24/0.48      nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 0.24/0.48      nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.24/0.48      nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.24/0.48      nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 0.24/0.48      nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.24/0.48      nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.24/0.48      nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 0.24/0.48      nth_real: list_real > nat > real ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.24/0.48      nth_set_nat: list_set_nat > nat > set_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 0.24/0.48      nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 0.24/0.48      product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 0.24/0.48      product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 0.24/0.48      product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
% 0.24/0.48      produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
% 0.24/0.48      product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
% 0.24/0.48      product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 0.24/0.48      produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 0.24/0.48      product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
% 0.24/0.48      product_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.24/0.48      product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
% 0.24/0.48      product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 0.24/0.48      produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
% 0.24/0.48      product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 0.24/0.48      product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
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% 0.24/0.48  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 0.24/0.48      produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.24/0.48      produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.24/0.48      produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 0.24/0.48      replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oupto,type,
% 0.24/0.48      upto: int > int > list_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oupto__aux,type,
% 0.24/0.48      upto_aux: int > int > list_int > list_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_List_Oupto__rel,type,
% 0.24/0.48      upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_OSuc,type,
% 0.24/0.48      suc: nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.24/0.48      compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 0.24/0.48      case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 0.24/0.48      case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.24/0.48      case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Onat_Opred,type,
% 0.24/0.48      pred: nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Complex__Ocomplex,type,
% 0.24/0.48      semiri3842193898606819883omplex: set_complex ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
% 0.24/0.48      semiring_1_Nats_int: set_int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 0.24/0.48      semiri4939895301339042750nteger: nat > code_integer ).
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% 0.24/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 0.24/0.48      semiri8010041392384452111omplex: nat > complex ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.24/0.48      semiri1314217659103216013at_int: nat > int ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.24/0.48      semiri1316708129612266289at_nat: nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.24/0.48      semiri5074537144036343181t_real: nat > real ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 0.24/0.48      size_size_list_o: list_o > nat ).
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
% 0.24/0.48      size_s3445333598471063425nteger: list_Code_integer > nat ).
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.24/0.48      size_s3451745648224563538omplex: list_complex > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 0.24/0.48      size_size_list_int: list_int > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.24/0.48      size_size_list_nat: list_nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 0.24/0.48      size_size_list_num: list_num > nat ).
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 0.24/0.48      size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 0.24/0.48      size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.24/0.48      size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_M_Eo_J_J,type,
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.24/0.48      size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
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% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 0.24/0.48      size_size_list_real: list_real > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.24/0.48      size_s3254054031482475050et_nat: list_set_nat > nat ).
% 0.24/0.48  
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% 0.24/0.48      size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 0.24/0.48      size_size_num: num > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
% 0.24/0.48      size_size_char: char > nat ).
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% 0.24/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
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% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat__Bijection_Olist__encode,type,
% 0.24/0.48      nat_list_encode: list_nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 0.24/0.48      nat_list_encode_rel: list_nat > list_nat > $o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.24/0.48      nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.24/0.48      nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 0.24/0.48      nat_prod_encode: product_prod_nat_nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat__Bijection_Oset__decode,type,
% 0.24/0.48      nat_set_decode: nat > set_nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.24/0.48      nat_set_encode: set_nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.24/0.48      nat_triangle: nat > nat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_NthRoot_Oroot,type,
% 0.24/0.48      root: nat > real > real ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_NthRoot_Osqrt,type,
% 0.24/0.48      sqrt: real > real ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Num_OBitM,type,
% 0.24/0.48      bitM: num > num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Num_Oinc,type,
% 0.24/0.48      inc: num > num ).
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% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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% 0.24/0.48      neg_nu7009210354673126013omplex: complex > complex ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
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% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 0.24/0.48      neg_numeral_dbl_rat: rat > rat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 0.24/0.48      neg_numeral_dbl_real: real > real ).
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% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
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% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 0.24/0.48      neg_nu3179335615603231917ec_rat: rat > rat ).
% 0.24/0.48  
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% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 0.24/0.48      neg_nu8557863876264182079omplex: complex > complex ).
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% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
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% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 0.24/0.48      neg_nu5219082963157363817nc_rat: rat > rat ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 0.24/0.48      neg_nu8295874005876285629c_real: real > real ).
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% 0.24/0.48  thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 0.24/0.48      neg_numeral_sub_int: num > num > int ).
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% 0.24/0.48  thf(sy_c_Num_Onum_OBit0,type,
% 0.24/0.48      bit0: num > num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Num_Onum_OBit1,type,
% 0.24/0.48      bit1: num > num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Num_Onum_OOne,type,
% 0.24/0.48      one: num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.24/0.48      case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 0.24/0.48  
% 0.24/0.48  thf(sy_c_Num_Onum_Osize__num,type,
% 0.24/0.48      size_num: num > nat ).
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% 0.24/0.48      num_of_nat: nat > num ).
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% 0.24/0.48  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
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% 0.24/0.48  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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% 0.24/0.48      pow: num > num > num ).
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% 0.24/0.49      pred_nat: set_Pr1261947904930325089at_nat ).
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% 0.24/0.49      member_option_num: option_num > set_option_num > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.24/0.49      member3954567711264315760at_nat: option4927543243414619207at_nat > set_op4508134149509766951at_nat > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.24/0.49      member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Rat__Orat,type,
% 0.24/0.49      member_rat: rat > set_rat > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Real__Oreal,type,
% 0.24/0.49      member_real: real > set_real > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
% 0.24/0.49      member_set_o: set_o > set_set_o > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.24/0.49      member_set_complex: set_complex > set_set_complex > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 0.24/0.49      member_set_int: set_int > set_set_int > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.24/0.49      member_set_nat: set_nat > set_set_nat > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.24/0.49      member_set_real: set_real > set_set_real > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.24/0.49      member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 0.24/0.49      member_set_VEBT_VEBT: set_VEBT_VEBT > set_set_VEBT_VEBT > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 0.24/0.49      member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_v_deg____,type,
% 0.24/0.49      deg: nat ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_v_m____,type,
% 0.24/0.49      m: nat ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_v_na____,type,
% 0.24/0.49      na: nat ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_v_summary____,type,
% 0.24/0.49      summary: vEBT_VEBT ).
% 0.24/0.49  
% 0.24/0.49  thf(sy_v_treeList____,type,
% 0.24/0.49      treeList: list_VEBT_VEBT ).
% 0.24/0.49  
% 0.24/0.49  % Relevant facts (10211)
% 0.24/0.49  thf(fact_0__C3_OIH_C_I2_J,axiom,
% 0.24/0.49      ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ summary ) )
% 0.24/0.49      = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % "3.IH"(2)
% 0.24/0.49  thf(fact_1__C3_Ohyps_C_I1_J,axiom,
% 0.24/0.49      vEBT_invar_vebt @ summary @ m ).
% 0.24/0.49  
% 0.24/0.49  % "3.hyps"(1)
% 0.24/0.49  thf(fact_2__C3_C,axiom,
% 0.24/0.49      ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ summary @ ( set_VEBT_VEBT2 @ treeList ) ) ) ) )
% 0.24/0.49      = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % "3"
% 0.24/0.49  thf(fact_3__C3_Ohyps_C_I5_J,axiom,
% 0.24/0.49      ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ summary @ X_1 ) ).
% 0.24/0.49  
% 0.24/0.49  % "3.hyps"(5)
% 0.24/0.49  thf(fact_4_ceiling__add__one,axiom,
% 0.24/0.49      ! [X: rat] :
% 0.24/0.49        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
% 0.24/0.49        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_add_one
% 0.24/0.49  thf(fact_5_ceiling__add__one,axiom,
% 0.24/0.49      ! [X: real] :
% 0.24/0.49        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
% 0.24/0.49        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_add_one
% 0.24/0.49  thf(fact_6_one__add__one,axiom,
% 0.24/0.49      ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 0.24/0.49      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_add_one
% 0.24/0.49  thf(fact_7_one__add__one,axiom,
% 0.24/0.49      ( ( plus_plus_real @ one_one_real @ one_one_real )
% 0.24/0.49      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_add_one
% 0.24/0.49  thf(fact_8_one__add__one,axiom,
% 0.24/0.49      ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 0.24/0.49      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_add_one
% 0.24/0.49  thf(fact_9_one__add__one,axiom,
% 0.24/0.49      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.24/0.49      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_add_one
% 0.24/0.49  thf(fact_10_one__add__one,axiom,
% 0.24/0.49      ( ( plus_plus_int @ one_one_int @ one_one_int )
% 0.24/0.49      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_add_one
% 0.24/0.49  thf(fact_11_ceiling__add__numeral,axiom,
% 0.24/0.49      ! [X: real,V: num] :
% 0.24/0.49        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 0.24/0.49        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_add_numeral
% 0.24/0.49  thf(fact_12_ceiling__add__numeral,axiom,
% 0.24/0.49      ! [X: rat,V: num] :
% 0.24/0.49        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 0.24/0.49        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_add_numeral
% 0.24/0.49  thf(fact_13_numeral__plus__one,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 0.24/0.49        = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_plus_one
% 0.24/0.49  thf(fact_14_numeral__plus__one,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.24/0.49        = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_plus_one
% 0.24/0.49  thf(fact_15_numeral__plus__one,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.24/0.49        = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_plus_one
% 0.24/0.49  thf(fact_16_numeral__plus__one,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.24/0.49        = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_plus_one
% 0.24/0.49  thf(fact_17_numeral__plus__one,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.24/0.49        = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_plus_one
% 0.24/0.49  thf(fact_18_one__plus__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 0.24/0.49        = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_plus_numeral
% 0.24/0.49  thf(fact_19_one__plus__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.24/0.49        = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_plus_numeral
% 0.24/0.49  thf(fact_20_one__plus__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.24/0.49        = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_plus_numeral
% 0.24/0.49  thf(fact_21_one__plus__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_plus_numeral
% 0.24/0.49  thf(fact_22_one__plus__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.24/0.49        = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_plus_numeral
% 0.24/0.49  thf(fact_23_ceiling__of__nat,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.49        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_of_nat
% 0.24/0.49  thf(fact_24_ceiling__of__nat,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( archim2889992004027027881ng_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.24/0.49        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_of_nat
% 0.24/0.49  thf(fact_25_ceiling__one,axiom,
% 0.24/0.49      ( ( archim7802044766580827645g_real @ one_one_real )
% 0.24/0.49      = one_one_int ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_one
% 0.24/0.49  thf(fact_26_ceiling__one,axiom,
% 0.24/0.49      ( ( archim2889992004027027881ng_rat @ one_one_rat )
% 0.24/0.49      = one_one_int ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_one
% 0.24/0.49  thf(fact_27_numeral__eq__one__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( ( numera6690914467698888265omplex @ N )
% 0.24/0.49          = one_one_complex )
% 0.24/0.49        = ( N = one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_eq_one_iff
% 0.24/0.49  thf(fact_28_numeral__eq__one__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( ( numeral_numeral_real @ N )
% 0.24/0.49          = one_one_real )
% 0.24/0.49        = ( N = one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_eq_one_iff
% 0.24/0.49  thf(fact_29_numeral__eq__one__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( ( numeral_numeral_rat @ N )
% 0.24/0.49          = one_one_rat )
% 0.24/0.49        = ( N = one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_eq_one_iff
% 0.24/0.49  thf(fact_30_numeral__eq__one__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( ( numeral_numeral_nat @ N )
% 0.24/0.49          = one_one_nat )
% 0.24/0.49        = ( N = one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_eq_one_iff
% 0.24/0.49  thf(fact_31_numeral__eq__one__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( ( numeral_numeral_int @ N )
% 0.24/0.49          = one_one_int )
% 0.24/0.49        = ( N = one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_eq_one_iff
% 0.24/0.49  thf(fact_32_one__eq__numeral__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( one_one_complex
% 0.24/0.49          = ( numera6690914467698888265omplex @ N ) )
% 0.24/0.49        = ( one = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_eq_numeral_iff
% 0.24/0.49  thf(fact_33_one__eq__numeral__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( one_one_real
% 0.24/0.49          = ( numeral_numeral_real @ N ) )
% 0.24/0.49        = ( one = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_eq_numeral_iff
% 0.24/0.49  thf(fact_34_one__eq__numeral__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( one_one_rat
% 0.24/0.49          = ( numeral_numeral_rat @ N ) )
% 0.24/0.49        = ( one = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_eq_numeral_iff
% 0.24/0.49  thf(fact_35_one__eq__numeral__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( one_one_nat
% 0.24/0.49          = ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( one = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_eq_numeral_iff
% 0.24/0.49  thf(fact_36_one__eq__numeral__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( one_one_int
% 0.24/0.49          = ( numeral_numeral_int @ N ) )
% 0.24/0.49        = ( one = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_eq_numeral_iff
% 0.24/0.49  thf(fact_37_ceiling__numeral,axiom,
% 0.24/0.49      ! [V: num] :
% 0.24/0.49        ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
% 0.24/0.49        = ( numeral_numeral_int @ V ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_numeral
% 0.24/0.49  thf(fact_38_ceiling__numeral,axiom,
% 0.24/0.49      ! [V: num] :
% 0.24/0.49        ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
% 0.24/0.49        = ( numeral_numeral_int @ V ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_numeral
% 0.24/0.49  thf(fact_39_of__nat__add,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.49        = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_add
% 0.24/0.49  thf(fact_40_of__nat__add,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.49        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_add
% 0.24/0.49  thf(fact_41_of__nat__add,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.49        = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_add
% 0.24/0.49  thf(fact_42_of__nat__add,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.49        = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_add
% 0.24/0.49  thf(fact_43_of__nat__add,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.49        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_add
% 0.24/0.49  thf(fact_44_of__nat__1,axiom,
% 0.24/0.49      ( ( semiri8010041392384452111omplex @ one_one_nat )
% 0.24/0.49      = one_one_complex ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_1
% 0.24/0.49  thf(fact_45_of__nat__1,axiom,
% 0.24/0.49      ( ( semiri5074537144036343181t_real @ one_one_nat )
% 0.24/0.49      = one_one_real ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_1
% 0.24/0.49  thf(fact_46_of__nat__1,axiom,
% 0.24/0.49      ( ( semiri681578069525770553at_rat @ one_one_nat )
% 0.24/0.49      = one_one_rat ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_1
% 0.24/0.49  thf(fact_47_of__nat__1,axiom,
% 0.24/0.49      ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.24/0.49      = one_one_nat ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_1
% 0.24/0.49  thf(fact_48_of__nat__1,axiom,
% 0.24/0.49      ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.24/0.49      = one_one_int ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_1
% 0.24/0.49  thf(fact_49__C3_Ohyps_C_I6_J,axiom,
% 0.24/0.49      ! [X2: vEBT_VEBT] :
% 0.24/0.49        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.24/0.49       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % "3.hyps"(6)
% 0.24/0.49  thf(fact_50_numeral__eq__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ( numera6690914467698888265omplex @ M )
% 0.24/0.49          = ( numera6690914467698888265omplex @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_eq_iff
% 0.24/0.49  thf(fact_51_numeral__eq__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ( numeral_numeral_real @ M )
% 0.24/0.49          = ( numeral_numeral_real @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_eq_iff
% 0.24/0.49  thf(fact_52_numeral__eq__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ( numeral_numeral_rat @ M )
% 0.24/0.49          = ( numeral_numeral_rat @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_eq_iff
% 0.24/0.49  thf(fact_53_numeral__eq__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ( numeral_numeral_nat @ M )
% 0.24/0.49          = ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_eq_iff
% 0.24/0.49  thf(fact_54_numeral__eq__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ( numeral_numeral_int @ M )
% 0.24/0.49          = ( numeral_numeral_int @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_eq_iff
% 0.24/0.49  thf(fact_55_of__nat__eq__iff,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ( semiri8010041392384452111omplex @ M )
% 0.24/0.49          = ( semiri8010041392384452111omplex @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_iff
% 0.24/0.49  thf(fact_56_of__nat__eq__iff,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ( semiri5074537144036343181t_real @ M )
% 0.24/0.49          = ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_iff
% 0.24/0.49  thf(fact_57_of__nat__eq__iff,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ( semiri681578069525770553at_rat @ M )
% 0.24/0.49          = ( semiri681578069525770553at_rat @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_iff
% 0.24/0.49  thf(fact_58_of__nat__eq__iff,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ( semiri1316708129612266289at_nat @ M )
% 0.24/0.49          = ( semiri1316708129612266289at_nat @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_iff
% 0.24/0.49  thf(fact_59_of__nat__eq__iff,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ( semiri1314217659103216013at_int @ M )
% 0.24/0.49          = ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_iff
% 0.24/0.49  thf(fact_60_of__nat__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( numera6690914467698888265omplex @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_numeral
% 0.24/0.49  thf(fact_61_of__nat__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( numeral_numeral_real @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_numeral
% 0.24/0.49  thf(fact_62_of__nat__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( numeral_numeral_rat @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_numeral
% 0.24/0.49  thf(fact_63_of__nat__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( numeral_numeral_nat @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_numeral
% 0.24/0.49  thf(fact_64_of__nat__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( numeral_numeral_int @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_numeral
% 0.24/0.49  thf(fact_65_of__nat__eq__1__iff,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( ( semiri8010041392384452111omplex @ N )
% 0.24/0.49          = one_one_complex )
% 0.24/0.49        = ( N = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_1_iff
% 0.24/0.49  thf(fact_66_of__nat__eq__1__iff,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( ( semiri5074537144036343181t_real @ N )
% 0.24/0.49          = one_one_real )
% 0.24/0.49        = ( N = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_1_iff
% 0.24/0.49  thf(fact_67_of__nat__eq__1__iff,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( ( semiri681578069525770553at_rat @ N )
% 0.24/0.49          = one_one_rat )
% 0.24/0.49        = ( N = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_1_iff
% 0.24/0.49  thf(fact_68_of__nat__eq__1__iff,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( ( semiri1316708129612266289at_nat @ N )
% 0.24/0.49          = one_one_nat )
% 0.24/0.49        = ( N = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_1_iff
% 0.24/0.49  thf(fact_69_of__nat__eq__1__iff,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( ( semiri1314217659103216013at_int @ N )
% 0.24/0.49          = one_one_int )
% 0.24/0.49        = ( N = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_1_iff
% 0.24/0.49  thf(fact_70_of__nat__1__eq__iff,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( one_one_complex
% 0.24/0.49          = ( semiri8010041392384452111omplex @ N ) )
% 0.24/0.49        = ( N = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_1_eq_iff
% 0.24/0.49  thf(fact_71_of__nat__1__eq__iff,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( one_one_real
% 0.24/0.49          = ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.49        = ( N = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_1_eq_iff
% 0.24/0.49  thf(fact_72_of__nat__1__eq__iff,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( one_one_rat
% 0.24/0.49          = ( semiri681578069525770553at_rat @ N ) )
% 0.24/0.49        = ( N = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_1_eq_iff
% 0.24/0.49  thf(fact_73_of__nat__1__eq__iff,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( one_one_nat
% 0.24/0.49          = ( semiri1316708129612266289at_nat @ N ) )
% 0.24/0.49        = ( N = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_1_eq_iff
% 0.24/0.49  thf(fact_74_of__nat__1__eq__iff,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( one_one_int
% 0.24/0.49          = ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.49        = ( N = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_1_eq_iff
% 0.24/0.49  thf(fact_75_add__numeral__left,axiom,
% 0.24/0.49      ! [V: num,W: num,Z: complex] :
% 0.24/0.49        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.24/0.49        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_numeral_left
% 0.24/0.49  thf(fact_76_add__numeral__left,axiom,
% 0.24/0.49      ! [V: num,W: num,Z: real] :
% 0.24/0.49        ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.24/0.49        = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_numeral_left
% 0.24/0.49  thf(fact_77_add__numeral__left,axiom,
% 0.24/0.49      ! [V: num,W: num,Z: rat] :
% 0.24/0.49        ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.24/0.49        = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_numeral_left
% 0.24/0.49  thf(fact_78_add__numeral__left,axiom,
% 0.24/0.49      ! [V: num,W: num,Z: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.24/0.49        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_numeral_left
% 0.24/0.49  thf(fact_79_add__numeral__left,axiom,
% 0.24/0.49      ! [V: num,W: num,Z: int] :
% 0.24/0.49        ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.24/0.49        = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_numeral_left
% 0.24/0.49  thf(fact_80_numeral__plus__numeral,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.24/0.49        = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_plus_numeral
% 0.24/0.49  thf(fact_81_numeral__plus__numeral,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.24/0.49        = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_plus_numeral
% 0.24/0.49  thf(fact_82_numeral__plus__numeral,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.24/0.49        = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_plus_numeral
% 0.24/0.49  thf(fact_83_numeral__plus__numeral,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_plus_numeral
% 0.24/0.49  thf(fact_84_numeral__plus__numeral,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.24/0.49        = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_plus_numeral
% 0.24/0.49  thf(fact_85__C3_Ohyps_C_I4_J,axiom,
% 0.24/0.49      ( deg
% 0.24/0.49      = ( plus_plus_nat @ na @ m ) ) ).
% 0.24/0.49  
% 0.24/0.49  % "3.hyps"(4)
% 0.24/0.49  thf(fact_86__C3_OIH_C_I1_J,axiom,
% 0.24/0.49      ! [X2: vEBT_VEBT] :
% 0.24/0.49        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.24/0.49       => ( ( vEBT_invar_vebt @ X2 @ na )
% 0.24/0.49          & ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X2 ) )
% 0.24/0.49            = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ na ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % "3.IH"(1)
% 0.24/0.49  thf(fact_87_numerals_I1_J,axiom,
% 0.24/0.49      ( ( numeral_numeral_nat @ one )
% 0.24/0.49      = one_one_nat ) ).
% 0.24/0.49  
% 0.24/0.49  % numerals(1)
% 0.24/0.49  thf(fact_88_nat__1__add__1,axiom,
% 0.24/0.49      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.24/0.49      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % nat_1_add_1
% 0.24/0.49  thf(fact_89_add__One__commute,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( plus_plus_num @ one @ N )
% 0.24/0.49        = ( plus_plus_num @ N @ one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_One_commute
% 0.24/0.49  thf(fact_90_VEBT__internal_Oheight_Osimps_I2_J,axiom,
% 0.24/0.49      ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.24/0.49        ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ Deg @ TreeList @ Summary ) )
% 0.24/0.49        = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % VEBT_internal.height.simps(2)
% 0.24/0.49  thf(fact_91_is__num__normalize_I1_J,axiom,
% 0.24/0.49      ! [A: real,B: real,C: real] :
% 0.24/0.49        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % is_num_normalize(1)
% 0.24/0.49  thf(fact_92_is__num__normalize_I1_J,axiom,
% 0.24/0.49      ! [A: rat,B: rat,C: rat] :
% 0.24/0.49        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % is_num_normalize(1)
% 0.24/0.49  thf(fact_93_is__num__normalize_I1_J,axiom,
% 0.24/0.49      ! [A: int,B: int,C: int] :
% 0.24/0.49        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % is_num_normalize(1)
% 0.24/0.49  thf(fact_94_is__num__normalize_I1_J,axiom,
% 0.24/0.49      ! [A: complex,B: complex,C: complex] :
% 0.24/0.49        ( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % is_num_normalize(1)
% 0.24/0.49  thf(fact_95_one__plus__numeral__commute,axiom,
% 0.24/0.49      ! [X: num] :
% 0.24/0.49        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
% 0.24/0.49        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_plus_numeral_commute
% 0.24/0.49  thf(fact_96_one__plus__numeral__commute,axiom,
% 0.24/0.49      ! [X: num] :
% 0.24/0.49        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 0.24/0.49        = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_plus_numeral_commute
% 0.24/0.49  thf(fact_97_one__plus__numeral__commute,axiom,
% 0.24/0.49      ! [X: num] :
% 0.24/0.49        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 0.24/0.49        = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_plus_numeral_commute
% 0.24/0.49  thf(fact_98_one__plus__numeral__commute,axiom,
% 0.24/0.49      ! [X: num] :
% 0.24/0.49        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 0.24/0.49        = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_plus_numeral_commute
% 0.24/0.49  thf(fact_99_one__plus__numeral__commute,axiom,
% 0.24/0.49      ! [X: num] :
% 0.24/0.49        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 0.24/0.49        = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_plus_numeral_commute
% 0.24/0.49  thf(fact_100_numeral__Bit0,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.24/0.49        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_Bit0
% 0.24/0.49  thf(fact_101_numeral__Bit0,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.24/0.49        = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_Bit0
% 0.24/0.49  thf(fact_102_numeral__Bit0,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.24/0.49        = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_Bit0
% 0.24/0.49  thf(fact_103_numeral__Bit0,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.24/0.49        = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_Bit0
% 0.24/0.49  thf(fact_104_numeral__Bit0,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.24/0.49        = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_Bit0
% 0.24/0.49  thf(fact_105_numeral__One,axiom,
% 0.24/0.49      ( ( numera6690914467698888265omplex @ one )
% 0.24/0.49      = one_one_complex ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_One
% 0.24/0.49  thf(fact_106_numeral__One,axiom,
% 0.24/0.49      ( ( numeral_numeral_real @ one )
% 0.24/0.49      = one_one_real ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_One
% 0.24/0.49  thf(fact_107_numeral__One,axiom,
% 0.24/0.49      ( ( numeral_numeral_rat @ one )
% 0.24/0.49      = one_one_rat ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_One
% 0.24/0.49  thf(fact_108_numeral__One,axiom,
% 0.24/0.49      ( ( numeral_numeral_nat @ one )
% 0.24/0.49      = one_one_nat ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_One
% 0.24/0.49  thf(fact_109_numeral__One,axiom,
% 0.24/0.49      ( ( numeral_numeral_int @ one )
% 0.24/0.49      = one_one_int ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_One
% 0.24/0.49  thf(fact_110_setceilmax,axiom,
% 0.24/0.49      ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
% 0.24/0.49        ( ( vEBT_invar_vebt @ S @ M )
% 0.24/0.49       => ( ! [X3: vEBT_VEBT] :
% 0.24/0.49              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
% 0.24/0.49             => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.24/0.49         => ( ( M
% 0.24/0.49              = ( suc @ N ) )
% 0.24/0.49           => ( ! [X3: vEBT_VEBT] :
% 0.24/0.49                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
% 0.24/0.49                 => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X3 ) )
% 0.24/0.49                    = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.24/0.49             => ( ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ S ) )
% 0.24/0.49                  = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) )
% 0.24/0.49               => ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ S @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
% 0.24/0.49                  = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % setceilmax
% 0.24/0.49  thf(fact_111_semiring__norm_I2_J,axiom,
% 0.24/0.49      ( ( plus_plus_num @ one @ one )
% 0.24/0.49      = ( bit0 @ one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % semiring_norm(2)
% 0.24/0.49  thf(fact_112_semiring__norm_I6_J,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.24/0.49        = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % semiring_norm(6)
% 0.24/0.49  thf(fact_113_int__eq__iff__numeral,axiom,
% 0.24/0.49      ! [M: nat,V: num] :
% 0.24/0.49        ( ( ( semiri1314217659103216013at_int @ M )
% 0.24/0.49          = ( numeral_numeral_int @ V ) )
% 0.24/0.49        = ( M
% 0.24/0.49          = ( numeral_numeral_nat @ V ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % int_eq_iff_numeral
% 0.24/0.49  thf(fact_114_image__insert,axiom,
% 0.24/0.49      ! [F: vEBT_VEBT > vEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( image_3375948659692109573T_VEBT @ F @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 0.24/0.49        = ( insert_VEBT_VEBT @ ( F @ A ) @ ( image_3375948659692109573T_VEBT @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_insert
% 0.24/0.49  thf(fact_115_image__insert,axiom,
% 0.24/0.49      ! [F: vEBT_VEBT > nat,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( image_VEBT_VEBT_nat @ F @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 0.24/0.49        = ( insert_nat @ ( F @ A ) @ ( image_VEBT_VEBT_nat @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_insert
% 0.24/0.49  thf(fact_116_image__insert,axiom,
% 0.24/0.49      ! [F: vEBT_VEBT > int,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( image_VEBT_VEBT_int @ F @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 0.24/0.49        = ( insert_int @ ( F @ A ) @ ( image_VEBT_VEBT_int @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_insert
% 0.24/0.49  thf(fact_117_image__insert,axiom,
% 0.24/0.49      ! [F: vEBT_VEBT > real,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( image_VEBT_VEBT_real @ F @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 0.24/0.49        = ( insert_real @ ( F @ A ) @ ( image_VEBT_VEBT_real @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_insert
% 0.24/0.49  thf(fact_118_image__insert,axiom,
% 0.24/0.49      ! [F: vEBT_VEBT > $o,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( image_VEBT_VEBT_o @ F @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 0.24/0.49        = ( insert_o @ ( F @ A ) @ ( image_VEBT_VEBT_o @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_insert
% 0.24/0.49  thf(fact_119_image__insert,axiom,
% 0.24/0.49      ! [F: nat > vEBT_VEBT,A: nat,B2: set_nat] :
% 0.24/0.49        ( ( image_nat_VEBT_VEBT @ F @ ( insert_nat @ A @ B2 ) )
% 0.24/0.49        = ( insert_VEBT_VEBT @ ( F @ A ) @ ( image_nat_VEBT_VEBT @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_insert
% 0.24/0.49  thf(fact_120_image__insert,axiom,
% 0.24/0.49      ! [F: nat > nat,A: nat,B2: set_nat] :
% 0.24/0.49        ( ( image_nat_nat @ F @ ( insert_nat @ A @ B2 ) )
% 0.24/0.49        = ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_insert
% 0.24/0.49  thf(fact_121_image__insert,axiom,
% 0.24/0.49      ! [F: nat > int,A: nat,B2: set_nat] :
% 0.24/0.49        ( ( image_nat_int @ F @ ( insert_nat @ A @ B2 ) )
% 0.24/0.49        = ( insert_int @ ( F @ A ) @ ( image_nat_int @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_insert
% 0.24/0.49  thf(fact_122_image__insert,axiom,
% 0.24/0.49      ! [F: nat > real,A: nat,B2: set_nat] :
% 0.24/0.49        ( ( image_nat_real @ F @ ( insert_nat @ A @ B2 ) )
% 0.24/0.49        = ( insert_real @ ( F @ A ) @ ( image_nat_real @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_insert
% 0.24/0.49  thf(fact_123_image__insert,axiom,
% 0.24/0.49      ! [F: nat > $o,A: nat,B2: set_nat] :
% 0.24/0.49        ( ( image_nat_o @ F @ ( insert_nat @ A @ B2 ) )
% 0.24/0.49        = ( insert_o @ ( F @ A ) @ ( image_nat_o @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_insert
% 0.24/0.49  thf(fact_124_insert__image,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,F: complex > vEBT_VEBT] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ( insert_VEBT_VEBT @ ( F @ X ) @ ( image_932796090930683071T_VEBT @ F @ A2 ) )
% 0.24/0.49          = ( image_932796090930683071T_VEBT @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_image
% 0.24/0.49  thf(fact_125_insert__image,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,F: complex > nat] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ( insert_nat @ ( F @ X ) @ ( image_complex_nat @ F @ A2 ) )
% 0.24/0.49          = ( image_complex_nat @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_image
% 0.24/0.49  thf(fact_126_insert__image,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,F: complex > int] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ( insert_int @ ( F @ X ) @ ( image_complex_int @ F @ A2 ) )
% 0.24/0.49          = ( image_complex_int @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_image
% 0.24/0.49  thf(fact_127_insert__image,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,F: complex > real] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ( insert_real @ ( F @ X ) @ ( image_complex_real @ F @ A2 ) )
% 0.24/0.49          = ( image_complex_real @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_image
% 0.24/0.49  thf(fact_128_insert__image,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,F: complex > $o] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ( insert_o @ ( F @ X ) @ ( image_complex_o @ F @ A2 ) )
% 0.24/0.49          = ( image_complex_o @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_image
% 0.24/0.49  thf(fact_129_insert__image,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,F: real > vEBT_VEBT] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( ( insert_VEBT_VEBT @ ( F @ X ) @ ( image_real_VEBT_VEBT @ F @ A2 ) )
% 0.24/0.49          = ( image_real_VEBT_VEBT @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_image
% 0.24/0.49  thf(fact_130_insert__image,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,F: real > nat] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( ( insert_nat @ ( F @ X ) @ ( image_real_nat @ F @ A2 ) )
% 0.24/0.49          = ( image_real_nat @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_image
% 0.24/0.49  thf(fact_131_insert__image,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,F: real > int] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( ( insert_int @ ( F @ X ) @ ( image_real_int @ F @ A2 ) )
% 0.24/0.49          = ( image_real_int @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_image
% 0.24/0.49  thf(fact_132_insert__image,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,F: real > real] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( ( insert_real @ ( F @ X ) @ ( image_real_real @ F @ A2 ) )
% 0.24/0.49          = ( image_real_real @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_image
% 0.24/0.49  thf(fact_133_insert__image,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,F: real > $o] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( ( insert_o @ ( F @ X ) @ ( image_real_o @ F @ A2 ) )
% 0.24/0.49          = ( image_real_o @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_image
% 0.24/0.49  thf(fact_134_semiring__norm_I83_J,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( one
% 0.24/0.49       != ( bit0 @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % semiring_norm(83)
% 0.24/0.49  thf(fact_135_semiring__norm_I85_J,axiom,
% 0.24/0.49      ! [M: num] :
% 0.24/0.49        ( ( bit0 @ M )
% 0.24/0.49       != one ) ).
% 0.24/0.49  
% 0.24/0.49  % semiring_norm(85)
% 0.24/0.49  thf(fact_136_int__ops_I2_J,axiom,
% 0.24/0.49      ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.24/0.49      = one_one_int ) ).
% 0.24/0.49  
% 0.24/0.49  % int_ops(2)
% 0.24/0.49  thf(fact_137_zadd__int__left,axiom,
% 0.24/0.49      ! [M: nat,N: nat,Z: int] :
% 0.24/0.49        ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 0.24/0.49        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % zadd_int_left
% 0.24/0.49  thf(fact_138_mem__Collect__eq,axiom,
% 0.24/0.49      ! [A: complex,P: complex > $o] :
% 0.24/0.49        ( ( member_complex @ A @ ( collect_complex @ P ) )
% 0.24/0.49        = ( P @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mem_Collect_eq
% 0.24/0.49  thf(fact_139_mem__Collect__eq,axiom,
% 0.24/0.49      ! [A: real,P: real > $o] :
% 0.24/0.49        ( ( member_real @ A @ ( collect_real @ P ) )
% 0.24/0.49        = ( P @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mem_Collect_eq
% 0.24/0.49  thf(fact_140_mem__Collect__eq,axiom,
% 0.24/0.49      ! [A: set_nat,P: set_nat > $o] :
% 0.24/0.49        ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 0.24/0.49        = ( P @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mem_Collect_eq
% 0.24/0.49  thf(fact_141_mem__Collect__eq,axiom,
% 0.24/0.49      ! [A: nat,P: nat > $o] :
% 0.24/0.49        ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.24/0.49        = ( P @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mem_Collect_eq
% 0.24/0.49  thf(fact_142_mem__Collect__eq,axiom,
% 0.24/0.49      ! [A: int,P: int > $o] :
% 0.24/0.49        ( ( member_int @ A @ ( collect_int @ P ) )
% 0.24/0.49        = ( P @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mem_Collect_eq
% 0.24/0.49  thf(fact_143_Collect__mem__eq,axiom,
% 0.24/0.49      ! [A2: set_complex] :
% 0.24/0.49        ( ( collect_complex
% 0.24/0.49          @ ^ [X4: complex] : ( member_complex @ X4 @ A2 ) )
% 0.24/0.49        = A2 ) ).
% 0.24/0.49  
% 0.24/0.49  % Collect_mem_eq
% 0.24/0.49  thf(fact_144_Collect__mem__eq,axiom,
% 0.24/0.49      ! [A2: set_real] :
% 0.24/0.49        ( ( collect_real
% 0.24/0.49          @ ^ [X4: real] : ( member_real @ X4 @ A2 ) )
% 0.24/0.49        = A2 ) ).
% 0.24/0.49  
% 0.24/0.49  % Collect_mem_eq
% 0.24/0.49  thf(fact_145_Collect__mem__eq,axiom,
% 0.24/0.49      ! [A2: set_set_nat] :
% 0.24/0.49        ( ( collect_set_nat
% 0.24/0.49          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A2 ) )
% 0.24/0.49        = A2 ) ).
% 0.24/0.49  
% 0.24/0.49  % Collect_mem_eq
% 0.24/0.49  thf(fact_146_Collect__mem__eq,axiom,
% 0.24/0.49      ! [A2: set_nat] :
% 0.24/0.49        ( ( collect_nat
% 0.24/0.49          @ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
% 0.24/0.49        = A2 ) ).
% 0.24/0.49  
% 0.24/0.49  % Collect_mem_eq
% 0.24/0.49  thf(fact_147_Collect__mem__eq,axiom,
% 0.24/0.49      ! [A2: set_int] :
% 0.24/0.49        ( ( collect_int
% 0.24/0.49          @ ^ [X4: int] : ( member_int @ X4 @ A2 ) )
% 0.24/0.49        = A2 ) ).
% 0.24/0.49  
% 0.24/0.49  % Collect_mem_eq
% 0.24/0.49  thf(fact_148_Collect__cong,axiom,
% 0.24/0.49      ! [P: complex > $o,Q: complex > $o] :
% 0.24/0.49        ( ! [X3: complex] :
% 0.24/0.49            ( ( P @ X3 )
% 0.24/0.49            = ( Q @ X3 ) )
% 0.24/0.49       => ( ( collect_complex @ P )
% 0.24/0.49          = ( collect_complex @ Q ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Collect_cong
% 0.24/0.49  thf(fact_149_Collect__cong,axiom,
% 0.24/0.49      ! [P: real > $o,Q: real > $o] :
% 0.24/0.49        ( ! [X3: real] :
% 0.24/0.49            ( ( P @ X3 )
% 0.24/0.49            = ( Q @ X3 ) )
% 0.24/0.49       => ( ( collect_real @ P )
% 0.24/0.49          = ( collect_real @ Q ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Collect_cong
% 0.24/0.49  thf(fact_150_Collect__cong,axiom,
% 0.24/0.49      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.24/0.49        ( ! [X3: set_nat] :
% 0.24/0.49            ( ( P @ X3 )
% 0.24/0.49            = ( Q @ X3 ) )
% 0.24/0.49       => ( ( collect_set_nat @ P )
% 0.24/0.49          = ( collect_set_nat @ Q ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Collect_cong
% 0.24/0.49  thf(fact_151_Collect__cong,axiom,
% 0.24/0.49      ! [P: nat > $o,Q: nat > $o] :
% 0.24/0.49        ( ! [X3: nat] :
% 0.24/0.49            ( ( P @ X3 )
% 0.24/0.49            = ( Q @ X3 ) )
% 0.24/0.49       => ( ( collect_nat @ P )
% 0.24/0.49          = ( collect_nat @ Q ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Collect_cong
% 0.24/0.49  thf(fact_152_Collect__cong,axiom,
% 0.24/0.49      ! [P: int > $o,Q: int > $o] :
% 0.24/0.49        ( ! [X3: int] :
% 0.24/0.49            ( ( P @ X3 )
% 0.24/0.49            = ( Q @ X3 ) )
% 0.24/0.49       => ( ( collect_int @ P )
% 0.24/0.49          = ( collect_int @ Q ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Collect_cong
% 0.24/0.49  thf(fact_153_int__plus,axiom,
% 0.24/0.49      ! [N: nat,M: nat] :
% 0.24/0.49        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 0.24/0.49        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % int_plus
% 0.24/0.49  thf(fact_154_int__ops_I5_J,axiom,
% 0.24/0.49      ! [A: nat,B: nat] :
% 0.24/0.49        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 0.24/0.49        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % int_ops(5)
% 0.24/0.49  thf(fact_155_even__odd__cases,axiom,
% 0.24/0.49      ! [X: nat] :
% 0.24/0.49        ( ! [N2: nat] :
% 0.24/0.49            ( X
% 0.24/0.49           != ( plus_plus_nat @ N2 @ N2 ) )
% 0.24/0.49       => ~ ! [N2: nat] :
% 0.24/0.49              ( X
% 0.24/0.49             != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % even_odd_cases
% 0.24/0.49  thf(fact_156__C3_Ohyps_C_I3_J,axiom,
% 0.24/0.49      ( m
% 0.24/0.49      = ( suc @ na ) ) ).
% 0.24/0.49  
% 0.24/0.49  % "3.hyps"(3)
% 0.24/0.49  thf(fact_157_verit__eq__simplify_I8_J,axiom,
% 0.24/0.49      ! [X22: num,Y2: num] :
% 0.24/0.49        ( ( ( bit0 @ X22 )
% 0.24/0.49          = ( bit0 @ Y2 ) )
% 0.24/0.49        = ( X22 = Y2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_eq_simplify(8)
% 0.24/0.49  thf(fact_158_semiring__norm_I87_J,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ( bit0 @ M )
% 0.24/0.49          = ( bit0 @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % semiring_norm(87)
% 0.24/0.49  thf(fact_159_image__eqI,axiom,
% 0.24/0.49      ! [B: complex,F: complex > complex,X: complex,A2: set_complex] :
% 0.24/0.49        ( ( B
% 0.24/0.49          = ( F @ X ) )
% 0.24/0.49       => ( ( member_complex @ X @ A2 )
% 0.24/0.49         => ( member_complex @ B @ ( image_1468599708987790691omplex @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_eqI
% 0.24/0.49  thf(fact_160_image__eqI,axiom,
% 0.24/0.49      ! [B: real,F: complex > real,X: complex,A2: set_complex] :
% 0.24/0.49        ( ( B
% 0.24/0.49          = ( F @ X ) )
% 0.24/0.49       => ( ( member_complex @ X @ A2 )
% 0.24/0.49         => ( member_real @ B @ ( image_complex_real @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_eqI
% 0.24/0.49  thf(fact_161_image__eqI,axiom,
% 0.24/0.49      ! [B: nat,F: complex > nat,X: complex,A2: set_complex] :
% 0.24/0.49        ( ( B
% 0.24/0.49          = ( F @ X ) )
% 0.24/0.49       => ( ( member_complex @ X @ A2 )
% 0.24/0.49         => ( member_nat @ B @ ( image_complex_nat @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_eqI
% 0.24/0.49  thf(fact_162_image__eqI,axiom,
% 0.24/0.49      ! [B: int,F: complex > int,X: complex,A2: set_complex] :
% 0.24/0.49        ( ( B
% 0.24/0.49          = ( F @ X ) )
% 0.24/0.49       => ( ( member_complex @ X @ A2 )
% 0.24/0.49         => ( member_int @ B @ ( image_complex_int @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_eqI
% 0.24/0.49  thf(fact_163_image__eqI,axiom,
% 0.24/0.49      ! [B: complex,F: real > complex,X: real,A2: set_real] :
% 0.24/0.49        ( ( B
% 0.24/0.49          = ( F @ X ) )
% 0.24/0.49       => ( ( member_real @ X @ A2 )
% 0.24/0.49         => ( member_complex @ B @ ( image_real_complex @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_eqI
% 0.24/0.49  thf(fact_164_image__eqI,axiom,
% 0.24/0.49      ! [B: real,F: real > real,X: real,A2: set_real] :
% 0.24/0.49        ( ( B
% 0.24/0.49          = ( F @ X ) )
% 0.24/0.49       => ( ( member_real @ X @ A2 )
% 0.24/0.49         => ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_eqI
% 0.24/0.49  thf(fact_165_image__eqI,axiom,
% 0.24/0.49      ! [B: nat,F: real > nat,X: real,A2: set_real] :
% 0.24/0.49        ( ( B
% 0.24/0.49          = ( F @ X ) )
% 0.24/0.49       => ( ( member_real @ X @ A2 )
% 0.24/0.49         => ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_eqI
% 0.24/0.49  thf(fact_166_image__eqI,axiom,
% 0.24/0.49      ! [B: int,F: real > int,X: real,A2: set_real] :
% 0.24/0.49        ( ( B
% 0.24/0.49          = ( F @ X ) )
% 0.24/0.49       => ( ( member_real @ X @ A2 )
% 0.24/0.49         => ( member_int @ B @ ( image_real_int @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_eqI
% 0.24/0.49  thf(fact_167_image__eqI,axiom,
% 0.24/0.49      ! [B: complex,F: nat > complex,X: nat,A2: set_nat] :
% 0.24/0.49        ( ( B
% 0.24/0.49          = ( F @ X ) )
% 0.24/0.49       => ( ( member_nat @ X @ A2 )
% 0.24/0.49         => ( member_complex @ B @ ( image_nat_complex @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_eqI
% 0.24/0.49  thf(fact_168_image__eqI,axiom,
% 0.24/0.49      ! [B: real,F: nat > real,X: nat,A2: set_nat] :
% 0.24/0.49        ( ( B
% 0.24/0.49          = ( F @ X ) )
% 0.24/0.49       => ( ( member_nat @ X @ A2 )
% 0.24/0.49         => ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_eqI
% 0.24/0.49  thf(fact_169_nat_Oinject,axiom,
% 0.24/0.49      ! [X22: nat,Y2: nat] :
% 0.24/0.49        ( ( ( suc @ X22 )
% 0.24/0.49          = ( suc @ Y2 ) )
% 0.24/0.49        = ( X22 = Y2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % nat.inject
% 0.24/0.49  thf(fact_170_old_Onat_Oinject,axiom,
% 0.24/0.49      ! [Nat: nat,Nat2: nat] :
% 0.24/0.49        ( ( ( suc @ Nat )
% 0.24/0.49          = ( suc @ Nat2 ) )
% 0.24/0.49        = ( Nat = Nat2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % old.nat.inject
% 0.24/0.49  thf(fact_171_insert__absorb2,axiom,
% 0.24/0.49      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( insert_VEBT_VEBT @ X @ ( insert_VEBT_VEBT @ X @ A2 ) )
% 0.24/0.49        = ( insert_VEBT_VEBT @ X @ A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb2
% 0.24/0.49  thf(fact_172_insert__absorb2,axiom,
% 0.24/0.49      ! [X: nat,A2: set_nat] :
% 0.24/0.49        ( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
% 0.24/0.49        = ( insert_nat @ X @ A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb2
% 0.24/0.49  thf(fact_173_insert__absorb2,axiom,
% 0.24/0.49      ! [X: int,A2: set_int] :
% 0.24/0.49        ( ( insert_int @ X @ ( insert_int @ X @ A2 ) )
% 0.24/0.49        = ( insert_int @ X @ A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb2
% 0.24/0.49  thf(fact_174_insert__absorb2,axiom,
% 0.24/0.49      ! [X: real,A2: set_real] :
% 0.24/0.49        ( ( insert_real @ X @ ( insert_real @ X @ A2 ) )
% 0.24/0.49        = ( insert_real @ X @ A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb2
% 0.24/0.49  thf(fact_175_insert__absorb2,axiom,
% 0.24/0.49      ! [X: $o,A2: set_o] :
% 0.24/0.49        ( ( insert_o @ X @ ( insert_o @ X @ A2 ) )
% 0.24/0.49        = ( insert_o @ X @ A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb2
% 0.24/0.49  thf(fact_176_insert__iff,axiom,
% 0.24/0.49      ! [A: vEBT_VEBT,B: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ A2 ) )
% 0.24/0.49        = ( ( A = B )
% 0.24/0.49          | ( member_VEBT_VEBT @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_iff
% 0.24/0.49  thf(fact_177_insert__iff,axiom,
% 0.24/0.49      ! [A: $o,B: $o,A2: set_o] :
% 0.24/0.49        ( ( member_o @ A @ ( insert_o @ B @ A2 ) )
% 0.24/0.49        = ( ( A = B )
% 0.24/0.49          | ( member_o @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_iff
% 0.24/0.49  thf(fact_178_insert__iff,axiom,
% 0.24/0.49      ! [A: complex,B: complex,A2: set_complex] :
% 0.24/0.49        ( ( member_complex @ A @ ( insert_complex @ B @ A2 ) )
% 0.24/0.49        = ( ( A = B )
% 0.24/0.49          | ( member_complex @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_iff
% 0.24/0.49  thf(fact_179_insert__iff,axiom,
% 0.24/0.49      ! [A: real,B: real,A2: set_real] :
% 0.24/0.49        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 0.24/0.49        = ( ( A = B )
% 0.24/0.49          | ( member_real @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_iff
% 0.24/0.49  thf(fact_180_insert__iff,axiom,
% 0.24/0.49      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 0.24/0.49        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 0.24/0.49        = ( ( A = B )
% 0.24/0.49          | ( member_set_nat @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_iff
% 0.24/0.49  thf(fact_181_insert__iff,axiom,
% 0.24/0.49      ! [A: nat,B: nat,A2: set_nat] :
% 0.24/0.49        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 0.24/0.49        = ( ( A = B )
% 0.24/0.49          | ( member_nat @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_iff
% 0.24/0.49  thf(fact_182_insert__iff,axiom,
% 0.24/0.49      ! [A: int,B: int,A2: set_int] :
% 0.24/0.49        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 0.24/0.49        = ( ( A = B )
% 0.24/0.49          | ( member_int @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_iff
% 0.24/0.49  thf(fact_183_insertCI,axiom,
% 0.24/0.49      ! [A: vEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
% 0.24/0.49        ( ( ~ ( member_VEBT_VEBT @ A @ B2 )
% 0.24/0.49         => ( A = B ) )
% 0.24/0.49       => ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertCI
% 0.24/0.49  thf(fact_184_insertCI,axiom,
% 0.24/0.49      ! [A: $o,B2: set_o,B: $o] :
% 0.24/0.49        ( ( ~ ( member_o @ A @ B2 )
% 0.24/0.49         => ( A = B ) )
% 0.24/0.49       => ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertCI
% 0.24/0.49  thf(fact_185_insertCI,axiom,
% 0.24/0.49      ! [A: complex,B2: set_complex,B: complex] :
% 0.24/0.49        ( ( ~ ( member_complex @ A @ B2 )
% 0.24/0.49         => ( A = B ) )
% 0.24/0.49       => ( member_complex @ A @ ( insert_complex @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertCI
% 0.24/0.49  thf(fact_186_insertCI,axiom,
% 0.24/0.49      ! [A: real,B2: set_real,B: real] :
% 0.24/0.49        ( ( ~ ( member_real @ A @ B2 )
% 0.24/0.49         => ( A = B ) )
% 0.24/0.49       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertCI
% 0.24/0.49  thf(fact_187_insertCI,axiom,
% 0.24/0.49      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 0.24/0.49        ( ( ~ ( member_set_nat @ A @ B2 )
% 0.24/0.49         => ( A = B ) )
% 0.24/0.49       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertCI
% 0.24/0.49  thf(fact_188_insertCI,axiom,
% 0.24/0.49      ! [A: nat,B2: set_nat,B: nat] :
% 0.24/0.49        ( ( ~ ( member_nat @ A @ B2 )
% 0.24/0.49         => ( A = B ) )
% 0.24/0.49       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertCI
% 0.24/0.49  thf(fact_189_insertCI,axiom,
% 0.24/0.49      ! [A: int,B2: set_int,B: int] :
% 0.24/0.49        ( ( ~ ( member_int @ A @ B2 )
% 0.24/0.49         => ( A = B ) )
% 0.24/0.49       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertCI
% 0.24/0.49  thf(fact_190_add__Suc__right,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.24/0.49        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_Suc_right
% 0.24/0.49  thf(fact_191_of__nat__Suc,axiom,
% 0.24/0.49      ! [M: nat] :
% 0.24/0.49        ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 0.24/0.49        = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_Suc
% 0.24/0.49  thf(fact_192_of__nat__Suc,axiom,
% 0.24/0.49      ! [M: nat] :
% 0.24/0.49        ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 0.24/0.49        = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_Suc
% 0.24/0.49  thf(fact_193_of__nat__Suc,axiom,
% 0.24/0.49      ! [M: nat] :
% 0.24/0.49        ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 0.24/0.49        = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_Suc
% 0.24/0.49  thf(fact_194_of__nat__Suc,axiom,
% 0.24/0.49      ! [M: nat] :
% 0.24/0.49        ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 0.24/0.49        = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_Suc
% 0.24/0.49  thf(fact_195_of__nat__Suc,axiom,
% 0.24/0.49      ! [M: nat] :
% 0.24/0.49        ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 0.24/0.49        = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_Suc
% 0.24/0.49  thf(fact_196_Suc__numeral,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( suc @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_numeral
% 0.24/0.49  thf(fact_197_add__2__eq__Suc,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.24/0.49        = ( suc @ ( suc @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_2_eq_Suc
% 0.24/0.49  thf(fact_198_add__2__eq__Suc_H,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.49        = ( suc @ ( suc @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_2_eq_Suc'
% 0.24/0.49  thf(fact_199_Suc__1,axiom,
% 0.24/0.49      ( ( suc @ one_one_nat )
% 0.24/0.49      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_1
% 0.24/0.49  thf(fact_200_Suc__inject,axiom,
% 0.24/0.49      ! [X: nat,Y: nat] :
% 0.24/0.49        ( ( ( suc @ X )
% 0.24/0.49          = ( suc @ Y ) )
% 0.24/0.49       => ( X = Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_inject
% 0.24/0.49  thf(fact_201_n__not__Suc__n,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( N
% 0.24/0.49       != ( suc @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % n_not_Suc_n
% 0.24/0.49  thf(fact_202_add__Suc__shift,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.24/0.49        = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_Suc_shift
% 0.24/0.49  thf(fact_203_add__Suc,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.24/0.49        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_Suc
% 0.24/0.49  thf(fact_204_nat__arith_Osuc1,axiom,
% 0.24/0.49      ! [A2: nat,K: nat,A: nat] :
% 0.24/0.49        ( ( A2
% 0.24/0.49          = ( plus_plus_nat @ K @ A ) )
% 0.24/0.49       => ( ( suc @ A2 )
% 0.24/0.49          = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % nat_arith.suc1
% 0.24/0.49  thf(fact_205_Suc__eq__plus1__left,axiom,
% 0.24/0.49      ( suc
% 0.24/0.49      = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_eq_plus1_left
% 0.24/0.49  thf(fact_206_plus__1__eq__Suc,axiom,
% 0.24/0.49      ( ( plus_plus_nat @ one_one_nat )
% 0.24/0.49      = suc ) ).
% 0.24/0.49  
% 0.24/0.49  % plus_1_eq_Suc
% 0.24/0.49  thf(fact_207_Suc__eq__plus1,axiom,
% 0.24/0.49      ( suc
% 0.24/0.49      = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_eq_plus1
% 0.24/0.49  thf(fact_208_int__ops_I4_J,axiom,
% 0.24/0.49      ! [A: nat] :
% 0.24/0.49        ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 0.24/0.49        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % int_ops(4)
% 0.24/0.49  thf(fact_209_int__Suc,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.24/0.49        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % int_Suc
% 0.24/0.49  thf(fact_210_rev__image__eqI,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,B: complex,F: complex > complex] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ( B
% 0.24/0.49            = ( F @ X ) )
% 0.24/0.49         => ( member_complex @ B @ ( image_1468599708987790691omplex @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % rev_image_eqI
% 0.24/0.49  thf(fact_211_rev__image__eqI,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,B: real,F: complex > real] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ( B
% 0.24/0.49            = ( F @ X ) )
% 0.24/0.49         => ( member_real @ B @ ( image_complex_real @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % rev_image_eqI
% 0.24/0.49  thf(fact_212_rev__image__eqI,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,B: nat,F: complex > nat] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ( B
% 0.24/0.49            = ( F @ X ) )
% 0.24/0.49         => ( member_nat @ B @ ( image_complex_nat @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % rev_image_eqI
% 0.24/0.49  thf(fact_213_rev__image__eqI,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,B: int,F: complex > int] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ( B
% 0.24/0.49            = ( F @ X ) )
% 0.24/0.49         => ( member_int @ B @ ( image_complex_int @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % rev_image_eqI
% 0.24/0.49  thf(fact_214_rev__image__eqI,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,B: complex,F: real > complex] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( ( B
% 0.24/0.49            = ( F @ X ) )
% 0.24/0.49         => ( member_complex @ B @ ( image_real_complex @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % rev_image_eqI
% 0.24/0.49  thf(fact_215_rev__image__eqI,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,B: real,F: real > real] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( ( B
% 0.24/0.49            = ( F @ X ) )
% 0.24/0.49         => ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % rev_image_eqI
% 0.24/0.49  thf(fact_216_rev__image__eqI,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,B: nat,F: real > nat] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( ( B
% 0.24/0.49            = ( F @ X ) )
% 0.24/0.49         => ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % rev_image_eqI
% 0.24/0.49  thf(fact_217_rev__image__eqI,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,B: int,F: real > int] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( ( B
% 0.24/0.49            = ( F @ X ) )
% 0.24/0.49         => ( member_int @ B @ ( image_real_int @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % rev_image_eqI
% 0.24/0.49  thf(fact_218_rev__image__eqI,axiom,
% 0.24/0.49      ! [X: nat,A2: set_nat,B: complex,F: nat > complex] :
% 0.24/0.49        ( ( member_nat @ X @ A2 )
% 0.24/0.49       => ( ( B
% 0.24/0.49            = ( F @ X ) )
% 0.24/0.49         => ( member_complex @ B @ ( image_nat_complex @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % rev_image_eqI
% 0.24/0.49  thf(fact_219_rev__image__eqI,axiom,
% 0.24/0.49      ! [X: nat,A2: set_nat,B: real,F: nat > real] :
% 0.24/0.49        ( ( member_nat @ X @ A2 )
% 0.24/0.49       => ( ( B
% 0.24/0.49            = ( F @ X ) )
% 0.24/0.49         => ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % rev_image_eqI
% 0.24/0.49  thf(fact_220_ball__imageD,axiom,
% 0.24/0.49      ! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
% 0.24/0.49        ( ! [X3: set_nat] :
% 0.24/0.49            ( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A2 ) )
% 0.24/0.49           => ( P @ X3 ) )
% 0.24/0.49       => ! [X2: nat] :
% 0.24/0.49            ( ( member_nat @ X2 @ A2 )
% 0.24/0.49           => ( P @ ( F @ X2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ball_imageD
% 0.24/0.49  thf(fact_221_ball__imageD,axiom,
% 0.24/0.49      ! [F: nat > nat,A2: set_nat,P: nat > $o] :
% 0.24/0.49        ( ! [X3: nat] :
% 0.24/0.49            ( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
% 0.24/0.49           => ( P @ X3 ) )
% 0.24/0.49       => ! [X2: nat] :
% 0.24/0.49            ( ( member_nat @ X2 @ A2 )
% 0.24/0.49           => ( P @ ( F @ X2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ball_imageD
% 0.24/0.49  thf(fact_222_ball__imageD,axiom,
% 0.24/0.49      ! [F: nat > int,A2: set_nat,P: int > $o] :
% 0.24/0.49        ( ! [X3: int] :
% 0.24/0.49            ( ( member_int @ X3 @ ( image_nat_int @ F @ A2 ) )
% 0.24/0.49           => ( P @ X3 ) )
% 0.24/0.49       => ! [X2: nat] :
% 0.24/0.49            ( ( member_nat @ X2 @ A2 )
% 0.24/0.49           => ( P @ ( F @ X2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ball_imageD
% 0.24/0.49  thf(fact_223_ball__imageD,axiom,
% 0.24/0.49      ! [F: int > nat,A2: set_int,P: nat > $o] :
% 0.24/0.49        ( ! [X3: nat] :
% 0.24/0.49            ( ( member_nat @ X3 @ ( image_int_nat @ F @ A2 ) )
% 0.24/0.49           => ( P @ X3 ) )
% 0.24/0.49       => ! [X2: int] :
% 0.24/0.49            ( ( member_int @ X2 @ A2 )
% 0.24/0.49           => ( P @ ( F @ X2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ball_imageD
% 0.24/0.49  thf(fact_224_ball__imageD,axiom,
% 0.24/0.49      ! [F: int > int,A2: set_int,P: int > $o] :
% 0.24/0.49        ( ! [X3: int] :
% 0.24/0.49            ( ( member_int @ X3 @ ( image_int_int @ F @ A2 ) )
% 0.24/0.49           => ( P @ X3 ) )
% 0.24/0.49       => ! [X2: int] :
% 0.24/0.49            ( ( member_int @ X2 @ A2 )
% 0.24/0.49           => ( P @ ( F @ X2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ball_imageD
% 0.24/0.49  thf(fact_225_image__cong,axiom,
% 0.24/0.49      ! [M2: set_nat,N4: set_nat,F: nat > set_nat,G: nat > set_nat] :
% 0.24/0.49        ( ( M2 = N4 )
% 0.24/0.49       => ( ! [X3: nat] :
% 0.24/0.49              ( ( member_nat @ X3 @ N4 )
% 0.24/0.49             => ( ( F @ X3 )
% 0.24/0.49                = ( G @ X3 ) ) )
% 0.24/0.49         => ( ( image_nat_set_nat @ F @ M2 )
% 0.24/0.49            = ( image_nat_set_nat @ G @ N4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_cong
% 0.24/0.49  thf(fact_226_image__cong,axiom,
% 0.24/0.49      ! [M2: set_nat,N4: set_nat,F: nat > nat,G: nat > nat] :
% 0.24/0.49        ( ( M2 = N4 )
% 0.24/0.49       => ( ! [X3: nat] :
% 0.24/0.49              ( ( member_nat @ X3 @ N4 )
% 0.24/0.49             => ( ( F @ X3 )
% 0.24/0.49                = ( G @ X3 ) ) )
% 0.24/0.49         => ( ( image_nat_nat @ F @ M2 )
% 0.24/0.49            = ( image_nat_nat @ G @ N4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_cong
% 0.24/0.49  thf(fact_227_image__cong,axiom,
% 0.24/0.49      ! [M2: set_nat,N4: set_nat,F: nat > int,G: nat > int] :
% 0.24/0.49        ( ( M2 = N4 )
% 0.24/0.49       => ( ! [X3: nat] :
% 0.24/0.49              ( ( member_nat @ X3 @ N4 )
% 0.24/0.49             => ( ( F @ X3 )
% 0.24/0.49                = ( G @ X3 ) ) )
% 0.24/0.49         => ( ( image_nat_int @ F @ M2 )
% 0.24/0.49            = ( image_nat_int @ G @ N4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_cong
% 0.24/0.49  thf(fact_228_image__cong,axiom,
% 0.24/0.49      ! [M2: set_int,N4: set_int,F: int > nat,G: int > nat] :
% 0.24/0.49        ( ( M2 = N4 )
% 0.24/0.49       => ( ! [X3: int] :
% 0.24/0.49              ( ( member_int @ X3 @ N4 )
% 0.24/0.49             => ( ( F @ X3 )
% 0.24/0.49                = ( G @ X3 ) ) )
% 0.24/0.49         => ( ( image_int_nat @ F @ M2 )
% 0.24/0.49            = ( image_int_nat @ G @ N4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_cong
% 0.24/0.49  thf(fact_229_image__cong,axiom,
% 0.24/0.49      ! [M2: set_int,N4: set_int,F: int > int,G: int > int] :
% 0.24/0.49        ( ( M2 = N4 )
% 0.24/0.49       => ( ! [X3: int] :
% 0.24/0.49              ( ( member_int @ X3 @ N4 )
% 0.24/0.49             => ( ( F @ X3 )
% 0.24/0.49                = ( G @ X3 ) ) )
% 0.24/0.49         => ( ( image_int_int @ F @ M2 )
% 0.24/0.49            = ( image_int_int @ G @ N4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_cong
% 0.24/0.49  thf(fact_230_bex__imageD,axiom,
% 0.24/0.49      ! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
% 0.24/0.49        ( ? [X2: set_nat] :
% 0.24/0.49            ( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A2 ) )
% 0.24/0.49            & ( P @ X2 ) )
% 0.24/0.49       => ? [X3: nat] :
% 0.24/0.49            ( ( member_nat @ X3 @ A2 )
% 0.24/0.49            & ( P @ ( F @ X3 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % bex_imageD
% 0.24/0.49  thf(fact_231_bex__imageD,axiom,
% 0.24/0.49      ! [F: nat > nat,A2: set_nat,P: nat > $o] :
% 0.24/0.49        ( ? [X2: nat] :
% 0.24/0.49            ( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
% 0.24/0.49            & ( P @ X2 ) )
% 0.24/0.49       => ? [X3: nat] :
% 0.24/0.49            ( ( member_nat @ X3 @ A2 )
% 0.24/0.49            & ( P @ ( F @ X3 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % bex_imageD
% 0.24/0.49  thf(fact_232_bex__imageD,axiom,
% 0.24/0.49      ! [F: nat > int,A2: set_nat,P: int > $o] :
% 0.24/0.49        ( ? [X2: int] :
% 0.24/0.49            ( ( member_int @ X2 @ ( image_nat_int @ F @ A2 ) )
% 0.24/0.49            & ( P @ X2 ) )
% 0.24/0.49       => ? [X3: nat] :
% 0.24/0.49            ( ( member_nat @ X3 @ A2 )
% 0.24/0.49            & ( P @ ( F @ X3 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % bex_imageD
% 0.24/0.49  thf(fact_233_bex__imageD,axiom,
% 0.24/0.49      ! [F: int > nat,A2: set_int,P: nat > $o] :
% 0.24/0.49        ( ? [X2: nat] :
% 0.24/0.49            ( ( member_nat @ X2 @ ( image_int_nat @ F @ A2 ) )
% 0.24/0.49            & ( P @ X2 ) )
% 0.24/0.49       => ? [X3: int] :
% 0.24/0.49            ( ( member_int @ X3 @ A2 )
% 0.24/0.49            & ( P @ ( F @ X3 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % bex_imageD
% 0.24/0.49  thf(fact_234_bex__imageD,axiom,
% 0.24/0.49      ! [F: int > int,A2: set_int,P: int > $o] :
% 0.24/0.49        ( ? [X2: int] :
% 0.24/0.49            ( ( member_int @ X2 @ ( image_int_int @ F @ A2 ) )
% 0.24/0.49            & ( P @ X2 ) )
% 0.24/0.49       => ? [X3: int] :
% 0.24/0.49            ( ( member_int @ X3 @ A2 )
% 0.24/0.49            & ( P @ ( F @ X3 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % bex_imageD
% 0.24/0.49  thf(fact_235_image__iff,axiom,
% 0.24/0.49      ! [Z: set_nat,F: nat > set_nat,A2: set_nat] :
% 0.24/0.49        ( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A2 ) )
% 0.24/0.49        = ( ? [X4: nat] :
% 0.24/0.49              ( ( member_nat @ X4 @ A2 )
% 0.24/0.49              & ( Z
% 0.24/0.49                = ( F @ X4 ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_iff
% 0.24/0.49  thf(fact_236_image__iff,axiom,
% 0.24/0.49      ! [Z: nat,F: nat > nat,A2: set_nat] :
% 0.24/0.49        ( ( member_nat @ Z @ ( image_nat_nat @ F @ A2 ) )
% 0.24/0.49        = ( ? [X4: nat] :
% 0.24/0.49              ( ( member_nat @ X4 @ A2 )
% 0.24/0.49              & ( Z
% 0.24/0.49                = ( F @ X4 ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_iff
% 0.24/0.49  thf(fact_237_image__iff,axiom,
% 0.24/0.49      ! [Z: nat,F: int > nat,A2: set_int] :
% 0.24/0.49        ( ( member_nat @ Z @ ( image_int_nat @ F @ A2 ) )
% 0.24/0.49        = ( ? [X4: int] :
% 0.24/0.49              ( ( member_int @ X4 @ A2 )
% 0.24/0.49              & ( Z
% 0.24/0.49                = ( F @ X4 ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_iff
% 0.24/0.49  thf(fact_238_image__iff,axiom,
% 0.24/0.49      ! [Z: int,F: nat > int,A2: set_nat] :
% 0.24/0.49        ( ( member_int @ Z @ ( image_nat_int @ F @ A2 ) )
% 0.24/0.49        = ( ? [X4: nat] :
% 0.24/0.49              ( ( member_nat @ X4 @ A2 )
% 0.24/0.49              & ( Z
% 0.24/0.49                = ( F @ X4 ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_iff
% 0.24/0.49  thf(fact_239_image__iff,axiom,
% 0.24/0.49      ! [Z: int,F: int > int,A2: set_int] :
% 0.24/0.49        ( ( member_int @ Z @ ( image_int_int @ F @ A2 ) )
% 0.24/0.49        = ( ? [X4: int] :
% 0.24/0.49              ( ( member_int @ X4 @ A2 )
% 0.24/0.49              & ( Z
% 0.24/0.49                = ( F @ X4 ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_iff
% 0.24/0.49  thf(fact_240_imageI,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,F: complex > complex] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( member_complex @ ( F @ X ) @ ( image_1468599708987790691omplex @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % imageI
% 0.24/0.49  thf(fact_241_imageI,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,F: complex > real] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( member_real @ ( F @ X ) @ ( image_complex_real @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % imageI
% 0.24/0.49  thf(fact_242_imageI,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,F: complex > nat] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( member_nat @ ( F @ X ) @ ( image_complex_nat @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % imageI
% 0.24/0.49  thf(fact_243_imageI,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,F: complex > int] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ( member_int @ ( F @ X ) @ ( image_complex_int @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % imageI
% 0.24/0.49  thf(fact_244_imageI,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,F: real > complex] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( member_complex @ ( F @ X ) @ ( image_real_complex @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % imageI
% 0.24/0.49  thf(fact_245_imageI,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,F: real > real] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % imageI
% 0.24/0.49  thf(fact_246_imageI,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,F: real > nat] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % imageI
% 0.24/0.49  thf(fact_247_imageI,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,F: real > int] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ( member_int @ ( F @ X ) @ ( image_real_int @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % imageI
% 0.24/0.49  thf(fact_248_imageI,axiom,
% 0.24/0.49      ! [X: nat,A2: set_nat,F: nat > complex] :
% 0.24/0.49        ( ( member_nat @ X @ A2 )
% 0.24/0.49       => ( member_complex @ ( F @ X ) @ ( image_nat_complex @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % imageI
% 0.24/0.49  thf(fact_249_imageI,axiom,
% 0.24/0.49      ! [X: nat,A2: set_nat,F: nat > real] :
% 0.24/0.49        ( ( member_nat @ X @ A2 )
% 0.24/0.49       => ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % imageI
% 0.24/0.49  thf(fact_250_Suc__nat__number__of__add,axiom,
% 0.24/0.49      ! [V: num,N: nat] :
% 0.24/0.49        ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 0.24/0.49        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_nat_number_of_add
% 0.24/0.49  thf(fact_251_mk__disjoint__insert,axiom,
% 0.24/0.49      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( member_VEBT_VEBT @ A @ A2 )
% 0.24/0.49       => ? [B3: set_VEBT_VEBT] :
% 0.24/0.49            ( ( A2
% 0.24/0.49              = ( insert_VEBT_VEBT @ A @ B3 ) )
% 0.24/0.49            & ~ ( member_VEBT_VEBT @ A @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mk_disjoint_insert
% 0.24/0.49  thf(fact_252_mk__disjoint__insert,axiom,
% 0.24/0.49      ! [A: $o,A2: set_o] :
% 0.24/0.49        ( ( member_o @ A @ A2 )
% 0.24/0.49       => ? [B3: set_o] :
% 0.24/0.49            ( ( A2
% 0.24/0.49              = ( insert_o @ A @ B3 ) )
% 0.24/0.49            & ~ ( member_o @ A @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mk_disjoint_insert
% 0.24/0.49  thf(fact_253_mk__disjoint__insert,axiom,
% 0.24/0.49      ! [A: complex,A2: set_complex] :
% 0.24/0.49        ( ( member_complex @ A @ A2 )
% 0.24/0.49       => ? [B3: set_complex] :
% 0.24/0.49            ( ( A2
% 0.24/0.49              = ( insert_complex @ A @ B3 ) )
% 0.24/0.49            & ~ ( member_complex @ A @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mk_disjoint_insert
% 0.24/0.49  thf(fact_254_mk__disjoint__insert,axiom,
% 0.24/0.49      ! [A: real,A2: set_real] :
% 0.24/0.49        ( ( member_real @ A @ A2 )
% 0.24/0.49       => ? [B3: set_real] :
% 0.24/0.49            ( ( A2
% 0.24/0.49              = ( insert_real @ A @ B3 ) )
% 0.24/0.49            & ~ ( member_real @ A @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mk_disjoint_insert
% 0.24/0.49  thf(fact_255_mk__disjoint__insert,axiom,
% 0.24/0.49      ! [A: set_nat,A2: set_set_nat] :
% 0.24/0.49        ( ( member_set_nat @ A @ A2 )
% 0.24/0.49       => ? [B3: set_set_nat] :
% 0.24/0.49            ( ( A2
% 0.24/0.49              = ( insert_set_nat @ A @ B3 ) )
% 0.24/0.49            & ~ ( member_set_nat @ A @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mk_disjoint_insert
% 0.24/0.49  thf(fact_256_mk__disjoint__insert,axiom,
% 0.24/0.49      ! [A: nat,A2: set_nat] :
% 0.24/0.49        ( ( member_nat @ A @ A2 )
% 0.24/0.49       => ? [B3: set_nat] :
% 0.24/0.49            ( ( A2
% 0.24/0.49              = ( insert_nat @ A @ B3 ) )
% 0.24/0.49            & ~ ( member_nat @ A @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mk_disjoint_insert
% 0.24/0.49  thf(fact_257_mk__disjoint__insert,axiom,
% 0.24/0.49      ! [A: int,A2: set_int] :
% 0.24/0.49        ( ( member_int @ A @ A2 )
% 0.24/0.49       => ? [B3: set_int] :
% 0.24/0.49            ( ( A2
% 0.24/0.49              = ( insert_int @ A @ B3 ) )
% 0.24/0.49            & ~ ( member_int @ A @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % mk_disjoint_insert
% 0.24/0.49  thf(fact_258_insert__commute,axiom,
% 0.24/0.49      ! [X: vEBT_VEBT,Y: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( insert_VEBT_VEBT @ X @ ( insert_VEBT_VEBT @ Y @ A2 ) )
% 0.24/0.49        = ( insert_VEBT_VEBT @ Y @ ( insert_VEBT_VEBT @ X @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_commute
% 0.24/0.49  thf(fact_259_insert__commute,axiom,
% 0.24/0.49      ! [X: nat,Y: nat,A2: set_nat] :
% 0.24/0.49        ( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
% 0.24/0.49        = ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_commute
% 0.24/0.49  thf(fact_260_insert__commute,axiom,
% 0.24/0.49      ! [X: int,Y: int,A2: set_int] :
% 0.24/0.49        ( ( insert_int @ X @ ( insert_int @ Y @ A2 ) )
% 0.24/0.49        = ( insert_int @ Y @ ( insert_int @ X @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_commute
% 0.24/0.49  thf(fact_261_insert__commute,axiom,
% 0.24/0.49      ! [X: real,Y: real,A2: set_real] :
% 0.24/0.49        ( ( insert_real @ X @ ( insert_real @ Y @ A2 ) )
% 0.24/0.49        = ( insert_real @ Y @ ( insert_real @ X @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_commute
% 0.24/0.49  thf(fact_262_insert__commute,axiom,
% 0.24/0.49      ! [X: $o,Y: $o,A2: set_o] :
% 0.24/0.49        ( ( insert_o @ X @ ( insert_o @ Y @ A2 ) )
% 0.24/0.49        = ( insert_o @ Y @ ( insert_o @ X @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_commute
% 0.24/0.49  thf(fact_263_insert__eq__iff,axiom,
% 0.24/0.49      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.24/0.49        ( ~ ( member_VEBT_VEBT @ A @ A2 )
% 0.24/0.49       => ( ~ ( member_VEBT_VEBT @ B @ B2 )
% 0.24/0.49         => ( ( ( insert_VEBT_VEBT @ A @ A2 )
% 0.24/0.49              = ( insert_VEBT_VEBT @ B @ B2 ) )
% 0.24/0.49            = ( ( ( A = B )
% 0.24/0.49               => ( A2 = B2 ) )
% 0.24/0.49              & ( ( A != B )
% 0.24/0.49               => ? [C2: set_VEBT_VEBT] :
% 0.24/0.49                    ( ( A2
% 0.24/0.49                      = ( insert_VEBT_VEBT @ B @ C2 ) )
% 0.24/0.49                    & ~ ( member_VEBT_VEBT @ B @ C2 )
% 0.24/0.49                    & ( B2
% 0.24/0.49                      = ( insert_VEBT_VEBT @ A @ C2 ) )
% 0.24/0.49                    & ~ ( member_VEBT_VEBT @ A @ C2 ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_eq_iff
% 0.24/0.49  thf(fact_264_insert__eq__iff,axiom,
% 0.24/0.49      ! [A: $o,A2: set_o,B: $o,B2: set_o] :
% 0.24/0.49        ( ~ ( member_o @ A @ A2 )
% 0.24/0.49       => ( ~ ( member_o @ B @ B2 )
% 0.24/0.49         => ( ( ( insert_o @ A @ A2 )
% 0.24/0.49              = ( insert_o @ B @ B2 ) )
% 0.24/0.49            = ( ( ( A = B )
% 0.24/0.49               => ( A2 = B2 ) )
% 0.24/0.49              & ( ( A = ( ~ B ) )
% 0.24/0.49               => ? [C2: set_o] :
% 0.24/0.49                    ( ( A2
% 0.24/0.49                      = ( insert_o @ B @ C2 ) )
% 0.24/0.49                    & ~ ( member_o @ B @ C2 )
% 0.24/0.49                    & ( B2
% 0.24/0.49                      = ( insert_o @ A @ C2 ) )
% 0.24/0.49                    & ~ ( member_o @ A @ C2 ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_eq_iff
% 0.24/0.49  thf(fact_265_insert__eq__iff,axiom,
% 0.24/0.49      ! [A: complex,A2: set_complex,B: complex,B2: set_complex] :
% 0.24/0.49        ( ~ ( member_complex @ A @ A2 )
% 0.24/0.49       => ( ~ ( member_complex @ B @ B2 )
% 0.24/0.49         => ( ( ( insert_complex @ A @ A2 )
% 0.24/0.49              = ( insert_complex @ B @ B2 ) )
% 0.24/0.49            = ( ( ( A = B )
% 0.24/0.49               => ( A2 = B2 ) )
% 0.24/0.49              & ( ( A != B )
% 0.24/0.49               => ? [C2: set_complex] :
% 0.24/0.49                    ( ( A2
% 0.24/0.49                      = ( insert_complex @ B @ C2 ) )
% 0.24/0.49                    & ~ ( member_complex @ B @ C2 )
% 0.24/0.49                    & ( B2
% 0.24/0.49                      = ( insert_complex @ A @ C2 ) )
% 0.24/0.49                    & ~ ( member_complex @ A @ C2 ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_eq_iff
% 0.24/0.49  thf(fact_266_insert__eq__iff,axiom,
% 0.24/0.49      ! [A: real,A2: set_real,B: real,B2: set_real] :
% 0.24/0.49        ( ~ ( member_real @ A @ A2 )
% 0.24/0.49       => ( ~ ( member_real @ B @ B2 )
% 0.24/0.49         => ( ( ( insert_real @ A @ A2 )
% 0.24/0.49              = ( insert_real @ B @ B2 ) )
% 0.24/0.49            = ( ( ( A = B )
% 0.24/0.49               => ( A2 = B2 ) )
% 0.24/0.49              & ( ( A != B )
% 0.24/0.49               => ? [C2: set_real] :
% 0.24/0.49                    ( ( A2
% 0.24/0.49                      = ( insert_real @ B @ C2 ) )
% 0.24/0.49                    & ~ ( member_real @ B @ C2 )
% 0.24/0.49                    & ( B2
% 0.24/0.49                      = ( insert_real @ A @ C2 ) )
% 0.24/0.49                    & ~ ( member_real @ A @ C2 ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_eq_iff
% 0.24/0.49  thf(fact_267_insert__eq__iff,axiom,
% 0.24/0.49      ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
% 0.24/0.49        ( ~ ( member_set_nat @ A @ A2 )
% 0.24/0.49       => ( ~ ( member_set_nat @ B @ B2 )
% 0.24/0.49         => ( ( ( insert_set_nat @ A @ A2 )
% 0.24/0.49              = ( insert_set_nat @ B @ B2 ) )
% 0.24/0.49            = ( ( ( A = B )
% 0.24/0.49               => ( A2 = B2 ) )
% 0.24/0.49              & ( ( A != B )
% 0.24/0.49               => ? [C2: set_set_nat] :
% 0.24/0.49                    ( ( A2
% 0.24/0.49                      = ( insert_set_nat @ B @ C2 ) )
% 0.24/0.49                    & ~ ( member_set_nat @ B @ C2 )
% 0.24/0.49                    & ( B2
% 0.24/0.49                      = ( insert_set_nat @ A @ C2 ) )
% 0.24/0.49                    & ~ ( member_set_nat @ A @ C2 ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_eq_iff
% 0.24/0.49  thf(fact_268_insert__eq__iff,axiom,
% 0.24/0.49      ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
% 0.24/0.49        ( ~ ( member_nat @ A @ A2 )
% 0.24/0.49       => ( ~ ( member_nat @ B @ B2 )
% 0.24/0.49         => ( ( ( insert_nat @ A @ A2 )
% 0.24/0.49              = ( insert_nat @ B @ B2 ) )
% 0.24/0.49            = ( ( ( A = B )
% 0.24/0.49               => ( A2 = B2 ) )
% 0.24/0.49              & ( ( A != B )
% 0.24/0.49               => ? [C2: set_nat] :
% 0.24/0.49                    ( ( A2
% 0.24/0.49                      = ( insert_nat @ B @ C2 ) )
% 0.24/0.49                    & ~ ( member_nat @ B @ C2 )
% 0.24/0.49                    & ( B2
% 0.24/0.49                      = ( insert_nat @ A @ C2 ) )
% 0.24/0.49                    & ~ ( member_nat @ A @ C2 ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_eq_iff
% 0.24/0.49  thf(fact_269_insert__eq__iff,axiom,
% 0.24/0.49      ! [A: int,A2: set_int,B: int,B2: set_int] :
% 0.24/0.49        ( ~ ( member_int @ A @ A2 )
% 0.24/0.49       => ( ~ ( member_int @ B @ B2 )
% 0.24/0.49         => ( ( ( insert_int @ A @ A2 )
% 0.24/0.49              = ( insert_int @ B @ B2 ) )
% 0.24/0.49            = ( ( ( A = B )
% 0.24/0.49               => ( A2 = B2 ) )
% 0.24/0.49              & ( ( A != B )
% 0.24/0.49               => ? [C2: set_int] :
% 0.24/0.49                    ( ( A2
% 0.24/0.49                      = ( insert_int @ B @ C2 ) )
% 0.24/0.49                    & ~ ( member_int @ B @ C2 )
% 0.24/0.49                    & ( B2
% 0.24/0.49                      = ( insert_int @ A @ C2 ) )
% 0.24/0.49                    & ~ ( member_int @ A @ C2 ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_eq_iff
% 0.24/0.49  thf(fact_270_insert__absorb,axiom,
% 0.24/0.49      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( member_VEBT_VEBT @ A @ A2 )
% 0.24/0.49       => ( ( insert_VEBT_VEBT @ A @ A2 )
% 0.24/0.49          = A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb
% 0.24/0.49  thf(fact_271_insert__absorb,axiom,
% 0.24/0.49      ! [A: $o,A2: set_o] :
% 0.24/0.49        ( ( member_o @ A @ A2 )
% 0.24/0.49       => ( ( insert_o @ A @ A2 )
% 0.24/0.49          = A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb
% 0.24/0.49  thf(fact_272_insert__absorb,axiom,
% 0.24/0.49      ! [A: complex,A2: set_complex] :
% 0.24/0.49        ( ( member_complex @ A @ A2 )
% 0.24/0.49       => ( ( insert_complex @ A @ A2 )
% 0.24/0.49          = A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb
% 0.24/0.49  thf(fact_273_insert__absorb,axiom,
% 0.24/0.49      ! [A: real,A2: set_real] :
% 0.24/0.49        ( ( member_real @ A @ A2 )
% 0.24/0.49       => ( ( insert_real @ A @ A2 )
% 0.24/0.49          = A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb
% 0.24/0.49  thf(fact_274_insert__absorb,axiom,
% 0.24/0.49      ! [A: set_nat,A2: set_set_nat] :
% 0.24/0.49        ( ( member_set_nat @ A @ A2 )
% 0.24/0.49       => ( ( insert_set_nat @ A @ A2 )
% 0.24/0.49          = A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb
% 0.24/0.49  thf(fact_275_insert__absorb,axiom,
% 0.24/0.49      ! [A: nat,A2: set_nat] :
% 0.24/0.49        ( ( member_nat @ A @ A2 )
% 0.24/0.49       => ( ( insert_nat @ A @ A2 )
% 0.24/0.49          = A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb
% 0.24/0.49  thf(fact_276_insert__absorb,axiom,
% 0.24/0.49      ! [A: int,A2: set_int] :
% 0.24/0.49        ( ( member_int @ A @ A2 )
% 0.24/0.49       => ( ( insert_int @ A @ A2 )
% 0.24/0.49          = A2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_absorb
% 0.24/0.49  thf(fact_277_insert__ident,axiom,
% 0.24/0.49      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.24/0.49        ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 0.24/0.49       => ( ~ ( member_VEBT_VEBT @ X @ B2 )
% 0.24/0.49         => ( ( ( insert_VEBT_VEBT @ X @ A2 )
% 0.24/0.49              = ( insert_VEBT_VEBT @ X @ B2 ) )
% 0.24/0.49            = ( A2 = B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_ident
% 0.24/0.49  thf(fact_278_insert__ident,axiom,
% 0.24/0.49      ! [X: $o,A2: set_o,B2: set_o] :
% 0.24/0.49        ( ~ ( member_o @ X @ A2 )
% 0.24/0.49       => ( ~ ( member_o @ X @ B2 )
% 0.24/0.49         => ( ( ( insert_o @ X @ A2 )
% 0.24/0.49              = ( insert_o @ X @ B2 ) )
% 0.24/0.49            = ( A2 = B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_ident
% 0.24/0.49  thf(fact_279_insert__ident,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,B2: set_complex] :
% 0.24/0.49        ( ~ ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ~ ( member_complex @ X @ B2 )
% 0.24/0.49         => ( ( ( insert_complex @ X @ A2 )
% 0.24/0.49              = ( insert_complex @ X @ B2 ) )
% 0.24/0.49            = ( A2 = B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_ident
% 0.24/0.49  thf(fact_280_insert__ident,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,B2: set_real] :
% 0.24/0.49        ( ~ ( member_real @ X @ A2 )
% 0.24/0.49       => ( ~ ( member_real @ X @ B2 )
% 0.24/0.49         => ( ( ( insert_real @ X @ A2 )
% 0.24/0.49              = ( insert_real @ X @ B2 ) )
% 0.24/0.49            = ( A2 = B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_ident
% 0.24/0.49  thf(fact_281_insert__ident,axiom,
% 0.24/0.49      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.24/0.49        ( ~ ( member_set_nat @ X @ A2 )
% 0.24/0.49       => ( ~ ( member_set_nat @ X @ B2 )
% 0.24/0.49         => ( ( ( insert_set_nat @ X @ A2 )
% 0.24/0.49              = ( insert_set_nat @ X @ B2 ) )
% 0.24/0.49            = ( A2 = B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_ident
% 0.24/0.49  thf(fact_282_insert__ident,axiom,
% 0.24/0.49      ! [X: nat,A2: set_nat,B2: set_nat] :
% 0.24/0.49        ( ~ ( member_nat @ X @ A2 )
% 0.24/0.49       => ( ~ ( member_nat @ X @ B2 )
% 0.24/0.49         => ( ( ( insert_nat @ X @ A2 )
% 0.24/0.49              = ( insert_nat @ X @ B2 ) )
% 0.24/0.49            = ( A2 = B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_ident
% 0.24/0.49  thf(fact_283_insert__ident,axiom,
% 0.24/0.49      ! [X: int,A2: set_int,B2: set_int] :
% 0.24/0.49        ( ~ ( member_int @ X @ A2 )
% 0.24/0.49       => ( ~ ( member_int @ X @ B2 )
% 0.24/0.49         => ( ( ( insert_int @ X @ A2 )
% 0.24/0.49              = ( insert_int @ X @ B2 ) )
% 0.24/0.49            = ( A2 = B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_ident
% 0.24/0.49  thf(fact_284_Set_Oset__insert,axiom,
% 0.24/0.49      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( member_VEBT_VEBT @ X @ A2 )
% 0.24/0.49       => ~ ! [B3: set_VEBT_VEBT] :
% 0.24/0.49              ( ( A2
% 0.24/0.49                = ( insert_VEBT_VEBT @ X @ B3 ) )
% 0.24/0.49             => ( member_VEBT_VEBT @ X @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Set.set_insert
% 0.24/0.49  thf(fact_285_Set_Oset__insert,axiom,
% 0.24/0.49      ! [X: $o,A2: set_o] :
% 0.24/0.49        ( ( member_o @ X @ A2 )
% 0.24/0.49       => ~ ! [B3: set_o] :
% 0.24/0.49              ( ( A2
% 0.24/0.49                = ( insert_o @ X @ B3 ) )
% 0.24/0.49             => ( member_o @ X @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Set.set_insert
% 0.24/0.49  thf(fact_286_Set_Oset__insert,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex] :
% 0.24/0.49        ( ( member_complex @ X @ A2 )
% 0.24/0.49       => ~ ! [B3: set_complex] :
% 0.24/0.49              ( ( A2
% 0.24/0.49                = ( insert_complex @ X @ B3 ) )
% 0.24/0.49             => ( member_complex @ X @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Set.set_insert
% 0.24/0.49  thf(fact_287_Set_Oset__insert,axiom,
% 0.24/0.49      ! [X: real,A2: set_real] :
% 0.24/0.49        ( ( member_real @ X @ A2 )
% 0.24/0.49       => ~ ! [B3: set_real] :
% 0.24/0.49              ( ( A2
% 0.24/0.49                = ( insert_real @ X @ B3 ) )
% 0.24/0.49             => ( member_real @ X @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Set.set_insert
% 0.24/0.49  thf(fact_288_Set_Oset__insert,axiom,
% 0.24/0.49      ! [X: set_nat,A2: set_set_nat] :
% 0.24/0.49        ( ( member_set_nat @ X @ A2 )
% 0.24/0.49       => ~ ! [B3: set_set_nat] :
% 0.24/0.49              ( ( A2
% 0.24/0.49                = ( insert_set_nat @ X @ B3 ) )
% 0.24/0.49             => ( member_set_nat @ X @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Set.set_insert
% 0.24/0.49  thf(fact_289_Set_Oset__insert,axiom,
% 0.24/0.49      ! [X: nat,A2: set_nat] :
% 0.24/0.49        ( ( member_nat @ X @ A2 )
% 0.24/0.49       => ~ ! [B3: set_nat] :
% 0.24/0.49              ( ( A2
% 0.24/0.49                = ( insert_nat @ X @ B3 ) )
% 0.24/0.49             => ( member_nat @ X @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Set.set_insert
% 0.24/0.49  thf(fact_290_Set_Oset__insert,axiom,
% 0.24/0.49      ! [X: int,A2: set_int] :
% 0.24/0.49        ( ( member_int @ X @ A2 )
% 0.24/0.49       => ~ ! [B3: set_int] :
% 0.24/0.49              ( ( A2
% 0.24/0.49                = ( insert_int @ X @ B3 ) )
% 0.24/0.49             => ( member_int @ X @ B3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Set.set_insert
% 0.24/0.49  thf(fact_291_insertI2,axiom,
% 0.24/0.49      ! [A: vEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
% 0.24/0.49        ( ( member_VEBT_VEBT @ A @ B2 )
% 0.24/0.49       => ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI2
% 0.24/0.49  thf(fact_292_insertI2,axiom,
% 0.24/0.49      ! [A: $o,B2: set_o,B: $o] :
% 0.24/0.49        ( ( member_o @ A @ B2 )
% 0.24/0.49       => ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI2
% 0.24/0.49  thf(fact_293_insertI2,axiom,
% 0.24/0.49      ! [A: complex,B2: set_complex,B: complex] :
% 0.24/0.49        ( ( member_complex @ A @ B2 )
% 0.24/0.49       => ( member_complex @ A @ ( insert_complex @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI2
% 0.24/0.49  thf(fact_294_insertI2,axiom,
% 0.24/0.49      ! [A: real,B2: set_real,B: real] :
% 0.24/0.49        ( ( member_real @ A @ B2 )
% 0.24/0.49       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI2
% 0.24/0.49  thf(fact_295_insertI2,axiom,
% 0.24/0.49      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 0.24/0.49        ( ( member_set_nat @ A @ B2 )
% 0.24/0.49       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI2
% 0.24/0.49  thf(fact_296_insertI2,axiom,
% 0.24/0.49      ! [A: nat,B2: set_nat,B: nat] :
% 0.24/0.49        ( ( member_nat @ A @ B2 )
% 0.24/0.49       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI2
% 0.24/0.49  thf(fact_297_insertI2,axiom,
% 0.24/0.49      ! [A: int,B2: set_int,B: int] :
% 0.24/0.49        ( ( member_int @ A @ B2 )
% 0.24/0.49       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI2
% 0.24/0.49  thf(fact_298_insertI1,axiom,
% 0.24/0.49      ! [A: vEBT_VEBT,B2: set_VEBT_VEBT] : ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI1
% 0.24/0.49  thf(fact_299_insertI1,axiom,
% 0.24/0.49      ! [A: $o,B2: set_o] : ( member_o @ A @ ( insert_o @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI1
% 0.24/0.49  thf(fact_300_insertI1,axiom,
% 0.24/0.49      ! [A: complex,B2: set_complex] : ( member_complex @ A @ ( insert_complex @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI1
% 0.24/0.49  thf(fact_301_insertI1,axiom,
% 0.24/0.49      ! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI1
% 0.24/0.49  thf(fact_302_insertI1,axiom,
% 0.24/0.49      ! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI1
% 0.24/0.49  thf(fact_303_insertI1,axiom,
% 0.24/0.49      ! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI1
% 0.24/0.49  thf(fact_304_insertI1,axiom,
% 0.24/0.49      ! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertI1
% 0.24/0.49  thf(fact_305_insertE,axiom,
% 0.24/0.49      ! [A: vEBT_VEBT,B: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ A2 ) )
% 0.24/0.49       => ( ( A != B )
% 0.24/0.49         => ( member_VEBT_VEBT @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertE
% 0.24/0.49  thf(fact_306_insertE,axiom,
% 0.24/0.49      ! [A: $o,B: $o,A2: set_o] :
% 0.24/0.49        ( ( member_o @ A @ ( insert_o @ B @ A2 ) )
% 0.24/0.49       => ( ( A = ( ~ B ) )
% 0.24/0.49         => ( member_o @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertE
% 0.24/0.49  thf(fact_307_insertE,axiom,
% 0.24/0.49      ! [A: complex,B: complex,A2: set_complex] :
% 0.24/0.49        ( ( member_complex @ A @ ( insert_complex @ B @ A2 ) )
% 0.24/0.49       => ( ( A != B )
% 0.24/0.49         => ( member_complex @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertE
% 0.24/0.49  thf(fact_308_insertE,axiom,
% 0.24/0.49      ! [A: real,B: real,A2: set_real] :
% 0.24/0.49        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 0.24/0.49       => ( ( A != B )
% 0.24/0.49         => ( member_real @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertE
% 0.24/0.49  thf(fact_309_insertE,axiom,
% 0.24/0.49      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 0.24/0.49        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 0.24/0.49       => ( ( A != B )
% 0.24/0.49         => ( member_set_nat @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertE
% 0.24/0.49  thf(fact_310_insertE,axiom,
% 0.24/0.49      ! [A: nat,B: nat,A2: set_nat] :
% 0.24/0.49        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 0.24/0.49       => ( ( A != B )
% 0.24/0.49         => ( member_nat @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertE
% 0.24/0.49  thf(fact_311_insertE,axiom,
% 0.24/0.49      ! [A: int,B: int,A2: set_int] :
% 0.24/0.49        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 0.24/0.49       => ( ( A != B )
% 0.24/0.49         => ( member_int @ A @ A2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insertE
% 0.24/0.49  thf(fact_312_nat__int__comparison_I1_J,axiom,
% 0.24/0.49      ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 0.24/0.49      = ( ^ [A3: nat,B4: nat] :
% 0.24/0.49            ( ( semiri1314217659103216013at_int @ A3 )
% 0.24/0.49            = ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % nat_int_comparison(1)
% 0.24/0.49  thf(fact_313_int__if,axiom,
% 0.24/0.49      ! [P: $o,A: nat,B: nat] :
% 0.24/0.49        ( ( P
% 0.24/0.49         => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 0.24/0.49            = ( semiri1314217659103216013at_int @ A ) ) )
% 0.24/0.49        & ( ~ P
% 0.24/0.49         => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 0.24/0.49            = ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % int_if
% 0.24/0.49  thf(fact_314_int__int__eq,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ( semiri1314217659103216013at_int @ M )
% 0.24/0.49          = ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.49        = ( M = N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % int_int_eq
% 0.24/0.49  thf(fact_315_verit__eq__simplify_I10_J,axiom,
% 0.24/0.49      ! [X22: num] :
% 0.24/0.49        ( one
% 0.24/0.49       != ( bit0 @ X22 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_eq_simplify(10)
% 0.24/0.49  thf(fact_316_int__ops_I3_J,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( numeral_numeral_int @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % int_ops(3)
% 0.24/0.49  thf(fact_317__C2_C,axiom,
% 0.24/0.49      ord_less_eq_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ na ) ) ) @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % "2"
% 0.24/0.49  thf(fact_318__C1_C,axiom,
% 0.24/0.49      ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ na ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ).
% 0.24/0.49  
% 0.24/0.49  % "1"
% 0.24/0.49  thf(fact_319__C00_C,axiom,
% 0.24/0.49      ( ( ord_less_eq_nat @ one_one_nat @ na )
% 0.24/0.49      & ( ( suc @ na )
% 0.24/0.49        = m ) ) ).
% 0.24/0.49  
% 0.24/0.49  % "00"
% 0.24/0.49  thf(fact_320__C0_C,axiom,
% 0.24/0.49      ord_less_eq_nat @ na @ m ).
% 0.24/0.49  
% 0.24/0.49  % "0"
% 0.24/0.49  thf(fact_321_dbl__simps_I3_J,axiom,
% 0.24/0.49      ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 0.24/0.49      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(3)
% 0.24/0.49  thf(fact_322_dbl__simps_I3_J,axiom,
% 0.24/0.49      ( ( neg_numeral_dbl_real @ one_one_real )
% 0.24/0.49      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(3)
% 0.24/0.49  thf(fact_323_dbl__simps_I3_J,axiom,
% 0.24/0.49      ( ( neg_numeral_dbl_rat @ one_one_rat )
% 0.24/0.49      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(3)
% 0.24/0.49  thf(fact_324_dbl__simps_I3_J,axiom,
% 0.24/0.49      ( ( neg_numeral_dbl_int @ one_one_int )
% 0.24/0.49      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(3)
% 0.24/0.49  thf(fact_325_VEBT_Oinject_I1_J,axiom,
% 0.24/0.49      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 0.24/0.49        ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.24/0.49          = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 0.24/0.49        = ( ( X11 = Y11 )
% 0.24/0.49          & ( X12 = Y12 )
% 0.24/0.49          & ( X13 = Y13 )
% 0.24/0.49          & ( X14 = Y14 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % VEBT.inject(1)
% 0.24/0.49  thf(fact_326__C3_Ohyps_C_I2_J,axiom,
% 0.24/0.49      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.24/0.49      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 0.24/0.49  
% 0.24/0.49  % "3.hyps"(2)
% 0.24/0.49  thf(fact_327_valid__eq2,axiom,
% 0.24/0.49      ! [T: vEBT_VEBT,D: nat] :
% 0.24/0.49        ( ( vEBT_VEBT_valid @ T @ D )
% 0.24/0.49       => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.24/0.49  
% 0.24/0.49  % valid_eq2
% 0.24/0.49  thf(fact_328_valid__eq1,axiom,
% 0.24/0.49      ! [T: vEBT_VEBT,D: nat] :
% 0.24/0.49        ( ( vEBT_invar_vebt @ T @ D )
% 0.24/0.49       => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.24/0.49  
% 0.24/0.49  % valid_eq1
% 0.24/0.49  thf(fact_329_valid__eq,axiom,
% 0.24/0.49      vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.24/0.49  
% 0.24/0.49  % valid_eq
% 0.24/0.49  thf(fact_330_add__left__cancel,axiom,
% 0.24/0.49      ! [A: real,B: real,C: real] :
% 0.24/0.49        ( ( ( plus_plus_real @ A @ B )
% 0.24/0.49          = ( plus_plus_real @ A @ C ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_cancel
% 0.24/0.49  thf(fact_331_add__left__cancel,axiom,
% 0.24/0.49      ! [A: rat,B: rat,C: rat] :
% 0.24/0.49        ( ( ( plus_plus_rat @ A @ B )
% 0.24/0.49          = ( plus_plus_rat @ A @ C ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_cancel
% 0.24/0.49  thf(fact_332_add__left__cancel,axiom,
% 0.24/0.49      ! [A: nat,B: nat,C: nat] :
% 0.24/0.49        ( ( ( plus_plus_nat @ A @ B )
% 0.24/0.49          = ( plus_plus_nat @ A @ C ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_cancel
% 0.24/0.49  thf(fact_333_add__left__cancel,axiom,
% 0.24/0.49      ! [A: int,B: int,C: int] :
% 0.24/0.49        ( ( ( plus_plus_int @ A @ B )
% 0.24/0.49          = ( plus_plus_int @ A @ C ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_cancel
% 0.24/0.49  thf(fact_334_add__left__cancel,axiom,
% 0.24/0.49      ! [A: complex,B: complex,C: complex] :
% 0.24/0.49        ( ( ( plus_plus_complex @ A @ B )
% 0.24/0.49          = ( plus_plus_complex @ A @ C ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_cancel
% 0.24/0.49  thf(fact_335_add__right__cancel,axiom,
% 0.24/0.49      ! [B: real,A: real,C: real] :
% 0.24/0.49        ( ( ( plus_plus_real @ B @ A )
% 0.24/0.49          = ( plus_plus_real @ C @ A ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_cancel
% 0.24/0.49  thf(fact_336_add__right__cancel,axiom,
% 0.24/0.49      ! [B: rat,A: rat,C: rat] :
% 0.24/0.49        ( ( ( plus_plus_rat @ B @ A )
% 0.24/0.49          = ( plus_plus_rat @ C @ A ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_cancel
% 0.24/0.49  thf(fact_337_add__right__cancel,axiom,
% 0.24/0.49      ! [B: nat,A: nat,C: nat] :
% 0.24/0.49        ( ( ( plus_plus_nat @ B @ A )
% 0.24/0.49          = ( plus_plus_nat @ C @ A ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_cancel
% 0.24/0.49  thf(fact_338_add__right__cancel,axiom,
% 0.24/0.49      ! [B: int,A: int,C: int] :
% 0.24/0.49        ( ( ( plus_plus_int @ B @ A )
% 0.24/0.49          = ( plus_plus_int @ C @ A ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_cancel
% 0.24/0.49  thf(fact_339_add__right__cancel,axiom,
% 0.24/0.49      ! [B: complex,A: complex,C: complex] :
% 0.24/0.49        ( ( ( plus_plus_complex @ B @ A )
% 0.24/0.49          = ( plus_plus_complex @ C @ A ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_cancel
% 0.24/0.49  thf(fact_340_numeral__le__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.24/0.49        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_le_iff
% 0.24/0.49  thf(fact_341_numeral__le__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.24/0.49        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_le_iff
% 0.24/0.49  thf(fact_342_numeral__le__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_le_iff
% 0.24/0.49  thf(fact_343_numeral__le__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.24/0.49        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_le_iff
% 0.24/0.49  thf(fact_344_add__le__cancel__right,axiom,
% 0.24/0.49      ! [A: real,C: real,B: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.24/0.49        = ( ord_less_eq_real @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_cancel_right
% 0.24/0.49  thf(fact_345_add__le__cancel__right,axiom,
% 0.24/0.49      ! [A: rat,C: rat,B: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.24/0.49        = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_cancel_right
% 0.24/0.49  thf(fact_346_add__le__cancel__right,axiom,
% 0.24/0.49      ! [A: nat,C: nat,B: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.24/0.49        = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_cancel_right
% 0.24/0.49  thf(fact_347_add__le__cancel__right,axiom,
% 0.24/0.49      ! [A: int,C: int,B: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.24/0.49        = ( ord_less_eq_int @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_cancel_right
% 0.24/0.49  thf(fact_348_add__le__cancel__left,axiom,
% 0.24/0.49      ! [C: real,A: real,B: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.24/0.49        = ( ord_less_eq_real @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_cancel_left
% 0.24/0.49  thf(fact_349_add__le__cancel__left,axiom,
% 0.24/0.49      ! [C: rat,A: rat,B: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.24/0.49        = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_cancel_left
% 0.24/0.49  thf(fact_350_add__le__cancel__left,axiom,
% 0.24/0.49      ! [C: nat,A: nat,B: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.24/0.49        = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_cancel_left
% 0.24/0.49  thf(fact_351_add__le__cancel__left,axiom,
% 0.24/0.49      ! [C: int,A: int,B: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.24/0.49        = ( ord_less_eq_int @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_cancel_left
% 0.24/0.49  thf(fact_352_Suc__le__mono,axiom,
% 0.24/0.49      ! [N: nat,M: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.24/0.49        = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_le_mono
% 0.24/0.49  thf(fact_353_nat__add__left__cancel__le,axiom,
% 0.24/0.49      ! [K: nat,M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % nat_add_left_cancel_le
% 0.24/0.49  thf(fact_354_set__n__deg__not__0,axiom,
% 0.24/0.49      ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
% 0.24/0.49        ( ! [X3: vEBT_VEBT] :
% 0.24/0.49            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.24/0.49           => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.24/0.49       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.24/0.49            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.24/0.49         => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % set_n_deg_not_0
% 0.24/0.49  thf(fact_355_of__nat__le__iff,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_iff
% 0.24/0.49  thf(fact_356_of__nat__le__iff,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_iff
% 0.24/0.49  thf(fact_357_of__nat__le__iff,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_iff
% 0.24/0.49  thf(fact_358_of__nat__le__iff,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_iff
% 0.24/0.49  thf(fact_359_dbl__simps_I5_J,axiom,
% 0.24/0.49      ! [K: num] :
% 0.24/0.49        ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.24/0.49        = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(5)
% 0.24/0.49  thf(fact_360_dbl__simps_I5_J,axiom,
% 0.24/0.49      ! [K: num] :
% 0.24/0.49        ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 0.24/0.49        = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(5)
% 0.24/0.49  thf(fact_361_dbl__simps_I5_J,axiom,
% 0.24/0.49      ! [K: num] :
% 0.24/0.49        ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 0.24/0.49        = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(5)
% 0.24/0.49  thf(fact_362_dbl__simps_I5_J,axiom,
% 0.24/0.49      ! [K: num] :
% 0.24/0.49        ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 0.24/0.49        = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(5)
% 0.24/0.49  thf(fact_363_numeral__le__one__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.24/0.49        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_le_one_iff
% 0.24/0.49  thf(fact_364_numeral__le__one__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.24/0.49        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_le_one_iff
% 0.24/0.49  thf(fact_365_numeral__le__one__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.24/0.49        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_le_one_iff
% 0.24/0.49  thf(fact_366_numeral__le__one__iff,axiom,
% 0.24/0.49      ! [N: num] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.24/0.49        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_le_one_iff
% 0.24/0.49  thf(fact_367_ceiling__le__numeral,axiom,
% 0.24/0.49      ! [X: real,V: num] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.24/0.49        = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_le_numeral
% 0.24/0.49  thf(fact_368_ceiling__le__numeral,axiom,
% 0.24/0.49      ! [X: rat,V: num] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.24/0.49        = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_le_numeral
% 0.24/0.49  thf(fact_369_ceiling__le__one,axiom,
% 0.24/0.49      ! [X: real] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 0.24/0.49        = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_le_one
% 0.24/0.49  thf(fact_370_ceiling__le__one,axiom,
% 0.24/0.49      ! [X: rat] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 0.24/0.49        = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_le_one
% 0.24/0.49  thf(fact_371_verit__la__disequality,axiom,
% 0.24/0.49      ! [A: rat,B: rat] :
% 0.24/0.49        ( ( A = B )
% 0.24/0.49        | ~ ( ord_less_eq_rat @ A @ B )
% 0.24/0.49        | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_la_disequality
% 0.24/0.49  thf(fact_372_verit__la__disequality,axiom,
% 0.24/0.49      ! [A: num,B: num] :
% 0.24/0.49        ( ( A = B )
% 0.24/0.49        | ~ ( ord_less_eq_num @ A @ B )
% 0.24/0.49        | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_la_disequality
% 0.24/0.49  thf(fact_373_verit__la__disequality,axiom,
% 0.24/0.49      ! [A: nat,B: nat] :
% 0.24/0.49        ( ( A = B )
% 0.24/0.49        | ~ ( ord_less_eq_nat @ A @ B )
% 0.24/0.49        | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_la_disequality
% 0.24/0.49  thf(fact_374_verit__la__disequality,axiom,
% 0.24/0.49      ! [A: int,B: int] :
% 0.24/0.49        ( ( A = B )
% 0.24/0.49        | ~ ( ord_less_eq_int @ A @ B )
% 0.24/0.49        | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_la_disequality
% 0.24/0.49  thf(fact_375_verit__la__generic,axiom,
% 0.24/0.49      ! [A: int,X: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ A @ X )
% 0.24/0.49        | ( A = X )
% 0.24/0.49        | ( ord_less_eq_int @ X @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_la_generic
% 0.24/0.49  thf(fact_376_verit__comp__simplify1_I2_J,axiom,
% 0.24/0.49      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_comp_simplify1(2)
% 0.24/0.49  thf(fact_377_verit__comp__simplify1_I2_J,axiom,
% 0.24/0.49      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_comp_simplify1(2)
% 0.24/0.49  thf(fact_378_verit__comp__simplify1_I2_J,axiom,
% 0.24/0.49      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_comp_simplify1(2)
% 0.24/0.49  thf(fact_379_verit__comp__simplify1_I2_J,axiom,
% 0.24/0.49      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_comp_simplify1(2)
% 0.24/0.49  thf(fact_380_verit__comp__simplify1_I2_J,axiom,
% 0.24/0.49      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % verit_comp_simplify1(2)
% 0.24/0.49  thf(fact_381_Nat_Oex__has__greatest__nat,axiom,
% 0.24/0.49      ! [P: nat > $o,K: nat,B: nat] :
% 0.24/0.49        ( ( P @ K )
% 0.24/0.49       => ( ! [Y4: nat] :
% 0.24/0.49              ( ( P @ Y4 )
% 0.24/0.49             => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.24/0.49         => ? [X3: nat] :
% 0.24/0.49              ( ( P @ X3 )
% 0.24/0.49              & ! [Y5: nat] :
% 0.24/0.49                  ( ( P @ Y5 )
% 0.24/0.49                 => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Nat.ex_has_greatest_nat
% 0.24/0.49  thf(fact_382_nat__le__linear,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.49        | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.24/0.49  
% 0.24/0.49  % nat_le_linear
% 0.24/0.49  thf(fact_383_le__antisym,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ M )
% 0.24/0.49         => ( M = N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_antisym
% 0.24/0.49  thf(fact_384_eq__imp__le,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( M = N )
% 0.24/0.49       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % eq_imp_le
% 0.24/0.49  thf(fact_385_le__trans,axiom,
% 0.24/0.49      ! [I: nat,J: nat,K: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49       => ( ( ord_less_eq_nat @ J @ K )
% 0.24/0.49         => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_trans
% 0.24/0.49  thf(fact_386_le__refl,axiom,
% 0.24/0.49      ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.24/0.49  
% 0.24/0.49  % le_refl
% 0.24/0.49  thf(fact_387_lift__Suc__antimono__le,axiom,
% 0.24/0.49      ! [F: nat > set_int,N: nat,N5: nat] :
% 0.24/0.49        ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.24/0.49         => ( ord_less_eq_set_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lift_Suc_antimono_le
% 0.24/0.49  thf(fact_388_lift__Suc__antimono__le,axiom,
% 0.24/0.49      ! [F: nat > rat,N: nat,N5: nat] :
% 0.24/0.49        ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.24/0.49         => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lift_Suc_antimono_le
% 0.24/0.49  thf(fact_389_lift__Suc__antimono__le,axiom,
% 0.24/0.49      ! [F: nat > num,N: nat,N5: nat] :
% 0.24/0.49        ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.24/0.49         => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lift_Suc_antimono_le
% 0.24/0.49  thf(fact_390_lift__Suc__antimono__le,axiom,
% 0.24/0.49      ! [F: nat > nat,N: nat,N5: nat] :
% 0.24/0.49        ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.24/0.49         => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lift_Suc_antimono_le
% 0.24/0.49  thf(fact_391_lift__Suc__antimono__le,axiom,
% 0.24/0.49      ! [F: nat > int,N: nat,N5: nat] :
% 0.24/0.49        ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.24/0.49         => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lift_Suc_antimono_le
% 0.24/0.49  thf(fact_392_lift__Suc__mono__le,axiom,
% 0.24/0.49      ! [F: nat > set_int,N: nat,N5: nat] :
% 0.24/0.49        ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.24/0.49         => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lift_Suc_mono_le
% 0.24/0.49  thf(fact_393_lift__Suc__mono__le,axiom,
% 0.24/0.49      ! [F: nat > rat,N: nat,N5: nat] :
% 0.24/0.49        ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.24/0.49         => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lift_Suc_mono_le
% 0.24/0.49  thf(fact_394_lift__Suc__mono__le,axiom,
% 0.24/0.49      ! [F: nat > num,N: nat,N5: nat] :
% 0.24/0.49        ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.24/0.49         => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lift_Suc_mono_le
% 0.24/0.49  thf(fact_395_lift__Suc__mono__le,axiom,
% 0.24/0.49      ! [F: nat > nat,N: nat,N5: nat] :
% 0.24/0.49        ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.24/0.49         => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lift_Suc_mono_le
% 0.24/0.49  thf(fact_396_lift__Suc__mono__le,axiom,
% 0.24/0.49      ! [F: nat > int,N: nat,N5: nat] :
% 0.24/0.49        ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.49       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.24/0.49         => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lift_Suc_mono_le
% 0.24/0.49  thf(fact_397_of__nat__mono,axiom,
% 0.24/0.49      ! [I: nat,J: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_mono
% 0.24/0.49  thf(fact_398_of__nat__mono,axiom,
% 0.24/0.49      ! [I: nat,J: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_mono
% 0.24/0.49  thf(fact_399_of__nat__mono,axiom,
% 0.24/0.49      ! [I: nat,J: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49       => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_mono
% 0.24/0.49  thf(fact_400_of__nat__mono,axiom,
% 0.24/0.49      ! [I: nat,J: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49       => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_mono
% 0.24/0.49  thf(fact_401_zle__int,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % zle_int
% 0.24/0.49  thf(fact_402_nat__int__comparison_I3_J,axiom,
% 0.24/0.49      ( ord_less_eq_nat
% 0.24/0.49      = ( ^ [A3: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % nat_int_comparison(3)
% 0.24/0.49  thf(fact_403_ceiling__mono,axiom,
% 0.24/0.49      ! [Y: real,X: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ Y @ X )
% 0.24/0.49       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_mono
% 0.24/0.49  thf(fact_404_ceiling__mono,axiom,
% 0.24/0.49      ! [Y: rat,X: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ Y @ X )
% 0.24/0.49       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_mono
% 0.24/0.49  thf(fact_405_add__le__imp__le__right,axiom,
% 0.24/0.49      ! [A: real,C: real,B: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.24/0.49       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_imp_le_right
% 0.24/0.49  thf(fact_406_add__le__imp__le__right,axiom,
% 0.24/0.49      ! [A: rat,C: rat,B: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.24/0.49       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_imp_le_right
% 0.24/0.49  thf(fact_407_add__le__imp__le__right,axiom,
% 0.24/0.49      ! [A: nat,C: nat,B: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.24/0.49       => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_imp_le_right
% 0.24/0.49  thf(fact_408_add__le__imp__le__right,axiom,
% 0.24/0.49      ! [A: int,C: int,B: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.24/0.49       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_imp_le_right
% 0.24/0.49  thf(fact_409_add__le__imp__le__left,axiom,
% 0.24/0.49      ! [C: real,A: real,B: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.24/0.49       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_imp_le_left
% 0.24/0.49  thf(fact_410_add__le__imp__le__left,axiom,
% 0.24/0.49      ! [C: rat,A: rat,B: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.24/0.49       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_imp_le_left
% 0.24/0.49  thf(fact_411_add__le__imp__le__left,axiom,
% 0.24/0.49      ! [C: nat,A: nat,B: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.24/0.49       => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_imp_le_left
% 0.24/0.49  thf(fact_412_add__le__imp__le__left,axiom,
% 0.24/0.49      ! [C: int,A: int,B: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.24/0.49       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_imp_le_left
% 0.24/0.49  thf(fact_413_le__iff__add,axiom,
% 0.24/0.49      ( ord_less_eq_nat
% 0.24/0.49      = ( ^ [A3: nat,B4: nat] :
% 0.24/0.49          ? [C3: nat] :
% 0.24/0.49            ( B4
% 0.24/0.49            = ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_iff_add
% 0.24/0.49  thf(fact_414_add__right__mono,axiom,
% 0.24/0.49      ! [A: real,B: real,C: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.49       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_mono
% 0.24/0.49  thf(fact_415_add__right__mono,axiom,
% 0.24/0.49      ! [A: rat,B: rat,C: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.49       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_mono
% 0.24/0.49  thf(fact_416_add__right__mono,axiom,
% 0.24/0.49      ! [A: nat,B: nat,C: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.49       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_mono
% 0.24/0.49  thf(fact_417_add__right__mono,axiom,
% 0.24/0.49      ! [A: int,B: int,C: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.49       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_mono
% 0.24/0.49  thf(fact_418_less__eqE,axiom,
% 0.24/0.49      ! [A: nat,B: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.49       => ~ ! [C4: nat] :
% 0.24/0.49              ( B
% 0.24/0.49             != ( plus_plus_nat @ A @ C4 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % less_eqE
% 0.24/0.49  thf(fact_419_add__left__mono,axiom,
% 0.24/0.49      ! [A: real,B: real,C: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.49       => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_mono
% 0.24/0.49  thf(fact_420_add__left__mono,axiom,
% 0.24/0.49      ! [A: rat,B: rat,C: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.49       => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_mono
% 0.24/0.49  thf(fact_421_add__left__mono,axiom,
% 0.24/0.49      ! [A: nat,B: nat,C: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.49       => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_mono
% 0.24/0.49  thf(fact_422_add__left__mono,axiom,
% 0.24/0.49      ! [A: int,B: int,C: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.49       => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_mono
% 0.24/0.49  thf(fact_423_add__mono,axiom,
% 0.24/0.49      ! [A: real,B: real,C: real,D: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.49       => ( ( ord_less_eq_real @ C @ D )
% 0.24/0.49         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono
% 0.24/0.49  thf(fact_424_add__mono,axiom,
% 0.24/0.49      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.49       => ( ( ord_less_eq_rat @ C @ D )
% 0.24/0.49         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono
% 0.24/0.49  thf(fact_425_add__mono,axiom,
% 0.24/0.49      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.49       => ( ( ord_less_eq_nat @ C @ D )
% 0.24/0.49         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono
% 0.24/0.49  thf(fact_426_add__mono,axiom,
% 0.24/0.49      ! [A: int,B: int,C: int,D: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.49       => ( ( ord_less_eq_int @ C @ D )
% 0.24/0.49         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono
% 0.24/0.49  thf(fact_427_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.24/0.49      ! [I: real,J: real,K: real,L: real] :
% 0.24/0.49        ( ( ( ord_less_eq_real @ I @ J )
% 0.24/0.49          & ( ord_less_eq_real @ K @ L ) )
% 0.24/0.49       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(1)
% 0.24/0.49  thf(fact_428_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.24/0.49      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.24/0.49        ( ( ( ord_less_eq_rat @ I @ J )
% 0.24/0.49          & ( ord_less_eq_rat @ K @ L ) )
% 0.24/0.49       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(1)
% 0.24/0.49  thf(fact_429_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.24/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.49        ( ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49          & ( ord_less_eq_nat @ K @ L ) )
% 0.24/0.49       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(1)
% 0.24/0.49  thf(fact_430_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.24/0.49      ! [I: int,J: int,K: int,L: int] :
% 0.24/0.49        ( ( ( ord_less_eq_int @ I @ J )
% 0.24/0.49          & ( ord_less_eq_int @ K @ L ) )
% 0.24/0.49       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(1)
% 0.24/0.49  thf(fact_431_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.24/0.49      ! [I: real,J: real,K: real,L: real] :
% 0.24/0.49        ( ( ( I = J )
% 0.24/0.49          & ( ord_less_eq_real @ K @ L ) )
% 0.24/0.49       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(2)
% 0.24/0.49  thf(fact_432_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.24/0.49      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.24/0.49        ( ( ( I = J )
% 0.24/0.49          & ( ord_less_eq_rat @ K @ L ) )
% 0.24/0.49       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(2)
% 0.24/0.49  thf(fact_433_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.24/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.49        ( ( ( I = J )
% 0.24/0.49          & ( ord_less_eq_nat @ K @ L ) )
% 0.24/0.49       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(2)
% 0.24/0.49  thf(fact_434_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.24/0.49      ! [I: int,J: int,K: int,L: int] :
% 0.24/0.49        ( ( ( I = J )
% 0.24/0.49          & ( ord_less_eq_int @ K @ L ) )
% 0.24/0.49       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(2)
% 0.24/0.49  thf(fact_435_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.24/0.49      ! [I: real,J: real,K: real,L: real] :
% 0.24/0.49        ( ( ( ord_less_eq_real @ I @ J )
% 0.24/0.49          & ( K = L ) )
% 0.24/0.49       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(3)
% 0.24/0.49  thf(fact_436_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.24/0.49      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.24/0.49        ( ( ( ord_less_eq_rat @ I @ J )
% 0.24/0.49          & ( K = L ) )
% 0.24/0.49       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(3)
% 0.24/0.49  thf(fact_437_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.24/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.49        ( ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49          & ( K = L ) )
% 0.24/0.49       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(3)
% 0.24/0.49  thf(fact_438_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.24/0.49      ! [I: int,J: int,K: int,L: int] :
% 0.24/0.49        ( ( ( ord_less_eq_int @ I @ J )
% 0.24/0.49          & ( K = L ) )
% 0.24/0.49       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(3)
% 0.24/0.49  thf(fact_439_size__neq__size__imp__neq,axiom,
% 0.24/0.49      ! [X: char,Y: char] :
% 0.24/0.49        ( ( ( size_size_char @ X )
% 0.24/0.49         != ( size_size_char @ Y ) )
% 0.24/0.49       => ( X != Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % size_neq_size_imp_neq
% 0.24/0.49  thf(fact_440_size__neq__size__imp__neq,axiom,
% 0.24/0.49      ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
% 0.24/0.49        ( ( ( size_s6755466524823107622T_VEBT @ X )
% 0.24/0.49         != ( size_s6755466524823107622T_VEBT @ Y ) )
% 0.24/0.49       => ( X != Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % size_neq_size_imp_neq
% 0.24/0.49  thf(fact_441_size__neq__size__imp__neq,axiom,
% 0.24/0.49      ! [X: list_o,Y: list_o] :
% 0.24/0.49        ( ( ( size_size_list_o @ X )
% 0.24/0.49         != ( size_size_list_o @ Y ) )
% 0.24/0.49       => ( X != Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % size_neq_size_imp_neq
% 0.24/0.49  thf(fact_442_size__neq__size__imp__neq,axiom,
% 0.24/0.49      ! [X: list_int,Y: list_int] :
% 0.24/0.49        ( ( ( size_size_list_int @ X )
% 0.24/0.49         != ( size_size_list_int @ Y ) )
% 0.24/0.49       => ( X != Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % size_neq_size_imp_neq
% 0.24/0.49  thf(fact_443_size__neq__size__imp__neq,axiom,
% 0.24/0.49      ! [X: num,Y: num] :
% 0.24/0.49        ( ( ( size_size_num @ X )
% 0.24/0.49         != ( size_size_num @ Y ) )
% 0.24/0.49       => ( X != Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % size_neq_size_imp_neq
% 0.24/0.49  thf(fact_444_le__numeral__extra_I4_J,axiom,
% 0.24/0.49      ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.24/0.49  
% 0.24/0.49  % le_numeral_extra(4)
% 0.24/0.49  thf(fact_445_le__numeral__extra_I4_J,axiom,
% 0.24/0.49      ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.24/0.49  
% 0.24/0.49  % le_numeral_extra(4)
% 0.24/0.49  thf(fact_446_le__numeral__extra_I4_J,axiom,
% 0.24/0.49      ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.24/0.49  
% 0.24/0.49  % le_numeral_extra(4)
% 0.24/0.49  thf(fact_447_le__numeral__extra_I4_J,axiom,
% 0.24/0.49      ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.24/0.49  
% 0.24/0.49  % le_numeral_extra(4)
% 0.24/0.49  thf(fact_448_transitive__stepwise__le,axiom,
% 0.24/0.49      ! [M: nat,N: nat,R: nat > nat > $o] :
% 0.24/0.49        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.49       => ( ! [X3: nat] : ( R @ X3 @ X3 )
% 0.24/0.49         => ( ! [X3: nat,Y4: nat,Z3: nat] :
% 0.24/0.49                ( ( R @ X3 @ Y4 )
% 0.24/0.49               => ( ( R @ Y4 @ Z3 )
% 0.24/0.49                 => ( R @ X3 @ Z3 ) ) )
% 0.24/0.49           => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
% 0.24/0.49             => ( R @ M @ N ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % transitive_stepwise_le
% 0.24/0.49  thf(fact_449_nat__induct__at__least,axiom,
% 0.24/0.49      ! [M: nat,N: nat,P: nat > $o] :
% 0.24/0.49        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.49       => ( ( P @ M )
% 0.24/0.49         => ( ! [N2: nat] :
% 0.24/0.49                ( ( ord_less_eq_nat @ M @ N2 )
% 0.24/0.49               => ( ( P @ N2 )
% 0.24/0.49                 => ( P @ ( suc @ N2 ) ) ) )
% 0.24/0.49           => ( P @ N ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % nat_induct_at_least
% 0.24/0.49  thf(fact_450_full__nat__induct,axiom,
% 0.24/0.49      ! [P: nat > $o,N: nat] :
% 0.24/0.49        ( ! [N2: nat] :
% 0.24/0.49            ( ! [M3: nat] :
% 0.24/0.49                ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
% 0.24/0.49               => ( P @ M3 ) )
% 0.24/0.49           => ( P @ N2 ) )
% 0.24/0.49       => ( P @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % full_nat_induct
% 0.24/0.49  thf(fact_451_not__less__eq__eq,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.24/0.49  
% 0.24/0.49  % not_less_eq_eq
% 0.24/0.49  thf(fact_452_Suc__n__not__le__n,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_n_not_le_n
% 0.24/0.49  thf(fact_453_le__Suc__eq,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.24/0.49        = ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.49          | ( M
% 0.24/0.49            = ( suc @ N ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_Suc_eq
% 0.24/0.49  thf(fact_454_Suc__le__D,axiom,
% 0.24/0.49      ! [N: nat,M4: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
% 0.24/0.49       => ? [M5: nat] :
% 0.24/0.49            ( M4
% 0.24/0.49            = ( suc @ M5 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_le_D
% 0.24/0.49  thf(fact_455_le__SucI,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.49       => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_SucI
% 0.24/0.49  thf(fact_456_le__SucE,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.24/0.49       => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.24/0.49         => ( M
% 0.24/0.49            = ( suc @ N ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_SucE
% 0.24/0.49  thf(fact_457_Suc__leD,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.24/0.49       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_leD
% 0.24/0.49  thf(fact_458_real__arch__simple,axiom,
% 0.24/0.49      ! [X: real] :
% 0.24/0.49      ? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % real_arch_simple
% 0.24/0.49  thf(fact_459_real__arch__simple,axiom,
% 0.24/0.49      ! [X: rat] :
% 0.24/0.49      ? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % real_arch_simple
% 0.24/0.49  thf(fact_460_add__leE,axiom,
% 0.24/0.49      ! [M: nat,K: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.24/0.49       => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.49           => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_leE
% 0.24/0.49  thf(fact_461_le__add1,axiom,
% 0.24/0.49      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_add1
% 0.24/0.49  thf(fact_462_le__add2,axiom,
% 0.24/0.49      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_add2
% 0.24/0.49  thf(fact_463_add__leD1,axiom,
% 0.24/0.49      ! [M: nat,K: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.24/0.49       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_leD1
% 0.24/0.49  thf(fact_464_add__leD2,axiom,
% 0.24/0.49      ! [M: nat,K: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.24/0.49       => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_leD2
% 0.24/0.49  thf(fact_465_le__Suc__ex,axiom,
% 0.24/0.49      ! [K: nat,L: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ K @ L )
% 0.24/0.49       => ? [N2: nat] :
% 0.24/0.49            ( L
% 0.24/0.49            = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_Suc_ex
% 0.24/0.49  thf(fact_466_add__le__mono,axiom,
% 0.24/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49       => ( ( ord_less_eq_nat @ K @ L )
% 0.24/0.49         => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_mono
% 0.24/0.49  thf(fact_467_add__le__mono1,axiom,
% 0.24/0.49      ! [I: nat,J: nat,K: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_mono1
% 0.24/0.49  thf(fact_468_trans__le__add1,axiom,
% 0.24/0.49      ! [I: nat,J: nat,M: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % trans_le_add1
% 0.24/0.49  thf(fact_469_trans__le__add2,axiom,
% 0.24/0.49      ! [I: nat,J: nat,M: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.49       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % trans_le_add2
% 0.24/0.49  thf(fact_470_nat__le__iff__add,axiom,
% 0.24/0.49      ( ord_less_eq_nat
% 0.24/0.49      = ( ^ [M6: nat,N3: nat] :
% 0.24/0.49          ? [K2: nat] :
% 0.24/0.49            ( N3
% 0.24/0.49            = ( plus_plus_nat @ M6 @ K2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % nat_le_iff_add
% 0.24/0.49  thf(fact_471_one__le__numeral,axiom,
% 0.24/0.49      ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_le_numeral
% 0.24/0.49  thf(fact_472_one__le__numeral,axiom,
% 0.24/0.49      ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_le_numeral
% 0.24/0.49  thf(fact_473_one__le__numeral,axiom,
% 0.24/0.49      ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_le_numeral
% 0.24/0.49  thf(fact_474_one__le__numeral,axiom,
% 0.24/0.49      ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_le_numeral
% 0.24/0.49  thf(fact_475_dbl__def,axiom,
% 0.24/0.49      ( neg_numeral_dbl_real
% 0.24/0.49      = ( ^ [X4: real] : ( plus_plus_real @ X4 @ X4 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_def
% 0.24/0.49  thf(fact_476_dbl__def,axiom,
% 0.24/0.49      ( neg_numeral_dbl_rat
% 0.24/0.49      = ( ^ [X4: rat] : ( plus_plus_rat @ X4 @ X4 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_def
% 0.24/0.49  thf(fact_477_dbl__def,axiom,
% 0.24/0.49      ( neg_numeral_dbl_int
% 0.24/0.49      = ( ^ [X4: int] : ( plus_plus_int @ X4 @ X4 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_def
% 0.24/0.49  thf(fact_478_dbl__def,axiom,
% 0.24/0.49      ( neg_nu7009210354673126013omplex
% 0.24/0.49      = ( ^ [X4: complex] : ( plus_plus_complex @ X4 @ X4 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_def
% 0.24/0.49  thf(fact_479_int__ge__induct,axiom,
% 0.24/0.49      ! [K: int,I: int,P: int > $o] :
% 0.24/0.49        ( ( ord_less_eq_int @ K @ I )
% 0.24/0.49       => ( ( P @ K )
% 0.24/0.49         => ( ! [I2: int] :
% 0.24/0.49                ( ( ord_less_eq_int @ K @ I2 )
% 0.24/0.49               => ( ( P @ I2 )
% 0.24/0.49                 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.24/0.49           => ( P @ I ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % int_ge_induct
% 0.24/0.49  thf(fact_480_zle__iff__zadd,axiom,
% 0.24/0.49      ( ord_less_eq_int
% 0.24/0.49      = ( ^ [W2: int,Z4: int] :
% 0.24/0.49          ? [N3: nat] :
% 0.24/0.49            ( Z4
% 0.24/0.49            = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % zle_iff_zadd
% 0.24/0.49  thf(fact_481_one__reorient,axiom,
% 0.24/0.49      ! [X: complex] :
% 0.24/0.49        ( ( one_one_complex = X )
% 0.24/0.49        = ( X = one_one_complex ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_reorient
% 0.24/0.49  thf(fact_482_one__reorient,axiom,
% 0.24/0.49      ! [X: real] :
% 0.24/0.49        ( ( one_one_real = X )
% 0.24/0.49        = ( X = one_one_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_reorient
% 0.24/0.49  thf(fact_483_one__reorient,axiom,
% 0.24/0.49      ! [X: rat] :
% 0.24/0.49        ( ( one_one_rat = X )
% 0.24/0.49        = ( X = one_one_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_reorient
% 0.24/0.49  thf(fact_484_one__reorient,axiom,
% 0.24/0.49      ! [X: nat] :
% 0.24/0.49        ( ( one_one_nat = X )
% 0.24/0.49        = ( X = one_one_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_reorient
% 0.24/0.49  thf(fact_485_one__reorient,axiom,
% 0.24/0.49      ! [X: int] :
% 0.24/0.49        ( ( one_one_int = X )
% 0.24/0.49        = ( X = one_one_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_reorient
% 0.24/0.49  thf(fact_486_add__right__imp__eq,axiom,
% 0.24/0.49      ! [B: real,A: real,C: real] :
% 0.24/0.49        ( ( ( plus_plus_real @ B @ A )
% 0.24/0.49          = ( plus_plus_real @ C @ A ) )
% 0.24/0.49       => ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_imp_eq
% 0.24/0.49  thf(fact_487_add__right__imp__eq,axiom,
% 0.24/0.49      ! [B: rat,A: rat,C: rat] :
% 0.24/0.49        ( ( ( plus_plus_rat @ B @ A )
% 0.24/0.49          = ( plus_plus_rat @ C @ A ) )
% 0.24/0.49       => ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_imp_eq
% 0.24/0.49  thf(fact_488_add__right__imp__eq,axiom,
% 0.24/0.49      ! [B: nat,A: nat,C: nat] :
% 0.24/0.49        ( ( ( plus_plus_nat @ B @ A )
% 0.24/0.49          = ( plus_plus_nat @ C @ A ) )
% 0.24/0.49       => ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_imp_eq
% 0.24/0.49  thf(fact_489_add__right__imp__eq,axiom,
% 0.24/0.49      ! [B: int,A: int,C: int] :
% 0.24/0.49        ( ( ( plus_plus_int @ B @ A )
% 0.24/0.49          = ( plus_plus_int @ C @ A ) )
% 0.24/0.49       => ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_imp_eq
% 0.24/0.49  thf(fact_490_add__right__imp__eq,axiom,
% 0.24/0.49      ! [B: complex,A: complex,C: complex] :
% 0.24/0.49        ( ( ( plus_plus_complex @ B @ A )
% 0.24/0.49          = ( plus_plus_complex @ C @ A ) )
% 0.24/0.49       => ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_right_imp_eq
% 0.24/0.49  thf(fact_491_add__left__imp__eq,axiom,
% 0.24/0.49      ! [A: real,B: real,C: real] :
% 0.24/0.49        ( ( ( plus_plus_real @ A @ B )
% 0.24/0.49          = ( plus_plus_real @ A @ C ) )
% 0.24/0.49       => ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_imp_eq
% 0.24/0.49  thf(fact_492_add__left__imp__eq,axiom,
% 0.24/0.49      ! [A: rat,B: rat,C: rat] :
% 0.24/0.49        ( ( ( plus_plus_rat @ A @ B )
% 0.24/0.49          = ( plus_plus_rat @ A @ C ) )
% 0.24/0.49       => ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_imp_eq
% 0.24/0.49  thf(fact_493_add__left__imp__eq,axiom,
% 0.24/0.49      ! [A: nat,B: nat,C: nat] :
% 0.24/0.49        ( ( ( plus_plus_nat @ A @ B )
% 0.24/0.49          = ( plus_plus_nat @ A @ C ) )
% 0.24/0.49       => ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_imp_eq
% 0.24/0.49  thf(fact_494_add__left__imp__eq,axiom,
% 0.24/0.49      ! [A: int,B: int,C: int] :
% 0.24/0.49        ( ( ( plus_plus_int @ A @ B )
% 0.24/0.49          = ( plus_plus_int @ A @ C ) )
% 0.24/0.49       => ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_imp_eq
% 0.24/0.49  thf(fact_495_add__left__imp__eq,axiom,
% 0.24/0.49      ! [A: complex,B: complex,C: complex] :
% 0.24/0.49        ( ( ( plus_plus_complex @ A @ B )
% 0.24/0.49          = ( plus_plus_complex @ A @ C ) )
% 0.24/0.49       => ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_left_imp_eq
% 0.24/0.49  thf(fact_496_add_Oleft__commute,axiom,
% 0.24/0.49      ! [B: real,A: real,C: real] :
% 0.24/0.49        ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.24/0.49        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.left_commute
% 0.24/0.49  thf(fact_497_add_Oleft__commute,axiom,
% 0.24/0.49      ! [B: rat,A: rat,C: rat] :
% 0.24/0.49        ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 0.24/0.49        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.left_commute
% 0.24/0.49  thf(fact_498_add_Oleft__commute,axiom,
% 0.24/0.49      ! [B: nat,A: nat,C: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.24/0.49        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.left_commute
% 0.24/0.49  thf(fact_499_add_Oleft__commute,axiom,
% 0.24/0.49      ! [B: int,A: int,C: int] :
% 0.24/0.49        ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.24/0.49        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.left_commute
% 0.24/0.49  thf(fact_500_add_Oleft__commute,axiom,
% 0.24/0.49      ! [B: complex,A: complex,C: complex] :
% 0.24/0.49        ( ( plus_plus_complex @ B @ ( plus_plus_complex @ A @ C ) )
% 0.24/0.49        = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.left_commute
% 0.24/0.49  thf(fact_501_add_Ocommute,axiom,
% 0.24/0.49      ( plus_plus_real
% 0.24/0.49      = ( ^ [A3: real,B4: real] : ( plus_plus_real @ B4 @ A3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.commute
% 0.24/0.49  thf(fact_502_add_Ocommute,axiom,
% 0.24/0.49      ( plus_plus_rat
% 0.24/0.49      = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ B4 @ A3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.commute
% 0.24/0.49  thf(fact_503_add_Ocommute,axiom,
% 0.24/0.49      ( plus_plus_nat
% 0.24/0.49      = ( ^ [A3: nat,B4: nat] : ( plus_plus_nat @ B4 @ A3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.commute
% 0.24/0.49  thf(fact_504_add_Ocommute,axiom,
% 0.24/0.49      ( plus_plus_int
% 0.24/0.49      = ( ^ [A3: int,B4: int] : ( plus_plus_int @ B4 @ A3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.commute
% 0.24/0.49  thf(fact_505_add_Ocommute,axiom,
% 0.24/0.49      ( plus_plus_complex
% 0.24/0.49      = ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ B4 @ A3 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.commute
% 0.24/0.49  thf(fact_506_add_Oright__cancel,axiom,
% 0.24/0.49      ! [B: real,A: real,C: real] :
% 0.24/0.49        ( ( ( plus_plus_real @ B @ A )
% 0.24/0.49          = ( plus_plus_real @ C @ A ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.right_cancel
% 0.24/0.49  thf(fact_507_add_Oright__cancel,axiom,
% 0.24/0.49      ! [B: rat,A: rat,C: rat] :
% 0.24/0.49        ( ( ( plus_plus_rat @ B @ A )
% 0.24/0.49          = ( plus_plus_rat @ C @ A ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.right_cancel
% 0.24/0.49  thf(fact_508_add_Oright__cancel,axiom,
% 0.24/0.49      ! [B: int,A: int,C: int] :
% 0.24/0.49        ( ( ( plus_plus_int @ B @ A )
% 0.24/0.49          = ( plus_plus_int @ C @ A ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.right_cancel
% 0.24/0.49  thf(fact_509_add_Oright__cancel,axiom,
% 0.24/0.49      ! [B: complex,A: complex,C: complex] :
% 0.24/0.49        ( ( ( plus_plus_complex @ B @ A )
% 0.24/0.49          = ( plus_plus_complex @ C @ A ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.right_cancel
% 0.24/0.49  thf(fact_510_add_Oleft__cancel,axiom,
% 0.24/0.49      ! [A: real,B: real,C: real] :
% 0.24/0.49        ( ( ( plus_plus_real @ A @ B )
% 0.24/0.49          = ( plus_plus_real @ A @ C ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.left_cancel
% 0.24/0.49  thf(fact_511_add_Oleft__cancel,axiom,
% 0.24/0.49      ! [A: rat,B: rat,C: rat] :
% 0.24/0.49        ( ( ( plus_plus_rat @ A @ B )
% 0.24/0.49          = ( plus_plus_rat @ A @ C ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.left_cancel
% 0.24/0.49  thf(fact_512_add_Oleft__cancel,axiom,
% 0.24/0.49      ! [A: int,B: int,C: int] :
% 0.24/0.49        ( ( ( plus_plus_int @ A @ B )
% 0.24/0.49          = ( plus_plus_int @ A @ C ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.left_cancel
% 0.24/0.49  thf(fact_513_add_Oleft__cancel,axiom,
% 0.24/0.49      ! [A: complex,B: complex,C: complex] :
% 0.24/0.49        ( ( ( plus_plus_complex @ A @ B )
% 0.24/0.49          = ( plus_plus_complex @ A @ C ) )
% 0.24/0.49        = ( B = C ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.left_cancel
% 0.24/0.49  thf(fact_514_add_Oassoc,axiom,
% 0.24/0.49      ! [A: real,B: real,C: real] :
% 0.24/0.49        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.assoc
% 0.24/0.49  thf(fact_515_add_Oassoc,axiom,
% 0.24/0.49      ! [A: rat,B: rat,C: rat] :
% 0.24/0.49        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.assoc
% 0.24/0.49  thf(fact_516_add_Oassoc,axiom,
% 0.24/0.49      ! [A: nat,B: nat,C: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.assoc
% 0.24/0.49  thf(fact_517_add_Oassoc,axiom,
% 0.24/0.49      ! [A: int,B: int,C: int] :
% 0.24/0.49        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.assoc
% 0.24/0.49  thf(fact_518_add_Oassoc,axiom,
% 0.24/0.49      ! [A: complex,B: complex,C: complex] :
% 0.24/0.49        ( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add.assoc
% 0.24/0.49  thf(fact_519_group__cancel_Oadd2,axiom,
% 0.24/0.49      ! [B2: real,K: real,B: real,A: real] :
% 0.24/0.49        ( ( B2
% 0.24/0.49          = ( plus_plus_real @ K @ B ) )
% 0.24/0.49       => ( ( plus_plus_real @ A @ B2 )
% 0.24/0.49          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % group_cancel.add2
% 0.24/0.49  thf(fact_520_group__cancel_Oadd2,axiom,
% 0.24/0.49      ! [B2: rat,K: rat,B: rat,A: rat] :
% 0.24/0.49        ( ( B2
% 0.24/0.49          = ( plus_plus_rat @ K @ B ) )
% 0.24/0.49       => ( ( plus_plus_rat @ A @ B2 )
% 0.24/0.49          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % group_cancel.add2
% 0.24/0.49  thf(fact_521_group__cancel_Oadd2,axiom,
% 0.24/0.49      ! [B2: nat,K: nat,B: nat,A: nat] :
% 0.24/0.49        ( ( B2
% 0.24/0.49          = ( plus_plus_nat @ K @ B ) )
% 0.24/0.49       => ( ( plus_plus_nat @ A @ B2 )
% 0.24/0.49          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % group_cancel.add2
% 0.24/0.49  thf(fact_522_group__cancel_Oadd2,axiom,
% 0.24/0.49      ! [B2: int,K: int,B: int,A: int] :
% 0.24/0.49        ( ( B2
% 0.24/0.49          = ( plus_plus_int @ K @ B ) )
% 0.24/0.49       => ( ( plus_plus_int @ A @ B2 )
% 0.24/0.49          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % group_cancel.add2
% 0.24/0.49  thf(fact_523_group__cancel_Oadd2,axiom,
% 0.24/0.49      ! [B2: complex,K: complex,B: complex,A: complex] :
% 0.24/0.49        ( ( B2
% 0.24/0.49          = ( plus_plus_complex @ K @ B ) )
% 0.24/0.49       => ( ( plus_plus_complex @ A @ B2 )
% 0.24/0.49          = ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % group_cancel.add2
% 0.24/0.49  thf(fact_524_group__cancel_Oadd1,axiom,
% 0.24/0.49      ! [A2: real,K: real,A: real,B: real] :
% 0.24/0.49        ( ( A2
% 0.24/0.49          = ( plus_plus_real @ K @ A ) )
% 0.24/0.49       => ( ( plus_plus_real @ A2 @ B )
% 0.24/0.49          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % group_cancel.add1
% 0.24/0.49  thf(fact_525_group__cancel_Oadd1,axiom,
% 0.24/0.49      ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.24/0.49        ( ( A2
% 0.24/0.49          = ( plus_plus_rat @ K @ A ) )
% 0.24/0.49       => ( ( plus_plus_rat @ A2 @ B )
% 0.24/0.49          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % group_cancel.add1
% 0.24/0.49  thf(fact_526_group__cancel_Oadd1,axiom,
% 0.24/0.49      ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.24/0.49        ( ( A2
% 0.24/0.49          = ( plus_plus_nat @ K @ A ) )
% 0.24/0.49       => ( ( plus_plus_nat @ A2 @ B )
% 0.24/0.49          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % group_cancel.add1
% 0.24/0.49  thf(fact_527_group__cancel_Oadd1,axiom,
% 0.24/0.49      ! [A2: int,K: int,A: int,B: int] :
% 0.24/0.49        ( ( A2
% 0.24/0.49          = ( plus_plus_int @ K @ A ) )
% 0.24/0.49       => ( ( plus_plus_int @ A2 @ B )
% 0.24/0.49          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % group_cancel.add1
% 0.24/0.49  thf(fact_528_group__cancel_Oadd1,axiom,
% 0.24/0.49      ! [A2: complex,K: complex,A: complex,B: complex] :
% 0.24/0.49        ( ( A2
% 0.24/0.49          = ( plus_plus_complex @ K @ A ) )
% 0.24/0.49       => ( ( plus_plus_complex @ A2 @ B )
% 0.24/0.49          = ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % group_cancel.add1
% 0.24/0.49  thf(fact_529_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.24/0.49      ! [I: real,J: real,K: real,L: real] :
% 0.24/0.49        ( ( ( I = J )
% 0.24/0.49          & ( K = L ) )
% 0.24/0.49       => ( ( plus_plus_real @ I @ K )
% 0.24/0.49          = ( plus_plus_real @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(4)
% 0.24/0.49  thf(fact_530_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.24/0.49      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.24/0.49        ( ( ( I = J )
% 0.24/0.49          & ( K = L ) )
% 0.24/0.49       => ( ( plus_plus_rat @ I @ K )
% 0.24/0.49          = ( plus_plus_rat @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(4)
% 0.24/0.49  thf(fact_531_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.24/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.49        ( ( ( I = J )
% 0.24/0.49          & ( K = L ) )
% 0.24/0.49       => ( ( plus_plus_nat @ I @ K )
% 0.24/0.49          = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(4)
% 0.24/0.49  thf(fact_532_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.24/0.49      ! [I: int,J: int,K: int,L: int] :
% 0.24/0.49        ( ( ( I = J )
% 0.24/0.49          & ( K = L ) )
% 0.24/0.49       => ( ( plus_plus_int @ I @ K )
% 0.24/0.49          = ( plus_plus_int @ J @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_mono_thms_linordered_semiring(4)
% 0.24/0.49  thf(fact_533_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.24/0.49      ! [A: real,B: real,C: real] :
% 0.24/0.49        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ab_semigroup_add_class.add_ac(1)
% 0.24/0.49  thf(fact_534_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.24/0.49      ! [A: rat,B: rat,C: rat] :
% 0.24/0.49        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ab_semigroup_add_class.add_ac(1)
% 0.24/0.49  thf(fact_535_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.24/0.49      ! [A: nat,B: nat,C: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ab_semigroup_add_class.add_ac(1)
% 0.24/0.49  thf(fact_536_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.24/0.49      ! [A: int,B: int,C: int] :
% 0.24/0.49        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ab_semigroup_add_class.add_ac(1)
% 0.24/0.49  thf(fact_537_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.24/0.49      ! [A: complex,B: complex,C: complex] :
% 0.24/0.49        ( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.24/0.49        = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ab_semigroup_add_class.add_ac(1)
% 0.24/0.49  thf(fact_538_ceiling__add__le,axiom,
% 0.24/0.49      ! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_add_le
% 0.24/0.49  thf(fact_539_ceiling__add__le,axiom,
% 0.24/0.49      ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_add_le
% 0.24/0.49  thf(fact_540_invar__vebt_Ointros_I2_J,axiom,
% 0.24/0.49      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.24/0.49        ( ! [X3: vEBT_VEBT] :
% 0.24/0.49            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.24/0.49           => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.24/0.49       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.24/0.49         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.24/0.49              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.24/0.49           => ( ( M = N )
% 0.24/0.49             => ( ( Deg
% 0.24/0.49                  = ( plus_plus_nat @ N @ M ) )
% 0.24/0.49               => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
% 0.24/0.49                 => ( ! [X3: vEBT_VEBT] :
% 0.24/0.49                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.24/0.49                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
% 0.24/0.49                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % invar_vebt.intros(2)
% 0.24/0.49  thf(fact_541_invar__vebt_Ointros_I3_J,axiom,
% 0.24/0.49      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.24/0.49        ( ! [X3: vEBT_VEBT] :
% 0.24/0.49            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.24/0.49           => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.24/0.49       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.24/0.49         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.24/0.49              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.24/0.49           => ( ( M
% 0.24/0.49                = ( suc @ N ) )
% 0.24/0.49             => ( ( Deg
% 0.24/0.49                  = ( plus_plus_nat @ N @ M ) )
% 0.24/0.49               => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
% 0.24/0.49                 => ( ! [X3: vEBT_VEBT] :
% 0.24/0.49                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.24/0.49                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
% 0.24/0.49                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % invar_vebt.intros(3)
% 0.24/0.49  thf(fact_542_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,I: num,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_numeral_power_cancel_iff
% 0.24/0.49  thf(fact_543_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,I: num,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_numeral_power_cancel_iff
% 0.24/0.49  thf(fact_544_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,I: num,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_numeral_power_cancel_iff
% 0.24/0.49  thf(fact_545_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,I: num,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_numeral_power_cancel_iff
% 0.24/0.49  thf(fact_546_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [I: num,N: nat,X: nat] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_power_le_of_nat_cancel_iff
% 0.24/0.49  thf(fact_547_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [I: num,N: nat,X: nat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_power_le_of_nat_cancel_iff
% 0.24/0.49  thf(fact_548_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [I: num,N: nat,X: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_power_le_of_nat_cancel_iff
% 0.24/0.49  thf(fact_549_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [I: num,N: nat,X: nat] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_power_le_of_nat_cancel_iff
% 0.24/0.49  thf(fact_550_numeral__le__real__of__nat__iff,axiom,
% 0.24/0.49      ! [N: num,M: nat] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_le_real_of_nat_iff
% 0.24/0.49  thf(fact_551_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.24/0.49      ! [Y: nat,X: num,N: nat] :
% 0.24/0.49        ( ( ( semiri8010041392384452111omplex @ Y )
% 0.24/0.49          = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 0.24/0.49        = ( Y
% 0.24/0.49          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % real_of_nat_eq_numeral_power_cancel_iff
% 0.24/0.49  thf(fact_552_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.24/0.49      ! [Y: nat,X: num,N: nat] :
% 0.24/0.49        ( ( ( semiri5074537144036343181t_real @ Y )
% 0.24/0.49          = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.24/0.49        = ( Y
% 0.24/0.49          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % real_of_nat_eq_numeral_power_cancel_iff
% 0.24/0.49  thf(fact_553_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.24/0.49      ! [Y: nat,X: num,N: nat] :
% 0.24/0.49        ( ( ( semiri681578069525770553at_rat @ Y )
% 0.24/0.49          = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.24/0.49        = ( Y
% 0.24/0.49          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % real_of_nat_eq_numeral_power_cancel_iff
% 0.24/0.49  thf(fact_554_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.24/0.49      ! [Y: nat,X: num,N: nat] :
% 0.24/0.49        ( ( ( semiri1316708129612266289at_nat @ Y )
% 0.24/0.49          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.24/0.49        = ( Y
% 0.24/0.49          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % real_of_nat_eq_numeral_power_cancel_iff
% 0.24/0.49  thf(fact_555_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.24/0.49      ! [Y: nat,X: num,N: nat] :
% 0.24/0.49        ( ( ( semiri1314217659103216013at_int @ Y )
% 0.24/0.49          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.24/0.49        = ( Y
% 0.24/0.49          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % real_of_nat_eq_numeral_power_cancel_iff
% 0.24/0.49  thf(fact_556_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: num,N: nat,Y: nat] :
% 0.24/0.49        ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 0.24/0.49          = ( semiri8010041392384452111omplex @ Y ) )
% 0.24/0.49        = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.24/0.49          = Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_power_eq_of_nat_cancel_iff
% 0.24/0.49  thf(fact_557_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: num,N: nat,Y: nat] :
% 0.24/0.49        ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.24/0.49          = ( semiri5074537144036343181t_real @ Y ) )
% 0.24/0.49        = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.24/0.49          = Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_power_eq_of_nat_cancel_iff
% 0.24/0.49  thf(fact_558_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: num,N: nat,Y: nat] :
% 0.24/0.49        ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 0.24/0.49          = ( semiri681578069525770553at_rat @ Y ) )
% 0.24/0.49        = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.24/0.49          = Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_power_eq_of_nat_cancel_iff
% 0.24/0.49  thf(fact_559_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: num,N: nat,Y: nat] :
% 0.24/0.49        ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.24/0.49          = ( semiri1316708129612266289at_nat @ Y ) )
% 0.24/0.49        = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.24/0.49          = Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_power_eq_of_nat_cancel_iff
% 0.24/0.49  thf(fact_560_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: num,N: nat,Y: nat] :
% 0.24/0.49        ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.24/0.49          = ( semiri1314217659103216013at_int @ Y ) )
% 0.24/0.49        = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.24/0.49          = Y ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_power_eq_of_nat_cancel_iff
% 0.24/0.49  thf(fact_561_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,B: nat,W: nat] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.24/0.49        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power_le_of_nat_cancel_iff
% 0.24/0.49  thf(fact_562_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,B: nat,W: nat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.24/0.49        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power_le_of_nat_cancel_iff
% 0.24/0.49  thf(fact_563_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,B: nat,W: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.24/0.49        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power_le_of_nat_cancel_iff
% 0.24/0.49  thf(fact_564_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,B: nat,W: nat] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.24/0.49        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power_le_of_nat_cancel_iff
% 0.24/0.49  thf(fact_565_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.24/0.49      ! [B: nat,W: nat,X: nat] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_of_nat_power_cancel_iff
% 0.24/0.49  thf(fact_566_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.24/0.49      ! [B: nat,W: nat,X: nat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_of_nat_power_cancel_iff
% 0.24/0.49  thf(fact_567_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.24/0.49      ! [B: nat,W: nat,X: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_of_nat_power_cancel_iff
% 0.24/0.49  thf(fact_568_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.24/0.49      ! [B: nat,W: nat,X: nat] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_le_of_nat_power_cancel_iff
% 0.24/0.49  thf(fact_569_ceiling__numeral__power,axiom,
% 0.24/0.49      ! [X: num,N: nat] :
% 0.24/0.49        ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.24/0.49        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_numeral_power
% 0.24/0.49  thf(fact_570_ceiling__numeral__power,axiom,
% 0.24/0.49      ! [X: num,N: nat] :
% 0.24/0.49        ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.24/0.49        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_numeral_power
% 0.24/0.49  thf(fact_571_le__log2__of__power,axiom,
% 0.24/0.49      ! [N: nat,M: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.24/0.49       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_log2_of_power
% 0.24/0.49  thf(fact_572_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,B: nat,W: nat] :
% 0.24/0.49        ( ( ( semiri8010041392384452111omplex @ X )
% 0.24/0.49          = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 0.24/0.49        = ( X
% 0.24/0.49          = ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power_eq_of_nat_cancel_iff
% 0.24/0.49  thf(fact_573_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,B: nat,W: nat] :
% 0.24/0.49        ( ( ( semiri5074537144036343181t_real @ X )
% 0.24/0.49          = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.24/0.49        = ( X
% 0.24/0.49          = ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power_eq_of_nat_cancel_iff
% 0.24/0.49  thf(fact_574_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,B: nat,W: nat] :
% 0.24/0.49        ( ( ( semiri681578069525770553at_rat @ X )
% 0.24/0.49          = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.24/0.49        = ( X
% 0.24/0.49          = ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power_eq_of_nat_cancel_iff
% 0.24/0.49  thf(fact_575_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,B: nat,W: nat] :
% 0.24/0.49        ( ( ( semiri1316708129612266289at_nat @ X )
% 0.24/0.49          = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.24/0.49        = ( X
% 0.24/0.49          = ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power_eq_of_nat_cancel_iff
% 0.24/0.49  thf(fact_576_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.24/0.49      ! [X: nat,B: nat,W: nat] :
% 0.24/0.49        ( ( ( semiri1314217659103216013at_int @ X )
% 0.24/0.49          = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.24/0.49        = ( X
% 0.24/0.49          = ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power_eq_of_nat_cancel_iff
% 0.24/0.49  thf(fact_577_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.24/0.49      ! [B: nat,W: nat,X: nat] :
% 0.24/0.49        ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 0.24/0.49          = ( semiri8010041392384452111omplex @ X ) )
% 0.24/0.49        = ( ( power_power_nat @ B @ W )
% 0.24/0.49          = X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_of_nat_power_cancel_iff
% 0.24/0.49  thf(fact_578_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.24/0.49      ! [B: nat,W: nat,X: nat] :
% 0.24/0.49        ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 0.24/0.49          = ( semiri5074537144036343181t_real @ X ) )
% 0.24/0.49        = ( ( power_power_nat @ B @ W )
% 0.24/0.49          = X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_of_nat_power_cancel_iff
% 0.24/0.49  thf(fact_579_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.24/0.49      ! [B: nat,W: nat,X: nat] :
% 0.24/0.49        ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
% 0.24/0.49          = ( semiri681578069525770553at_rat @ X ) )
% 0.24/0.49        = ( ( power_power_nat @ B @ W )
% 0.24/0.49          = X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_of_nat_power_cancel_iff
% 0.24/0.49  thf(fact_580_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.24/0.49      ! [B: nat,W: nat,X: nat] :
% 0.24/0.49        ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 0.24/0.49          = ( semiri1316708129612266289at_nat @ X ) )
% 0.24/0.49        = ( ( power_power_nat @ B @ W )
% 0.24/0.49          = X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_of_nat_power_cancel_iff
% 0.24/0.49  thf(fact_581_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.24/0.49      ! [B: nat,W: nat,X: nat] :
% 0.24/0.49        ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 0.24/0.49          = ( semiri1314217659103216013at_int @ X ) )
% 0.24/0.49        = ( ( power_power_nat @ B @ W )
% 0.24/0.49          = X ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_eq_of_nat_power_cancel_iff
% 0.24/0.49  thf(fact_582_of__nat__power,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 0.24/0.49        = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power
% 0.24/0.49  thf(fact_583_of__nat__power,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 0.24/0.49        = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power
% 0.24/0.49  thf(fact_584_of__nat__power,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 0.24/0.49        = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power
% 0.24/0.49  thf(fact_585_of__nat__power,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 0.24/0.49        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power
% 0.24/0.49  thf(fact_586_of__nat__power,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 0.24/0.49        = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_nat_power
% 0.24/0.49  thf(fact_587_insert__subset,axiom,
% 0.24/0.49      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.24/0.49        ( ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B2 )
% 0.24/0.49        = ( ( member_VEBT_VEBT @ X @ B2 )
% 0.24/0.49          & ( ord_le4337996190870823476T_VEBT @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_subset
% 0.24/0.49  thf(fact_588_insert__subset,axiom,
% 0.24/0.49      ! [X: $o,A2: set_o,B2: set_o] :
% 0.24/0.49        ( ( ord_less_eq_set_o @ ( insert_o @ X @ A2 ) @ B2 )
% 0.24/0.49        = ( ( member_o @ X @ B2 )
% 0.24/0.49          & ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_subset
% 0.24/0.49  thf(fact_589_insert__subset,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,B2: set_complex] :
% 0.24/0.49        ( ( ord_le211207098394363844omplex @ ( insert_complex @ X @ A2 ) @ B2 )
% 0.24/0.49        = ( ( member_complex @ X @ B2 )
% 0.24/0.49          & ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_subset
% 0.24/0.49  thf(fact_590_insert__subset,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,B2: set_real] :
% 0.24/0.49        ( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 0.24/0.49        = ( ( member_real @ X @ B2 )
% 0.24/0.49          & ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_subset
% 0.24/0.49  thf(fact_591_insert__subset,axiom,
% 0.24/0.49      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.24/0.49        ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 0.24/0.49        = ( ( member_set_nat @ X @ B2 )
% 0.24/0.49          & ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_subset
% 0.24/0.49  thf(fact_592_insert__subset,axiom,
% 0.24/0.49      ! [X: nat,A2: set_nat,B2: set_nat] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 0.24/0.49        = ( ( member_nat @ X @ B2 )
% 0.24/0.49          & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_subset
% 0.24/0.49  thf(fact_593_insert__subset,axiom,
% 0.24/0.49      ! [X: int,A2: set_int,B2: set_int] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 0.24/0.49        = ( ( member_int @ X @ B2 )
% 0.24/0.49          & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_subset
% 0.24/0.49  thf(fact_594_power__one,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( power_power_rat @ one_one_rat @ N )
% 0.24/0.49        = one_one_rat ) ).
% 0.24/0.49  
% 0.24/0.49  % power_one
% 0.24/0.49  thf(fact_595_power__one,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( power_power_nat @ one_one_nat @ N )
% 0.24/0.49        = one_one_nat ) ).
% 0.24/0.49  
% 0.24/0.49  % power_one
% 0.24/0.49  thf(fact_596_power__one,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( power_power_real @ one_one_real @ N )
% 0.24/0.49        = one_one_real ) ).
% 0.24/0.49  
% 0.24/0.49  % power_one
% 0.24/0.49  thf(fact_597_power__one,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( power_power_int @ one_one_int @ N )
% 0.24/0.49        = one_one_int ) ).
% 0.24/0.49  
% 0.24/0.49  % power_one
% 0.24/0.49  thf(fact_598_power__one,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( power_power_complex @ one_one_complex @ N )
% 0.24/0.49        = one_one_complex ) ).
% 0.24/0.49  
% 0.24/0.49  % power_one
% 0.24/0.49  thf(fact_599_power__one__right,axiom,
% 0.24/0.49      ! [A: nat] :
% 0.24/0.49        ( ( power_power_nat @ A @ one_one_nat )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % power_one_right
% 0.24/0.49  thf(fact_600_power__one__right,axiom,
% 0.24/0.49      ! [A: real] :
% 0.24/0.49        ( ( power_power_real @ A @ one_one_nat )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % power_one_right
% 0.24/0.49  thf(fact_601_power__one__right,axiom,
% 0.24/0.49      ! [A: int] :
% 0.24/0.49        ( ( power_power_int @ A @ one_one_nat )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % power_one_right
% 0.24/0.49  thf(fact_602_power__one__right,axiom,
% 0.24/0.49      ! [A: complex] :
% 0.24/0.49        ( ( power_power_complex @ A @ one_one_nat )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % power_one_right
% 0.24/0.49  thf(fact_603_semiring__norm_I71_J,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.24/0.49        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % semiring_norm(71)
% 0.24/0.49  thf(fact_604_semiring__norm_I68_J,axiom,
% 0.24/0.49      ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 0.24/0.49  
% 0.24/0.49  % semiring_norm(68)
% 0.24/0.49  thf(fact_605_semiring__norm_I69_J,axiom,
% 0.24/0.49      ! [M: num] :
% 0.24/0.49        ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 0.24/0.49  
% 0.24/0.49  % semiring_norm(69)
% 0.24/0.49  thf(fact_606_le__num__One__iff,axiom,
% 0.24/0.49      ! [X: num] :
% 0.24/0.49        ( ( ord_less_eq_num @ X @ one )
% 0.24/0.49        = ( X = one ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_num_One_iff
% 0.24/0.49  thf(fact_607_subset__image__iff,axiom,
% 0.24/0.49      ! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
% 0.24/0.49        ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
% 0.24/0.49        = ( ? [AA: set_nat] :
% 0.24/0.49              ( ( ord_less_eq_set_nat @ AA @ A2 )
% 0.24/0.49              & ( B2
% 0.24/0.49                = ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_image_iff
% 0.24/0.49  thf(fact_608_subset__image__iff,axiom,
% 0.24/0.49      ! [B2: set_nat,F: nat > nat,A2: set_nat] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
% 0.24/0.49        = ( ? [AA: set_nat] :
% 0.24/0.49              ( ( ord_less_eq_set_nat @ AA @ A2 )
% 0.24/0.49              & ( B2
% 0.24/0.49                = ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_image_iff
% 0.24/0.49  thf(fact_609_subset__image__iff,axiom,
% 0.24/0.49      ! [B2: set_nat,F: int > nat,A2: set_int] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F @ A2 ) )
% 0.24/0.49        = ( ? [AA: set_int] :
% 0.24/0.49              ( ( ord_less_eq_set_int @ AA @ A2 )
% 0.24/0.49              & ( B2
% 0.24/0.49                = ( image_int_nat @ F @ AA ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_image_iff
% 0.24/0.49  thf(fact_610_subset__image__iff,axiom,
% 0.24/0.49      ! [B2: set_int,F: nat > int,A2: set_nat] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A2 ) )
% 0.24/0.49        = ( ? [AA: set_nat] :
% 0.24/0.49              ( ( ord_less_eq_set_nat @ AA @ A2 )
% 0.24/0.49              & ( B2
% 0.24/0.49                = ( image_nat_int @ F @ AA ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_image_iff
% 0.24/0.49  thf(fact_611_subset__image__iff,axiom,
% 0.24/0.49      ! [B2: set_int,F: int > int,A2: set_int] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ B2 @ ( image_int_int @ F @ A2 ) )
% 0.24/0.49        = ( ? [AA: set_int] :
% 0.24/0.49              ( ( ord_less_eq_set_int @ AA @ A2 )
% 0.24/0.49              & ( B2
% 0.24/0.49                = ( image_int_int @ F @ AA ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_image_iff
% 0.24/0.49  thf(fact_612_image__subset__iff,axiom,
% 0.24/0.49      ! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
% 0.24/0.49        ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
% 0.24/0.49        = ( ! [X4: nat] :
% 0.24/0.49              ( ( member_nat @ X4 @ A2 )
% 0.24/0.49             => ( member_set_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subset_iff
% 0.24/0.49  thf(fact_613_image__subset__iff,axiom,
% 0.24/0.49      ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
% 0.24/0.49        = ( ! [X4: nat] :
% 0.24/0.49              ( ( member_nat @ X4 @ A2 )
% 0.24/0.49             => ( member_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subset_iff
% 0.24/0.49  thf(fact_614_image__subset__iff,axiom,
% 0.24/0.49      ! [F: int > nat,A2: set_int,B2: set_nat] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A2 ) @ B2 )
% 0.24/0.49        = ( ! [X4: int] :
% 0.24/0.49              ( ( member_int @ X4 @ A2 )
% 0.24/0.49             => ( member_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subset_iff
% 0.24/0.49  thf(fact_615_image__subset__iff,axiom,
% 0.24/0.49      ! [F: nat > int,A2: set_nat,B2: set_int] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B2 )
% 0.24/0.49        = ( ! [X4: nat] :
% 0.24/0.49              ( ( member_nat @ X4 @ A2 )
% 0.24/0.49             => ( member_int @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subset_iff
% 0.24/0.49  thf(fact_616_image__subset__iff,axiom,
% 0.24/0.49      ! [F: int > int,A2: set_int,B2: set_int] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A2 ) @ B2 )
% 0.24/0.49        = ( ! [X4: int] :
% 0.24/0.49              ( ( member_int @ X4 @ A2 )
% 0.24/0.49             => ( member_int @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subset_iff
% 0.24/0.49  thf(fact_617_subset__imageE,axiom,
% 0.24/0.49      ! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
% 0.24/0.49        ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
% 0.24/0.49       => ~ ! [C5: set_nat] :
% 0.24/0.49              ( ( ord_less_eq_set_nat @ C5 @ A2 )
% 0.24/0.49             => ( B2
% 0.24/0.49               != ( image_nat_set_nat @ F @ C5 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_imageE
% 0.24/0.49  thf(fact_618_subset__imageE,axiom,
% 0.24/0.49      ! [B2: set_nat,F: nat > nat,A2: set_nat] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
% 0.24/0.49       => ~ ! [C5: set_nat] :
% 0.24/0.49              ( ( ord_less_eq_set_nat @ C5 @ A2 )
% 0.24/0.49             => ( B2
% 0.24/0.49               != ( image_nat_nat @ F @ C5 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_imageE
% 0.24/0.49  thf(fact_619_subset__imageE,axiom,
% 0.24/0.49      ! [B2: set_nat,F: int > nat,A2: set_int] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F @ A2 ) )
% 0.24/0.49       => ~ ! [C5: set_int] :
% 0.24/0.49              ( ( ord_less_eq_set_int @ C5 @ A2 )
% 0.24/0.49             => ( B2
% 0.24/0.49               != ( image_int_nat @ F @ C5 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_imageE
% 0.24/0.49  thf(fact_620_subset__imageE,axiom,
% 0.24/0.49      ! [B2: set_int,F: nat > int,A2: set_nat] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A2 ) )
% 0.24/0.49       => ~ ! [C5: set_nat] :
% 0.24/0.49              ( ( ord_less_eq_set_nat @ C5 @ A2 )
% 0.24/0.49             => ( B2
% 0.24/0.49               != ( image_nat_int @ F @ C5 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_imageE
% 0.24/0.49  thf(fact_621_subset__imageE,axiom,
% 0.24/0.49      ! [B2: set_int,F: int > int,A2: set_int] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ B2 @ ( image_int_int @ F @ A2 ) )
% 0.24/0.49       => ~ ! [C5: set_int] :
% 0.24/0.49              ( ( ord_less_eq_set_int @ C5 @ A2 )
% 0.24/0.49             => ( B2
% 0.24/0.49               != ( image_int_int @ F @ C5 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_imageE
% 0.24/0.49  thf(fact_622_image__subsetI,axiom,
% 0.24/0.49      ! [A2: set_complex,F: complex > complex,B2: set_complex] :
% 0.24/0.49        ( ! [X3: complex] :
% 0.24/0.49            ( ( member_complex @ X3 @ A2 )
% 0.24/0.49           => ( member_complex @ ( F @ X3 ) @ B2 ) )
% 0.24/0.49       => ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F @ A2 ) @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subsetI
% 0.24/0.49  thf(fact_623_image__subsetI,axiom,
% 0.24/0.49      ! [A2: set_complex,F: complex > real,B2: set_real] :
% 0.24/0.49        ( ! [X3: complex] :
% 0.24/0.49            ( ( member_complex @ X3 @ A2 )
% 0.24/0.49           => ( member_real @ ( F @ X3 ) @ B2 ) )
% 0.24/0.49       => ( ord_less_eq_set_real @ ( image_complex_real @ F @ A2 ) @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subsetI
% 0.24/0.49  thf(fact_624_image__subsetI,axiom,
% 0.24/0.49      ! [A2: set_complex,F: complex > nat,B2: set_nat] :
% 0.24/0.49        ( ! [X3: complex] :
% 0.24/0.49            ( ( member_complex @ X3 @ A2 )
% 0.24/0.49           => ( member_nat @ ( F @ X3 ) @ B2 ) )
% 0.24/0.49       => ( ord_less_eq_set_nat @ ( image_complex_nat @ F @ A2 ) @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subsetI
% 0.24/0.49  thf(fact_625_image__subsetI,axiom,
% 0.24/0.49      ! [A2: set_real,F: real > complex,B2: set_complex] :
% 0.24/0.49        ( ! [X3: real] :
% 0.24/0.49            ( ( member_real @ X3 @ A2 )
% 0.24/0.49           => ( member_complex @ ( F @ X3 ) @ B2 ) )
% 0.24/0.49       => ( ord_le211207098394363844omplex @ ( image_real_complex @ F @ A2 ) @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subsetI
% 0.24/0.49  thf(fact_626_image__subsetI,axiom,
% 0.24/0.49      ! [A2: set_real,F: real > real,B2: set_real] :
% 0.24/0.49        ( ! [X3: real] :
% 0.24/0.49            ( ( member_real @ X3 @ A2 )
% 0.24/0.49           => ( member_real @ ( F @ X3 ) @ B2 ) )
% 0.24/0.49       => ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subsetI
% 0.24/0.49  thf(fact_627_image__subsetI,axiom,
% 0.24/0.49      ! [A2: set_real,F: real > nat,B2: set_nat] :
% 0.24/0.49        ( ! [X3: real] :
% 0.24/0.49            ( ( member_real @ X3 @ A2 )
% 0.24/0.49           => ( member_nat @ ( F @ X3 ) @ B2 ) )
% 0.24/0.49       => ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A2 ) @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subsetI
% 0.24/0.49  thf(fact_628_image__subsetI,axiom,
% 0.24/0.49      ! [A2: set_nat,F: nat > complex,B2: set_complex] :
% 0.24/0.49        ( ! [X3: nat] :
% 0.24/0.49            ( ( member_nat @ X3 @ A2 )
% 0.24/0.49           => ( member_complex @ ( F @ X3 ) @ B2 ) )
% 0.24/0.49       => ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A2 ) @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subsetI
% 0.24/0.49  thf(fact_629_image__subsetI,axiom,
% 0.24/0.49      ! [A2: set_nat,F: nat > real,B2: set_real] :
% 0.24/0.49        ( ! [X3: nat] :
% 0.24/0.49            ( ( member_nat @ X3 @ A2 )
% 0.24/0.49           => ( member_real @ ( F @ X3 ) @ B2 ) )
% 0.24/0.49       => ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subsetI
% 0.24/0.49  thf(fact_630_image__subsetI,axiom,
% 0.24/0.49      ! [A2: set_nat,F: nat > nat,B2: set_nat] :
% 0.24/0.49        ( ! [X3: nat] :
% 0.24/0.49            ( ( member_nat @ X3 @ A2 )
% 0.24/0.49           => ( member_nat @ ( F @ X3 ) @ B2 ) )
% 0.24/0.49       => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subsetI
% 0.24/0.49  thf(fact_631_image__subsetI,axiom,
% 0.24/0.49      ! [A2: set_int,F: int > complex,B2: set_complex] :
% 0.24/0.49        ( ! [X3: int] :
% 0.24/0.49            ( ( member_int @ X3 @ A2 )
% 0.24/0.49           => ( member_complex @ ( F @ X3 ) @ B2 ) )
% 0.24/0.49       => ( ord_le211207098394363844omplex @ ( image_int_complex @ F @ A2 ) @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_subsetI
% 0.24/0.49  thf(fact_632_image__mono,axiom,
% 0.24/0.49      ! [A2: set_nat,B2: set_nat,F: nat > set_nat] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.24/0.49       => ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_mono
% 0.24/0.49  thf(fact_633_image__mono,axiom,
% 0.24/0.49      ! [A2: set_nat,B2: set_nat,F: nat > nat] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.24/0.49       => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_mono
% 0.24/0.49  thf(fact_634_image__mono,axiom,
% 0.24/0.49      ! [A2: set_nat,B2: set_nat,F: nat > int] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.24/0.49       => ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_mono
% 0.24/0.49  thf(fact_635_image__mono,axiom,
% 0.24/0.49      ! [A2: set_int,B2: set_int,F: int > nat] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.49       => ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A2 ) @ ( image_int_nat @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_mono
% 0.24/0.49  thf(fact_636_image__mono,axiom,
% 0.24/0.49      ! [A2: set_int,B2: set_int,F: int > int] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.49       => ( ord_less_eq_set_int @ ( image_int_int @ F @ A2 ) @ ( image_int_int @ F @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % image_mono
% 0.24/0.49  thf(fact_637_subset__insertI2,axiom,
% 0.24/0.49      ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
% 0.24/0.49        ( ( ord_le4337996190870823476T_VEBT @ A2 @ B2 )
% 0.24/0.49       => ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insertI2
% 0.24/0.49  thf(fact_638_subset__insertI2,axiom,
% 0.24/0.49      ! [A2: set_nat,B2: set_nat,B: nat] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.24/0.49       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insertI2
% 0.24/0.49  thf(fact_639_subset__insertI2,axiom,
% 0.24/0.49      ! [A2: set_real,B2: set_real,B: real] :
% 0.24/0.49        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.24/0.49       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insertI2
% 0.24/0.49  thf(fact_640_subset__insertI2,axiom,
% 0.24/0.49      ! [A2: set_o,B2: set_o,B: $o] :
% 0.24/0.49        ( ( ord_less_eq_set_o @ A2 @ B2 )
% 0.24/0.49       => ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insertI2
% 0.24/0.49  thf(fact_641_subset__insertI2,axiom,
% 0.24/0.49      ! [A2: set_int,B2: set_int,B: int] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.49       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insertI2
% 0.24/0.49  thf(fact_642_subset__insertI,axiom,
% 0.24/0.49      ! [B2: set_VEBT_VEBT,A: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ B2 @ ( insert_VEBT_VEBT @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insertI
% 0.24/0.49  thf(fact_643_subset__insertI,axiom,
% 0.24/0.49      ! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insertI
% 0.24/0.49  thf(fact_644_subset__insertI,axiom,
% 0.24/0.49      ! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insertI
% 0.24/0.49  thf(fact_645_subset__insertI,axiom,
% 0.24/0.49      ! [B2: set_o,A: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insertI
% 0.24/0.49  thf(fact_646_subset__insertI,axiom,
% 0.24/0.49      ! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insertI
% 0.24/0.49  thf(fact_647_subset__insert,axiom,
% 0.24/0.49      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.24/0.49        ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 0.24/0.49       => ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B2 ) )
% 0.24/0.49          = ( ord_le4337996190870823476T_VEBT @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insert
% 0.24/0.49  thf(fact_648_subset__insert,axiom,
% 0.24/0.49      ! [X: $o,A2: set_o,B2: set_o] :
% 0.24/0.49        ( ~ ( member_o @ X @ A2 )
% 0.24/0.49       => ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
% 0.24/0.49          = ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insert
% 0.24/0.49  thf(fact_649_subset__insert,axiom,
% 0.24/0.49      ! [X: complex,A2: set_complex,B2: set_complex] :
% 0.24/0.49        ( ~ ( member_complex @ X @ A2 )
% 0.24/0.49       => ( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B2 ) )
% 0.24/0.49          = ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insert
% 0.24/0.49  thf(fact_650_subset__insert,axiom,
% 0.24/0.49      ! [X: real,A2: set_real,B2: set_real] :
% 0.24/0.49        ( ~ ( member_real @ X @ A2 )
% 0.24/0.49       => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 0.24/0.49          = ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insert
% 0.24/0.49  thf(fact_651_subset__insert,axiom,
% 0.24/0.49      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.24/0.49        ( ~ ( member_set_nat @ X @ A2 )
% 0.24/0.49       => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 0.24/0.49          = ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insert
% 0.24/0.49  thf(fact_652_subset__insert,axiom,
% 0.24/0.49      ! [X: nat,A2: set_nat,B2: set_nat] :
% 0.24/0.49        ( ~ ( member_nat @ X @ A2 )
% 0.24/0.49       => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 0.24/0.49          = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insert
% 0.24/0.49  thf(fact_653_subset__insert,axiom,
% 0.24/0.49      ! [X: int,A2: set_int,B2: set_int] :
% 0.24/0.49        ( ~ ( member_int @ X @ A2 )
% 0.24/0.49       => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 0.24/0.49          = ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_insert
% 0.24/0.49  thf(fact_654_insert__mono,axiom,
% 0.24/0.49      ! [C6: set_VEBT_VEBT,D2: set_VEBT_VEBT,A: vEBT_VEBT] :
% 0.24/0.49        ( ( ord_le4337996190870823476T_VEBT @ C6 @ D2 )
% 0.24/0.49       => ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ A @ C6 ) @ ( insert_VEBT_VEBT @ A @ D2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_mono
% 0.24/0.49  thf(fact_655_insert__mono,axiom,
% 0.24/0.49      ! [C6: set_nat,D2: set_nat,A: nat] :
% 0.24/0.49        ( ( ord_less_eq_set_nat @ C6 @ D2 )
% 0.24/0.49       => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C6 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_mono
% 0.24/0.49  thf(fact_656_insert__mono,axiom,
% 0.24/0.49      ! [C6: set_real,D2: set_real,A: real] :
% 0.24/0.49        ( ( ord_less_eq_set_real @ C6 @ D2 )
% 0.24/0.49       => ( ord_less_eq_set_real @ ( insert_real @ A @ C6 ) @ ( insert_real @ A @ D2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_mono
% 0.24/0.49  thf(fact_657_insert__mono,axiom,
% 0.24/0.49      ! [C6: set_o,D2: set_o,A: $o] :
% 0.24/0.49        ( ( ord_less_eq_set_o @ C6 @ D2 )
% 0.24/0.49       => ( ord_less_eq_set_o @ ( insert_o @ A @ C6 ) @ ( insert_o @ A @ D2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_mono
% 0.24/0.49  thf(fact_658_insert__mono,axiom,
% 0.24/0.49      ! [C6: set_int,D2: set_int,A: int] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ C6 @ D2 )
% 0.24/0.49       => ( ord_less_eq_set_int @ ( insert_int @ A @ C6 ) @ ( insert_int @ A @ D2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % insert_mono
% 0.24/0.49  thf(fact_659_two__realpow__ge__one,axiom,
% 0.24/0.49      ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % two_realpow_ge_one
% 0.24/0.49  thf(fact_660_complete__real,axiom,
% 0.24/0.49      ! [S2: set_real] :
% 0.24/0.49        ( ? [X2: real] : ( member_real @ X2 @ S2 )
% 0.24/0.49       => ( ? [Z5: real] :
% 0.24/0.49            ! [X3: real] :
% 0.24/0.49              ( ( member_real @ X3 @ S2 )
% 0.24/0.49             => ( ord_less_eq_real @ X3 @ Z5 ) )
% 0.24/0.49         => ? [Y4: real] :
% 0.24/0.49              ( ! [X2: real] :
% 0.24/0.49                  ( ( member_real @ X2 @ S2 )
% 0.24/0.49                 => ( ord_less_eq_real @ X2 @ Y4 ) )
% 0.24/0.49              & ! [Z5: real] :
% 0.24/0.49                  ( ! [X3: real] :
% 0.24/0.49                      ( ( member_real @ X3 @ S2 )
% 0.24/0.49                     => ( ord_less_eq_real @ X3 @ Z5 ) )
% 0.24/0.49                 => ( ord_less_eq_real @ Y4 @ Z5 ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % complete_real
% 0.24/0.49  thf(fact_661_one__le__power,axiom,
% 0.24/0.49      ! [A: real,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_real @ one_one_real @ A )
% 0.24/0.49       => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_le_power
% 0.24/0.49  thf(fact_662_one__le__power,axiom,
% 0.24/0.49      ! [A: rat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.24/0.49       => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_le_power
% 0.24/0.49  thf(fact_663_one__le__power,axiom,
% 0.24/0.49      ! [A: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.24/0.49       => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_le_power
% 0.24/0.49  thf(fact_664_one__le__power,axiom,
% 0.24/0.49      ! [A: int,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_int @ one_one_int @ A )
% 0.24/0.49       => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % one_le_power
% 0.24/0.49  thf(fact_665_power__increasing,axiom,
% 0.24/0.49      ! [N: nat,N4: nat,A: real] :
% 0.24/0.49        ( ( ord_less_eq_nat @ N @ N4 )
% 0.24/0.49       => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.24/0.49         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power_increasing
% 0.24/0.49  thf(fact_666_power__increasing,axiom,
% 0.24/0.49      ! [N: nat,N4: nat,A: rat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ N @ N4 )
% 0.24/0.49       => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.24/0.49         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power_increasing
% 0.24/0.49  thf(fact_667_power__increasing,axiom,
% 0.24/0.49      ! [N: nat,N4: nat,A: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ N @ N4 )
% 0.24/0.49       => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.24/0.49         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power_increasing
% 0.24/0.49  thf(fact_668_power__increasing,axiom,
% 0.24/0.49      ! [N: nat,N4: nat,A: int] :
% 0.24/0.49        ( ( ord_less_eq_nat @ N @ N4 )
% 0.24/0.49       => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.24/0.49         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power_increasing
% 0.24/0.49  thf(fact_669_one__power2,axiom,
% 0.24/0.49      ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.49      = one_one_rat ) ).
% 0.24/0.49  
% 0.24/0.49  % one_power2
% 0.24/0.49  thf(fact_670_one__power2,axiom,
% 0.24/0.49      ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.49      = one_one_nat ) ).
% 0.24/0.49  
% 0.24/0.49  % one_power2
% 0.24/0.49  thf(fact_671_one__power2,axiom,
% 0.24/0.49      ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.49      = one_one_real ) ).
% 0.24/0.49  
% 0.24/0.49  % one_power2
% 0.24/0.49  thf(fact_672_one__power2,axiom,
% 0.24/0.49      ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.49      = one_one_int ) ).
% 0.24/0.49  
% 0.24/0.49  % one_power2
% 0.24/0.49  thf(fact_673_one__power2,axiom,
% 0.24/0.49      ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.49      = one_one_complex ) ).
% 0.24/0.49  
% 0.24/0.49  % one_power2
% 0.24/0.49  thf(fact_674_self__le__ge2__pow,axiom,
% 0.24/0.49      ! [K: nat,M: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.24/0.49       => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % self_le_ge2_pow
% 0.24/0.49  thf(fact_675_power2__nat__le__eq__le,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.49        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power2_nat_le_eq_le
% 0.24/0.49  thf(fact_676_power2__nat__le__imp__le,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.24/0.49       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power2_nat_le_imp_le
% 0.24/0.49  thf(fact_677_log2__of__power__eq,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( M
% 0.24/0.49          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.24/0.49       => ( ( semiri5074537144036343181t_real @ N )
% 0.24/0.49          = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % log2_of_power_eq
% 0.24/0.49  thf(fact_678_enat__ord__number_I1_J,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.24/0.49        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % enat_ord_number(1)
% 0.24/0.49  thf(fact_679_log__ceil__idem,axiom,
% 0.24/0.49      ! [X: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ one_one_real @ X )
% 0.24/0.49       => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.24/0.49          = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % log_ceil_idem
% 0.24/0.49  thf(fact_680_ceiling__log__nat__eq__if,axiom,
% 0.24/0.49      ! [B: nat,N: nat,K: nat] :
% 0.24/0.49        ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.24/0.49       => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.24/0.49         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.24/0.49           => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.24/0.49              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_log_nat_eq_if
% 0.24/0.49  thf(fact_681_pow__sum,axiom,
% 0.24/0.49      ! [A: nat,B: nat] :
% 0.24/0.49        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.24/0.49        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % pow_sum
% 0.24/0.49  thf(fact_682_power__numeral,axiom,
% 0.24/0.49      ! [K: num,L: num] :
% 0.24/0.49        ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.24/0.49        = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power_numeral
% 0.24/0.49  thf(fact_683_power__numeral,axiom,
% 0.24/0.49      ! [K: num,L: num] :
% 0.24/0.49        ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.24/0.49        = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power_numeral
% 0.24/0.49  thf(fact_684_power__numeral,axiom,
% 0.24/0.49      ! [K: num,L: num] :
% 0.24/0.49        ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.24/0.49        = ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power_numeral
% 0.24/0.49  thf(fact_685_power__numeral,axiom,
% 0.24/0.49      ! [K: num,L: num] :
% 0.24/0.49        ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.24/0.49        = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power_numeral
% 0.24/0.49  thf(fact_686_power__numeral,axiom,
% 0.24/0.49      ! [K: num,L: num] :
% 0.24/0.49        ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.24/0.49        = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % power_numeral
% 0.24/0.49  thf(fact_687_dual__order_Orefl,axiom,
% 0.24/0.49      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % dual_order.refl
% 0.24/0.49  thf(fact_688_dual__order_Orefl,axiom,
% 0.24/0.49      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % dual_order.refl
% 0.24/0.49  thf(fact_689_dual__order_Orefl,axiom,
% 0.24/0.49      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % dual_order.refl
% 0.24/0.49  thf(fact_690_dual__order_Orefl,axiom,
% 0.24/0.49      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % dual_order.refl
% 0.24/0.49  thf(fact_691_dual__order_Orefl,axiom,
% 0.24/0.49      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % dual_order.refl
% 0.24/0.49  thf(fact_692_order__refl,axiom,
% 0.24/0.49      ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% 0.24/0.49  
% 0.24/0.49  % order_refl
% 0.24/0.49  thf(fact_693_order__refl,axiom,
% 0.24/0.49      ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 0.24/0.49  
% 0.24/0.49  % order_refl
% 0.24/0.49  thf(fact_694_order__refl,axiom,
% 0.24/0.49      ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 0.24/0.49  
% 0.24/0.49  % order_refl
% 0.24/0.49  thf(fact_695_order__refl,axiom,
% 0.24/0.49      ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.24/0.49  
% 0.24/0.49  % order_refl
% 0.24/0.49  thf(fact_696_order__refl,axiom,
% 0.24/0.49      ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.24/0.49  
% 0.24/0.49  % order_refl
% 0.24/0.49  thf(fact_697_ex__power__ivl2,axiom,
% 0.24/0.49      ! [B: nat,K: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.24/0.49       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.24/0.49         => ? [N2: nat] :
% 0.24/0.49              ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.24/0.49              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ex_power_ivl2
% 0.24/0.49  thf(fact_698_ex__power__ivl1,axiom,
% 0.24/0.49      ! [B: nat,K: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.24/0.49       => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 0.24/0.49         => ? [N2: nat] :
% 0.24/0.49              ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.24/0.49              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ex_power_ivl1
% 0.24/0.49  thf(fact_699_ceiling__log__nat__eq__powr__iff,axiom,
% 0.24/0.49      ! [B: nat,K: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.24/0.49       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.24/0.49         => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.24/0.49              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 0.24/0.49            = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.24/0.49              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_log_nat_eq_powr_iff
% 0.24/0.49  thf(fact_700_deg__not__0,axiom,
% 0.24/0.49      ! [T: vEBT_VEBT,N: nat] :
% 0.24/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.24/0.49       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % deg_not_0
% 0.24/0.49  thf(fact_701_subset__antisym,axiom,
% 0.24/0.49      ! [A2: set_int,B2: set_int] :
% 0.24/0.49        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.49       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.24/0.49         => ( A2 = B2 ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subset_antisym
% 0.24/0.49  thf(fact_702_subsetI,axiom,
% 0.24/0.49      ! [A2: set_complex,B2: set_complex] :
% 0.24/0.49        ( ! [X3: complex] :
% 0.24/0.49            ( ( member_complex @ X3 @ A2 )
% 0.24/0.49           => ( member_complex @ X3 @ B2 ) )
% 0.24/0.49       => ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subsetI
% 0.24/0.49  thf(fact_703_subsetI,axiom,
% 0.24/0.49      ! [A2: set_real,B2: set_real] :
% 0.24/0.49        ( ! [X3: real] :
% 0.24/0.49            ( ( member_real @ X3 @ A2 )
% 0.24/0.49           => ( member_real @ X3 @ B2 ) )
% 0.24/0.49       => ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subsetI
% 0.24/0.49  thf(fact_704_subsetI,axiom,
% 0.24/0.49      ! [A2: set_set_nat,B2: set_set_nat] :
% 0.24/0.49        ( ! [X3: set_nat] :
% 0.24/0.49            ( ( member_set_nat @ X3 @ A2 )
% 0.24/0.49           => ( member_set_nat @ X3 @ B2 ) )
% 0.24/0.49       => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subsetI
% 0.24/0.49  thf(fact_705_subsetI,axiom,
% 0.24/0.49      ! [A2: set_nat,B2: set_nat] :
% 0.24/0.49        ( ! [X3: nat] :
% 0.24/0.49            ( ( member_nat @ X3 @ A2 )
% 0.24/0.49           => ( member_nat @ X3 @ B2 ) )
% 0.24/0.49       => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subsetI
% 0.24/0.49  thf(fact_706_subsetI,axiom,
% 0.24/0.49      ! [A2: set_int,B2: set_int] :
% 0.24/0.49        ( ! [X3: int] :
% 0.24/0.49            ( ( member_int @ X3 @ A2 )
% 0.24/0.49           => ( member_int @ X3 @ B2 ) )
% 0.24/0.49       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.24/0.49  
% 0.24/0.49  % subsetI
% 0.24/0.49  thf(fact_707_of__int__eq__iff,axiom,
% 0.24/0.49      ! [W: int,Z: int] :
% 0.24/0.49        ( ( ( ring_1_of_int_real @ W )
% 0.24/0.49          = ( ring_1_of_int_real @ Z ) )
% 0.24/0.49        = ( W = Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_eq_iff
% 0.24/0.49  thf(fact_708_of__int__eq__iff,axiom,
% 0.24/0.49      ! [W: int,Z: int] :
% 0.24/0.49        ( ( ( ring_1_of_int_rat @ W )
% 0.24/0.49          = ( ring_1_of_int_rat @ Z ) )
% 0.24/0.49        = ( W = Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_eq_iff
% 0.24/0.49  thf(fact_709_of__int__eq__iff,axiom,
% 0.24/0.49      ! [W: int,Z: int] :
% 0.24/0.49        ( ( ( ring_18347121197199848620nteger @ W )
% 0.24/0.49          = ( ring_18347121197199848620nteger @ Z ) )
% 0.24/0.49        = ( W = Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_eq_iff
% 0.24/0.49  thf(fact_710_of__int__eq__iff,axiom,
% 0.24/0.49      ! [W: int,Z: int] :
% 0.24/0.49        ( ( ( ring_17405671764205052669omplex @ W )
% 0.24/0.49          = ( ring_17405671764205052669omplex @ Z ) )
% 0.24/0.49        = ( W = Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_eq_iff
% 0.24/0.49  thf(fact_711_le__zero__eq,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.24/0.49        = ( N = zero_zero_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_zero_eq
% 0.24/0.49  thf(fact_712_not__gr__zero,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.24/0.49        = ( N = zero_zero_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % not_gr_zero
% 0.24/0.49  thf(fact_713_numeral__less__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.24/0.49        = ( ord_less_num @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_less_iff
% 0.24/0.49  thf(fact_714_numeral__less__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.24/0.49        = ( ord_less_num @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_less_iff
% 0.24/0.49  thf(fact_715_numeral__less__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.24/0.49        = ( ord_less_num @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_less_iff
% 0.24/0.49  thf(fact_716_numeral__less__iff,axiom,
% 0.24/0.49      ! [M: num,N: num] :
% 0.24/0.49        ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.24/0.49        = ( ord_less_num @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % numeral_less_iff
% 0.24/0.49  thf(fact_717_add__0,axiom,
% 0.24/0.49      ! [A: complex] :
% 0.24/0.49        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % add_0
% 0.24/0.49  thf(fact_718_add__0,axiom,
% 0.24/0.49      ! [A: real] :
% 0.24/0.49        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % add_0
% 0.24/0.49  thf(fact_719_add__0,axiom,
% 0.24/0.49      ! [A: rat] :
% 0.24/0.49        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % add_0
% 0.24/0.49  thf(fact_720_add__0,axiom,
% 0.24/0.49      ! [A: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % add_0
% 0.24/0.49  thf(fact_721_add__0,axiom,
% 0.24/0.49      ! [A: int] :
% 0.24/0.49        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % add_0
% 0.24/0.49  thf(fact_722_zero__eq__add__iff__both__eq__0,axiom,
% 0.24/0.49      ! [X: nat,Y: nat] :
% 0.24/0.49        ( ( zero_zero_nat
% 0.24/0.49          = ( plus_plus_nat @ X @ Y ) )
% 0.24/0.49        = ( ( X = zero_zero_nat )
% 0.24/0.49          & ( Y = zero_zero_nat ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % zero_eq_add_iff_both_eq_0
% 0.24/0.49  thf(fact_723_add__eq__0__iff__both__eq__0,axiom,
% 0.24/0.49      ! [X: nat,Y: nat] :
% 0.24/0.49        ( ( ( plus_plus_nat @ X @ Y )
% 0.24/0.49          = zero_zero_nat )
% 0.24/0.49        = ( ( X = zero_zero_nat )
% 0.24/0.49          & ( Y = zero_zero_nat ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_eq_0_iff_both_eq_0
% 0.24/0.49  thf(fact_724_add__cancel__right__right,axiom,
% 0.24/0.49      ! [A: complex,B: complex] :
% 0.24/0.49        ( ( A
% 0.24/0.49          = ( plus_plus_complex @ A @ B ) )
% 0.24/0.49        = ( B = zero_zero_complex ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_right_right
% 0.24/0.49  thf(fact_725_add__cancel__right__right,axiom,
% 0.24/0.49      ! [A: real,B: real] :
% 0.24/0.49        ( ( A
% 0.24/0.49          = ( plus_plus_real @ A @ B ) )
% 0.24/0.49        = ( B = zero_zero_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_right_right
% 0.24/0.49  thf(fact_726_add__cancel__right__right,axiom,
% 0.24/0.49      ! [A: rat,B: rat] :
% 0.24/0.49        ( ( A
% 0.24/0.49          = ( plus_plus_rat @ A @ B ) )
% 0.24/0.49        = ( B = zero_zero_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_right_right
% 0.24/0.49  thf(fact_727_add__cancel__right__right,axiom,
% 0.24/0.49      ! [A: nat,B: nat] :
% 0.24/0.49        ( ( A
% 0.24/0.49          = ( plus_plus_nat @ A @ B ) )
% 0.24/0.49        = ( B = zero_zero_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_right_right
% 0.24/0.49  thf(fact_728_add__cancel__right__right,axiom,
% 0.24/0.49      ! [A: int,B: int] :
% 0.24/0.49        ( ( A
% 0.24/0.49          = ( plus_plus_int @ A @ B ) )
% 0.24/0.49        = ( B = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_right_right
% 0.24/0.49  thf(fact_729_add__cancel__right__left,axiom,
% 0.24/0.49      ! [A: complex,B: complex] :
% 0.24/0.49        ( ( A
% 0.24/0.49          = ( plus_plus_complex @ B @ A ) )
% 0.24/0.49        = ( B = zero_zero_complex ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_right_left
% 0.24/0.49  thf(fact_730_add__cancel__right__left,axiom,
% 0.24/0.49      ! [A: real,B: real] :
% 0.24/0.49        ( ( A
% 0.24/0.49          = ( plus_plus_real @ B @ A ) )
% 0.24/0.49        = ( B = zero_zero_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_right_left
% 0.24/0.49  thf(fact_731_add__cancel__right__left,axiom,
% 0.24/0.49      ! [A: rat,B: rat] :
% 0.24/0.49        ( ( A
% 0.24/0.49          = ( plus_plus_rat @ B @ A ) )
% 0.24/0.49        = ( B = zero_zero_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_right_left
% 0.24/0.49  thf(fact_732_add__cancel__right__left,axiom,
% 0.24/0.49      ! [A: nat,B: nat] :
% 0.24/0.49        ( ( A
% 0.24/0.49          = ( plus_plus_nat @ B @ A ) )
% 0.24/0.49        = ( B = zero_zero_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_right_left
% 0.24/0.49  thf(fact_733_add__cancel__right__left,axiom,
% 0.24/0.49      ! [A: int,B: int] :
% 0.24/0.49        ( ( A
% 0.24/0.49          = ( plus_plus_int @ B @ A ) )
% 0.24/0.49        = ( B = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_right_left
% 0.24/0.49  thf(fact_734_add__cancel__left__right,axiom,
% 0.24/0.49      ! [A: complex,B: complex] :
% 0.24/0.49        ( ( ( plus_plus_complex @ A @ B )
% 0.24/0.49          = A )
% 0.24/0.49        = ( B = zero_zero_complex ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_left_right
% 0.24/0.49  thf(fact_735_add__cancel__left__right,axiom,
% 0.24/0.49      ! [A: real,B: real] :
% 0.24/0.49        ( ( ( plus_plus_real @ A @ B )
% 0.24/0.49          = A )
% 0.24/0.49        = ( B = zero_zero_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_left_right
% 0.24/0.49  thf(fact_736_add__cancel__left__right,axiom,
% 0.24/0.49      ! [A: rat,B: rat] :
% 0.24/0.49        ( ( ( plus_plus_rat @ A @ B )
% 0.24/0.49          = A )
% 0.24/0.49        = ( B = zero_zero_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_left_right
% 0.24/0.49  thf(fact_737_add__cancel__left__right,axiom,
% 0.24/0.49      ! [A: nat,B: nat] :
% 0.24/0.49        ( ( ( plus_plus_nat @ A @ B )
% 0.24/0.49          = A )
% 0.24/0.49        = ( B = zero_zero_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_left_right
% 0.24/0.49  thf(fact_738_add__cancel__left__right,axiom,
% 0.24/0.49      ! [A: int,B: int] :
% 0.24/0.49        ( ( ( plus_plus_int @ A @ B )
% 0.24/0.49          = A )
% 0.24/0.49        = ( B = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_left_right
% 0.24/0.49  thf(fact_739_add__cancel__left__left,axiom,
% 0.24/0.49      ! [B: complex,A: complex] :
% 0.24/0.49        ( ( ( plus_plus_complex @ B @ A )
% 0.24/0.49          = A )
% 0.24/0.49        = ( B = zero_zero_complex ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_left_left
% 0.24/0.49  thf(fact_740_add__cancel__left__left,axiom,
% 0.24/0.49      ! [B: real,A: real] :
% 0.24/0.49        ( ( ( plus_plus_real @ B @ A )
% 0.24/0.49          = A )
% 0.24/0.49        = ( B = zero_zero_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_left_left
% 0.24/0.49  thf(fact_741_add__cancel__left__left,axiom,
% 0.24/0.49      ! [B: rat,A: rat] :
% 0.24/0.49        ( ( ( plus_plus_rat @ B @ A )
% 0.24/0.49          = A )
% 0.24/0.49        = ( B = zero_zero_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_left_left
% 0.24/0.49  thf(fact_742_add__cancel__left__left,axiom,
% 0.24/0.49      ! [B: nat,A: nat] :
% 0.24/0.49        ( ( ( plus_plus_nat @ B @ A )
% 0.24/0.49          = A )
% 0.24/0.49        = ( B = zero_zero_nat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_left_left
% 0.24/0.49  thf(fact_743_add__cancel__left__left,axiom,
% 0.24/0.49      ! [B: int,A: int] :
% 0.24/0.49        ( ( ( plus_plus_int @ B @ A )
% 0.24/0.49          = A )
% 0.24/0.49        = ( B = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_cancel_left_left
% 0.24/0.49  thf(fact_744_double__zero__sym,axiom,
% 0.24/0.49      ! [A: real] :
% 0.24/0.49        ( ( zero_zero_real
% 0.24/0.49          = ( plus_plus_real @ A @ A ) )
% 0.24/0.49        = ( A = zero_zero_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % double_zero_sym
% 0.24/0.49  thf(fact_745_double__zero__sym,axiom,
% 0.24/0.49      ! [A: rat] :
% 0.24/0.49        ( ( zero_zero_rat
% 0.24/0.49          = ( plus_plus_rat @ A @ A ) )
% 0.24/0.49        = ( A = zero_zero_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % double_zero_sym
% 0.24/0.49  thf(fact_746_double__zero__sym,axiom,
% 0.24/0.49      ! [A: int] :
% 0.24/0.49        ( ( zero_zero_int
% 0.24/0.49          = ( plus_plus_int @ A @ A ) )
% 0.24/0.49        = ( A = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % double_zero_sym
% 0.24/0.49  thf(fact_747_add_Oright__neutral,axiom,
% 0.24/0.49      ! [A: complex] :
% 0.24/0.49        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % add.right_neutral
% 0.24/0.49  thf(fact_748_add_Oright__neutral,axiom,
% 0.24/0.49      ! [A: real] :
% 0.24/0.49        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % add.right_neutral
% 0.24/0.49  thf(fact_749_add_Oright__neutral,axiom,
% 0.24/0.49      ! [A: rat] :
% 0.24/0.49        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % add.right_neutral
% 0.24/0.49  thf(fact_750_add_Oright__neutral,axiom,
% 0.24/0.49      ! [A: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % add.right_neutral
% 0.24/0.49  thf(fact_751_add_Oright__neutral,axiom,
% 0.24/0.49      ! [A: int] :
% 0.24/0.49        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.24/0.49        = A ) ).
% 0.24/0.49  
% 0.24/0.49  % add.right_neutral
% 0.24/0.49  thf(fact_752_double__eq__0__iff,axiom,
% 0.24/0.49      ! [A: real] :
% 0.24/0.49        ( ( ( plus_plus_real @ A @ A )
% 0.24/0.49          = zero_zero_real )
% 0.24/0.49        = ( A = zero_zero_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % double_eq_0_iff
% 0.24/0.49  thf(fact_753_double__eq__0__iff,axiom,
% 0.24/0.49      ! [A: rat] :
% 0.24/0.49        ( ( ( plus_plus_rat @ A @ A )
% 0.24/0.49          = zero_zero_rat )
% 0.24/0.49        = ( A = zero_zero_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % double_eq_0_iff
% 0.24/0.49  thf(fact_754_double__eq__0__iff,axiom,
% 0.24/0.49      ! [A: int] :
% 0.24/0.49        ( ( ( plus_plus_int @ A @ A )
% 0.24/0.49          = zero_zero_int )
% 0.24/0.49        = ( A = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % double_eq_0_iff
% 0.24/0.49  thf(fact_755_add__less__cancel__right,axiom,
% 0.24/0.49      ! [A: real,C: real,B: real] :
% 0.24/0.49        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.24/0.49        = ( ord_less_real @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_less_cancel_right
% 0.24/0.49  thf(fact_756_add__less__cancel__right,axiom,
% 0.24/0.49      ! [A: rat,C: rat,B: rat] :
% 0.24/0.49        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.24/0.49        = ( ord_less_rat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_less_cancel_right
% 0.24/0.49  thf(fact_757_add__less__cancel__right,axiom,
% 0.24/0.49      ! [A: nat,C: nat,B: nat] :
% 0.24/0.49        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.24/0.49        = ( ord_less_nat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_less_cancel_right
% 0.24/0.49  thf(fact_758_add__less__cancel__right,axiom,
% 0.24/0.49      ! [A: int,C: int,B: int] :
% 0.24/0.49        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.24/0.49        = ( ord_less_int @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_less_cancel_right
% 0.24/0.49  thf(fact_759_add__less__cancel__left,axiom,
% 0.24/0.49      ! [C: real,A: real,B: real] :
% 0.24/0.49        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.24/0.49        = ( ord_less_real @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_less_cancel_left
% 0.24/0.49  thf(fact_760_add__less__cancel__left,axiom,
% 0.24/0.49      ! [C: rat,A: rat,B: rat] :
% 0.24/0.49        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.24/0.49        = ( ord_less_rat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_less_cancel_left
% 0.24/0.49  thf(fact_761_add__less__cancel__left,axiom,
% 0.24/0.49      ! [C: nat,A: nat,B: nat] :
% 0.24/0.49        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.24/0.49        = ( ord_less_nat @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_less_cancel_left
% 0.24/0.49  thf(fact_762_add__less__cancel__left,axiom,
% 0.24/0.49      ! [C: int,A: int,B: int] :
% 0.24/0.49        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.24/0.49        = ( ord_less_int @ A @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_less_cancel_left
% 0.24/0.49  thf(fact_763_Suc__less__eq,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.24/0.49        = ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_less_eq
% 0.24/0.49  thf(fact_764_Suc__mono,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_nat @ M @ N )
% 0.24/0.49       => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % Suc_mono
% 0.24/0.49  thf(fact_765_lessI,axiom,
% 0.24/0.49      ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % lessI
% 0.24/0.49  thf(fact_766_less__nat__zero__code,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.24/0.49  
% 0.24/0.49  % less_nat_zero_code
% 0.24/0.49  thf(fact_767_neq0__conv,axiom,
% 0.24/0.49      ! [N: nat] :
% 0.24/0.49        ( ( N != zero_zero_nat )
% 0.24/0.49        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % neq0_conv
% 0.24/0.49  thf(fact_768_bot__nat__0_Onot__eq__extremum,axiom,
% 0.24/0.49      ! [A: nat] :
% 0.24/0.49        ( ( A != zero_zero_nat )
% 0.24/0.49        = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % bot_nat_0.not_eq_extremum
% 0.24/0.49  thf(fact_769_of__int__eq__0__iff,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( ( ring_1_of_int_int @ Z )
% 0.24/0.49          = zero_zero_int )
% 0.24/0.49        = ( Z = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_eq_0_iff
% 0.24/0.49  thf(fact_770_of__int__eq__0__iff,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( ( ring_1_of_int_real @ Z )
% 0.24/0.49          = zero_zero_real )
% 0.24/0.49        = ( Z = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_eq_0_iff
% 0.24/0.49  thf(fact_771_of__int__eq__0__iff,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( ( ring_1_of_int_rat @ Z )
% 0.24/0.49          = zero_zero_rat )
% 0.24/0.49        = ( Z = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_eq_0_iff
% 0.24/0.49  thf(fact_772_of__int__eq__0__iff,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( ( ring_18347121197199848620nteger @ Z )
% 0.24/0.49          = zero_z3403309356797280102nteger )
% 0.24/0.49        = ( Z = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_eq_0_iff
% 0.24/0.49  thf(fact_773_of__int__eq__0__iff,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( ( ring_17405671764205052669omplex @ Z )
% 0.24/0.49          = zero_zero_complex )
% 0.24/0.49        = ( Z = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_eq_0_iff
% 0.24/0.49  thf(fact_774_of__int__0__eq__iff,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( zero_zero_int
% 0.24/0.49          = ( ring_1_of_int_int @ Z ) )
% 0.24/0.49        = ( Z = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_0_eq_iff
% 0.24/0.49  thf(fact_775_of__int__0__eq__iff,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( zero_zero_real
% 0.24/0.49          = ( ring_1_of_int_real @ Z ) )
% 0.24/0.49        = ( Z = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_0_eq_iff
% 0.24/0.49  thf(fact_776_of__int__0__eq__iff,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( zero_zero_rat
% 0.24/0.49          = ( ring_1_of_int_rat @ Z ) )
% 0.24/0.49        = ( Z = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_0_eq_iff
% 0.24/0.49  thf(fact_777_of__int__0__eq__iff,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( zero_z3403309356797280102nteger
% 0.24/0.49          = ( ring_18347121197199848620nteger @ Z ) )
% 0.24/0.49        = ( Z = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_0_eq_iff
% 0.24/0.49  thf(fact_778_of__int__0__eq__iff,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( zero_zero_complex
% 0.24/0.49          = ( ring_17405671764205052669omplex @ Z ) )
% 0.24/0.49        = ( Z = zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_0_eq_iff
% 0.24/0.49  thf(fact_779_of__int__0,axiom,
% 0.24/0.49      ( ( ring_1_of_int_int @ zero_zero_int )
% 0.24/0.49      = zero_zero_int ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_0
% 0.24/0.49  thf(fact_780_of__int__0,axiom,
% 0.24/0.49      ( ( ring_1_of_int_real @ zero_zero_int )
% 0.24/0.49      = zero_zero_real ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_0
% 0.24/0.49  thf(fact_781_of__int__0,axiom,
% 0.24/0.49      ( ( ring_1_of_int_rat @ zero_zero_int )
% 0.24/0.49      = zero_zero_rat ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_0
% 0.24/0.49  thf(fact_782_of__int__0,axiom,
% 0.24/0.49      ( ( ring_18347121197199848620nteger @ zero_zero_int )
% 0.24/0.49      = zero_z3403309356797280102nteger ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_0
% 0.24/0.49  thf(fact_783_of__int__0,axiom,
% 0.24/0.49      ( ( ring_17405671764205052669omplex @ zero_zero_int )
% 0.24/0.49      = zero_zero_complex ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_0
% 0.24/0.49  thf(fact_784_le0,axiom,
% 0.24/0.49      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.24/0.49  
% 0.24/0.49  % le0
% 0.24/0.49  thf(fact_785_bot__nat__0_Oextremum,axiom,
% 0.24/0.49      ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.24/0.49  
% 0.24/0.49  % bot_nat_0.extremum
% 0.24/0.49  thf(fact_786_of__int__less__iff,axiom,
% 0.24/0.49      ! [W: int,Z: int] :
% 0.24/0.49        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) )
% 0.24/0.49        = ( ord_less_int @ W @ Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_less_iff
% 0.24/0.49  thf(fact_787_of__int__less__iff,axiom,
% 0.24/0.49      ! [W: int,Z: int] :
% 0.24/0.49        ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.24/0.49        = ( ord_less_int @ W @ Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_less_iff
% 0.24/0.49  thf(fact_788_of__int__less__iff,axiom,
% 0.24/0.49      ! [W: int,Z: int] :
% 0.24/0.49        ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.24/0.49        = ( ord_less_int @ W @ Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_less_iff
% 0.24/0.49  thf(fact_789_of__int__less__iff,axiom,
% 0.24/0.49      ! [W: int,Z: int] :
% 0.24/0.49        ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.24/0.49        = ( ord_less_int @ W @ Z ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_less_iff
% 0.24/0.49  thf(fact_790_add__is__0,axiom,
% 0.24/0.49      ! [M: nat,N: nat] :
% 0.24/0.49        ( ( ( plus_plus_nat @ M @ N )
% 0.24/0.49          = zero_zero_nat )
% 0.24/0.49        = ( ( M = zero_zero_nat )
% 0.24/0.49          & ( N = zero_zero_nat ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_is_0
% 0.24/0.49  thf(fact_791_Nat_Oadd__0__right,axiom,
% 0.24/0.49      ! [M: nat] :
% 0.24/0.49        ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.24/0.49        = M ) ).
% 0.24/0.49  
% 0.24/0.49  % Nat.add_0_right
% 0.24/0.49  thf(fact_792_nat__add__left__cancel__less,axiom,
% 0.24/0.49      ! [K: nat,M: nat,N: nat] :
% 0.24/0.49        ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.24/0.49        = ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.49  
% 0.24/0.49  % nat_add_left_cancel_less
% 0.24/0.49  thf(fact_793_ceiling__zero,axiom,
% 0.24/0.49      ( ( archim7802044766580827645g_real @ zero_zero_real )
% 0.24/0.49      = zero_zero_int ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_zero
% 0.24/0.49  thf(fact_794_ceiling__zero,axiom,
% 0.24/0.49      ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 0.24/0.49      = zero_zero_int ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_zero
% 0.24/0.49  thf(fact_795_ceiling__of__int,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( archim7802044766580827645g_real @ ( ring_1_of_int_real @ Z ) )
% 0.24/0.49        = Z ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_of_int
% 0.24/0.49  thf(fact_796_ceiling__of__int,axiom,
% 0.24/0.49      ! [Z: int] :
% 0.24/0.49        ( ( archim2889992004027027881ng_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.24/0.49        = Z ) ).
% 0.24/0.49  
% 0.24/0.49  % ceiling_of_int
% 0.24/0.49  thf(fact_797_of__int__ceiling__cancel,axiom,
% 0.24/0.49      ! [X: real] :
% 0.24/0.49        ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
% 0.24/0.49          = X )
% 0.24/0.49        = ( ? [N3: int] :
% 0.24/0.49              ( X
% 0.24/0.49              = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_ceiling_cancel
% 0.24/0.49  thf(fact_798_of__int__ceiling__cancel,axiom,
% 0.24/0.49      ! [X: rat] :
% 0.24/0.49        ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
% 0.24/0.49          = X )
% 0.24/0.49        = ( ? [N3: int] :
% 0.24/0.49              ( X
% 0.24/0.49              = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 0.24/0.49  
% 0.24/0.49  % of_int_ceiling_cancel
% 0.24/0.49  thf(fact_799_dbl__simps_I2_J,axiom,
% 0.24/0.49      ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 0.24/0.49      = zero_zero_complex ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(2)
% 0.24/0.49  thf(fact_800_dbl__simps_I2_J,axiom,
% 0.24/0.49      ( ( neg_numeral_dbl_real @ zero_zero_real )
% 0.24/0.49      = zero_zero_real ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(2)
% 0.24/0.49  thf(fact_801_dbl__simps_I2_J,axiom,
% 0.24/0.49      ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 0.24/0.49      = zero_zero_rat ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(2)
% 0.24/0.49  thf(fact_802_dbl__simps_I2_J,axiom,
% 0.24/0.49      ( ( neg_numeral_dbl_int @ zero_zero_int )
% 0.24/0.49      = zero_zero_int ) ).
% 0.24/0.49  
% 0.24/0.49  % dbl_simps(2)
% 0.24/0.49  thf(fact_803_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.24/0.49      ! [A: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.24/0.49        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % zero_le_double_add_iff_zero_le_single_add
% 0.24/0.49  thf(fact_804_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.24/0.49      ! [A: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.24/0.49        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % zero_le_double_add_iff_zero_le_single_add
% 0.24/0.49  thf(fact_805_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.24/0.49      ! [A: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.24/0.49        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.24/0.49  
% 0.24/0.49  % zero_le_double_add_iff_zero_le_single_add
% 0.24/0.49  thf(fact_806_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.24/0.49      ! [A: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.24/0.49        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % double_add_le_zero_iff_single_add_le_zero
% 0.24/0.49  thf(fact_807_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.24/0.49      ! [A: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.24/0.49        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % double_add_le_zero_iff_single_add_le_zero
% 0.24/0.49  thf(fact_808_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.24/0.49      ! [A: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.24/0.49        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.24/0.49  
% 0.24/0.49  % double_add_le_zero_iff_single_add_le_zero
% 0.24/0.49  thf(fact_809_le__add__same__cancel2,axiom,
% 0.24/0.49      ! [A: real,B: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.24/0.49        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_add_same_cancel2
% 0.24/0.49  thf(fact_810_le__add__same__cancel2,axiom,
% 0.24/0.49      ! [A: rat,B: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.24/0.49        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_add_same_cancel2
% 0.24/0.49  thf(fact_811_le__add__same__cancel2,axiom,
% 0.24/0.49      ! [A: nat,B: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.24/0.49        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_add_same_cancel2
% 0.24/0.49  thf(fact_812_le__add__same__cancel2,axiom,
% 0.24/0.49      ! [A: int,B: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.24/0.49        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_add_same_cancel2
% 0.24/0.49  thf(fact_813_le__add__same__cancel1,axiom,
% 0.24/0.49      ! [A: real,B: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.24/0.49        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_add_same_cancel1
% 0.24/0.49  thf(fact_814_le__add__same__cancel1,axiom,
% 0.24/0.49      ! [A: rat,B: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.24/0.49        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_add_same_cancel1
% 0.24/0.49  thf(fact_815_le__add__same__cancel1,axiom,
% 0.24/0.49      ! [A: nat,B: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.24/0.49        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_add_same_cancel1
% 0.24/0.49  thf(fact_816_le__add__same__cancel1,axiom,
% 0.24/0.49      ! [A: int,B: int] :
% 0.24/0.49        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.24/0.49        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.24/0.49  
% 0.24/0.49  % le_add_same_cancel1
% 0.24/0.49  thf(fact_817_add__le__same__cancel2,axiom,
% 0.24/0.49      ! [A: real,B: real] :
% 0.24/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.24/0.49        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_same_cancel2
% 0.24/0.49  thf(fact_818_add__le__same__cancel2,axiom,
% 0.24/0.49      ! [A: rat,B: rat] :
% 0.24/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.24/0.49        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.24/0.49  
% 0.24/0.49  % add_le_same_cancel2
% 0.24/0.49  thf(fact_819_add__le__same__cancel2,axiom,
% 0.24/0.49      ! [A: nat,B: nat] :
% 0.24/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.24/0.50        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_le_same_cancel2
% 0.24/0.50  thf(fact_820_add__le__same__cancel2,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.24/0.50        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_le_same_cancel2
% 0.24/0.50  thf(fact_821_add__le__same__cancel1,axiom,
% 0.24/0.50      ! [B: real,A: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.24/0.50        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_le_same_cancel1
% 0.24/0.50  thf(fact_822_add__le__same__cancel1,axiom,
% 0.24/0.50      ! [B: rat,A: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.24/0.50        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_le_same_cancel1
% 0.24/0.50  thf(fact_823_add__le__same__cancel1,axiom,
% 0.24/0.50      ! [B: nat,A: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.24/0.50        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_le_same_cancel1
% 0.24/0.50  thf(fact_824_add__le__same__cancel1,axiom,
% 0.24/0.50      ! [B: int,A: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.24/0.50        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_le_same_cancel1
% 0.24/0.50  thf(fact_825_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.24/0.50      ! [A: real] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.24/0.50        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_double_add_iff_zero_less_single_add
% 0.24/0.50  thf(fact_826_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.24/0.50      ! [A: rat] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.24/0.50        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_double_add_iff_zero_less_single_add
% 0.24/0.50  thf(fact_827_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.24/0.50      ! [A: int] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.24/0.50        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_double_add_iff_zero_less_single_add
% 0.24/0.50  thf(fact_828_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.24/0.50      ! [A: real] :
% 0.24/0.50        ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.24/0.50        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.24/0.50  
% 0.24/0.50  % double_add_less_zero_iff_single_add_less_zero
% 0.24/0.50  thf(fact_829_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.24/0.50      ! [A: rat] :
% 0.24/0.50        ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.24/0.50        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % double_add_less_zero_iff_single_add_less_zero
% 0.24/0.50  thf(fact_830_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.24/0.50      ! [A: int] :
% 0.24/0.50        ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.24/0.50        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % double_add_less_zero_iff_single_add_less_zero
% 0.24/0.50  thf(fact_831_less__add__same__cancel2,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.24/0.50        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_add_same_cancel2
% 0.24/0.50  thf(fact_832_less__add__same__cancel2,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.24/0.50        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_add_same_cancel2
% 0.24/0.50  thf(fact_833_less__add__same__cancel2,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.24/0.50        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_add_same_cancel2
% 0.24/0.50  thf(fact_834_less__add__same__cancel2,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.24/0.50        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_add_same_cancel2
% 0.24/0.50  thf(fact_835_less__add__same__cancel1,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.24/0.50        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_add_same_cancel1
% 0.24/0.50  thf(fact_836_less__add__same__cancel1,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.24/0.50        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_add_same_cancel1
% 0.24/0.50  thf(fact_837_less__add__same__cancel1,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.24/0.50        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_add_same_cancel1
% 0.24/0.50  thf(fact_838_less__add__same__cancel1,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.24/0.50        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_add_same_cancel1
% 0.24/0.50  thf(fact_839_add__less__same__cancel2,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.24/0.50        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_less_same_cancel2
% 0.24/0.50  thf(fact_840_add__less__same__cancel2,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.24/0.50        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_less_same_cancel2
% 0.24/0.50  thf(fact_841_add__less__same__cancel2,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.24/0.50        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_less_same_cancel2
% 0.24/0.50  thf(fact_842_add__less__same__cancel2,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.24/0.50        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_less_same_cancel2
% 0.24/0.50  thf(fact_843_add__less__same__cancel1,axiom,
% 0.24/0.50      ! [B: real,A: real] :
% 0.24/0.50        ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.24/0.50        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_less_same_cancel1
% 0.24/0.50  thf(fact_844_add__less__same__cancel1,axiom,
% 0.24/0.50      ! [B: rat,A: rat] :
% 0.24/0.50        ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.24/0.50        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_less_same_cancel1
% 0.24/0.50  thf(fact_845_add__less__same__cancel1,axiom,
% 0.24/0.50      ! [B: nat,A: nat] :
% 0.24/0.50        ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.24/0.50        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_less_same_cancel1
% 0.24/0.50  thf(fact_846_add__less__same__cancel1,axiom,
% 0.24/0.50      ! [B: int,A: int] :
% 0.24/0.50        ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.24/0.50        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_less_same_cancel1
% 0.24/0.50  thf(fact_847_image__add__0,axiom,
% 0.24/0.50      ! [S2: set_complex] :
% 0.24/0.50        ( ( image_1468599708987790691omplex @ ( plus_plus_complex @ zero_zero_complex ) @ S2 )
% 0.24/0.50        = S2 ) ).
% 0.24/0.50  
% 0.24/0.50  % image_add_0
% 0.24/0.50  thf(fact_848_image__add__0,axiom,
% 0.24/0.50      ! [S2: set_real] :
% 0.24/0.50        ( ( image_real_real @ ( plus_plus_real @ zero_zero_real ) @ S2 )
% 0.24/0.50        = S2 ) ).
% 0.24/0.50  
% 0.24/0.50  % image_add_0
% 0.24/0.50  thf(fact_849_image__add__0,axiom,
% 0.24/0.50      ! [S2: set_rat] :
% 0.24/0.50        ( ( image_rat_rat @ ( plus_plus_rat @ zero_zero_rat ) @ S2 )
% 0.24/0.50        = S2 ) ).
% 0.24/0.50  
% 0.24/0.50  % image_add_0
% 0.24/0.50  thf(fact_850_image__add__0,axiom,
% 0.24/0.50      ! [S2: set_nat] :
% 0.24/0.50        ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S2 )
% 0.24/0.50        = S2 ) ).
% 0.24/0.50  
% 0.24/0.50  % image_add_0
% 0.24/0.50  thf(fact_851_image__add__0,axiom,
% 0.24/0.50      ! [S2: set_int] :
% 0.24/0.50        ( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S2 )
% 0.24/0.50        = S2 ) ).
% 0.24/0.50  
% 0.24/0.50  % image_add_0
% 0.24/0.50  thf(fact_852_power__inject__exp,axiom,
% 0.24/0.50      ! [A: real,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.50       => ( ( ( power_power_real @ A @ M )
% 0.24/0.50            = ( power_power_real @ A @ N ) )
% 0.24/0.50          = ( M = N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_inject_exp
% 0.24/0.50  thf(fact_853_power__inject__exp,axiom,
% 0.24/0.50      ! [A: rat,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ one_one_rat @ A )
% 0.24/0.50       => ( ( ( power_power_rat @ A @ M )
% 0.24/0.50            = ( power_power_rat @ A @ N ) )
% 0.24/0.50          = ( M = N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_inject_exp
% 0.24/0.50  thf(fact_854_power__inject__exp,axiom,
% 0.24/0.50      ! [A: nat,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ one_one_nat @ A )
% 0.24/0.50       => ( ( ( power_power_nat @ A @ M )
% 0.24/0.50            = ( power_power_nat @ A @ N ) )
% 0.24/0.50          = ( M = N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_inject_exp
% 0.24/0.50  thf(fact_855_power__inject__exp,axiom,
% 0.24/0.50      ! [A: int,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_int @ one_one_int @ A )
% 0.24/0.50       => ( ( ( power_power_int @ A @ M )
% 0.24/0.50            = ( power_power_int @ A @ N ) )
% 0.24/0.50          = ( M = N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_inject_exp
% 0.24/0.50  thf(fact_856_power__0__Suc,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 0.24/0.50        = zero_zero_rat ) ).
% 0.24/0.50  
% 0.24/0.50  % power_0_Suc
% 0.24/0.50  thf(fact_857_power__0__Suc,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.24/0.50        = zero_zero_nat ) ).
% 0.24/0.50  
% 0.24/0.50  % power_0_Suc
% 0.24/0.50  thf(fact_858_power__0__Suc,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.24/0.50        = zero_zero_real ) ).
% 0.24/0.50  
% 0.24/0.50  % power_0_Suc
% 0.24/0.50  thf(fact_859_power__0__Suc,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.24/0.50        = zero_zero_int ) ).
% 0.24/0.50  
% 0.24/0.50  % power_0_Suc
% 0.24/0.50  thf(fact_860_power__0__Suc,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 0.24/0.50        = zero_zero_complex ) ).
% 0.24/0.50  
% 0.24/0.50  % power_0_Suc
% 0.24/0.50  thf(fact_861_power__zero__numeral,axiom,
% 0.24/0.50      ! [K: num] :
% 0.24/0.50        ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.24/0.50        = zero_zero_rat ) ).
% 0.24/0.50  
% 0.24/0.50  % power_zero_numeral
% 0.24/0.50  thf(fact_862_power__zero__numeral,axiom,
% 0.24/0.50      ! [K: num] :
% 0.24/0.50        ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.24/0.50        = zero_zero_nat ) ).
% 0.24/0.50  
% 0.24/0.50  % power_zero_numeral
% 0.24/0.50  thf(fact_863_power__zero__numeral,axiom,
% 0.24/0.50      ! [K: num] :
% 0.24/0.50        ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.24/0.50        = zero_zero_real ) ).
% 0.24/0.50  
% 0.24/0.50  % power_zero_numeral
% 0.24/0.50  thf(fact_864_power__zero__numeral,axiom,
% 0.24/0.50      ! [K: num] :
% 0.24/0.50        ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.24/0.50        = zero_zero_int ) ).
% 0.24/0.50  
% 0.24/0.50  % power_zero_numeral
% 0.24/0.50  thf(fact_865_power__zero__numeral,axiom,
% 0.24/0.50      ! [K: num] :
% 0.24/0.50        ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.24/0.50        = zero_zero_complex ) ).
% 0.24/0.50  
% 0.24/0.50  % power_zero_numeral
% 0.24/0.50  thf(fact_866_of__nat__0,axiom,
% 0.24/0.50      ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 0.24/0.50      = zero_zero_complex ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0
% 0.24/0.50  thf(fact_867_of__nat__0,axiom,
% 0.24/0.50      ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 0.24/0.50      = zero_zero_real ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0
% 0.24/0.50  thf(fact_868_of__nat__0,axiom,
% 0.24/0.50      ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 0.24/0.50      = zero_zero_rat ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0
% 0.24/0.50  thf(fact_869_of__nat__0,axiom,
% 0.24/0.50      ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 0.24/0.50      = zero_zero_nat ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0
% 0.24/0.50  thf(fact_870_of__nat__0,axiom,
% 0.24/0.50      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.24/0.50      = zero_zero_int ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0
% 0.24/0.50  thf(fact_871_of__nat__0__eq__iff,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( zero_zero_complex
% 0.24/0.50          = ( semiri8010041392384452111omplex @ N ) )
% 0.24/0.50        = ( zero_zero_nat = N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0_eq_iff
% 0.24/0.50  thf(fact_872_of__nat__0__eq__iff,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( zero_zero_real
% 0.24/0.50          = ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.50        = ( zero_zero_nat = N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0_eq_iff
% 0.24/0.50  thf(fact_873_of__nat__0__eq__iff,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( zero_zero_rat
% 0.24/0.50          = ( semiri681578069525770553at_rat @ N ) )
% 0.24/0.50        = ( zero_zero_nat = N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0_eq_iff
% 0.24/0.50  thf(fact_874_of__nat__0__eq__iff,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( zero_zero_nat
% 0.24/0.50          = ( semiri1316708129612266289at_nat @ N ) )
% 0.24/0.50        = ( zero_zero_nat = N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0_eq_iff
% 0.24/0.50  thf(fact_875_of__nat__0__eq__iff,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( zero_zero_int
% 0.24/0.50          = ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.50        = ( zero_zero_nat = N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0_eq_iff
% 0.24/0.50  thf(fact_876_of__nat__eq__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ( ( ( semiri8010041392384452111omplex @ M )
% 0.24/0.50          = zero_zero_complex )
% 0.24/0.50        = ( M = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_eq_0_iff
% 0.24/0.50  thf(fact_877_of__nat__eq__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ( ( ( semiri5074537144036343181t_real @ M )
% 0.24/0.50          = zero_zero_real )
% 0.24/0.50        = ( M = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_eq_0_iff
% 0.24/0.50  thf(fact_878_of__nat__eq__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ( ( ( semiri681578069525770553at_rat @ M )
% 0.24/0.50          = zero_zero_rat )
% 0.24/0.50        = ( M = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_eq_0_iff
% 0.24/0.50  thf(fact_879_of__nat__eq__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ( ( ( semiri1316708129612266289at_nat @ M )
% 0.24/0.50          = zero_zero_nat )
% 0.24/0.50        = ( M = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_eq_0_iff
% 0.24/0.50  thf(fact_880_of__nat__eq__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ( ( ( semiri1314217659103216013at_int @ M )
% 0.24/0.50          = zero_zero_int )
% 0.24/0.50        = ( M = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_eq_0_iff
% 0.24/0.50  thf(fact_881_of__int__less__0__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ zero_z3403309356797280102nteger )
% 0.24/0.50        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_0_iff
% 0.24/0.50  thf(fact_882_of__int__less__0__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.24/0.50        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_0_iff
% 0.24/0.50  thf(fact_883_of__int__less__0__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.24/0.50        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_0_iff
% 0.24/0.50  thf(fact_884_of__int__less__0__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.24/0.50        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_0_iff
% 0.24/0.50  thf(fact_885_of__int__0__less__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) )
% 0.24/0.50        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_0_less_iff
% 0.24/0.50  thf(fact_886_of__int__0__less__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.24/0.50        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_0_less_iff
% 0.24/0.50  thf(fact_887_of__int__0__less__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.24/0.50        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_0_less_iff
% 0.24/0.50  thf(fact_888_of__int__0__less__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.24/0.50        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_0_less_iff
% 0.24/0.50  thf(fact_889_of__nat__less__iff,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.50        = ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_iff
% 0.24/0.50  thf(fact_890_of__nat__less__iff,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.24/0.50        = ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_iff
% 0.24/0.50  thf(fact_891_of__nat__less__iff,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.24/0.50        = ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_iff
% 0.24/0.50  thf(fact_892_of__nat__less__iff,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.50        = ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_iff
% 0.24/0.50  thf(fact_893_power__Suc0__right,axiom,
% 0.24/0.50      ! [A: nat] :
% 0.24/0.50        ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.24/0.50        = A ) ).
% 0.24/0.50  
% 0.24/0.50  % power_Suc0_right
% 0.24/0.50  thf(fact_894_power__Suc0__right,axiom,
% 0.24/0.50      ! [A: real] :
% 0.24/0.50        ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.24/0.50        = A ) ).
% 0.24/0.50  
% 0.24/0.50  % power_Suc0_right
% 0.24/0.50  thf(fact_895_power__Suc0__right,axiom,
% 0.24/0.50      ! [A: int] :
% 0.24/0.50        ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.24/0.50        = A ) ).
% 0.24/0.50  
% 0.24/0.50  % power_Suc0_right
% 0.24/0.50  thf(fact_896_power__Suc0__right,axiom,
% 0.24/0.50      ! [A: complex] :
% 0.24/0.50        ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 0.24/0.50        = A ) ).
% 0.24/0.50  
% 0.24/0.50  % power_Suc0_right
% 0.24/0.50  thf(fact_897_zero__less__Suc,axiom,
% 0.24/0.50      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_Suc
% 0.24/0.50  thf(fact_898_less__Suc0,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.24/0.50        = ( N = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_Suc0
% 0.24/0.50  thf(fact_899_add__gr__0,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.50        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.24/0.50          | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_gr_0
% 0.24/0.50  thf(fact_900_zero__less__ceiling,axiom,
% 0.24/0.50      ! [X: real] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 0.24/0.50        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_ceiling
% 0.24/0.50  thf(fact_901_zero__less__ceiling,axiom,
% 0.24/0.50      ! [X: rat] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.24/0.50        = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_ceiling
% 0.24/0.50  thf(fact_902_of__int__le__iff,axiom,
% 0.24/0.50      ! [W: int,Z: int] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_iff
% 0.24/0.50  thf(fact_903_of__int__le__iff,axiom,
% 0.24/0.50      ! [W: int,Z: int] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_iff
% 0.24/0.50  thf(fact_904_of__int__le__iff,axiom,
% 0.24/0.50      ! [W: int,Z: int] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_iff
% 0.24/0.50  thf(fact_905_of__int__le__iff,axiom,
% 0.24/0.50      ! [W: int,Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_iff
% 0.24/0.50  thf(fact_906_of__int__eq__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ( ring_18347121197199848620nteger @ Z )
% 0.24/0.50          = ( numera6620942414471956472nteger @ N ) )
% 0.24/0.50        = ( Z
% 0.24/0.50          = ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_numeral_iff
% 0.24/0.50  thf(fact_907_of__int__eq__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ( ring_17405671764205052669omplex @ Z )
% 0.24/0.50          = ( numera6690914467698888265omplex @ N ) )
% 0.24/0.50        = ( Z
% 0.24/0.50          = ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_numeral_iff
% 0.24/0.50  thf(fact_908_of__int__eq__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ( ring_1_of_int_real @ Z )
% 0.24/0.50          = ( numeral_numeral_real @ N ) )
% 0.24/0.50        = ( Z
% 0.24/0.50          = ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_numeral_iff
% 0.24/0.50  thf(fact_909_of__int__eq__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ( ring_1_of_int_rat @ Z )
% 0.24/0.50          = ( numeral_numeral_rat @ N ) )
% 0.24/0.50        = ( Z
% 0.24/0.50          = ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_numeral_iff
% 0.24/0.50  thf(fact_910_of__int__eq__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ( ring_1_of_int_int @ Z )
% 0.24/0.50          = ( numeral_numeral_int @ N ) )
% 0.24/0.50        = ( Z
% 0.24/0.50          = ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_numeral_iff
% 0.24/0.50  thf(fact_911_of__int__numeral,axiom,
% 0.24/0.50      ! [K: num] :
% 0.24/0.50        ( ( ring_18347121197199848620nteger @ ( numeral_numeral_int @ K ) )
% 0.24/0.50        = ( numera6620942414471956472nteger @ K ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral
% 0.24/0.50  thf(fact_912_of__int__numeral,axiom,
% 0.24/0.50      ! [K: num] :
% 0.24/0.50        ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 0.24/0.50        = ( numera6690914467698888265omplex @ K ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral
% 0.24/0.50  thf(fact_913_of__int__numeral,axiom,
% 0.24/0.50      ! [K: num] :
% 0.24/0.50        ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.24/0.50        = ( numeral_numeral_real @ K ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral
% 0.24/0.50  thf(fact_914_of__int__numeral,axiom,
% 0.24/0.50      ! [K: num] :
% 0.24/0.50        ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 0.24/0.50        = ( numeral_numeral_rat @ K ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral
% 0.24/0.50  thf(fact_915_of__int__numeral,axiom,
% 0.24/0.50      ! [K: num] :
% 0.24/0.50        ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.24/0.50        = ( numeral_numeral_int @ K ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral
% 0.24/0.50  thf(fact_916_less__one,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ N @ one_one_nat )
% 0.24/0.50        = ( N = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_one
% 0.24/0.50  thf(fact_917_nat__power__eq__Suc__0__iff,axiom,
% 0.24/0.50      ! [X: nat,M: nat] :
% 0.24/0.50        ( ( ( power_power_nat @ X @ M )
% 0.24/0.50          = ( suc @ zero_zero_nat ) )
% 0.24/0.50        = ( ( M = zero_zero_nat )
% 0.24/0.50          | ( X
% 0.24/0.50            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nat_power_eq_Suc_0_iff
% 0.24/0.50  thf(fact_918_power__Suc__0,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.24/0.50        = ( suc @ zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_Suc_0
% 0.24/0.50  thf(fact_919_of__int__eq__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ( ring_1_of_int_int @ Z )
% 0.24/0.50          = one_one_int )
% 0.24/0.50        = ( Z = one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_1_iff
% 0.24/0.50  thf(fact_920_of__int__eq__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ( ring_1_of_int_real @ Z )
% 0.24/0.50          = one_one_real )
% 0.24/0.50        = ( Z = one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_1_iff
% 0.24/0.50  thf(fact_921_of__int__eq__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ( ring_1_of_int_rat @ Z )
% 0.24/0.50          = one_one_rat )
% 0.24/0.50        = ( Z = one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_1_iff
% 0.24/0.50  thf(fact_922_of__int__eq__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ( ring_18347121197199848620nteger @ Z )
% 0.24/0.50          = one_one_Code_integer )
% 0.24/0.50        = ( Z = one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_1_iff
% 0.24/0.50  thf(fact_923_of__int__eq__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ( ring_17405671764205052669omplex @ Z )
% 0.24/0.50          = one_one_complex )
% 0.24/0.50        = ( Z = one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_1_iff
% 0.24/0.50  thf(fact_924_of__int__1,axiom,
% 0.24/0.50      ( ( ring_1_of_int_int @ one_one_int )
% 0.24/0.50      = one_one_int ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1
% 0.24/0.50  thf(fact_925_of__int__1,axiom,
% 0.24/0.50      ( ( ring_1_of_int_real @ one_one_int )
% 0.24/0.50      = one_one_real ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1
% 0.24/0.50  thf(fact_926_of__int__1,axiom,
% 0.24/0.50      ( ( ring_1_of_int_rat @ one_one_int )
% 0.24/0.50      = one_one_rat ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1
% 0.24/0.50  thf(fact_927_of__int__1,axiom,
% 0.24/0.50      ( ( ring_18347121197199848620nteger @ one_one_int )
% 0.24/0.50      = one_one_Code_integer ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1
% 0.24/0.50  thf(fact_928_of__int__1,axiom,
% 0.24/0.50      ( ( ring_17405671764205052669omplex @ one_one_int )
% 0.24/0.50      = one_one_complex ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1
% 0.24/0.50  thf(fact_929_nat__zero__less__power__iff,axiom,
% 0.24/0.50      ! [X: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 0.24/0.50        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.24/0.50          | ( N = zero_zero_nat ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nat_zero_less_power_iff
% 0.24/0.50  thf(fact_930_of__int__add,axiom,
% 0.24/0.50      ! [W: int,Z: int] :
% 0.24/0.50        ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 0.24/0.50        = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_add
% 0.24/0.50  thf(fact_931_of__int__add,axiom,
% 0.24/0.50      ! [W: int,Z: int] :
% 0.24/0.50        ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 0.24/0.50        = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_add
% 0.24/0.50  thf(fact_932_of__int__add,axiom,
% 0.24/0.50      ! [W: int,Z: int] :
% 0.24/0.50        ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 0.24/0.50        = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_add
% 0.24/0.50  thf(fact_933_of__int__add,axiom,
% 0.24/0.50      ! [W: int,Z: int] :
% 0.24/0.50        ( ( ring_18347121197199848620nteger @ ( plus_plus_int @ W @ Z ) )
% 0.24/0.50        = ( plus_p5714425477246183910nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_add
% 0.24/0.50  thf(fact_934_of__int__add,axiom,
% 0.24/0.50      ! [W: int,Z: int] :
% 0.24/0.50        ( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W @ Z ) )
% 0.24/0.50        = ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_add
% 0.24/0.50  thf(fact_935_of__int__of__nat__eq,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ring_18347121197199848620nteger @ ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.50        = ( semiri4939895301339042750nteger @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_of_nat_eq
% 0.24/0.50  thf(fact_936_of__int__of__nat__eq,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.50        = ( semiri8010041392384452111omplex @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_of_nat_eq
% 0.24/0.50  thf(fact_937_of__int__of__nat__eq,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.50        = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_of_nat_eq
% 0.24/0.50  thf(fact_938_of__int__of__nat__eq,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.50        = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_of_nat_eq
% 0.24/0.50  thf(fact_939_of__int__of__nat__eq,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.50        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_of_nat_eq
% 0.24/0.50  thf(fact_940_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ( ring_1_of_int_rat @ X )
% 0.24/0.50          = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.24/0.50        = ( X
% 0.24/0.50          = ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_eq_of_int_cancel_iff
% 0.24/0.50  thf(fact_941_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ( ring_18347121197199848620nteger @ X )
% 0.24/0.50          = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
% 0.24/0.50        = ( X
% 0.24/0.50          = ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_eq_of_int_cancel_iff
% 0.24/0.50  thf(fact_942_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ( ring_1_of_int_real @ X )
% 0.24/0.50          = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.24/0.50        = ( X
% 0.24/0.50          = ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_eq_of_int_cancel_iff
% 0.24/0.50  thf(fact_943_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ( ring_1_of_int_int @ X )
% 0.24/0.50          = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.24/0.50        = ( X
% 0.24/0.50          = ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_eq_of_int_cancel_iff
% 0.24/0.50  thf(fact_944_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ( ring_17405671764205052669omplex @ X )
% 0.24/0.50          = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 0.24/0.50        = ( X
% 0.24/0.50          = ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_eq_of_int_cancel_iff
% 0.24/0.50  thf(fact_945_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 0.24/0.50          = ( ring_1_of_int_rat @ X ) )
% 0.24/0.50        = ( ( power_power_int @ B @ W )
% 0.24/0.50          = X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_946_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W )
% 0.24/0.50          = ( ring_18347121197199848620nteger @ X ) )
% 0.24/0.50        = ( ( power_power_int @ B @ W )
% 0.24/0.50          = X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_947_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 0.24/0.50          = ( ring_1_of_int_real @ X ) )
% 0.24/0.50        = ( ( power_power_int @ B @ W )
% 0.24/0.50          = X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_948_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 0.24/0.50          = ( ring_1_of_int_int @ X ) )
% 0.24/0.50        = ( ( power_power_int @ B @ W )
% 0.24/0.50          = X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_949_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 0.24/0.50          = ( ring_17405671764205052669omplex @ X ) )
% 0.24/0.50        = ( ( power_power_int @ B @ W )
% 0.24/0.50          = X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_950_of__int__power,axiom,
% 0.24/0.50      ! [Z: int,N: nat] :
% 0.24/0.50        ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
% 0.24/0.50        = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power
% 0.24/0.50  thf(fact_951_of__int__power,axiom,
% 0.24/0.50      ! [Z: int,N: nat] :
% 0.24/0.50        ( ( ring_18347121197199848620nteger @ ( power_power_int @ Z @ N ) )
% 0.24/0.50        = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ Z ) @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power
% 0.24/0.50  thf(fact_952_of__int__power,axiom,
% 0.24/0.50      ! [Z: int,N: nat] :
% 0.24/0.50        ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 0.24/0.50        = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power
% 0.24/0.50  thf(fact_953_of__int__power,axiom,
% 0.24/0.50      ! [Z: int,N: nat] :
% 0.24/0.50        ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 0.24/0.50        = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power
% 0.24/0.50  thf(fact_954_of__int__power,axiom,
% 0.24/0.50      ! [Z: int,N: nat] :
% 0.24/0.50        ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
% 0.24/0.50        = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power
% 0.24/0.50  thf(fact_955_one__less__numeral__iff,axiom,
% 0.24/0.50      ! [N: num] :
% 0.24/0.50        ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.24/0.50        = ( ord_less_num @ one @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % one_less_numeral_iff
% 0.24/0.50  thf(fact_956_one__less__numeral__iff,axiom,
% 0.24/0.50      ! [N: num] :
% 0.24/0.50        ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.24/0.50        = ( ord_less_num @ one @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % one_less_numeral_iff
% 0.24/0.50  thf(fact_957_one__less__numeral__iff,axiom,
% 0.24/0.50      ! [N: num] :
% 0.24/0.50        ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.24/0.50        = ( ord_less_num @ one @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % one_less_numeral_iff
% 0.24/0.50  thf(fact_958_one__less__numeral__iff,axiom,
% 0.24/0.50      ! [N: num] :
% 0.24/0.50        ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.24/0.50        = ( ord_less_num @ one @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % one_less_numeral_iff
% 0.24/0.50  thf(fact_959_of__nat__le__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 0.24/0.50        = ( M = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_le_0_iff
% 0.24/0.50  thf(fact_960_of__nat__le__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 0.24/0.50        = ( M = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_le_0_iff
% 0.24/0.50  thf(fact_961_of__nat__le__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 0.24/0.50        = ( M = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_le_0_iff
% 0.24/0.50  thf(fact_962_of__nat__le__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 0.24/0.50        = ( M = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_le_0_iff
% 0.24/0.50  thf(fact_963_power__strict__increasing__iff,axiom,
% 0.24/0.50      ! [B: real,X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_real @ one_one_real @ B )
% 0.24/0.50       => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 0.24/0.50          = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_increasing_iff
% 0.24/0.50  thf(fact_964_power__strict__increasing__iff,axiom,
% 0.24/0.50      ! [B: rat,X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_rat @ one_one_rat @ B )
% 0.24/0.50       => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 0.24/0.50          = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_increasing_iff
% 0.24/0.50  thf(fact_965_power__strict__increasing__iff,axiom,
% 0.24/0.50      ! [B: nat,X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_nat @ one_one_nat @ B )
% 0.24/0.50       => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 0.24/0.50          = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_increasing_iff
% 0.24/0.50  thf(fact_966_power__strict__increasing__iff,axiom,
% 0.24/0.50      ! [B: int,X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_int @ one_one_int @ B )
% 0.24/0.50       => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 0.24/0.50          = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_increasing_iff
% 0.24/0.50  thf(fact_967_power__eq__0__iff,axiom,
% 0.24/0.50      ! [A: rat,N: nat] :
% 0.24/0.50        ( ( ( power_power_rat @ A @ N )
% 0.24/0.50          = zero_zero_rat )
% 0.24/0.50        = ( ( A = zero_zero_rat )
% 0.24/0.50          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_eq_0_iff
% 0.24/0.50  thf(fact_968_power__eq__0__iff,axiom,
% 0.24/0.50      ! [A: nat,N: nat] :
% 0.24/0.50        ( ( ( power_power_nat @ A @ N )
% 0.24/0.50          = zero_zero_nat )
% 0.24/0.50        = ( ( A = zero_zero_nat )
% 0.24/0.50          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_eq_0_iff
% 0.24/0.50  thf(fact_969_power__eq__0__iff,axiom,
% 0.24/0.50      ! [A: real,N: nat] :
% 0.24/0.50        ( ( ( power_power_real @ A @ N )
% 0.24/0.50          = zero_zero_real )
% 0.24/0.50        = ( ( A = zero_zero_real )
% 0.24/0.50          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_eq_0_iff
% 0.24/0.50  thf(fact_970_power__eq__0__iff,axiom,
% 0.24/0.50      ! [A: int,N: nat] :
% 0.24/0.50        ( ( ( power_power_int @ A @ N )
% 0.24/0.50          = zero_zero_int )
% 0.24/0.50        = ( ( A = zero_zero_int )
% 0.24/0.50          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_eq_0_iff
% 0.24/0.50  thf(fact_971_power__eq__0__iff,axiom,
% 0.24/0.50      ! [A: complex,N: nat] :
% 0.24/0.50        ( ( ( power_power_complex @ A @ N )
% 0.24/0.50          = zero_zero_complex )
% 0.24/0.50        = ( ( A = zero_zero_complex )
% 0.24/0.50          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_eq_0_iff
% 0.24/0.50  thf(fact_972_of__int__le__0__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_0_iff
% 0.24/0.50  thf(fact_973_of__int__le__0__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ zero_z3403309356797280102nteger )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_0_iff
% 0.24/0.50  thf(fact_974_of__int__le__0__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_0_iff
% 0.24/0.50  thf(fact_975_of__int__le__0__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_0_iff
% 0.24/0.50  thf(fact_976_of__int__0__le__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_0_le_iff
% 0.24/0.50  thf(fact_977_of__int__0__le__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_0_le_iff
% 0.24/0.50  thf(fact_978_of__int__0__le__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_0_le_iff
% 0.24/0.50  thf(fact_979_of__int__0__le__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_0_le_iff
% 0.24/0.50  thf(fact_980_of__int__less__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
% 0.24/0.50        = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_numeral_iff
% 0.24/0.50  thf(fact_981_of__int__less__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.24/0.50        = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_numeral_iff
% 0.24/0.50  thf(fact_982_of__int__less__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.24/0.50        = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_numeral_iff
% 0.24/0.50  thf(fact_983_of__int__less__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.24/0.50        = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_numeral_iff
% 0.24/0.50  thf(fact_984_of__int__numeral__less__iff,axiom,
% 0.24/0.50      ! [N: num,Z: int] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
% 0.24/0.50        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral_less_iff
% 0.24/0.50  thf(fact_985_of__int__numeral__less__iff,axiom,
% 0.24/0.50      ! [N: num,Z: int] :
% 0.24/0.50        ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.24/0.50        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral_less_iff
% 0.24/0.50  thf(fact_986_of__int__numeral__less__iff,axiom,
% 0.24/0.50      ! [N: num,Z: int] :
% 0.24/0.50        ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.24/0.50        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral_less_iff
% 0.24/0.50  thf(fact_987_of__int__numeral__less__iff,axiom,
% 0.24/0.50      ! [N: num,Z: int] :
% 0.24/0.50        ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.24/0.50        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral_less_iff
% 0.24/0.50  thf(fact_988_of__int__less__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
% 0.24/0.50        = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_1_iff
% 0.24/0.50  thf(fact_989_of__int__less__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.24/0.50        = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_1_iff
% 0.24/0.50  thf(fact_990_of__int__less__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.24/0.50        = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_1_iff
% 0.24/0.50  thf(fact_991_of__int__less__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.24/0.50        = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_1_iff
% 0.24/0.50  thf(fact_992_of__int__1__less__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
% 0.24/0.50        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1_less_iff
% 0.24/0.50  thf(fact_993_of__int__1__less__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.24/0.50        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1_less_iff
% 0.24/0.50  thf(fact_994_of__int__1__less__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.24/0.50        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1_less_iff
% 0.24/0.50  thf(fact_995_of__int__1__less__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.24/0.50        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1_less_iff
% 0.24/0.50  thf(fact_996_ceiling__le__zero,axiom,
% 0.24/0.50      ! [X: real] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 0.24/0.50        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ceiling_le_zero
% 0.24/0.50  thf(fact_997_ceiling__le__zero,axiom,
% 0.24/0.50      ! [X: rat] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 0.24/0.50        = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ceiling_le_zero
% 0.24/0.50  thf(fact_998_numeral__less__ceiling,axiom,
% 0.24/0.50      ! [V: num,X: real] :
% 0.24/0.50        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 0.24/0.50        = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_less_ceiling
% 0.24/0.50  thf(fact_999_numeral__less__ceiling,axiom,
% 0.24/0.50      ! [V: num,X: rat] :
% 0.24/0.50        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.24/0.50        = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_less_ceiling
% 0.24/0.50  thf(fact_1000_one__less__ceiling,axiom,
% 0.24/0.50      ! [X: real] :
% 0.24/0.50        ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 0.24/0.50        = ( ord_less_real @ one_one_real @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % one_less_ceiling
% 0.24/0.50  thf(fact_1001_one__less__ceiling,axiom,
% 0.24/0.50      ! [X: rat] :
% 0.24/0.50        ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.24/0.50        = ( ord_less_rat @ one_one_rat @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % one_less_ceiling
% 0.24/0.50  thf(fact_1002_of__int__less__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) @ ( ring_18347121197199848620nteger @ X ) )
% 0.24/0.50        = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_1003_of__int__less__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.24/0.50        = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_1004_of__int__less__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 0.24/0.50        = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_1005_of__int__less__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.24/0.50        = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_1006_of__int__power__less__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_less_of_int_cancel_iff
% 0.24/0.50  thf(fact_1007_of__int__power__less__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_less_of_int_cancel_iff
% 0.24/0.50  thf(fact_1008_of__int__power__less__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_less_of_int_cancel_iff
% 0.24/0.50  thf(fact_1009_of__int__power__less__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_less_of_int_cancel_iff
% 0.24/0.50  thf(fact_1010_ceiling__add__of__int,axiom,
% 0.24/0.50      ! [X: real,Z: int] :
% 0.24/0.50        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 0.24/0.50        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ceiling_add_of_int
% 0.24/0.50  thf(fact_1011_ceiling__add__of__int,axiom,
% 0.24/0.50      ! [X: rat,Z: int] :
% 0.24/0.50        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
% 0.24/0.50        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ceiling_add_of_int
% 0.24/0.50  thf(fact_1012_power__strict__decreasing__iff,axiom,
% 0.24/0.50      ! [B: real,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ B )
% 0.24/0.50       => ( ( ord_less_real @ B @ one_one_real )
% 0.24/0.50         => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.24/0.50            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_decreasing_iff
% 0.24/0.50  thf(fact_1013_power__strict__decreasing__iff,axiom,
% 0.24/0.50      ! [B: rat,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.24/0.50       => ( ( ord_less_rat @ B @ one_one_rat )
% 0.24/0.50         => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.24/0.50            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_decreasing_iff
% 0.24/0.50  thf(fact_1014_power__strict__decreasing__iff,axiom,
% 0.24/0.50      ! [B: nat,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.24/0.50       => ( ( ord_less_nat @ B @ one_one_nat )
% 0.24/0.50         => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.24/0.50            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_decreasing_iff
% 0.24/0.50  thf(fact_1015_power__strict__decreasing__iff,axiom,
% 0.24/0.50      ! [B: int,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ B )
% 0.24/0.50       => ( ( ord_less_int @ B @ one_one_int )
% 0.24/0.50         => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.24/0.50            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_decreasing_iff
% 0.24/0.50  thf(fact_1016_power__increasing__iff,axiom,
% 0.24/0.50      ! [B: real,X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_real @ one_one_real @ B )
% 0.24/0.50       => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 0.24/0.50          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_increasing_iff
% 0.24/0.50  thf(fact_1017_power__increasing__iff,axiom,
% 0.24/0.50      ! [B: rat,X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_rat @ one_one_rat @ B )
% 0.24/0.50       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 0.24/0.50          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_increasing_iff
% 0.24/0.50  thf(fact_1018_power__increasing__iff,axiom,
% 0.24/0.50      ! [B: nat,X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_nat @ one_one_nat @ B )
% 0.24/0.50       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 0.24/0.50          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_increasing_iff
% 0.24/0.50  thf(fact_1019_power__increasing__iff,axiom,
% 0.24/0.50      ! [B: int,X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_int @ one_one_int @ B )
% 0.24/0.50       => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 0.24/0.50          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_increasing_iff
% 0.24/0.50  thf(fact_1020_power__mono__iff,axiom,
% 0.24/0.50      ! [A: real,B: real,N: nat] :
% 0.24/0.50        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.50       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.50         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50           => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.24/0.50              = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_mono_iff
% 0.24/0.50  thf(fact_1021_power__mono__iff,axiom,
% 0.24/0.50      ! [A: rat,B: rat,N: nat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.50       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.50         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50           => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.24/0.50              = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_mono_iff
% 0.24/0.50  thf(fact_1022_power__mono__iff,axiom,
% 0.24/0.50      ! [A: nat,B: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.50       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.24/0.50         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50           => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.24/0.50              = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_mono_iff
% 0.24/0.50  thf(fact_1023_power__mono__iff,axiom,
% 0.24/0.50      ! [A: int,B: int,N: nat] :
% 0.24/0.50        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.50       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.24/0.50         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50           => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.24/0.50              = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_mono_iff
% 0.24/0.50  thf(fact_1024_zero__eq__power2,axiom,
% 0.24/0.50      ! [A: rat] :
% 0.24/0.50        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.50          = zero_zero_rat )
% 0.24/0.50        = ( A = zero_zero_rat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_eq_power2
% 0.24/0.50  thf(fact_1025_zero__eq__power2,axiom,
% 0.24/0.50      ! [A: nat] :
% 0.24/0.50        ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.50          = zero_zero_nat )
% 0.24/0.50        = ( A = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_eq_power2
% 0.24/0.50  thf(fact_1026_zero__eq__power2,axiom,
% 0.24/0.50      ! [A: real] :
% 0.24/0.50        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.50          = zero_zero_real )
% 0.24/0.50        = ( A = zero_zero_real ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_eq_power2
% 0.24/0.50  thf(fact_1027_zero__eq__power2,axiom,
% 0.24/0.50      ! [A: int] :
% 0.24/0.50        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.50          = zero_zero_int )
% 0.24/0.50        = ( A = zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_eq_power2
% 0.24/0.50  thf(fact_1028_zero__eq__power2,axiom,
% 0.24/0.50      ! [A: complex] :
% 0.24/0.50        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.50          = zero_zero_complex )
% 0.24/0.50        = ( A = zero_zero_complex ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_eq_power2
% 0.24/0.50  thf(fact_1029_of__nat__0__less__iff,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.50        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0_less_iff
% 0.24/0.50  thf(fact_1030_of__nat__0__less__iff,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.24/0.50        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0_less_iff
% 0.24/0.50  thf(fact_1031_of__nat__0__less__iff,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 0.24/0.50        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0_less_iff
% 0.24/0.50  thf(fact_1032_of__nat__0__less__iff,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.50        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_0_less_iff
% 0.24/0.50  thf(fact_1033_of__int__le__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_numeral_iff
% 0.24/0.50  thf(fact_1034_of__int__le__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_numeral_iff
% 0.24/0.50  thf(fact_1035_of__int__le__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_numeral_iff
% 0.24/0.50  thf(fact_1036_of__int__le__numeral__iff,axiom,
% 0.24/0.50      ! [Z: int,N: num] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_numeral_iff
% 0.24/0.50  thf(fact_1037_of__int__numeral__le__iff,axiom,
% 0.24/0.50      ! [N: num,Z: int] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral_le_iff
% 0.24/0.50  thf(fact_1038_of__int__numeral__le__iff,axiom,
% 0.24/0.50      ! [N: num,Z: int] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral_le_iff
% 0.24/0.50  thf(fact_1039_of__int__numeral__le__iff,axiom,
% 0.24/0.50      ! [N: num,Z: int] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral_le_iff
% 0.24/0.50  thf(fact_1040_of__int__numeral__le__iff,axiom,
% 0.24/0.50      ! [N: num,Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_numeral_le_iff
% 0.24/0.50  thf(fact_1041_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.24/0.50      ! [X: nat,B: nat,W: nat] :
% 0.24/0.50        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_power_less_of_nat_cancel_iff
% 0.24/0.50  thf(fact_1042_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.24/0.50      ! [X: nat,B: nat,W: nat] :
% 0.24/0.50        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_power_less_of_nat_cancel_iff
% 0.24/0.50  thf(fact_1043_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.24/0.50      ! [X: nat,B: nat,W: nat] :
% 0.24/0.50        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_power_less_of_nat_cancel_iff
% 0.24/0.50  thf(fact_1044_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.24/0.50      ! [X: nat,B: nat,W: nat] :
% 0.24/0.50        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_power_less_of_nat_cancel_iff
% 0.24/0.50  thf(fact_1045_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: nat,W: nat,X: nat] :
% 0.24/0.50        ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.24/0.50        = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_of_nat_power_cancel_iff
% 0.24/0.50  thf(fact_1046_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: nat,W: nat,X: nat] :
% 0.24/0.50        ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.24/0.50        = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_of_nat_power_cancel_iff
% 0.24/0.50  thf(fact_1047_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: nat,W: nat,X: nat] :
% 0.24/0.50        ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.24/0.50        = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_of_nat_power_cancel_iff
% 0.24/0.50  thf(fact_1048_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: nat,W: nat,X: nat] :
% 0.24/0.50        ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.24/0.50        = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_of_nat_power_cancel_iff
% 0.24/0.50  thf(fact_1049_of__int__le__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_1_iff
% 0.24/0.50  thf(fact_1050_of__int__le__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_1_iff
% 0.24/0.50  thf(fact_1051_of__int__le__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_1_iff
% 0.24/0.50  thf(fact_1052_of__int__le__1__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.24/0.50        = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_1_iff
% 0.24/0.50  thf(fact_1053_of__int__1__le__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1_le_iff
% 0.24/0.50  thf(fact_1054_of__int__1__le__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1_le_iff
% 0.24/0.50  thf(fact_1055_of__int__1__le__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1_le_iff
% 0.24/0.50  thf(fact_1056_of__int__1__le__iff,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.24/0.50        = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_1_le_iff
% 0.24/0.50  thf(fact_1057_one__le__ceiling,axiom,
% 0.24/0.50      ! [X: real] :
% 0.24/0.50        ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 0.24/0.50        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % one_le_ceiling
% 0.24/0.50  thf(fact_1058_one__le__ceiling,axiom,
% 0.24/0.50      ! [X: rat] :
% 0.24/0.50        ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.24/0.50        = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % one_le_ceiling
% 0.24/0.50  thf(fact_1059_of__int__power__le__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_le_of_int_cancel_iff
% 0.24/0.50  thf(fact_1060_of__int__power__le__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_le_of_int_cancel_iff
% 0.24/0.50  thf(fact_1061_of__int__power__le__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_le_of_int_cancel_iff
% 0.24/0.50  thf(fact_1062_of__int__power__le__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: int,B: int,W: nat] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.24/0.50        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_power_le_of_int_cancel_iff
% 0.24/0.50  thf(fact_1063_of__int__le__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) @ ( ring_18347121197199848620nteger @ X ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_1064_of__int__le__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_1065_of__int__le__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_1066_of__int__le__of__int__power__cancel__iff,axiom,
% 0.24/0.50      ! [B: int,W: nat,X: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_of_int_power_cancel_iff
% 0.24/0.50  thf(fact_1067_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,Y: int] :
% 0.24/0.50        ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
% 0.24/0.50          = ( ring_18347121197199848620nteger @ Y ) )
% 0.24/0.50        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.24/0.50          = Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_eq_of_int_cancel_iff
% 0.24/0.50  thf(fact_1068_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,Y: int] :
% 0.24/0.50        ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 0.24/0.50          = ( ring_17405671764205052669omplex @ Y ) )
% 0.24/0.50        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.24/0.50          = Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_eq_of_int_cancel_iff
% 0.24/0.50  thf(fact_1069_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,Y: int] :
% 0.24/0.50        ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.24/0.50          = ( ring_1_of_int_real @ Y ) )
% 0.24/0.50        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.24/0.50          = Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_eq_of_int_cancel_iff
% 0.24/0.50  thf(fact_1070_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,Y: int] :
% 0.24/0.50        ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 0.24/0.50          = ( ring_1_of_int_rat @ Y ) )
% 0.24/0.50        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.24/0.50          = Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_eq_of_int_cancel_iff
% 0.24/0.50  thf(fact_1071_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,Y: int] :
% 0.24/0.50        ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.24/0.50          = ( ring_1_of_int_int @ Y ) )
% 0.24/0.50        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.24/0.50          = Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_eq_of_int_cancel_iff
% 0.24/0.50  thf(fact_1072_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [Y: int,X: num,N: nat] :
% 0.24/0.50        ( ( ( ring_18347121197199848620nteger @ Y )
% 0.24/0.50          = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
% 0.24/0.50        = ( Y
% 0.24/0.50          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1073_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [Y: int,X: num,N: nat] :
% 0.24/0.50        ( ( ( ring_17405671764205052669omplex @ Y )
% 0.24/0.50          = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 0.24/0.50        = ( Y
% 0.24/0.50          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1074_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [Y: int,X: num,N: nat] :
% 0.24/0.50        ( ( ( ring_1_of_int_real @ Y )
% 0.24/0.50          = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.24/0.50        = ( Y
% 0.24/0.50          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1075_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [Y: int,X: num,N: nat] :
% 0.24/0.50        ( ( ( ring_1_of_int_rat @ Y )
% 0.24/0.50          = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.24/0.50        = ( Y
% 0.24/0.50          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1076_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [Y: int,X: num,N: nat] :
% 0.24/0.50        ( ( ( ring_1_of_int_int @ Y )
% 0.24/0.50          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.24/0.50        = ( Y
% 0.24/0.50          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_eq_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1077_power__decreasing__iff,axiom,
% 0.24/0.50      ! [B: real,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ B )
% 0.24/0.50       => ( ( ord_less_real @ B @ one_one_real )
% 0.24/0.50         => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.24/0.50            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_decreasing_iff
% 0.24/0.50  thf(fact_1078_power__decreasing__iff,axiom,
% 0.24/0.50      ! [B: rat,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.24/0.50       => ( ( ord_less_rat @ B @ one_one_rat )
% 0.24/0.50         => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.24/0.50            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_decreasing_iff
% 0.24/0.50  thf(fact_1079_power__decreasing__iff,axiom,
% 0.24/0.50      ! [B: nat,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.24/0.50       => ( ( ord_less_nat @ B @ one_one_nat )
% 0.24/0.50         => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.24/0.50            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_decreasing_iff
% 0.24/0.50  thf(fact_1080_power__decreasing__iff,axiom,
% 0.24/0.50      ! [B: int,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ B )
% 0.24/0.50       => ( ( ord_less_int @ B @ one_one_int )
% 0.24/0.50         => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.24/0.50            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_decreasing_iff
% 0.24/0.50  thf(fact_1081_power2__less__eq__zero__iff,axiom,
% 0.24/0.50      ! [A: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.24/0.50        = ( A = zero_zero_real ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power2_less_eq_zero_iff
% 0.24/0.50  thf(fact_1082_power2__less__eq__zero__iff,axiom,
% 0.24/0.50      ! [A: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.24/0.50        = ( A = zero_zero_rat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power2_less_eq_zero_iff
% 0.24/0.50  thf(fact_1083_power2__less__eq__zero__iff,axiom,
% 0.24/0.50      ! [A: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.24/0.50        = ( A = zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power2_less_eq_zero_iff
% 0.24/0.50  thf(fact_1084_power2__eq__iff__nonneg,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.50       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.24/0.50         => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.50              = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.50            = ( X = Y ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power2_eq_iff_nonneg
% 0.24/0.50  thf(fact_1085_power2__eq__iff__nonneg,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.24/0.50       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.24/0.50         => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.50              = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.50            = ( X = Y ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power2_eq_iff_nonneg
% 0.24/0.50  thf(fact_1086_power2__eq__iff__nonneg,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.24/0.50       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.24/0.50         => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.50              = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.50            = ( X = Y ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power2_eq_iff_nonneg
% 0.24/0.50  thf(fact_1087_power2__eq__iff__nonneg,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.24/0.50       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.24/0.50         => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.50              = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.50            = ( X = Y ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power2_eq_iff_nonneg
% 0.24/0.50  thf(fact_1088_zero__less__power2,axiom,
% 0.24/0.50      ! [A: real] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.50        = ( A != zero_zero_real ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_power2
% 0.24/0.50  thf(fact_1089_zero__less__power2,axiom,
% 0.24/0.50      ! [A: rat] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.50        = ( A != zero_zero_rat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_power2
% 0.24/0.50  thf(fact_1090_zero__less__power2,axiom,
% 0.24/0.50      ! [A: int] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.50        = ( A != zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_power2
% 0.24/0.50  thf(fact_1091_sum__power2__eq__zero__iff,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.50          = zero_zero_rat )
% 0.24/0.50        = ( ( X = zero_zero_rat )
% 0.24/0.50          & ( Y = zero_zero_rat ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % sum_power2_eq_zero_iff
% 0.24/0.50  thf(fact_1092_sum__power2__eq__zero__iff,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.50          = zero_zero_real )
% 0.24/0.50        = ( ( X = zero_zero_real )
% 0.24/0.50          & ( Y = zero_zero_real ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % sum_power2_eq_zero_iff
% 0.24/0.50  thf(fact_1093_sum__power2__eq__zero__iff,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.50          = zero_zero_int )
% 0.24/0.50        = ( ( X = zero_zero_int )
% 0.24/0.50          & ( Y = zero_zero_int ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % sum_power2_eq_zero_iff
% 0.24/0.50  thf(fact_1094_of__nat__zero__less__power__iff,axiom,
% 0.24/0.50      ! [X: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
% 0.24/0.50        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.24/0.50          | ( N = zero_zero_nat ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_zero_less_power_iff
% 0.24/0.50  thf(fact_1095_of__nat__zero__less__power__iff,axiom,
% 0.24/0.50      ! [X: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
% 0.24/0.50        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.24/0.50          | ( N = zero_zero_nat ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_zero_less_power_iff
% 0.24/0.50  thf(fact_1096_of__nat__zero__less__power__iff,axiom,
% 0.24/0.50      ! [X: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
% 0.24/0.50        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.24/0.50          | ( N = zero_zero_nat ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_zero_less_power_iff
% 0.24/0.50  thf(fact_1097_of__nat__zero__less__power__iff,axiom,
% 0.24/0.50      ! [X: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
% 0.24/0.50        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.24/0.50          | ( N = zero_zero_nat ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_zero_less_power_iff
% 0.24/0.50  thf(fact_1098_numeral__power__less__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,A: int] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.24/0.50        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_less_of_int_cancel_iff
% 0.24/0.50  thf(fact_1099_numeral__power__less__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,A: int] :
% 0.24/0.50        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.24/0.50        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_less_of_int_cancel_iff
% 0.24/0.50  thf(fact_1100_numeral__power__less__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,A: int] :
% 0.24/0.50        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.24/0.50        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_less_of_int_cancel_iff
% 0.24/0.50  thf(fact_1101_numeral__power__less__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,A: int] :
% 0.24/0.50        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.24/0.50        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_less_of_int_cancel_iff
% 0.24/0.50  thf(fact_1102_of__int__less__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [A: int,X: num,N: nat] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
% 0.24/0.50        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1103_of__int__less__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [A: int,X: num,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.24/0.50        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1104_of__int__less__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [A: int,X: num,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.24/0.50        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1105_of__int__less__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [A: int,X: num,N: nat] :
% 0.24/0.50        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.24/0.50        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_less_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1106_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [X: nat,I: num,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.24/0.50        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1107_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [X: nat,I: num,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.24/0.50        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1108_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [X: nat,I: num,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.24/0.50        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1109_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [X: nat,I: num,N: nat] :
% 0.24/0.50        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.24/0.50        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1110_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.24/0.50      ! [I: num,N: nat,X: nat] :
% 0.24/0.50        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.24/0.50        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_less_of_nat_cancel_iff
% 0.24/0.50  thf(fact_1111_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.24/0.50      ! [I: num,N: nat,X: nat] :
% 0.24/0.50        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.24/0.50        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_less_of_nat_cancel_iff
% 0.24/0.50  thf(fact_1112_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.24/0.50      ! [I: num,N: nat,X: nat] :
% 0.24/0.50        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.24/0.50        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_less_of_nat_cancel_iff
% 0.24/0.50  thf(fact_1113_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.24/0.50      ! [I: num,N: nat,X: nat] :
% 0.24/0.50        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.24/0.50        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_less_of_nat_cancel_iff
% 0.24/0.50  thf(fact_1114_numeral__power__le__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,A: int] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_le_of_int_cancel_iff
% 0.24/0.50  thf(fact_1115_numeral__power__le__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,A: int] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_le_of_int_cancel_iff
% 0.24/0.50  thf(fact_1116_numeral__power__le__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,A: int] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_le_of_int_cancel_iff
% 0.24/0.50  thf(fact_1117_numeral__power__le__of__int__cancel__iff,axiom,
% 0.24/0.50      ! [X: num,N: nat,A: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.24/0.50        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % numeral_power_le_of_int_cancel_iff
% 0.24/0.50  thf(fact_1118_of__int__le__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [A: int,X: num,N: nat] :
% 0.24/0.50        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
% 0.24/0.50        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1119_of__int__le__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [A: int,X: num,N: nat] :
% 0.24/0.50        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.24/0.50        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1120_of__int__le__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [A: int,X: num,N: nat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.24/0.50        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1121_of__int__le__numeral__power__cancel__iff,axiom,
% 0.24/0.50      ! [A: int,X: num,N: nat] :
% 0.24/0.50        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.24/0.50        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_le_numeral_power_cancel_iff
% 0.24/0.50  thf(fact_1122_Collect__mono__iff,axiom,
% 0.24/0.50      ! [P: complex > $o,Q: complex > $o] :
% 0.24/0.50        ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
% 0.24/0.50        = ( ! [X4: complex] :
% 0.24/0.50              ( ( P @ X4 )
% 0.24/0.50             => ( Q @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Collect_mono_iff
% 0.24/0.50  thf(fact_1123_Collect__mono__iff,axiom,
% 0.24/0.50      ! [P: real > $o,Q: real > $o] :
% 0.24/0.50        ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 0.24/0.50        = ( ! [X4: real] :
% 0.24/0.50              ( ( P @ X4 )
% 0.24/0.50             => ( Q @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Collect_mono_iff
% 0.24/0.50  thf(fact_1124_Collect__mono__iff,axiom,
% 0.24/0.50      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.24/0.50        ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 0.24/0.50        = ( ! [X4: set_nat] :
% 0.24/0.50              ( ( P @ X4 )
% 0.24/0.50             => ( Q @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Collect_mono_iff
% 0.24/0.50  thf(fact_1125_Collect__mono__iff,axiom,
% 0.24/0.50      ! [P: nat > $o,Q: nat > $o] :
% 0.24/0.50        ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 0.24/0.50        = ( ! [X4: nat] :
% 0.24/0.50              ( ( P @ X4 )
% 0.24/0.50             => ( Q @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Collect_mono_iff
% 0.24/0.50  thf(fact_1126_Collect__mono__iff,axiom,
% 0.24/0.50      ! [P: int > $o,Q: int > $o] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 0.24/0.50        = ( ! [X4: int] :
% 0.24/0.50              ( ( P @ X4 )
% 0.24/0.50             => ( Q @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Collect_mono_iff
% 0.24/0.50  thf(fact_1127_set__eq__subset,axiom,
% 0.24/0.50      ( ( ^ [Y3: set_int,Z2: set_int] : ( Y3 = Z2 ) )
% 0.24/0.50      = ( ^ [A4: set_int,B5: set_int] :
% 0.24/0.50            ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.24/0.50            & ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % set_eq_subset
% 0.24/0.50  thf(fact_1128_subset__trans,axiom,
% 0.24/0.50      ! [A2: set_int,B2: set_int,C6: set_int] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.50       => ( ( ord_less_eq_set_int @ B2 @ C6 )
% 0.24/0.50         => ( ord_less_eq_set_int @ A2 @ C6 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_trans
% 0.24/0.50  thf(fact_1129_Collect__mono,axiom,
% 0.24/0.50      ! [P: complex > $o,Q: complex > $o] :
% 0.24/0.50        ( ! [X3: complex] :
% 0.24/0.50            ( ( P @ X3 )
% 0.24/0.50           => ( Q @ X3 ) )
% 0.24/0.50       => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Collect_mono
% 0.24/0.50  thf(fact_1130_Collect__mono,axiom,
% 0.24/0.50      ! [P: real > $o,Q: real > $o] :
% 0.24/0.50        ( ! [X3: real] :
% 0.24/0.50            ( ( P @ X3 )
% 0.24/0.50           => ( Q @ X3 ) )
% 0.24/0.50       => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Collect_mono
% 0.24/0.50  thf(fact_1131_Collect__mono,axiom,
% 0.24/0.50      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.24/0.50        ( ! [X3: set_nat] :
% 0.24/0.50            ( ( P @ X3 )
% 0.24/0.50           => ( Q @ X3 ) )
% 0.24/0.50       => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Collect_mono
% 0.24/0.50  thf(fact_1132_Collect__mono,axiom,
% 0.24/0.50      ! [P: nat > $o,Q: nat > $o] :
% 0.24/0.50        ( ! [X3: nat] :
% 0.24/0.50            ( ( P @ X3 )
% 0.24/0.50           => ( Q @ X3 ) )
% 0.24/0.50       => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Collect_mono
% 0.24/0.50  thf(fact_1133_Collect__mono,axiom,
% 0.24/0.50      ! [P: int > $o,Q: int > $o] :
% 0.24/0.50        ( ! [X3: int] :
% 0.24/0.50            ( ( P @ X3 )
% 0.24/0.50           => ( Q @ X3 ) )
% 0.24/0.50       => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Collect_mono
% 0.24/0.50  thf(fact_1134_subset__refl,axiom,
% 0.24/0.50      ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_refl
% 0.24/0.50  thf(fact_1135_subset__iff,axiom,
% 0.24/0.50      ( ord_le211207098394363844omplex
% 0.24/0.50      = ( ^ [A4: set_complex,B5: set_complex] :
% 0.24/0.50          ! [T2: complex] :
% 0.24/0.50            ( ( member_complex @ T2 @ A4 )
% 0.24/0.50           => ( member_complex @ T2 @ B5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_iff
% 0.24/0.50  thf(fact_1136_subset__iff,axiom,
% 0.24/0.50      ( ord_less_eq_set_real
% 0.24/0.50      = ( ^ [A4: set_real,B5: set_real] :
% 0.24/0.50          ! [T2: real] :
% 0.24/0.50            ( ( member_real @ T2 @ A4 )
% 0.24/0.50           => ( member_real @ T2 @ B5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_iff
% 0.24/0.50  thf(fact_1137_subset__iff,axiom,
% 0.24/0.50      ( ord_le6893508408891458716et_nat
% 0.24/0.50      = ( ^ [A4: set_set_nat,B5: set_set_nat] :
% 0.24/0.50          ! [T2: set_nat] :
% 0.24/0.50            ( ( member_set_nat @ T2 @ A4 )
% 0.24/0.50           => ( member_set_nat @ T2 @ B5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_iff
% 0.24/0.50  thf(fact_1138_subset__iff,axiom,
% 0.24/0.50      ( ord_less_eq_set_nat
% 0.24/0.50      = ( ^ [A4: set_nat,B5: set_nat] :
% 0.24/0.50          ! [T2: nat] :
% 0.24/0.50            ( ( member_nat @ T2 @ A4 )
% 0.24/0.50           => ( member_nat @ T2 @ B5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_iff
% 0.24/0.50  thf(fact_1139_subset__iff,axiom,
% 0.24/0.50      ( ord_less_eq_set_int
% 0.24/0.50      = ( ^ [A4: set_int,B5: set_int] :
% 0.24/0.50          ! [T2: int] :
% 0.24/0.50            ( ( member_int @ T2 @ A4 )
% 0.24/0.50           => ( member_int @ T2 @ B5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_iff
% 0.24/0.50  thf(fact_1140_equalityD2,axiom,
% 0.24/0.50      ! [A2: set_int,B2: set_int] :
% 0.24/0.50        ( ( A2 = B2 )
% 0.24/0.50       => ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 0.24/0.50  
% 0.24/0.50  % equalityD2
% 0.24/0.50  thf(fact_1141_equalityD1,axiom,
% 0.24/0.50      ! [A2: set_int,B2: set_int] :
% 0.24/0.50        ( ( A2 = B2 )
% 0.24/0.50       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.24/0.50  
% 0.24/0.50  % equalityD1
% 0.24/0.50  thf(fact_1142_subset__eq,axiom,
% 0.24/0.50      ( ord_le211207098394363844omplex
% 0.24/0.50      = ( ^ [A4: set_complex,B5: set_complex] :
% 0.24/0.50          ! [X4: complex] :
% 0.24/0.50            ( ( member_complex @ X4 @ A4 )
% 0.24/0.50           => ( member_complex @ X4 @ B5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_eq
% 0.24/0.50  thf(fact_1143_subset__eq,axiom,
% 0.24/0.50      ( ord_less_eq_set_real
% 0.24/0.50      = ( ^ [A4: set_real,B5: set_real] :
% 0.24/0.50          ! [X4: real] :
% 0.24/0.50            ( ( member_real @ X4 @ A4 )
% 0.24/0.50           => ( member_real @ X4 @ B5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_eq
% 0.24/0.50  thf(fact_1144_subset__eq,axiom,
% 0.24/0.50      ( ord_le6893508408891458716et_nat
% 0.24/0.50      = ( ^ [A4: set_set_nat,B5: set_set_nat] :
% 0.24/0.50          ! [X4: set_nat] :
% 0.24/0.50            ( ( member_set_nat @ X4 @ A4 )
% 0.24/0.50           => ( member_set_nat @ X4 @ B5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_eq
% 0.24/0.50  thf(fact_1145_subset__eq,axiom,
% 0.24/0.50      ( ord_less_eq_set_nat
% 0.24/0.50      = ( ^ [A4: set_nat,B5: set_nat] :
% 0.24/0.50          ! [X4: nat] :
% 0.24/0.50            ( ( member_nat @ X4 @ A4 )
% 0.24/0.50           => ( member_nat @ X4 @ B5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_eq
% 0.24/0.50  thf(fact_1146_subset__eq,axiom,
% 0.24/0.50      ( ord_less_eq_set_int
% 0.24/0.50      = ( ^ [A4: set_int,B5: set_int] :
% 0.24/0.50          ! [X4: int] :
% 0.24/0.50            ( ( member_int @ X4 @ A4 )
% 0.24/0.50           => ( member_int @ X4 @ B5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subset_eq
% 0.24/0.50  thf(fact_1147_equalityE,axiom,
% 0.24/0.50      ! [A2: set_int,B2: set_int] :
% 0.24/0.50        ( ( A2 = B2 )
% 0.24/0.50       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.50           => ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % equalityE
% 0.24/0.50  thf(fact_1148_subsetD,axiom,
% 0.24/0.50      ! [A2: set_complex,B2: set_complex,C: complex] :
% 0.24/0.50        ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.24/0.50       => ( ( member_complex @ C @ A2 )
% 0.24/0.50         => ( member_complex @ C @ B2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subsetD
% 0.24/0.50  thf(fact_1149_subsetD,axiom,
% 0.24/0.50      ! [A2: set_real,B2: set_real,C: real] :
% 0.24/0.50        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.24/0.50       => ( ( member_real @ C @ A2 )
% 0.24/0.50         => ( member_real @ C @ B2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subsetD
% 0.24/0.50  thf(fact_1150_subsetD,axiom,
% 0.24/0.50      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 0.24/0.50        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 0.24/0.50       => ( ( member_set_nat @ C @ A2 )
% 0.24/0.50         => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subsetD
% 0.24/0.50  thf(fact_1151_subsetD,axiom,
% 0.24/0.50      ! [A2: set_nat,B2: set_nat,C: nat] :
% 0.24/0.50        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.24/0.50       => ( ( member_nat @ C @ A2 )
% 0.24/0.50         => ( member_nat @ C @ B2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subsetD
% 0.24/0.50  thf(fact_1152_subsetD,axiom,
% 0.24/0.50      ! [A2: set_int,B2: set_int,C: int] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.50       => ( ( member_int @ C @ A2 )
% 0.24/0.50         => ( member_int @ C @ B2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % subsetD
% 0.24/0.50  thf(fact_1153_in__mono,axiom,
% 0.24/0.50      ! [A2: set_complex,B2: set_complex,X: complex] :
% 0.24/0.50        ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.24/0.50       => ( ( member_complex @ X @ A2 )
% 0.24/0.50         => ( member_complex @ X @ B2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % in_mono
% 0.24/0.50  thf(fact_1154_in__mono,axiom,
% 0.24/0.50      ! [A2: set_real,B2: set_real,X: real] :
% 0.24/0.50        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.24/0.50       => ( ( member_real @ X @ A2 )
% 0.24/0.50         => ( member_real @ X @ B2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % in_mono
% 0.24/0.50  thf(fact_1155_in__mono,axiom,
% 0.24/0.50      ! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
% 0.24/0.50        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 0.24/0.50       => ( ( member_set_nat @ X @ A2 )
% 0.24/0.50         => ( member_set_nat @ X @ B2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % in_mono
% 0.24/0.50  thf(fact_1156_in__mono,axiom,
% 0.24/0.50      ! [A2: set_nat,B2: set_nat,X: nat] :
% 0.24/0.50        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.24/0.50       => ( ( member_nat @ X @ A2 )
% 0.24/0.50         => ( member_nat @ X @ B2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % in_mono
% 0.24/0.50  thf(fact_1157_in__mono,axiom,
% 0.24/0.50      ! [A2: set_int,B2: set_int,X: int] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.50       => ( ( member_int @ X @ A2 )
% 0.24/0.50         => ( member_int @ X @ B2 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % in_mono
% 0.24/0.50  thf(fact_1158_zero__power,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50       => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.24/0.50          = zero_zero_rat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_power
% 0.24/0.50  thf(fact_1159_zero__power,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50       => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.24/0.50          = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_power
% 0.24/0.50  thf(fact_1160_zero__power,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50       => ( ( power_power_real @ zero_zero_real @ N )
% 0.24/0.50          = zero_zero_real ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_power
% 0.24/0.50  thf(fact_1161_zero__power,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50       => ( ( power_power_int @ zero_zero_int @ N )
% 0.24/0.50          = zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_power
% 0.24/0.50  thf(fact_1162_zero__power,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50       => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.24/0.50          = zero_zero_complex ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_power
% 0.24/0.50  thf(fact_1163_field__lbound__gt__zero,axiom,
% 0.24/0.50      ! [D1: real,D22: real] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.24/0.50       => ( ( ord_less_real @ zero_zero_real @ D22 )
% 0.24/0.50         => ? [E: real] :
% 0.24/0.50              ( ( ord_less_real @ zero_zero_real @ E )
% 0.24/0.50              & ( ord_less_real @ E @ D1 )
% 0.24/0.50              & ( ord_less_real @ E @ D22 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % field_lbound_gt_zero
% 0.24/0.50  thf(fact_1164_field__lbound__gt__zero,axiom,
% 0.24/0.50      ! [D1: rat,D22: rat] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.24/0.50       => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 0.24/0.50         => ? [E: rat] :
% 0.24/0.50              ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.24/0.50              & ( ord_less_rat @ E @ D1 )
% 0.24/0.50              & ( ord_less_rat @ E @ D22 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % field_lbound_gt_zero
% 0.24/0.50  thf(fact_1165_of__int__pos,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.24/0.50       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_pos
% 0.24/0.50  thf(fact_1166_of__int__pos,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.24/0.50       => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_pos
% 0.24/0.50  thf(fact_1167_of__int__pos,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.24/0.50       => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_pos
% 0.24/0.50  thf(fact_1168_of__int__pos,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.24/0.50       => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_pos
% 0.24/0.50  thf(fact_1169_verit__comp__simplify1_I1_J,axiom,
% 0.24/0.50      ! [A: real] :
% 0.24/0.50        ~ ( ord_less_real @ A @ A ) ).
% 0.24/0.50  
% 0.24/0.50  % verit_comp_simplify1(1)
% 0.24/0.50  thf(fact_1170_verit__comp__simplify1_I1_J,axiom,
% 0.24/0.50      ! [A: rat] :
% 0.24/0.50        ~ ( ord_less_rat @ A @ A ) ).
% 0.24/0.50  
% 0.24/0.50  % verit_comp_simplify1(1)
% 0.24/0.50  thf(fact_1171_verit__comp__simplify1_I1_J,axiom,
% 0.24/0.50      ! [A: num] :
% 0.24/0.50        ~ ( ord_less_num @ A @ A ) ).
% 0.24/0.50  
% 0.24/0.50  % verit_comp_simplify1(1)
% 0.24/0.50  thf(fact_1172_verit__comp__simplify1_I1_J,axiom,
% 0.24/0.50      ! [A: nat] :
% 0.24/0.50        ~ ( ord_less_nat @ A @ A ) ).
% 0.24/0.50  
% 0.24/0.50  % verit_comp_simplify1(1)
% 0.24/0.50  thf(fact_1173_verit__comp__simplify1_I1_J,axiom,
% 0.24/0.50      ! [A: int] :
% 0.24/0.50        ~ ( ord_less_int @ A @ A ) ).
% 0.24/0.50  
% 0.24/0.50  % verit_comp_simplify1(1)
% 0.24/0.50  thf(fact_1174_lt__ex,axiom,
% 0.24/0.50      ! [X: real] :
% 0.24/0.50      ? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% 0.24/0.50  
% 0.24/0.50  % lt_ex
% 0.24/0.50  thf(fact_1175_lt__ex,axiom,
% 0.24/0.50      ! [X: rat] :
% 0.24/0.50      ? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).
% 0.24/0.50  
% 0.24/0.50  % lt_ex
% 0.24/0.50  thf(fact_1176_lt__ex,axiom,
% 0.24/0.50      ! [X: int] :
% 0.24/0.50      ? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% 0.24/0.50  
% 0.24/0.50  % lt_ex
% 0.24/0.50  thf(fact_1177_gt__ex,axiom,
% 0.24/0.50      ! [X: real] :
% 0.24/0.50      ? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).
% 0.24/0.50  
% 0.24/0.50  % gt_ex
% 0.24/0.50  thf(fact_1178_gt__ex,axiom,
% 0.24/0.50      ! [X: rat] :
% 0.24/0.50      ? [X_12: rat] : ( ord_less_rat @ X @ X_12 ) ).
% 0.24/0.50  
% 0.24/0.50  % gt_ex
% 0.24/0.50  thf(fact_1179_gt__ex,axiom,
% 0.24/0.50      ! [X: nat] :
% 0.24/0.50      ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% 0.24/0.50  
% 0.24/0.50  % gt_ex
% 0.24/0.50  thf(fact_1180_gt__ex,axiom,
% 0.24/0.50      ! [X: int] :
% 0.24/0.50      ? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).
% 0.24/0.50  
% 0.24/0.50  % gt_ex
% 0.24/0.50  thf(fact_1181_dense,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ? [Z3: real] :
% 0.24/0.50            ( ( ord_less_real @ X @ Z3 )
% 0.24/0.50            & ( ord_less_real @ Z3 @ Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dense
% 0.24/0.50  thf(fact_1182_dense,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ? [Z3: rat] :
% 0.24/0.50            ( ( ord_less_rat @ X @ Z3 )
% 0.24/0.50            & ( ord_less_rat @ Z3 @ Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dense
% 0.24/0.50  thf(fact_1183_less__imp__neq,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( X != Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_imp_neq
% 0.24/0.50  thf(fact_1184_less__imp__neq,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( X != Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_imp_neq
% 0.24/0.50  thf(fact_1185_less__imp__neq,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50       => ( X != Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_imp_neq
% 0.24/0.50  thf(fact_1186_less__imp__neq,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ( X != Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_imp_neq
% 0.24/0.50  thf(fact_1187_less__imp__neq,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50       => ( X != Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_imp_neq
% 0.24/0.50  thf(fact_1188_order_Oasym,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ~ ( ord_less_real @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.asym
% 0.24/0.50  thf(fact_1189_order_Oasym,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.asym
% 0.24/0.50  thf(fact_1190_order_Oasym,axiom,
% 0.24/0.50      ! [A: num,B: num] :
% 0.24/0.50        ( ( ord_less_num @ A @ B )
% 0.24/0.50       => ~ ( ord_less_num @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.asym
% 0.24/0.50  thf(fact_1191_order_Oasym,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.asym
% 0.24/0.50  thf(fact_1192_order_Oasym,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_int @ A @ B )
% 0.24/0.50       => ~ ( ord_less_int @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.asym
% 0.24/0.50  thf(fact_1193_ord__eq__less__trans,axiom,
% 0.24/0.50      ! [A: real,B: real,C: real] :
% 0.24/0.50        ( ( A = B )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ord_less_real @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_trans
% 0.24/0.50  thf(fact_1194_ord__eq__less__trans,axiom,
% 0.24/0.50      ! [A: rat,B: rat,C: rat] :
% 0.24/0.50        ( ( A = B )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_trans
% 0.24/0.50  thf(fact_1195_ord__eq__less__trans,axiom,
% 0.24/0.50      ! [A: num,B: num,C: num] :
% 0.24/0.50        ( ( A = B )
% 0.24/0.50       => ( ( ord_less_num @ B @ C )
% 0.24/0.50         => ( ord_less_num @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_trans
% 0.24/0.50  thf(fact_1196_ord__eq__less__trans,axiom,
% 0.24/0.50      ! [A: nat,B: nat,C: nat] :
% 0.24/0.50        ( ( A = B )
% 0.24/0.50       => ( ( ord_less_nat @ B @ C )
% 0.24/0.50         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_trans
% 0.24/0.50  thf(fact_1197_ord__eq__less__trans,axiom,
% 0.24/0.50      ! [A: int,B: int,C: int] :
% 0.24/0.50        ( ( A = B )
% 0.24/0.50       => ( ( ord_less_int @ B @ C )
% 0.24/0.50         => ( ord_less_int @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_trans
% 0.24/0.50  thf(fact_1198_ord__less__eq__trans,axiom,
% 0.24/0.50      ! [A: real,B: real,C: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( B = C )
% 0.24/0.50         => ( ord_less_real @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_trans
% 0.24/0.50  thf(fact_1199_ord__less__eq__trans,axiom,
% 0.24/0.50      ! [A: rat,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( B = C )
% 0.24/0.50         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_trans
% 0.24/0.50  thf(fact_1200_ord__less__eq__trans,axiom,
% 0.24/0.50      ! [A: num,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_num @ A @ B )
% 0.24/0.50       => ( ( B = C )
% 0.24/0.50         => ( ord_less_num @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_trans
% 0.24/0.50  thf(fact_1201_ord__less__eq__trans,axiom,
% 0.24/0.50      ! [A: nat,B: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ( ( B = C )
% 0.24/0.50         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_trans
% 0.24/0.50  thf(fact_1202_ord__less__eq__trans,axiom,
% 0.24/0.50      ! [A: int,B: int,C: int] :
% 0.24/0.50        ( ( ord_less_int @ A @ B )
% 0.24/0.50       => ( ( B = C )
% 0.24/0.50         => ( ord_less_int @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_trans
% 0.24/0.50  thf(fact_1203_less__induct,axiom,
% 0.24/0.50      ! [P: nat > $o,A: nat] :
% 0.24/0.50        ( ! [X3: nat] :
% 0.24/0.50            ( ! [Y5: nat] :
% 0.24/0.50                ( ( ord_less_nat @ Y5 @ X3 )
% 0.24/0.50               => ( P @ Y5 ) )
% 0.24/0.50           => ( P @ X3 ) )
% 0.24/0.50       => ( P @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_induct
% 0.24/0.50  thf(fact_1204_antisym__conv3,axiom,
% 0.24/0.50      ! [Y: real,X: real] :
% 0.24/0.50        ( ~ ( ord_less_real @ Y @ X )
% 0.24/0.50       => ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv3
% 0.24/0.50  thf(fact_1205_antisym__conv3,axiom,
% 0.24/0.50      ! [Y: rat,X: rat] :
% 0.24/0.50        ( ~ ( ord_less_rat @ Y @ X )
% 0.24/0.50       => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv3
% 0.24/0.50  thf(fact_1206_antisym__conv3,axiom,
% 0.24/0.50      ! [Y: num,X: num] :
% 0.24/0.50        ( ~ ( ord_less_num @ Y @ X )
% 0.24/0.50       => ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv3
% 0.24/0.50  thf(fact_1207_antisym__conv3,axiom,
% 0.24/0.50      ! [Y: nat,X: nat] :
% 0.24/0.50        ( ~ ( ord_less_nat @ Y @ X )
% 0.24/0.50       => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv3
% 0.24/0.50  thf(fact_1208_antisym__conv3,axiom,
% 0.24/0.50      ! [Y: int,X: int] :
% 0.24/0.50        ( ~ ( ord_less_int @ Y @ X )
% 0.24/0.50       => ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv3
% 0.24/0.50  thf(fact_1209_linorder__cases,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ~ ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( ( X != Y )
% 0.24/0.50         => ( ord_less_real @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_cases
% 0.24/0.50  thf(fact_1210_linorder__cases,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ~ ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( ( X != Y )
% 0.24/0.50         => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_cases
% 0.24/0.50  thf(fact_1211_linorder__cases,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ~ ( ord_less_num @ X @ Y )
% 0.24/0.50       => ( ( X != Y )
% 0.24/0.50         => ( ord_less_num @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_cases
% 0.24/0.50  thf(fact_1212_linorder__cases,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ~ ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ( ( X != Y )
% 0.24/0.50         => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_cases
% 0.24/0.50  thf(fact_1213_linorder__cases,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ~ ( ord_less_int @ X @ Y )
% 0.24/0.50       => ( ( X != Y )
% 0.24/0.50         => ( ord_less_int @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_cases
% 0.24/0.50  thf(fact_1214_dual__order_Oasym,axiom,
% 0.24/0.50      ! [B: real,A: real] :
% 0.24/0.50        ( ( ord_less_real @ B @ A )
% 0.24/0.50       => ~ ( ord_less_real @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.asym
% 0.24/0.50  thf(fact_1215_dual__order_Oasym,axiom,
% 0.24/0.50      ! [B: rat,A: rat] :
% 0.24/0.50        ( ( ord_less_rat @ B @ A )
% 0.24/0.50       => ~ ( ord_less_rat @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.asym
% 0.24/0.50  thf(fact_1216_dual__order_Oasym,axiom,
% 0.24/0.50      ! [B: num,A: num] :
% 0.24/0.50        ( ( ord_less_num @ B @ A )
% 0.24/0.50       => ~ ( ord_less_num @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.asym
% 0.24/0.50  thf(fact_1217_dual__order_Oasym,axiom,
% 0.24/0.50      ! [B: nat,A: nat] :
% 0.24/0.50        ( ( ord_less_nat @ B @ A )
% 0.24/0.50       => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.asym
% 0.24/0.50  thf(fact_1218_dual__order_Oasym,axiom,
% 0.24/0.50      ! [B: int,A: int] :
% 0.24/0.50        ( ( ord_less_int @ B @ A )
% 0.24/0.50       => ~ ( ord_less_int @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.asym
% 0.24/0.50  thf(fact_1219_dual__order_Oirrefl,axiom,
% 0.24/0.50      ! [A: real] :
% 0.24/0.50        ~ ( ord_less_real @ A @ A ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.irrefl
% 0.24/0.50  thf(fact_1220_dual__order_Oirrefl,axiom,
% 0.24/0.50      ! [A: rat] :
% 0.24/0.50        ~ ( ord_less_rat @ A @ A ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.irrefl
% 0.24/0.50  thf(fact_1221_dual__order_Oirrefl,axiom,
% 0.24/0.50      ! [A: num] :
% 0.24/0.50        ~ ( ord_less_num @ A @ A ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.irrefl
% 0.24/0.50  thf(fact_1222_dual__order_Oirrefl,axiom,
% 0.24/0.50      ! [A: nat] :
% 0.24/0.50        ~ ( ord_less_nat @ A @ A ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.irrefl
% 0.24/0.50  thf(fact_1223_dual__order_Oirrefl,axiom,
% 0.24/0.50      ! [A: int] :
% 0.24/0.50        ~ ( ord_less_int @ A @ A ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.irrefl
% 0.24/0.50  thf(fact_1224_exists__least__iff,axiom,
% 0.24/0.50      ( ( ^ [P2: nat > $o] :
% 0.24/0.50          ? [X5: nat] : ( P2 @ X5 ) )
% 0.24/0.50      = ( ^ [P3: nat > $o] :
% 0.24/0.50          ? [N3: nat] :
% 0.24/0.50            ( ( P3 @ N3 )
% 0.24/0.50            & ! [M6: nat] :
% 0.24/0.50                ( ( ord_less_nat @ M6 @ N3 )
% 0.24/0.50               => ~ ( P3 @ M6 ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % exists_least_iff
% 0.24/0.50  thf(fact_1225_linorder__less__wlog,axiom,
% 0.24/0.50      ! [P: real > real > $o,A: real,B: real] :
% 0.24/0.50        ( ! [A5: real,B6: real] :
% 0.24/0.50            ( ( ord_less_real @ A5 @ B6 )
% 0.24/0.50           => ( P @ A5 @ B6 ) )
% 0.24/0.50       => ( ! [A5: real] : ( P @ A5 @ A5 )
% 0.24/0.50         => ( ! [A5: real,B6: real] :
% 0.24/0.50                ( ( P @ B6 @ A5 )
% 0.24/0.50               => ( P @ A5 @ B6 ) )
% 0.24/0.50           => ( P @ A @ B ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_less_wlog
% 0.24/0.50  thf(fact_1226_linorder__less__wlog,axiom,
% 0.24/0.50      ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.24/0.50        ( ! [A5: rat,B6: rat] :
% 0.24/0.50            ( ( ord_less_rat @ A5 @ B6 )
% 0.24/0.50           => ( P @ A5 @ B6 ) )
% 0.24/0.50       => ( ! [A5: rat] : ( P @ A5 @ A5 )
% 0.24/0.50         => ( ! [A5: rat,B6: rat] :
% 0.24/0.50                ( ( P @ B6 @ A5 )
% 0.24/0.50               => ( P @ A5 @ B6 ) )
% 0.24/0.50           => ( P @ A @ B ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_less_wlog
% 0.24/0.50  thf(fact_1227_linorder__less__wlog,axiom,
% 0.24/0.50      ! [P: num > num > $o,A: num,B: num] :
% 0.24/0.50        ( ! [A5: num,B6: num] :
% 0.24/0.50            ( ( ord_less_num @ A5 @ B6 )
% 0.24/0.50           => ( P @ A5 @ B6 ) )
% 0.24/0.50       => ( ! [A5: num] : ( P @ A5 @ A5 )
% 0.24/0.50         => ( ! [A5: num,B6: num] :
% 0.24/0.50                ( ( P @ B6 @ A5 )
% 0.24/0.50               => ( P @ A5 @ B6 ) )
% 0.24/0.50           => ( P @ A @ B ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_less_wlog
% 0.24/0.50  thf(fact_1228_linorder__less__wlog,axiom,
% 0.24/0.50      ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.24/0.50        ( ! [A5: nat,B6: nat] :
% 0.24/0.50            ( ( ord_less_nat @ A5 @ B6 )
% 0.24/0.50           => ( P @ A5 @ B6 ) )
% 0.24/0.50       => ( ! [A5: nat] : ( P @ A5 @ A5 )
% 0.24/0.50         => ( ! [A5: nat,B6: nat] :
% 0.24/0.50                ( ( P @ B6 @ A5 )
% 0.24/0.50               => ( P @ A5 @ B6 ) )
% 0.24/0.50           => ( P @ A @ B ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_less_wlog
% 0.24/0.50  thf(fact_1229_linorder__less__wlog,axiom,
% 0.24/0.50      ! [P: int > int > $o,A: int,B: int] :
% 0.24/0.50        ( ! [A5: int,B6: int] :
% 0.24/0.50            ( ( ord_less_int @ A5 @ B6 )
% 0.24/0.50           => ( P @ A5 @ B6 ) )
% 0.24/0.50       => ( ! [A5: int] : ( P @ A5 @ A5 )
% 0.24/0.50         => ( ! [A5: int,B6: int] :
% 0.24/0.50                ( ( P @ B6 @ A5 )
% 0.24/0.50               => ( P @ A5 @ B6 ) )
% 0.24/0.50           => ( P @ A @ B ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_less_wlog
% 0.24/0.50  thf(fact_1230_order_Ostrict__trans,axiom,
% 0.24/0.50      ! [A: real,B: real,C: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ord_less_real @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans
% 0.24/0.50  thf(fact_1231_order_Ostrict__trans,axiom,
% 0.24/0.50      ! [A: rat,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans
% 0.24/0.50  thf(fact_1232_order_Ostrict__trans,axiom,
% 0.24/0.50      ! [A: num,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_num @ A @ B )
% 0.24/0.50       => ( ( ord_less_num @ B @ C )
% 0.24/0.50         => ( ord_less_num @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans
% 0.24/0.50  thf(fact_1233_order_Ostrict__trans,axiom,
% 0.24/0.50      ! [A: nat,B: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ( ( ord_less_nat @ B @ C )
% 0.24/0.50         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans
% 0.24/0.50  thf(fact_1234_order_Ostrict__trans,axiom,
% 0.24/0.50      ! [A: int,B: int,C: int] :
% 0.24/0.50        ( ( ord_less_int @ A @ B )
% 0.24/0.50       => ( ( ord_less_int @ B @ C )
% 0.24/0.50         => ( ord_less_int @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans
% 0.24/0.50  thf(fact_1235_not__less__iff__gr__or__eq,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.24/0.50        = ( ( ord_less_real @ Y @ X )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_less_iff_gr_or_eq
% 0.24/0.50  thf(fact_1236_not__less__iff__gr__or__eq,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.24/0.50        = ( ( ord_less_rat @ Y @ X )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_less_iff_gr_or_eq
% 0.24/0.50  thf(fact_1237_not__less__iff__gr__or__eq,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.24/0.50        = ( ( ord_less_num @ Y @ X )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_less_iff_gr_or_eq
% 0.24/0.50  thf(fact_1238_not__less__iff__gr__or__eq,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.24/0.50        = ( ( ord_less_nat @ Y @ X )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_less_iff_gr_or_eq
% 0.24/0.50  thf(fact_1239_not__less__iff__gr__or__eq,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.24/0.50        = ( ( ord_less_int @ Y @ X )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_less_iff_gr_or_eq
% 0.24/0.50  thf(fact_1240_dual__order_Ostrict__trans,axiom,
% 0.24/0.50      ! [B: real,A: real,C: real] :
% 0.24/0.50        ( ( ord_less_real @ B @ A )
% 0.24/0.50       => ( ( ord_less_real @ C @ B )
% 0.24/0.50         => ( ord_less_real @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans
% 0.24/0.50  thf(fact_1241_dual__order_Ostrict__trans,axiom,
% 0.24/0.50      ! [B: rat,A: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ B @ A )
% 0.24/0.50       => ( ( ord_less_rat @ C @ B )
% 0.24/0.50         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans
% 0.24/0.50  thf(fact_1242_dual__order_Ostrict__trans,axiom,
% 0.24/0.50      ! [B: num,A: num,C: num] :
% 0.24/0.50        ( ( ord_less_num @ B @ A )
% 0.24/0.50       => ( ( ord_less_num @ C @ B )
% 0.24/0.50         => ( ord_less_num @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans
% 0.24/0.50  thf(fact_1243_dual__order_Ostrict__trans,axiom,
% 0.24/0.50      ! [B: nat,A: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_nat @ B @ A )
% 0.24/0.50       => ( ( ord_less_nat @ C @ B )
% 0.24/0.50         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans
% 0.24/0.50  thf(fact_1244_dual__order_Ostrict__trans,axiom,
% 0.24/0.50      ! [B: int,A: int,C: int] :
% 0.24/0.50        ( ( ord_less_int @ B @ A )
% 0.24/0.50       => ( ( ord_less_int @ C @ B )
% 0.24/0.50         => ( ord_less_int @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans
% 0.24/0.50  thf(fact_1245_order_Ostrict__implies__not__eq,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( A != B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_not_eq
% 0.24/0.50  thf(fact_1246_order_Ostrict__implies__not__eq,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( A != B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_not_eq
% 0.24/0.50  thf(fact_1247_order_Ostrict__implies__not__eq,axiom,
% 0.24/0.50      ! [A: num,B: num] :
% 0.24/0.50        ( ( ord_less_num @ A @ B )
% 0.24/0.50       => ( A != B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_not_eq
% 0.24/0.50  thf(fact_1248_order_Ostrict__implies__not__eq,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ( A != B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_not_eq
% 0.24/0.50  thf(fact_1249_order_Ostrict__implies__not__eq,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_int @ A @ B )
% 0.24/0.50       => ( A != B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_not_eq
% 0.24/0.50  thf(fact_1250_dual__order_Ostrict__implies__not__eq,axiom,
% 0.24/0.50      ! [B: real,A: real] :
% 0.24/0.50        ( ( ord_less_real @ B @ A )
% 0.24/0.50       => ( A != B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_not_eq
% 0.24/0.50  thf(fact_1251_dual__order_Ostrict__implies__not__eq,axiom,
% 0.24/0.50      ! [B: rat,A: rat] :
% 0.24/0.50        ( ( ord_less_rat @ B @ A )
% 0.24/0.50       => ( A != B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_not_eq
% 0.24/0.50  thf(fact_1252_dual__order_Ostrict__implies__not__eq,axiom,
% 0.24/0.50      ! [B: num,A: num] :
% 0.24/0.50        ( ( ord_less_num @ B @ A )
% 0.24/0.50       => ( A != B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_not_eq
% 0.24/0.50  thf(fact_1253_dual__order_Ostrict__implies__not__eq,axiom,
% 0.24/0.50      ! [B: nat,A: nat] :
% 0.24/0.50        ( ( ord_less_nat @ B @ A )
% 0.24/0.50       => ( A != B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_not_eq
% 0.24/0.50  thf(fact_1254_dual__order_Ostrict__implies__not__eq,axiom,
% 0.24/0.50      ! [B: int,A: int] :
% 0.24/0.50        ( ( ord_less_int @ B @ A )
% 0.24/0.50       => ( A != B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_not_eq
% 0.24/0.50  thf(fact_1255_linorder__neqE,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50       => ( ~ ( ord_less_real @ X @ Y )
% 0.24/0.50         => ( ord_less_real @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neqE
% 0.24/0.50  thf(fact_1256_linorder__neqE,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50       => ( ~ ( ord_less_rat @ X @ Y )
% 0.24/0.50         => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neqE
% 0.24/0.50  thf(fact_1257_linorder__neqE,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50       => ( ~ ( ord_less_num @ X @ Y )
% 0.24/0.50         => ( ord_less_num @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neqE
% 0.24/0.50  thf(fact_1258_linorder__neqE,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50       => ( ~ ( ord_less_nat @ X @ Y )
% 0.24/0.50         => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neqE
% 0.24/0.50  thf(fact_1259_linorder__neqE,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50       => ( ~ ( ord_less_int @ X @ Y )
% 0.24/0.50         => ( ord_less_int @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neqE
% 0.24/0.50  thf(fact_1260_order__less__asym,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_real @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_asym
% 0.24/0.50  thf(fact_1261_order__less__asym,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_asym
% 0.24/0.50  thf(fact_1262_order__less__asym,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_num @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_asym
% 0.24/0.50  thf(fact_1263_order__less__asym,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_asym
% 0.24/0.50  thf(fact_1264_order__less__asym,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_asym
% 0.24/0.50  thf(fact_1265_linorder__neq__iff,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50        = ( ( ord_less_real @ X @ Y )
% 0.24/0.50          | ( ord_less_real @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neq_iff
% 0.24/0.50  thf(fact_1266_linorder__neq__iff,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50        = ( ( ord_less_rat @ X @ Y )
% 0.24/0.50          | ( ord_less_rat @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neq_iff
% 0.24/0.50  thf(fact_1267_linorder__neq__iff,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50        = ( ( ord_less_num @ X @ Y )
% 0.24/0.50          | ( ord_less_num @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neq_iff
% 0.24/0.50  thf(fact_1268_linorder__neq__iff,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50        = ( ( ord_less_nat @ X @ Y )
% 0.24/0.50          | ( ord_less_nat @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neq_iff
% 0.24/0.50  thf(fact_1269_linorder__neq__iff,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50        = ( ( ord_less_int @ X @ Y )
% 0.24/0.50          | ( ord_less_int @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neq_iff
% 0.24/0.50  thf(fact_1270_order__less__asym_H,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ~ ( ord_less_real @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_asym'
% 0.24/0.50  thf(fact_1271_order__less__asym_H,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_asym'
% 0.24/0.50  thf(fact_1272_order__less__asym_H,axiom,
% 0.24/0.50      ! [A: num,B: num] :
% 0.24/0.50        ( ( ord_less_num @ A @ B )
% 0.24/0.50       => ~ ( ord_less_num @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_asym'
% 0.24/0.50  thf(fact_1273_order__less__asym_H,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_asym'
% 0.24/0.50  thf(fact_1274_order__less__asym_H,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_int @ A @ B )
% 0.24/0.50       => ~ ( ord_less_int @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_asym'
% 0.24/0.50  thf(fact_1275_order__less__trans,axiom,
% 0.24/0.50      ! [X: real,Y: real,Z: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( ( ord_less_real @ Y @ Z )
% 0.24/0.50         => ( ord_less_real @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_trans
% 0.24/0.50  thf(fact_1276_order__less__trans,axiom,
% 0.24/0.50      ! [X: rat,Y: rat,Z: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_rat @ Y @ Z )
% 0.24/0.50         => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_trans
% 0.24/0.50  thf(fact_1277_order__less__trans,axiom,
% 0.24/0.50      ! [X: num,Y: num,Z: num] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50       => ( ( ord_less_num @ Y @ Z )
% 0.24/0.50         => ( ord_less_num @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_trans
% 0.24/0.50  thf(fact_1278_order__less__trans,axiom,
% 0.24/0.50      ! [X: nat,Y: nat,Z: nat] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_nat @ Y @ Z )
% 0.24/0.50         => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_trans
% 0.24/0.50  thf(fact_1279_order__less__trans,axiom,
% 0.24/0.50      ! [X: int,Y: int,Z: int] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50       => ( ( ord_less_int @ Y @ Z )
% 0.24/0.50         => ( ord_less_int @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_trans
% 0.24/0.50  thf(fact_1280_ord__eq__less__subst,axiom,
% 0.24/0.50      ! [A: real,F: real > real,B: real,C: real] :
% 0.24/0.50        ( ( A
% 0.24/0.50          = ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_subst
% 0.24/0.50  thf(fact_1281_ord__eq__less__subst,axiom,
% 0.24/0.50      ! [A: rat,F: real > rat,B: real,C: real] :
% 0.24/0.50        ( ( A
% 0.24/0.50          = ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_subst
% 0.24/0.50  thf(fact_1282_ord__eq__less__subst,axiom,
% 0.24/0.50      ! [A: num,F: real > num,B: real,C: real] :
% 0.24/0.50        ( ( A
% 0.24/0.50          = ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_subst
% 0.24/0.50  thf(fact_1283_ord__eq__less__subst,axiom,
% 0.24/0.50      ! [A: nat,F: real > nat,B: real,C: real] :
% 0.24/0.50        ( ( A
% 0.24/0.50          = ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_subst
% 0.24/0.50  thf(fact_1284_ord__eq__less__subst,axiom,
% 0.24/0.50      ! [A: int,F: real > int,B: real,C: real] :
% 0.24/0.50        ( ( A
% 0.24/0.50          = ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_subst
% 0.24/0.50  thf(fact_1285_ord__eq__less__subst,axiom,
% 0.24/0.50      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.24/0.50        ( ( A
% 0.24/0.50          = ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_subst
% 0.24/0.50  thf(fact_1286_ord__eq__less__subst,axiom,
% 0.24/0.50      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.24/0.50        ( ( A
% 0.24/0.50          = ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_subst
% 0.24/0.50  thf(fact_1287_ord__eq__less__subst,axiom,
% 0.24/0.50      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.24/0.50        ( ( A
% 0.24/0.50          = ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_subst
% 0.24/0.50  thf(fact_1288_ord__eq__less__subst,axiom,
% 0.24/0.50      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.24/0.50        ( ( A
% 0.24/0.50          = ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_subst
% 0.24/0.50  thf(fact_1289_ord__eq__less__subst,axiom,
% 0.24/0.50      ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.24/0.50        ( ( A
% 0.24/0.50          = ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_eq_less_subst
% 0.24/0.50  thf(fact_1290_ord__less__eq__subst,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > real,C: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ( F @ B )
% 0.24/0.50            = C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_subst
% 0.24/0.50  thf(fact_1291_ord__less__eq__subst,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ( F @ B )
% 0.24/0.50            = C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_subst
% 0.24/0.50  thf(fact_1292_ord__less__eq__subst,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > num,C: num] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ( F @ B )
% 0.24/0.50            = C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_subst
% 0.24/0.50  thf(fact_1293_ord__less__eq__subst,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > nat,C: nat] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ( F @ B )
% 0.24/0.50            = C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_subst
% 0.24/0.50  thf(fact_1294_ord__less__eq__subst,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > int,C: int] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ( F @ B )
% 0.24/0.50            = C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_subst
% 0.24/0.50  thf(fact_1295_ord__less__eq__subst,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ( F @ B )
% 0.24/0.50            = C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_subst
% 0.24/0.50  thf(fact_1296_ord__less__eq__subst,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ( F @ B )
% 0.24/0.50            = C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_subst
% 0.24/0.50  thf(fact_1297_ord__less__eq__subst,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ( F @ B )
% 0.24/0.50            = C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_subst
% 0.24/0.50  thf(fact_1298_ord__less__eq__subst,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ( F @ B )
% 0.24/0.50            = C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_subst
% 0.24/0.50  thf(fact_1299_ord__less__eq__subst,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ( F @ B )
% 0.24/0.50            = C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ord_less_eq_subst
% 0.24/0.50  thf(fact_1300_order__less__irrefl,axiom,
% 0.24/0.50      ! [X: real] :
% 0.24/0.50        ~ ( ord_less_real @ X @ X ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_irrefl
% 0.24/0.50  thf(fact_1301_order__less__irrefl,axiom,
% 0.24/0.50      ! [X: rat] :
% 0.24/0.50        ~ ( ord_less_rat @ X @ X ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_irrefl
% 0.24/0.50  thf(fact_1302_order__less__irrefl,axiom,
% 0.24/0.50      ! [X: num] :
% 0.24/0.50        ~ ( ord_less_num @ X @ X ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_irrefl
% 0.24/0.50  thf(fact_1303_order__less__irrefl,axiom,
% 0.24/0.50      ! [X: nat] :
% 0.24/0.50        ~ ( ord_less_nat @ X @ X ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_irrefl
% 0.24/0.50  thf(fact_1304_order__less__irrefl,axiom,
% 0.24/0.50      ! [X: int] :
% 0.24/0.50        ~ ( ord_less_int @ X @ X ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_irrefl
% 0.24/0.50  thf(fact_1305_order__less__subst1,axiom,
% 0.24/0.50      ! [A: real,F: real > real,B: real,C: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst1
% 0.24/0.50  thf(fact_1306_order__less__subst1,axiom,
% 0.24/0.50      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst1
% 0.24/0.50  thf(fact_1307_order__less__subst1,axiom,
% 0.24/0.50      ! [A: real,F: num > real,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_num @ B @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst1
% 0.24/0.50  thf(fact_1308_order__less__subst1,axiom,
% 0.24/0.50      ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_nat @ B @ C )
% 0.24/0.50         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.50                ( ( ord_less_nat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst1
% 0.24/0.50  thf(fact_1309_order__less__subst1,axiom,
% 0.24/0.50      ! [A: real,F: int > real,B: int,C: int] :
% 0.24/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_int @ B @ C )
% 0.24/0.50         => ( ! [X3: int,Y4: int] :
% 0.24/0.50                ( ( ord_less_int @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst1
% 0.24/0.50  thf(fact_1310_order__less__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: real > rat,B: real,C: real] :
% 0.24/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst1
% 0.24/0.50  thf(fact_1311_order__less__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst1
% 0.24/0.50  thf(fact_1312_order__less__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_num @ B @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst1
% 0.24/0.50  thf(fact_1313_order__less__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_nat @ B @ C )
% 0.24/0.50         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.50                ( ( ord_less_nat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst1
% 0.24/0.50  thf(fact_1314_order__less__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: int > rat,B: int,C: int] :
% 0.24/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_int @ B @ C )
% 0.24/0.50         => ( ! [X3: int,Y4: int] :
% 0.24/0.50                ( ( ord_less_int @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst1
% 0.24/0.50  thf(fact_1315_order__less__subst2,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > real,C: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst2
% 0.24/0.50  thf(fact_1316_order__less__subst2,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst2
% 0.24/0.50  thf(fact_1317_order__less__subst2,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > num,C: num] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst2
% 0.24/0.50  thf(fact_1318_order__less__subst2,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > nat,C: nat] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst2
% 0.24/0.50  thf(fact_1319_order__less__subst2,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > int,C: int] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst2
% 0.24/0.50  thf(fact_1320_order__less__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst2
% 0.24/0.50  thf(fact_1321_order__less__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst2
% 0.24/0.50  thf(fact_1322_order__less__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst2
% 0.24/0.50  thf(fact_1323_order__less__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst2
% 0.24/0.50  thf(fact_1324_order__less__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_subst2
% 0.24/0.50  thf(fact_1325_order__less__not__sym,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_real @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_not_sym
% 0.24/0.50  thf(fact_1326_order__less__not__sym,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_not_sym
% 0.24/0.50  thf(fact_1327_order__less__not__sym,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_num @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_not_sym
% 0.24/0.50  thf(fact_1328_order__less__not__sym,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_not_sym
% 0.24/0.50  thf(fact_1329_order__less__not__sym,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_not_sym
% 0.24/0.50  thf(fact_1330_order__less__imp__triv,axiom,
% 0.24/0.50      ! [X: real,Y: real,P: $o] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( ( ord_less_real @ Y @ X )
% 0.24/0.50         => P ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_triv
% 0.24/0.50  thf(fact_1331_order__less__imp__triv,axiom,
% 0.24/0.50      ! [X: rat,Y: rat,P: $o] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_rat @ Y @ X )
% 0.24/0.50         => P ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_triv
% 0.24/0.50  thf(fact_1332_order__less__imp__triv,axiom,
% 0.24/0.50      ! [X: num,Y: num,P: $o] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50       => ( ( ord_less_num @ Y @ X )
% 0.24/0.50         => P ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_triv
% 0.24/0.50  thf(fact_1333_order__less__imp__triv,axiom,
% 0.24/0.50      ! [X: nat,Y: nat,P: $o] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_nat @ Y @ X )
% 0.24/0.50         => P ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_triv
% 0.24/0.50  thf(fact_1334_order__less__imp__triv,axiom,
% 0.24/0.50      ! [X: int,Y: int,P: $o] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50       => ( ( ord_less_int @ Y @ X )
% 0.24/0.50         => P ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_triv
% 0.24/0.50  thf(fact_1335_linorder__less__linear,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50        | ( X = Y )
% 0.24/0.50        | ( ord_less_real @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_less_linear
% 0.24/0.50  thf(fact_1336_linorder__less__linear,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50        | ( X = Y )
% 0.24/0.50        | ( ord_less_rat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_less_linear
% 0.24/0.50  thf(fact_1337_linorder__less__linear,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50        | ( X = Y )
% 0.24/0.50        | ( ord_less_num @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_less_linear
% 0.24/0.50  thf(fact_1338_linorder__less__linear,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50        | ( X = Y )
% 0.24/0.50        | ( ord_less_nat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_less_linear
% 0.24/0.50  thf(fact_1339_linorder__less__linear,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50        | ( X = Y )
% 0.24/0.50        | ( ord_less_int @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_less_linear
% 0.24/0.50  thf(fact_1340_order__less__imp__not__eq,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( X != Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_eq
% 0.24/0.50  thf(fact_1341_order__less__imp__not__eq,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( X != Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_eq
% 0.24/0.50  thf(fact_1342_order__less__imp__not__eq,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50       => ( X != Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_eq
% 0.24/0.50  thf(fact_1343_order__less__imp__not__eq,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ( X != Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_eq
% 0.24/0.50  thf(fact_1344_order__less__imp__not__eq,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50       => ( X != Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_eq
% 0.24/0.50  thf(fact_1345_order__less__imp__not__eq2,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( Y != X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_eq2
% 0.24/0.50  thf(fact_1346_order__less__imp__not__eq2,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( Y != X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_eq2
% 0.24/0.50  thf(fact_1347_order__less__imp__not__eq2,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50       => ( Y != X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_eq2
% 0.24/0.50  thf(fact_1348_order__less__imp__not__eq2,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ( Y != X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_eq2
% 0.24/0.50  thf(fact_1349_order__less__imp__not__eq2,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50       => ( Y != X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_eq2
% 0.24/0.50  thf(fact_1350_order__less__imp__not__less,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_real @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_less
% 0.24/0.50  thf(fact_1351_order__less__imp__not__less,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_less
% 0.24/0.50  thf(fact_1352_order__less__imp__not__less,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_num @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_less
% 0.24/0.50  thf(fact_1353_order__less__imp__not__less,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_less
% 0.24/0.50  thf(fact_1354_order__less__imp__not__less,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50       => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_not_less
% 0.24/0.50  thf(fact_1355_less__ceiling__iff,axiom,
% 0.24/0.50      ! [Z: int,X: real] :
% 0.24/0.50        ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
% 0.24/0.50        = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_ceiling_iff
% 0.24/0.50  thf(fact_1356_less__ceiling__iff,axiom,
% 0.24/0.50      ! [Z: int,X: rat] :
% 0.24/0.50        ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
% 0.24/0.50        = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_ceiling_iff
% 0.24/0.50  thf(fact_1357_ex__of__int__less,axiom,
% 0.24/0.50      ! [X: real] :
% 0.24/0.50      ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).
% 0.24/0.50  
% 0.24/0.50  % ex_of_int_less
% 0.24/0.50  thf(fact_1358_ex__of__int__less,axiom,
% 0.24/0.50      ! [X: rat] :
% 0.24/0.50      ? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X ) ).
% 0.24/0.50  
% 0.24/0.50  % ex_of_int_less
% 0.24/0.50  thf(fact_1359_ex__less__of__int,axiom,
% 0.24/0.50      ! [X: real] :
% 0.24/0.50      ? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ex_less_of_int
% 0.24/0.50  thf(fact_1360_ex__less__of__int,axiom,
% 0.24/0.50      ! [X: rat] :
% 0.24/0.50      ? [Z3: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ex_less_of_int
% 0.24/0.50  thf(fact_1361_linorder__neqE__nat,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( X != Y )
% 0.24/0.50       => ( ~ ( ord_less_nat @ X @ Y )
% 0.24/0.50         => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_neqE_nat
% 0.24/0.50  thf(fact_1362_infinite__descent0,axiom,
% 0.24/0.50      ! [P: nat > $o,N: nat] :
% 0.24/0.50        ( ( P @ zero_zero_nat )
% 0.24/0.50       => ( ! [N2: nat] :
% 0.24/0.50              ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.24/0.50             => ( ~ ( P @ N2 )
% 0.24/0.50               => ? [M3: nat] :
% 0.24/0.50                    ( ( ord_less_nat @ M3 @ N2 )
% 0.24/0.50                    & ~ ( P @ M3 ) ) ) )
% 0.24/0.50         => ( P @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % infinite_descent0
% 0.24/0.50  thf(fact_1363_infinite__descent,axiom,
% 0.24/0.50      ! [P: nat > $o,N: nat] :
% 0.24/0.50        ( ! [N2: nat] :
% 0.24/0.50            ( ~ ( P @ N2 )
% 0.24/0.50           => ? [M3: nat] :
% 0.24/0.50                ( ( ord_less_nat @ M3 @ N2 )
% 0.24/0.50                & ~ ( P @ M3 ) ) )
% 0.24/0.50       => ( P @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % infinite_descent
% 0.24/0.50  thf(fact_1364_nat__less__induct,axiom,
% 0.24/0.50      ! [P: nat > $o,N: nat] :
% 0.24/0.50        ( ! [N2: nat] :
% 0.24/0.50            ( ! [M3: nat] :
% 0.24/0.50                ( ( ord_less_nat @ M3 @ N2 )
% 0.24/0.50               => ( P @ M3 ) )
% 0.24/0.50           => ( P @ N2 ) )
% 0.24/0.50       => ( P @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nat_less_induct
% 0.24/0.50  thf(fact_1365_less__irrefl__nat,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ~ ( ord_less_nat @ N @ N ) ).
% 0.24/0.50  
% 0.24/0.50  % less_irrefl_nat
% 0.24/0.50  thf(fact_1366_gr__implies__not0,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ M @ N )
% 0.24/0.50       => ( N != zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % gr_implies_not0
% 0.24/0.50  thf(fact_1367_less__not__refl3,axiom,
% 0.24/0.50      ! [S: nat,T: nat] :
% 0.24/0.50        ( ( ord_less_nat @ S @ T )
% 0.24/0.50       => ( S != T ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_not_refl3
% 0.24/0.50  thf(fact_1368_less__not__refl2,axiom,
% 0.24/0.50      ! [N: nat,M: nat] :
% 0.24/0.50        ( ( ord_less_nat @ N @ M )
% 0.24/0.50       => ( M != N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_not_refl2
% 0.24/0.50  thf(fact_1369_less__not__refl,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ~ ( ord_less_nat @ N @ N ) ).
% 0.24/0.50  
% 0.24/0.50  % less_not_refl
% 0.24/0.50  thf(fact_1370_nat__neq__iff,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( M != N )
% 0.24/0.50        = ( ( ord_less_nat @ M @ N )
% 0.24/0.50          | ( ord_less_nat @ N @ M ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nat_neq_iff
% 0.24/0.50  thf(fact_1371_less__zeroE,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.24/0.50  
% 0.24/0.50  % less_zeroE
% 0.24/0.50  thf(fact_1372_not__less0,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.24/0.50  
% 0.24/0.50  % not_less0
% 0.24/0.50  thf(fact_1373_not__gr0,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.24/0.50        = ( N = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_gr0
% 0.24/0.50  thf(fact_1374_gr0I,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( N != zero_zero_nat )
% 0.24/0.50       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % gr0I
% 0.24/0.50  thf(fact_1375_bot__nat__0_Oextremum__strict,axiom,
% 0.24/0.50      ! [A: nat] :
% 0.24/0.50        ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.24/0.50  
% 0.24/0.50  % bot_nat_0.extremum_strict
% 0.24/0.50  thf(fact_1376_less__numeral__extra_I3_J,axiom,
% 0.24/0.50      ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.24/0.50  
% 0.24/0.50  % less_numeral_extra(3)
% 0.24/0.50  thf(fact_1377_less__numeral__extra_I3_J,axiom,
% 0.24/0.50      ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.24/0.50  
% 0.24/0.50  % less_numeral_extra(3)
% 0.24/0.50  thf(fact_1378_less__numeral__extra_I3_J,axiom,
% 0.24/0.50      ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.24/0.50  
% 0.24/0.50  % less_numeral_extra(3)
% 0.24/0.50  thf(fact_1379_less__numeral__extra_I3_J,axiom,
% 0.24/0.50      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.24/0.50  
% 0.24/0.50  % less_numeral_extra(3)
% 0.24/0.50  thf(fact_1380_zero__reorient,axiom,
% 0.24/0.50      ! [X: complex] :
% 0.24/0.50        ( ( zero_zero_complex = X )
% 0.24/0.50        = ( X = zero_zero_complex ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_reorient
% 0.24/0.50  thf(fact_1381_zero__reorient,axiom,
% 0.24/0.50      ! [X: real] :
% 0.24/0.50        ( ( zero_zero_real = X )
% 0.24/0.50        = ( X = zero_zero_real ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_reorient
% 0.24/0.50  thf(fact_1382_zero__reorient,axiom,
% 0.24/0.50      ! [X: rat] :
% 0.24/0.50        ( ( zero_zero_rat = X )
% 0.24/0.50        = ( X = zero_zero_rat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_reorient
% 0.24/0.50  thf(fact_1383_zero__reorient,axiom,
% 0.24/0.50      ! [X: nat] :
% 0.24/0.50        ( ( zero_zero_nat = X )
% 0.24/0.50        = ( X = zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_reorient
% 0.24/0.50  thf(fact_1384_zero__reorient,axiom,
% 0.24/0.50      ! [X: int] :
% 0.24/0.50        ( ( zero_zero_int = X )
% 0.24/0.50        = ( X = zero_zero_int ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_reorient
% 0.24/0.50  thf(fact_1385_zero__less__iff__neq__zero,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50        = ( N != zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_iff_neq_zero
% 0.24/0.50  thf(fact_1386_gr__implies__not__zero,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ M @ N )
% 0.24/0.50       => ( N != zero_zero_nat ) ) ).
% 0.24/0.50  
% 0.24/0.50  % gr_implies_not_zero
% 0.24/0.50  thf(fact_1387_not__less__zero,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.24/0.50  
% 0.24/0.50  % not_less_zero
% 0.24/0.50  thf(fact_1388_gr__zeroI,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( N != zero_zero_nat )
% 0.24/0.50       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % gr_zeroI
% 0.24/0.50  thf(fact_1389_power__strict__mono,axiom,
% 0.24/0.50      ! [A: real,B: real,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.50         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50           => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_mono
% 0.24/0.50  thf(fact_1390_power__strict__mono,axiom,
% 0.24/0.50      ! [A: rat,B: rat,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.50         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50           => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_mono
% 0.24/0.50  thf(fact_1391_power__strict__mono,axiom,
% 0.24/0.50      ! [A: nat,B: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.50         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50           => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_mono
% 0.24/0.50  thf(fact_1392_power__strict__mono,axiom,
% 0.24/0.50      ! [A: int,B: int,N: nat] :
% 0.24/0.50        ( ( ord_less_int @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.50         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50           => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_mono
% 0.24/0.50  thf(fact_1393_lift__Suc__mono__less__iff,axiom,
% 0.24/0.50      ! [F: nat > real,N: nat,M: nat] :
% 0.24/0.50        ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.50       => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.24/0.50          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % lift_Suc_mono_less_iff
% 0.24/0.50  thf(fact_1394_lift__Suc__mono__less__iff,axiom,
% 0.24/0.50      ! [F: nat > rat,N: nat,M: nat] :
% 0.24/0.50        ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.50       => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 0.24/0.50          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % lift_Suc_mono_less_iff
% 0.24/0.50  thf(fact_1395_lift__Suc__mono__less__iff,axiom,
% 0.24/0.50      ! [F: nat > num,N: nat,M: nat] :
% 0.24/0.50        ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.50       => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 0.24/0.50          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % lift_Suc_mono_less_iff
% 0.24/0.50  thf(fact_1396_lift__Suc__mono__less__iff,axiom,
% 0.24/0.50      ! [F: nat > nat,N: nat,M: nat] :
% 0.24/0.50        ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.50       => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.24/0.50          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % lift_Suc_mono_less_iff
% 0.24/0.50  thf(fact_1397_lift__Suc__mono__less__iff,axiom,
% 0.24/0.50      ! [F: nat > int,N: nat,M: nat] :
% 0.24/0.50        ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.50       => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.24/0.50          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % lift_Suc_mono_less_iff
% 0.24/0.50  thf(fact_1398_lift__Suc__mono__less,axiom,
% 0.24/0.50      ! [F: nat > real,N: nat,N5: nat] :
% 0.24/0.50        ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.50       => ( ( ord_less_nat @ N @ N5 )
% 0.24/0.50         => ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % lift_Suc_mono_less
% 0.24/0.50  thf(fact_1399_lift__Suc__mono__less,axiom,
% 0.24/0.50      ! [F: nat > rat,N: nat,N5: nat] :
% 0.24/0.50        ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.50       => ( ( ord_less_nat @ N @ N5 )
% 0.24/0.50         => ( ord_less_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % lift_Suc_mono_less
% 0.24/0.50  thf(fact_1400_lift__Suc__mono__less,axiom,
% 0.24/0.50      ! [F: nat > num,N: nat,N5: nat] :
% 0.24/0.50        ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.50       => ( ( ord_less_nat @ N @ N5 )
% 0.24/0.50         => ( ord_less_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % lift_Suc_mono_less
% 0.24/0.50  thf(fact_1401_lift__Suc__mono__less,axiom,
% 0.24/0.50      ! [F: nat > nat,N: nat,N5: nat] :
% 0.24/0.50        ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.50       => ( ( ord_less_nat @ N @ N5 )
% 0.24/0.50         => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % lift_Suc_mono_less
% 0.24/0.50  thf(fact_1402_lift__Suc__mono__less,axiom,
% 0.24/0.50      ! [F: nat > int,N: nat,N5: nat] :
% 0.24/0.50        ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.24/0.50       => ( ( ord_less_nat @ N @ N5 )
% 0.24/0.50         => ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % lift_Suc_mono_less
% 0.24/0.50  thf(fact_1403_zero__less__numeral,axiom,
% 0.24/0.50      ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_numeral
% 0.24/0.50  thf(fact_1404_zero__less__numeral,axiom,
% 0.24/0.50      ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_numeral
% 0.24/0.50  thf(fact_1405_zero__less__numeral,axiom,
% 0.24/0.50      ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_numeral
% 0.24/0.50  thf(fact_1406_zero__less__numeral,axiom,
% 0.24/0.50      ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_numeral
% 0.24/0.50  thf(fact_1407_not__numeral__less__zero,axiom,
% 0.24/0.50      ! [N: num] :
% 0.24/0.50        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.24/0.50  
% 0.24/0.50  % not_numeral_less_zero
% 0.24/0.50  thf(fact_1408_not__numeral__less__zero,axiom,
% 0.24/0.50      ! [N: num] :
% 0.24/0.50        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.24/0.50  
% 0.24/0.50  % not_numeral_less_zero
% 0.24/0.50  thf(fact_1409_not__numeral__less__zero,axiom,
% 0.24/0.50      ! [N: num] :
% 0.24/0.50        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.24/0.50  
% 0.24/0.50  % not_numeral_less_zero
% 0.24/0.50  thf(fact_1410_not__numeral__less__zero,axiom,
% 0.24/0.50      ! [N: num] :
% 0.24/0.50        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.24/0.50  
% 0.24/0.50  % not_numeral_less_zero
% 0.24/0.50  thf(fact_1411_less__numeral__extra_I1_J,axiom,
% 0.24/0.50      ord_less_real @ zero_zero_real @ one_one_real ).
% 0.24/0.50  
% 0.24/0.50  % less_numeral_extra(1)
% 0.24/0.50  thf(fact_1412_less__numeral__extra_I1_J,axiom,
% 0.24/0.50      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.24/0.50  
% 0.24/0.50  % less_numeral_extra(1)
% 0.24/0.50  thf(fact_1413_less__numeral__extra_I1_J,axiom,
% 0.24/0.50      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.24/0.50  
% 0.24/0.50  % less_numeral_extra(1)
% 0.24/0.50  thf(fact_1414_less__numeral__extra_I1_J,axiom,
% 0.24/0.50      ord_less_int @ zero_zero_int @ one_one_int ).
% 0.24/0.50  
% 0.24/0.50  % less_numeral_extra(1)
% 0.24/0.50  thf(fact_1415_pos__add__strict,axiom,
% 0.24/0.50      ! [A: real,B: real,C: real] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % pos_add_strict
% 0.24/0.50  thf(fact_1416_pos__add__strict,axiom,
% 0.24/0.50      ! [A: rat,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % pos_add_strict
% 0.24/0.50  thf(fact_1417_pos__add__strict,axiom,
% 0.24/0.50      ! [A: nat,B: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.24/0.50       => ( ( ord_less_nat @ B @ C )
% 0.24/0.50         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % pos_add_strict
% 0.24/0.50  thf(fact_1418_pos__add__strict,axiom,
% 0.24/0.50      ! [A: int,B: int,C: int] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ A )
% 0.24/0.50       => ( ( ord_less_int @ B @ C )
% 0.24/0.50         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % pos_add_strict
% 0.24/0.50  thf(fact_1419_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ~ ! [C4: nat] :
% 0.24/0.50              ( ( B
% 0.24/0.50                = ( plus_plus_nat @ A @ C4 ) )
% 0.24/0.50             => ( C4 = zero_zero_nat ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % canonically_ordered_monoid_add_class.lessE
% 0.24/0.50  thf(fact_1420_add__pos__pos,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.50       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.24/0.50         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_pos_pos
% 0.24/0.50  thf(fact_1421_add__pos__pos,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.50       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.24/0.50         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_pos_pos
% 0.24/0.50  thf(fact_1422_add__pos__pos,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.24/0.50       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.24/0.50         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_pos_pos
% 0.24/0.50  thf(fact_1423_add__pos__pos,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ A )
% 0.24/0.50       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.24/0.50         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_pos_pos
% 0.24/0.50  thf(fact_1424_add__neg__neg,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.50       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.24/0.50         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_neg_neg
% 0.24/0.50  thf(fact_1425_add__neg__neg,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.50       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.24/0.50         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_neg_neg
% 0.24/0.50  thf(fact_1426_add__neg__neg,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.24/0.50       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.24/0.50         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_neg_neg
% 0.24/0.50  thf(fact_1427_add__neg__neg,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_int @ A @ zero_zero_int )
% 0.24/0.50       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.24/0.50         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % add_neg_neg
% 0.24/0.50  thf(fact_1428_less__imp__of__nat__less,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ M @ N )
% 0.24/0.50       => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_imp_of_nat_less
% 0.24/0.50  thf(fact_1429_less__imp__of__nat__less,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ M @ N )
% 0.24/0.50       => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_imp_of_nat_less
% 0.24/0.50  thf(fact_1430_less__imp__of__nat__less,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ M @ N )
% 0.24/0.50       => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_imp_of_nat_less
% 0.24/0.50  thf(fact_1431_less__imp__of__nat__less,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ M @ N )
% 0.24/0.50       => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_imp_of_nat_less
% 0.24/0.50  thf(fact_1432_of__nat__less__imp__less,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.50       => ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_imp_less
% 0.24/0.50  thf(fact_1433_of__nat__less__imp__less,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.24/0.50       => ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_imp_less
% 0.24/0.50  thf(fact_1434_of__nat__less__imp__less,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.24/0.50       => ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_imp_less
% 0.24/0.50  thf(fact_1435_of__nat__less__imp__less,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.50       => ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_imp_less
% 0.24/0.50  thf(fact_1436_zero__less__power,axiom,
% 0.24/0.50      ! [A: real,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.50       => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_power
% 0.24/0.50  thf(fact_1437_zero__less__power,axiom,
% 0.24/0.50      ! [A: rat,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.50       => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_power
% 0.24/0.50  thf(fact_1438_zero__less__power,axiom,
% 0.24/0.50      ! [A: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.24/0.50       => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_power
% 0.24/0.50  thf(fact_1439_zero__less__power,axiom,
% 0.24/0.50      ! [A: int,N: nat] :
% 0.24/0.50        ( ( ord_less_int @ zero_zero_int @ A )
% 0.24/0.50       => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % zero_less_power
% 0.24/0.50  thf(fact_1440_of__nat__less__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_0_iff
% 0.24/0.50  thf(fact_1441_of__nat__less__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_0_iff
% 0.24/0.50  thf(fact_1442_of__nat__less__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_0_iff
% 0.24/0.50  thf(fact_1443_of__nat__less__0__iff,axiom,
% 0.24/0.50      ! [M: nat] :
% 0.24/0.50        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_0_iff
% 0.24/0.50  thf(fact_1444_less__Suc__eq__0__disj,axiom,
% 0.24/0.50      ! [M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.24/0.50        = ( ( M = zero_zero_nat )
% 0.24/0.50          | ? [J2: nat] :
% 0.24/0.50              ( ( M
% 0.24/0.50                = ( suc @ J2 ) )
% 0.24/0.50              & ( ord_less_nat @ J2 @ N ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_Suc_eq_0_disj
% 0.24/0.50  thf(fact_1445_gr0__implies__Suc,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50       => ? [M5: nat] :
% 0.24/0.50            ( N
% 0.24/0.50            = ( suc @ M5 ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % gr0_implies_Suc
% 0.24/0.50  thf(fact_1446_All__less__Suc2,axiom,
% 0.24/0.50      ! [N: nat,P: nat > $o] :
% 0.24/0.50        ( ( ! [I3: nat] :
% 0.24/0.50              ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.24/0.50             => ( P @ I3 ) ) )
% 0.24/0.50        = ( ( P @ zero_zero_nat )
% 0.24/0.50          & ! [I3: nat] :
% 0.24/0.50              ( ( ord_less_nat @ I3 @ N )
% 0.24/0.50             => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % All_less_Suc2
% 0.24/0.50  thf(fact_1447_gr0__conv__Suc,axiom,
% 0.24/0.50      ! [N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50        = ( ? [M6: nat] :
% 0.24/0.50              ( N
% 0.24/0.50              = ( suc @ M6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % gr0_conv_Suc
% 0.24/0.50  thf(fact_1448_Ex__less__Suc2,axiom,
% 0.24/0.50      ! [N: nat,P: nat > $o] :
% 0.24/0.50        ( ( ? [I3: nat] :
% 0.24/0.50              ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.24/0.50              & ( P @ I3 ) ) )
% 0.24/0.50        = ( ( P @ zero_zero_nat )
% 0.24/0.50          | ? [I3: nat] :
% 0.24/0.50              ( ( ord_less_nat @ I3 @ N )
% 0.24/0.50              & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % Ex_less_Suc2
% 0.24/0.50  thf(fact_1449_ex__least__nat__le,axiom,
% 0.24/0.50      ! [P: nat > $o,N: nat] :
% 0.24/0.50        ( ( P @ N )
% 0.24/0.50       => ( ~ ( P @ zero_zero_nat )
% 0.24/0.50         => ? [K3: nat] :
% 0.24/0.50              ( ( ord_less_eq_nat @ K3 @ N )
% 0.24/0.50              & ! [I4: nat] :
% 0.24/0.50                  ( ( ord_less_nat @ I4 @ K3 )
% 0.24/0.50                 => ~ ( P @ I4 ) )
% 0.24/0.50              & ( P @ K3 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ex_least_nat_le
% 0.24/0.50  thf(fact_1450_less__imp__add__positive,axiom,
% 0.24/0.50      ! [I: nat,J: nat] :
% 0.24/0.50        ( ( ord_less_nat @ I @ J )
% 0.24/0.50       => ? [K3: nat] :
% 0.24/0.50            ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.24/0.50            & ( ( plus_plus_nat @ I @ K3 )
% 0.24/0.50              = J ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_imp_add_positive
% 0.24/0.50  thf(fact_1451_leD,axiom,
% 0.24/0.50      ! [Y: real,X: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ Y @ X )
% 0.24/0.50       => ~ ( ord_less_real @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leD
% 0.24/0.50  thf(fact_1452_leD,axiom,
% 0.24/0.50      ! [Y: set_int,X: set_int] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ Y @ X )
% 0.24/0.50       => ~ ( ord_less_set_int @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leD
% 0.24/0.50  thf(fact_1453_leD,axiom,
% 0.24/0.50      ! [Y: rat,X: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ Y @ X )
% 0.24/0.50       => ~ ( ord_less_rat @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leD
% 0.24/0.50  thf(fact_1454_leD,axiom,
% 0.24/0.50      ! [Y: num,X: num] :
% 0.24/0.50        ( ( ord_less_eq_num @ Y @ X )
% 0.24/0.50       => ~ ( ord_less_num @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leD
% 0.24/0.50  thf(fact_1455_leD,axiom,
% 0.24/0.50      ! [Y: nat,X: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ Y @ X )
% 0.24/0.50       => ~ ( ord_less_nat @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leD
% 0.24/0.50  thf(fact_1456_leD,axiom,
% 0.24/0.50      ! [Y: int,X: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ Y @ X )
% 0.24/0.50       => ~ ( ord_less_int @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leD
% 0.24/0.50  thf(fact_1457_leI,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ~ ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_real @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leI
% 0.24/0.50  thf(fact_1458_leI,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ~ ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leI
% 0.24/0.50  thf(fact_1459_leI,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ~ ( ord_less_num @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_num @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leI
% 0.24/0.50  thf(fact_1460_leI,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ~ ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leI
% 0.24/0.50  thf(fact_1461_leI,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ~ ( ord_less_int @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_int @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % leI
% 0.24/0.50  thf(fact_1462_nless__le,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ~ ( ord_less_real @ A @ B ) )
% 0.24/0.50        = ( ~ ( ord_less_eq_real @ A @ B )
% 0.24/0.50          | ( A = B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nless_le
% 0.24/0.50  thf(fact_1463_nless__le,axiom,
% 0.24/0.50      ! [A: set_int,B: set_int] :
% 0.24/0.50        ( ( ~ ( ord_less_set_int @ A @ B ) )
% 0.24/0.50        = ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.24/0.50          | ( A = B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nless_le
% 0.24/0.50  thf(fact_1464_nless__le,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ~ ( ord_less_rat @ A @ B ) )
% 0.24/0.50        = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.24/0.50          | ( A = B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nless_le
% 0.24/0.50  thf(fact_1465_nless__le,axiom,
% 0.24/0.50      ! [A: num,B: num] :
% 0.24/0.50        ( ( ~ ( ord_less_num @ A @ B ) )
% 0.24/0.50        = ( ~ ( ord_less_eq_num @ A @ B )
% 0.24/0.50          | ( A = B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nless_le
% 0.24/0.50  thf(fact_1466_nless__le,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ~ ( ord_less_nat @ A @ B ) )
% 0.24/0.50        = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.24/0.50          | ( A = B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nless_le
% 0.24/0.50  thf(fact_1467_nless__le,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ~ ( ord_less_int @ A @ B ) )
% 0.24/0.50        = ( ~ ( ord_less_eq_int @ A @ B )
% 0.24/0.50          | ( A = B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nless_le
% 0.24/0.50  thf(fact_1468_antisym__conv1,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ~ ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_real @ X @ Y )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv1
% 0.24/0.50  thf(fact_1469_antisym__conv1,axiom,
% 0.24/0.50      ! [X: set_int,Y: set_int] :
% 0.24/0.50        ( ~ ( ord_less_set_int @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_set_int @ X @ Y )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv1
% 0.24/0.50  thf(fact_1470_antisym__conv1,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ~ ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv1
% 0.24/0.50  thf(fact_1471_antisym__conv1,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ~ ( ord_less_num @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_num @ X @ Y )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv1
% 0.24/0.50  thf(fact_1472_antisym__conv1,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ~ ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_nat @ X @ Y )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv1
% 0.24/0.50  thf(fact_1473_antisym__conv1,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ~ ( ord_less_int @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_int @ X @ Y )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv1
% 0.24/0.50  thf(fact_1474_antisym__conv2,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ X @ Y )
% 0.24/0.50       => ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv2
% 0.24/0.50  thf(fact_1475_antisym__conv2,axiom,
% 0.24/0.50      ! [X: set_int,Y: set_int] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ X @ Y )
% 0.24/0.50       => ( ( ~ ( ord_less_set_int @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv2
% 0.24/0.50  thf(fact_1476_antisym__conv2,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.50       => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv2
% 0.24/0.50  thf(fact_1477_antisym__conv2,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_eq_num @ X @ Y )
% 0.24/0.50       => ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv2
% 0.24/0.50  thf(fact_1478_antisym__conv2,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ X @ Y )
% 0.24/0.50       => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv2
% 0.24/0.50  thf(fact_1479_antisym__conv2,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ X @ Y )
% 0.24/0.50       => ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.24/0.50          = ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % antisym_conv2
% 0.24/0.50  thf(fact_1480_dense__ge,axiom,
% 0.24/0.50      ! [Z: real,Y: real] :
% 0.24/0.50        ( ! [X3: real] :
% 0.24/0.50            ( ( ord_less_real @ Z @ X3 )
% 0.24/0.50           => ( ord_less_eq_real @ Y @ X3 ) )
% 0.24/0.50       => ( ord_less_eq_real @ Y @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dense_ge
% 0.24/0.50  thf(fact_1481_dense__ge,axiom,
% 0.24/0.50      ! [Z: rat,Y: rat] :
% 0.24/0.50        ( ! [X3: rat] :
% 0.24/0.50            ( ( ord_less_rat @ Z @ X3 )
% 0.24/0.50           => ( ord_less_eq_rat @ Y @ X3 ) )
% 0.24/0.50       => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dense_ge
% 0.24/0.50  thf(fact_1482_dense__le,axiom,
% 0.24/0.50      ! [Y: real,Z: real] :
% 0.24/0.50        ( ! [X3: real] :
% 0.24/0.50            ( ( ord_less_real @ X3 @ Y )
% 0.24/0.50           => ( ord_less_eq_real @ X3 @ Z ) )
% 0.24/0.50       => ( ord_less_eq_real @ Y @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dense_le
% 0.24/0.50  thf(fact_1483_dense__le,axiom,
% 0.24/0.50      ! [Y: rat,Z: rat] :
% 0.24/0.50        ( ! [X3: rat] :
% 0.24/0.50            ( ( ord_less_rat @ X3 @ Y )
% 0.24/0.50           => ( ord_less_eq_rat @ X3 @ Z ) )
% 0.24/0.50       => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dense_le
% 0.24/0.50  thf(fact_1484_less__le__not__le,axiom,
% 0.24/0.50      ( ord_less_real
% 0.24/0.50      = ( ^ [X4: real,Y6: real] :
% 0.24/0.50            ( ( ord_less_eq_real @ X4 @ Y6 )
% 0.24/0.50            & ~ ( ord_less_eq_real @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_le_not_le
% 0.24/0.50  thf(fact_1485_less__le__not__le,axiom,
% 0.24/0.50      ( ord_less_set_int
% 0.24/0.50      = ( ^ [X4: set_int,Y6: set_int] :
% 0.24/0.50            ( ( ord_less_eq_set_int @ X4 @ Y6 )
% 0.24/0.50            & ~ ( ord_less_eq_set_int @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_le_not_le
% 0.24/0.50  thf(fact_1486_less__le__not__le,axiom,
% 0.24/0.50      ( ord_less_rat
% 0.24/0.50      = ( ^ [X4: rat,Y6: rat] :
% 0.24/0.50            ( ( ord_less_eq_rat @ X4 @ Y6 )
% 0.24/0.50            & ~ ( ord_less_eq_rat @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_le_not_le
% 0.24/0.50  thf(fact_1487_less__le__not__le,axiom,
% 0.24/0.50      ( ord_less_num
% 0.24/0.50      = ( ^ [X4: num,Y6: num] :
% 0.24/0.50            ( ( ord_less_eq_num @ X4 @ Y6 )
% 0.24/0.50            & ~ ( ord_less_eq_num @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_le_not_le
% 0.24/0.50  thf(fact_1488_less__le__not__le,axiom,
% 0.24/0.50      ( ord_less_nat
% 0.24/0.50      = ( ^ [X4: nat,Y6: nat] :
% 0.24/0.50            ( ( ord_less_eq_nat @ X4 @ Y6 )
% 0.24/0.50            & ~ ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_le_not_le
% 0.24/0.50  thf(fact_1489_less__le__not__le,axiom,
% 0.24/0.50      ( ord_less_int
% 0.24/0.50      = ( ^ [X4: int,Y6: int] :
% 0.24/0.50            ( ( ord_less_eq_int @ X4 @ Y6 )
% 0.24/0.50            & ~ ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % less_le_not_le
% 0.24/0.50  thf(fact_1490_not__le__imp__less,axiom,
% 0.24/0.50      ! [Y: real,X: real] :
% 0.24/0.50        ( ~ ( ord_less_eq_real @ Y @ X )
% 0.24/0.50       => ( ord_less_real @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_le_imp_less
% 0.24/0.50  thf(fact_1491_not__le__imp__less,axiom,
% 0.24/0.50      ! [Y: rat,X: rat] :
% 0.24/0.50        ( ~ ( ord_less_eq_rat @ Y @ X )
% 0.24/0.50       => ( ord_less_rat @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_le_imp_less
% 0.24/0.50  thf(fact_1492_not__le__imp__less,axiom,
% 0.24/0.50      ! [Y: num,X: num] :
% 0.24/0.50        ( ~ ( ord_less_eq_num @ Y @ X )
% 0.24/0.50       => ( ord_less_num @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_le_imp_less
% 0.24/0.50  thf(fact_1493_not__le__imp__less,axiom,
% 0.24/0.50      ! [Y: nat,X: nat] :
% 0.24/0.50        ( ~ ( ord_less_eq_nat @ Y @ X )
% 0.24/0.50       => ( ord_less_nat @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_le_imp_less
% 0.24/0.50  thf(fact_1494_not__le__imp__less,axiom,
% 0.24/0.50      ! [Y: int,X: int] :
% 0.24/0.50        ( ~ ( ord_less_eq_int @ Y @ X )
% 0.24/0.50       => ( ord_less_int @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % not_le_imp_less
% 0.24/0.50  thf(fact_1495_order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_real
% 0.24/0.50      = ( ^ [A3: real,B4: real] :
% 0.24/0.50            ( ( ord_less_real @ A3 @ B4 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.order_iff_strict
% 0.24/0.50  thf(fact_1496_order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_set_int
% 0.24/0.50      = ( ^ [A3: set_int,B4: set_int] :
% 0.24/0.50            ( ( ord_less_set_int @ A3 @ B4 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.order_iff_strict
% 0.24/0.50  thf(fact_1497_order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_rat
% 0.24/0.50      = ( ^ [A3: rat,B4: rat] :
% 0.24/0.50            ( ( ord_less_rat @ A3 @ B4 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.order_iff_strict
% 0.24/0.50  thf(fact_1498_order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_num
% 0.24/0.50      = ( ^ [A3: num,B4: num] :
% 0.24/0.50            ( ( ord_less_num @ A3 @ B4 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.order_iff_strict
% 0.24/0.50  thf(fact_1499_order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_nat
% 0.24/0.50      = ( ^ [A3: nat,B4: nat] :
% 0.24/0.50            ( ( ord_less_nat @ A3 @ B4 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.order_iff_strict
% 0.24/0.50  thf(fact_1500_order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_int
% 0.24/0.50      = ( ^ [A3: int,B4: int] :
% 0.24/0.50            ( ( ord_less_int @ A3 @ B4 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.order_iff_strict
% 0.24/0.50  thf(fact_1501_order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_real
% 0.24/0.50      = ( ^ [A3: real,B4: real] :
% 0.24/0.50            ( ( ord_less_eq_real @ A3 @ B4 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_order
% 0.24/0.50  thf(fact_1502_order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_set_int
% 0.24/0.50      = ( ^ [A3: set_int,B4: set_int] :
% 0.24/0.50            ( ( ord_less_eq_set_int @ A3 @ B4 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_order
% 0.24/0.50  thf(fact_1503_order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_rat
% 0.24/0.50      = ( ^ [A3: rat,B4: rat] :
% 0.24/0.50            ( ( ord_less_eq_rat @ A3 @ B4 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_order
% 0.24/0.50  thf(fact_1504_order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_num
% 0.24/0.50      = ( ^ [A3: num,B4: num] :
% 0.24/0.50            ( ( ord_less_eq_num @ A3 @ B4 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_order
% 0.24/0.50  thf(fact_1505_order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_nat
% 0.24/0.50      = ( ^ [A3: nat,B4: nat] :
% 0.24/0.50            ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_order
% 0.24/0.50  thf(fact_1506_order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_int
% 0.24/0.50      = ( ^ [A3: int,B4: int] :
% 0.24/0.50            ( ( ord_less_eq_int @ A3 @ B4 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_order
% 0.24/0.50  thf(fact_1507_order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [A: real,B: real,C: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ord_less_real @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans1
% 0.24/0.50  thf(fact_1508_order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [A: set_int,B: set_int,C: set_int] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ A @ B )
% 0.24/0.50       => ( ( ord_less_set_int @ B @ C )
% 0.24/0.50         => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans1
% 0.24/0.50  thf(fact_1509_order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [A: rat,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans1
% 0.24/0.50  thf(fact_1510_order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [A: num,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.50       => ( ( ord_less_num @ B @ C )
% 0.24/0.50         => ( ord_less_num @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans1
% 0.24/0.50  thf(fact_1511_order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [A: nat,B: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.50       => ( ( ord_less_nat @ B @ C )
% 0.24/0.50         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans1
% 0.24/0.50  thf(fact_1512_order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [A: int,B: int,C: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.50       => ( ( ord_less_int @ B @ C )
% 0.24/0.50         => ( ord_less_int @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans1
% 0.24/0.50  thf(fact_1513_order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [A: real,B: real,C: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_real @ B @ C )
% 0.24/0.50         => ( ord_less_real @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans2
% 0.24/0.50  thf(fact_1514_order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [A: set_int,B: set_int,C: set_int] :
% 0.24/0.50        ( ( ord_less_set_int @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_set_int @ B @ C )
% 0.24/0.50         => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans2
% 0.24/0.50  thf(fact_1515_order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.50         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans2
% 0.24/0.50  thf(fact_1516_order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [A: num,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_num @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.50         => ( ord_less_num @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans2
% 0.24/0.50  thf(fact_1517_order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [A: nat,B: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_nat @ B @ C )
% 0.24/0.50         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans2
% 0.24/0.50  thf(fact_1518_order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [A: int,B: int,C: int] :
% 0.24/0.50        ( ( ord_less_int @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_int @ B @ C )
% 0.24/0.50         => ( ord_less_int @ A @ C ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_trans2
% 0.24/0.50  thf(fact_1519_order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_real
% 0.24/0.50      = ( ^ [A3: real,B4: real] :
% 0.24/0.50            ( ( ord_less_eq_real @ A3 @ B4 )
% 0.24/0.50            & ~ ( ord_less_eq_real @ B4 @ A3 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_not
% 0.24/0.50  thf(fact_1520_order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_set_int
% 0.24/0.50      = ( ^ [A3: set_int,B4: set_int] :
% 0.24/0.50            ( ( ord_less_eq_set_int @ A3 @ B4 )
% 0.24/0.50            & ~ ( ord_less_eq_set_int @ B4 @ A3 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_not
% 0.24/0.50  thf(fact_1521_order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_rat
% 0.24/0.50      = ( ^ [A3: rat,B4: rat] :
% 0.24/0.50            ( ( ord_less_eq_rat @ A3 @ B4 )
% 0.24/0.50            & ~ ( ord_less_eq_rat @ B4 @ A3 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_not
% 0.24/0.50  thf(fact_1522_order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_num
% 0.24/0.50      = ( ^ [A3: num,B4: num] :
% 0.24/0.50            ( ( ord_less_eq_num @ A3 @ B4 )
% 0.24/0.50            & ~ ( ord_less_eq_num @ B4 @ A3 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_not
% 0.24/0.50  thf(fact_1523_order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_nat
% 0.24/0.50      = ( ^ [A3: nat,B4: nat] :
% 0.24/0.50            ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.24/0.50            & ~ ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_not
% 0.24/0.50  thf(fact_1524_order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_int
% 0.24/0.50      = ( ^ [A3: int,B4: int] :
% 0.24/0.50            ( ( ord_less_eq_int @ A3 @ B4 )
% 0.24/0.50            & ~ ( ord_less_eq_int @ B4 @ A3 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_iff_not
% 0.24/0.50  thf(fact_1525_dense__ge__bounded,axiom,
% 0.24/0.50      ! [Z: real,X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_real @ Z @ X )
% 0.24/0.50       => ( ! [W3: real] :
% 0.24/0.50              ( ( ord_less_real @ Z @ W3 )
% 0.24/0.50             => ( ( ord_less_real @ W3 @ X )
% 0.24/0.50               => ( ord_less_eq_real @ Y @ W3 ) ) )
% 0.24/0.50         => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dense_ge_bounded
% 0.24/0.50  thf(fact_1526_dense__ge__bounded,axiom,
% 0.24/0.50      ! [Z: rat,X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_rat @ Z @ X )
% 0.24/0.50       => ( ! [W3: rat] :
% 0.24/0.50              ( ( ord_less_rat @ Z @ W3 )
% 0.24/0.50             => ( ( ord_less_rat @ W3 @ X )
% 0.24/0.50               => ( ord_less_eq_rat @ Y @ W3 ) ) )
% 0.24/0.50         => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dense_ge_bounded
% 0.24/0.50  thf(fact_1527_dense__le__bounded,axiom,
% 0.24/0.50      ! [X: real,Y: real,Z: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( ! [W3: real] :
% 0.24/0.50              ( ( ord_less_real @ X @ W3 )
% 0.24/0.50             => ( ( ord_less_real @ W3 @ Y )
% 0.24/0.50               => ( ord_less_eq_real @ W3 @ Z ) ) )
% 0.24/0.50         => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dense_le_bounded
% 0.24/0.50  thf(fact_1528_dense__le__bounded,axiom,
% 0.24/0.50      ! [X: rat,Y: rat,Z: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( ! [W3: rat] :
% 0.24/0.50              ( ( ord_less_rat @ X @ W3 )
% 0.24/0.50             => ( ( ord_less_rat @ W3 @ Y )
% 0.24/0.50               => ( ord_less_eq_rat @ W3 @ Z ) ) )
% 0.24/0.50         => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dense_le_bounded
% 0.24/0.50  thf(fact_1529_dual__order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_real
% 0.24/0.50      = ( ^ [B4: real,A3: real] :
% 0.24/0.50            ( ( ord_less_real @ B4 @ A3 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.order_iff_strict
% 0.24/0.50  thf(fact_1530_dual__order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_set_int
% 0.24/0.50      = ( ^ [B4: set_int,A3: set_int] :
% 0.24/0.50            ( ( ord_less_set_int @ B4 @ A3 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.order_iff_strict
% 0.24/0.50  thf(fact_1531_dual__order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_rat
% 0.24/0.50      = ( ^ [B4: rat,A3: rat] :
% 0.24/0.50            ( ( ord_less_rat @ B4 @ A3 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.order_iff_strict
% 0.24/0.50  thf(fact_1532_dual__order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_num
% 0.24/0.50      = ( ^ [B4: num,A3: num] :
% 0.24/0.50            ( ( ord_less_num @ B4 @ A3 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.order_iff_strict
% 0.24/0.50  thf(fact_1533_dual__order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_nat
% 0.24/0.50      = ( ^ [B4: nat,A3: nat] :
% 0.24/0.50            ( ( ord_less_nat @ B4 @ A3 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.order_iff_strict
% 0.24/0.50  thf(fact_1534_dual__order_Oorder__iff__strict,axiom,
% 0.24/0.50      ( ord_less_eq_int
% 0.24/0.50      = ( ^ [B4: int,A3: int] :
% 0.24/0.50            ( ( ord_less_int @ B4 @ A3 )
% 0.24/0.50            | ( A3 = B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.order_iff_strict
% 0.24/0.50  thf(fact_1535_dual__order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_real
% 0.24/0.50      = ( ^ [B4: real,A3: real] :
% 0.24/0.50            ( ( ord_less_eq_real @ B4 @ A3 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_order
% 0.24/0.50  thf(fact_1536_dual__order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_set_int
% 0.24/0.50      = ( ^ [B4: set_int,A3: set_int] :
% 0.24/0.50            ( ( ord_less_eq_set_int @ B4 @ A3 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_order
% 0.24/0.50  thf(fact_1537_dual__order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_rat
% 0.24/0.50      = ( ^ [B4: rat,A3: rat] :
% 0.24/0.50            ( ( ord_less_eq_rat @ B4 @ A3 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_order
% 0.24/0.50  thf(fact_1538_dual__order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_num
% 0.24/0.50      = ( ^ [B4: num,A3: num] :
% 0.24/0.50            ( ( ord_less_eq_num @ B4 @ A3 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_order
% 0.24/0.50  thf(fact_1539_dual__order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_nat
% 0.24/0.50      = ( ^ [B4: nat,A3: nat] :
% 0.24/0.50            ( ( ord_less_eq_nat @ B4 @ A3 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_order
% 0.24/0.50  thf(fact_1540_dual__order_Ostrict__iff__order,axiom,
% 0.24/0.50      ( ord_less_int
% 0.24/0.50      = ( ^ [B4: int,A3: int] :
% 0.24/0.50            ( ( ord_less_eq_int @ B4 @ A3 )
% 0.24/0.50            & ( A3 != B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_order
% 0.24/0.50  thf(fact_1541_dual__order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [B: real,A: real,C: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ B @ A )
% 0.24/0.50       => ( ( ord_less_real @ C @ B )
% 0.24/0.50         => ( ord_less_real @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans1
% 0.24/0.50  thf(fact_1542_dual__order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [B: set_int,A: set_int,C: set_int] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ B @ A )
% 0.24/0.50       => ( ( ord_less_set_int @ C @ B )
% 0.24/0.50         => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans1
% 0.24/0.50  thf(fact_1543_dual__order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [B: rat,A: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ B @ A )
% 0.24/0.50       => ( ( ord_less_rat @ C @ B )
% 0.24/0.50         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans1
% 0.24/0.50  thf(fact_1544_dual__order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [B: num,A: num,C: num] :
% 0.24/0.50        ( ( ord_less_eq_num @ B @ A )
% 0.24/0.50       => ( ( ord_less_num @ C @ B )
% 0.24/0.50         => ( ord_less_num @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans1
% 0.24/0.50  thf(fact_1545_dual__order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [B: nat,A: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ B @ A )
% 0.24/0.50       => ( ( ord_less_nat @ C @ B )
% 0.24/0.50         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans1
% 0.24/0.50  thf(fact_1546_dual__order_Ostrict__trans1,axiom,
% 0.24/0.50      ! [B: int,A: int,C: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ B @ A )
% 0.24/0.50       => ( ( ord_less_int @ C @ B )
% 0.24/0.50         => ( ord_less_int @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans1
% 0.24/0.50  thf(fact_1547_dual__order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [B: real,A: real,C: real] :
% 0.24/0.50        ( ( ord_less_real @ B @ A )
% 0.24/0.50       => ( ( ord_less_eq_real @ C @ B )
% 0.24/0.50         => ( ord_less_real @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans2
% 0.24/0.50  thf(fact_1548_dual__order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [B: set_int,A: set_int,C: set_int] :
% 0.24/0.50        ( ( ord_less_set_int @ B @ A )
% 0.24/0.50       => ( ( ord_less_eq_set_int @ C @ B )
% 0.24/0.50         => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans2
% 0.24/0.50  thf(fact_1549_dual__order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [B: rat,A: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ B @ A )
% 0.24/0.50       => ( ( ord_less_eq_rat @ C @ B )
% 0.24/0.50         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans2
% 0.24/0.50  thf(fact_1550_dual__order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [B: num,A: num,C: num] :
% 0.24/0.50        ( ( ord_less_num @ B @ A )
% 0.24/0.50       => ( ( ord_less_eq_num @ C @ B )
% 0.24/0.50         => ( ord_less_num @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans2
% 0.24/0.50  thf(fact_1551_dual__order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [B: nat,A: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_nat @ B @ A )
% 0.24/0.50       => ( ( ord_less_eq_nat @ C @ B )
% 0.24/0.50         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans2
% 0.24/0.50  thf(fact_1552_dual__order_Ostrict__trans2,axiom,
% 0.24/0.50      ! [B: int,A: int,C: int] :
% 0.24/0.50        ( ( ord_less_int @ B @ A )
% 0.24/0.50       => ( ( ord_less_eq_int @ C @ B )
% 0.24/0.50         => ( ord_less_int @ C @ A ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_trans2
% 0.24/0.50  thf(fact_1553_dual__order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_real
% 0.24/0.50      = ( ^ [B4: real,A3: real] :
% 0.24/0.50            ( ( ord_less_eq_real @ B4 @ A3 )
% 0.24/0.50            & ~ ( ord_less_eq_real @ A3 @ B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_not
% 0.24/0.50  thf(fact_1554_dual__order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_set_int
% 0.24/0.50      = ( ^ [B4: set_int,A3: set_int] :
% 0.24/0.50            ( ( ord_less_eq_set_int @ B4 @ A3 )
% 0.24/0.50            & ~ ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_not
% 0.24/0.50  thf(fact_1555_dual__order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_rat
% 0.24/0.50      = ( ^ [B4: rat,A3: rat] :
% 0.24/0.50            ( ( ord_less_eq_rat @ B4 @ A3 )
% 0.24/0.50            & ~ ( ord_less_eq_rat @ A3 @ B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_not
% 0.24/0.50  thf(fact_1556_dual__order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_num
% 0.24/0.50      = ( ^ [B4: num,A3: num] :
% 0.24/0.50            ( ( ord_less_eq_num @ B4 @ A3 )
% 0.24/0.50            & ~ ( ord_less_eq_num @ A3 @ B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_not
% 0.24/0.50  thf(fact_1557_dual__order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_nat
% 0.24/0.50      = ( ^ [B4: nat,A3: nat] :
% 0.24/0.50            ( ( ord_less_eq_nat @ B4 @ A3 )
% 0.24/0.50            & ~ ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_not
% 0.24/0.50  thf(fact_1558_dual__order_Ostrict__iff__not,axiom,
% 0.24/0.50      ( ord_less_int
% 0.24/0.50      = ( ^ [B4: int,A3: int] :
% 0.24/0.50            ( ( ord_less_eq_int @ B4 @ A3 )
% 0.24/0.50            & ~ ( ord_less_eq_int @ A3 @ B4 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_iff_not
% 0.24/0.50  thf(fact_1559_order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_order
% 0.24/0.50  thf(fact_1560_order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [A: set_int,B: set_int] :
% 0.24/0.50        ( ( ord_less_set_int @ A @ B )
% 0.24/0.50       => ( ord_less_eq_set_int @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_order
% 0.24/0.50  thf(fact_1561_order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_order
% 0.24/0.50  thf(fact_1562_order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [A: num,B: num] :
% 0.24/0.50        ( ( ord_less_num @ A @ B )
% 0.24/0.50       => ( ord_less_eq_num @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_order
% 0.24/0.50  thf(fact_1563_order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_order
% 0.24/0.50  thf(fact_1564_order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_int @ A @ B )
% 0.24/0.50       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order.strict_implies_order
% 0.24/0.50  thf(fact_1565_dual__order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [B: real,A: real] :
% 0.24/0.50        ( ( ord_less_real @ B @ A )
% 0.24/0.50       => ( ord_less_eq_real @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_order
% 0.24/0.50  thf(fact_1566_dual__order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [B: set_int,A: set_int] :
% 0.24/0.50        ( ( ord_less_set_int @ B @ A )
% 0.24/0.50       => ( ord_less_eq_set_int @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_order
% 0.24/0.50  thf(fact_1567_dual__order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [B: rat,A: rat] :
% 0.24/0.50        ( ( ord_less_rat @ B @ A )
% 0.24/0.50       => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_order
% 0.24/0.50  thf(fact_1568_dual__order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [B: num,A: num] :
% 0.24/0.50        ( ( ord_less_num @ B @ A )
% 0.24/0.50       => ( ord_less_eq_num @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_order
% 0.24/0.50  thf(fact_1569_dual__order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [B: nat,A: nat] :
% 0.24/0.50        ( ( ord_less_nat @ B @ A )
% 0.24/0.50       => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_order
% 0.24/0.50  thf(fact_1570_dual__order_Ostrict__implies__order,axiom,
% 0.24/0.50      ! [B: int,A: int] :
% 0.24/0.50        ( ( ord_less_int @ B @ A )
% 0.24/0.50       => ( ord_less_eq_int @ B @ A ) ) ).
% 0.24/0.50  
% 0.24/0.50  % dual_order.strict_implies_order
% 0.24/0.50  thf(fact_1571_order__le__less,axiom,
% 0.24/0.50      ( ord_less_eq_real
% 0.24/0.50      = ( ^ [X4: real,Y6: real] :
% 0.24/0.50            ( ( ord_less_real @ X4 @ Y6 )
% 0.24/0.50            | ( X4 = Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less
% 0.24/0.50  thf(fact_1572_order__le__less,axiom,
% 0.24/0.50      ( ord_less_eq_set_int
% 0.24/0.50      = ( ^ [X4: set_int,Y6: set_int] :
% 0.24/0.50            ( ( ord_less_set_int @ X4 @ Y6 )
% 0.24/0.50            | ( X4 = Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less
% 0.24/0.50  thf(fact_1573_order__le__less,axiom,
% 0.24/0.50      ( ord_less_eq_rat
% 0.24/0.50      = ( ^ [X4: rat,Y6: rat] :
% 0.24/0.50            ( ( ord_less_rat @ X4 @ Y6 )
% 0.24/0.50            | ( X4 = Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less
% 0.24/0.50  thf(fact_1574_order__le__less,axiom,
% 0.24/0.50      ( ord_less_eq_num
% 0.24/0.50      = ( ^ [X4: num,Y6: num] :
% 0.24/0.50            ( ( ord_less_num @ X4 @ Y6 )
% 0.24/0.50            | ( X4 = Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less
% 0.24/0.50  thf(fact_1575_order__le__less,axiom,
% 0.24/0.50      ( ord_less_eq_nat
% 0.24/0.50      = ( ^ [X4: nat,Y6: nat] :
% 0.24/0.50            ( ( ord_less_nat @ X4 @ Y6 )
% 0.24/0.50            | ( X4 = Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less
% 0.24/0.50  thf(fact_1576_order__le__less,axiom,
% 0.24/0.50      ( ord_less_eq_int
% 0.24/0.50      = ( ^ [X4: int,Y6: int] :
% 0.24/0.50            ( ( ord_less_int @ X4 @ Y6 )
% 0.24/0.50            | ( X4 = Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less
% 0.24/0.50  thf(fact_1577_order__less__le,axiom,
% 0.24/0.50      ( ord_less_real
% 0.24/0.50      = ( ^ [X4: real,Y6: real] :
% 0.24/0.50            ( ( ord_less_eq_real @ X4 @ Y6 )
% 0.24/0.50            & ( X4 != Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le
% 0.24/0.50  thf(fact_1578_order__less__le,axiom,
% 0.24/0.50      ( ord_less_set_int
% 0.24/0.50      = ( ^ [X4: set_int,Y6: set_int] :
% 0.24/0.50            ( ( ord_less_eq_set_int @ X4 @ Y6 )
% 0.24/0.50            & ( X4 != Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le
% 0.24/0.50  thf(fact_1579_order__less__le,axiom,
% 0.24/0.50      ( ord_less_rat
% 0.24/0.50      = ( ^ [X4: rat,Y6: rat] :
% 0.24/0.50            ( ( ord_less_eq_rat @ X4 @ Y6 )
% 0.24/0.50            & ( X4 != Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le
% 0.24/0.50  thf(fact_1580_order__less__le,axiom,
% 0.24/0.50      ( ord_less_num
% 0.24/0.50      = ( ^ [X4: num,Y6: num] :
% 0.24/0.50            ( ( ord_less_eq_num @ X4 @ Y6 )
% 0.24/0.50            & ( X4 != Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le
% 0.24/0.50  thf(fact_1581_order__less__le,axiom,
% 0.24/0.50      ( ord_less_nat
% 0.24/0.50      = ( ^ [X4: nat,Y6: nat] :
% 0.24/0.50            ( ( ord_less_eq_nat @ X4 @ Y6 )
% 0.24/0.50            & ( X4 != Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le
% 0.24/0.50  thf(fact_1582_order__less__le,axiom,
% 0.24/0.50      ( ord_less_int
% 0.24/0.50      = ( ^ [X4: int,Y6: int] :
% 0.24/0.50            ( ( ord_less_eq_int @ X4 @ Y6 )
% 0.24/0.50            & ( X4 != Y6 ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le
% 0.24/0.50  thf(fact_1583_linorder__not__le,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ~ ( ord_less_eq_real @ X @ Y ) )
% 0.24/0.50        = ( ord_less_real @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_not_le
% 0.24/0.50  thf(fact_1584_linorder__not__le,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
% 0.24/0.50        = ( ord_less_rat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_not_le
% 0.24/0.50  thf(fact_1585_linorder__not__le,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ~ ( ord_less_eq_num @ X @ Y ) )
% 0.24/0.50        = ( ord_less_num @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_not_le
% 0.24/0.50  thf(fact_1586_linorder__not__le,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
% 0.24/0.50        = ( ord_less_nat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_not_le
% 0.24/0.50  thf(fact_1587_linorder__not__le,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ~ ( ord_less_eq_int @ X @ Y ) )
% 0.24/0.50        = ( ord_less_int @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_not_le
% 0.24/0.50  thf(fact_1588_linorder__not__less,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.24/0.50        = ( ord_less_eq_real @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_not_less
% 0.24/0.50  thf(fact_1589_linorder__not__less,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.24/0.50        = ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_not_less
% 0.24/0.50  thf(fact_1590_linorder__not__less,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.24/0.50        = ( ord_less_eq_num @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_not_less
% 0.24/0.50  thf(fact_1591_linorder__not__less,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.24/0.50        = ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_not_less
% 0.24/0.50  thf(fact_1592_linorder__not__less,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.24/0.50        = ( ord_less_eq_int @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_not_less
% 0.24/0.50  thf(fact_1593_order__less__imp__le,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_le
% 0.24/0.50  thf(fact_1594_order__less__imp__le,axiom,
% 0.24/0.50      ! [X: set_int,Y: set_int] :
% 0.24/0.50        ( ( ord_less_set_int @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_set_int @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_le
% 0.24/0.50  thf(fact_1595_order__less__imp__le,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_le
% 0.24/0.50  thf(fact_1596_order__less__imp__le,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_num @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_le
% 0.24/0.50  thf(fact_1597_order__less__imp__le,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_le
% 0.24/0.50  thf(fact_1598_order__less__imp__le,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50       => ( ord_less_eq_int @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_imp_le
% 0.24/0.50  thf(fact_1599_order__le__neq__trans,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.50       => ( ( A != B )
% 0.24/0.50         => ( ord_less_real @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_neq_trans
% 0.24/0.50  thf(fact_1600_order__le__neq__trans,axiom,
% 0.24/0.50      ! [A: set_int,B: set_int] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ A @ B )
% 0.24/0.50       => ( ( A != B )
% 0.24/0.50         => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_neq_trans
% 0.24/0.50  thf(fact_1601_order__le__neq__trans,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.50       => ( ( A != B )
% 0.24/0.50         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_neq_trans
% 0.24/0.50  thf(fact_1602_order__le__neq__trans,axiom,
% 0.24/0.50      ! [A: num,B: num] :
% 0.24/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.50       => ( ( A != B )
% 0.24/0.50         => ( ord_less_num @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_neq_trans
% 0.24/0.50  thf(fact_1603_order__le__neq__trans,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.50       => ( ( A != B )
% 0.24/0.50         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_neq_trans
% 0.24/0.50  thf(fact_1604_order__le__neq__trans,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.50       => ( ( A != B )
% 0.24/0.50         => ( ord_less_int @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_neq_trans
% 0.24/0.50  thf(fact_1605_order__neq__le__trans,axiom,
% 0.24/0.50      ! [A: real,B: real] :
% 0.24/0.50        ( ( A != B )
% 0.24/0.50       => ( ( ord_less_eq_real @ A @ B )
% 0.24/0.50         => ( ord_less_real @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_neq_le_trans
% 0.24/0.50  thf(fact_1606_order__neq__le__trans,axiom,
% 0.24/0.50      ! [A: set_int,B: set_int] :
% 0.24/0.50        ( ( A != B )
% 0.24/0.50       => ( ( ord_less_eq_set_int @ A @ B )
% 0.24/0.50         => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_neq_le_trans
% 0.24/0.50  thf(fact_1607_order__neq__le__trans,axiom,
% 0.24/0.50      ! [A: rat,B: rat] :
% 0.24/0.50        ( ( A != B )
% 0.24/0.50       => ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.50         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_neq_le_trans
% 0.24/0.50  thf(fact_1608_order__neq__le__trans,axiom,
% 0.24/0.50      ! [A: num,B: num] :
% 0.24/0.50        ( ( A != B )
% 0.24/0.50       => ( ( ord_less_eq_num @ A @ B )
% 0.24/0.50         => ( ord_less_num @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_neq_le_trans
% 0.24/0.50  thf(fact_1609_order__neq__le__trans,axiom,
% 0.24/0.50      ! [A: nat,B: nat] :
% 0.24/0.50        ( ( A != B )
% 0.24/0.50       => ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.50         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_neq_le_trans
% 0.24/0.50  thf(fact_1610_order__neq__le__trans,axiom,
% 0.24/0.50      ! [A: int,B: int] :
% 0.24/0.50        ( ( A != B )
% 0.24/0.50       => ( ( ord_less_eq_int @ A @ B )
% 0.24/0.50         => ( ord_less_int @ A @ B ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_neq_le_trans
% 0.24/0.50  thf(fact_1611_order__le__less__trans,axiom,
% 0.24/0.50      ! [X: real,Y: real,Z: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ X @ Y )
% 0.24/0.50       => ( ( ord_less_real @ Y @ Z )
% 0.24/0.50         => ( ord_less_real @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_trans
% 0.24/0.50  thf(fact_1612_order__le__less__trans,axiom,
% 0.24/0.50      ! [X: set_int,Y: set_int,Z: set_int] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ X @ Y )
% 0.24/0.50       => ( ( ord_less_set_int @ Y @ Z )
% 0.24/0.50         => ( ord_less_set_int @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_trans
% 0.24/0.50  thf(fact_1613_order__le__less__trans,axiom,
% 0.24/0.50      ! [X: rat,Y: rat,Z: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_rat @ Y @ Z )
% 0.24/0.50         => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_trans
% 0.24/0.50  thf(fact_1614_order__le__less__trans,axiom,
% 0.24/0.50      ! [X: num,Y: num,Z: num] :
% 0.24/0.50        ( ( ord_less_eq_num @ X @ Y )
% 0.24/0.50       => ( ( ord_less_num @ Y @ Z )
% 0.24/0.50         => ( ord_less_num @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_trans
% 0.24/0.50  thf(fact_1615_order__le__less__trans,axiom,
% 0.24/0.50      ! [X: nat,Y: nat,Z: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_nat @ Y @ Z )
% 0.24/0.50         => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_trans
% 0.24/0.50  thf(fact_1616_order__le__less__trans,axiom,
% 0.24/0.50      ! [X: int,Y: int,Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ X @ Y )
% 0.24/0.50       => ( ( ord_less_int @ Y @ Z )
% 0.24/0.50         => ( ord_less_int @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_trans
% 0.24/0.50  thf(fact_1617_order__less__le__trans,axiom,
% 0.24/0.50      ! [X: real,Y: real,Z: real] :
% 0.24/0.50        ( ( ord_less_real @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_real @ Y @ Z )
% 0.24/0.50         => ( ord_less_real @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_trans
% 0.24/0.50  thf(fact_1618_order__less__le__trans,axiom,
% 0.24/0.50      ! [X: set_int,Y: set_int,Z: set_int] :
% 0.24/0.50        ( ( ord_less_set_int @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_set_int @ Y @ Z )
% 0.24/0.50         => ( ord_less_set_int @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_trans
% 0.24/0.50  thf(fact_1619_order__less__le__trans,axiom,
% 0.24/0.50      ! [X: rat,Y: rat,Z: rat] :
% 0.24/0.50        ( ( ord_less_rat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_rat @ Y @ Z )
% 0.24/0.50         => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_trans
% 0.24/0.50  thf(fact_1620_order__less__le__trans,axiom,
% 0.24/0.50      ! [X: num,Y: num,Z: num] :
% 0.24/0.50        ( ( ord_less_num @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_num @ Y @ Z )
% 0.24/0.50         => ( ord_less_num @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_trans
% 0.24/0.50  thf(fact_1621_order__less__le__trans,axiom,
% 0.24/0.50      ! [X: nat,Y: nat,Z: nat] :
% 0.24/0.50        ( ( ord_less_nat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_nat @ Y @ Z )
% 0.24/0.50         => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_trans
% 0.24/0.50  thf(fact_1622_order__less__le__trans,axiom,
% 0.24/0.50      ! [X: int,Y: int,Z: int] :
% 0.24/0.50        ( ( ord_less_int @ X @ Y )
% 0.24/0.50       => ( ( ord_less_eq_int @ Y @ Z )
% 0.24/0.50         => ( ord_less_int @ X @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_trans
% 0.24/0.50  thf(fact_1623_order__le__less__subst1,axiom,
% 0.24/0.50      ! [A: real,F: real > real,B: real,C: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst1
% 0.24/0.50  thf(fact_1624_order__le__less__subst1,axiom,
% 0.24/0.50      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst1
% 0.24/0.50  thf(fact_1625_order__le__less__subst1,axiom,
% 0.24/0.50      ! [A: real,F: num > real,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_num @ B @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst1
% 0.24/0.50  thf(fact_1626_order__le__less__subst1,axiom,
% 0.24/0.50      ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_nat @ B @ C )
% 0.24/0.50         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.50                ( ( ord_less_nat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst1
% 0.24/0.50  thf(fact_1627_order__le__less__subst1,axiom,
% 0.24/0.50      ! [A: real,F: int > real,B: int,C: int] :
% 0.24/0.50        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_int @ B @ C )
% 0.24/0.50         => ( ! [X3: int,Y4: int] :
% 0.24/0.50                ( ( ord_less_int @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst1
% 0.24/0.50  thf(fact_1628_order__le__less__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: real > rat,B: real,C: real] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_real @ B @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst1
% 0.24/0.50  thf(fact_1629_order__le__less__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst1
% 0.24/0.50  thf(fact_1630_order__le__less__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_num @ B @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst1
% 0.24/0.50  thf(fact_1631_order__le__less__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_nat @ B @ C )
% 0.24/0.50         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.50                ( ( ord_less_nat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst1
% 0.24/0.50  thf(fact_1632_order__le__less__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: int > rat,B: int,C: int] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_int @ B @ C )
% 0.24/0.50         => ( ! [X3: int,Y4: int] :
% 0.24/0.50                ( ( ord_less_int @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst1
% 0.24/0.50  thf(fact_1633_order__le__less__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst2
% 0.24/0.50  thf(fact_1634_order__le__less__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst2
% 0.24/0.50  thf(fact_1635_order__le__less__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst2
% 0.24/0.50  thf(fact_1636_order__le__less__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst2
% 0.24/0.50  thf(fact_1637_order__le__less__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.24/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst2
% 0.24/0.50  thf(fact_1638_order__le__less__subst2,axiom,
% 0.24/0.50      ! [A: num,B: num,F: num > real,C: real] :
% 0.24/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.50       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst2
% 0.24/0.50  thf(fact_1639_order__le__less__subst2,axiom,
% 0.24/0.50      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.50       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst2
% 0.24/0.50  thf(fact_1640_order__le__less__subst2,axiom,
% 0.24/0.50      ! [A: num,B: num,F: num > num,C: num] :
% 0.24/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.50       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst2
% 0.24/0.50  thf(fact_1641_order__le__less__subst2,axiom,
% 0.24/0.50      ! [A: num,B: num,F: num > nat,C: nat] :
% 0.24/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.50       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst2
% 0.24/0.50  thf(fact_1642_order__le__less__subst2,axiom,
% 0.24/0.50      ! [A: num,B: num,F: num > int,C: int] :
% 0.24/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.50       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_less_subst2
% 0.24/0.50  thf(fact_1643_order__less__le__subst1,axiom,
% 0.24/0.50      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst1
% 0.24/0.50  thf(fact_1644_order__less__le__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst1
% 0.24/0.50  thf(fact_1645_order__less__le__subst1,axiom,
% 0.24/0.50      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_num @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst1
% 0.24/0.50  thf(fact_1646_order__less__le__subst1,axiom,
% 0.24/0.50      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst1
% 0.24/0.50  thf(fact_1647_order__less__le__subst1,axiom,
% 0.24/0.50      ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.24/0.50        ( ( ord_less_int @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst1
% 0.24/0.50  thf(fact_1648_order__less__le__subst1,axiom,
% 0.24/0.50      ! [A: real,F: num > real,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst1
% 0.24/0.50  thf(fact_1649_order__less__le__subst1,axiom,
% 0.24/0.50      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst1
% 0.24/0.50  thf(fact_1650_order__less__le__subst1,axiom,
% 0.24/0.50      ! [A: num,F: num > num,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_num @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst1
% 0.24/0.50  thf(fact_1651_order__less__le__subst1,axiom,
% 0.24/0.50      ! [A: nat,F: num > nat,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst1
% 0.24/0.50  thf(fact_1652_order__less__le__subst1,axiom,
% 0.24/0.50      ! [A: int,F: num > int,B: num,C: num] :
% 0.24/0.50        ( ( ord_less_int @ A @ ( F @ B ) )
% 0.24/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst1
% 0.24/0.50  thf(fact_1653_order__less__le__subst2,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > real,C: real] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst2
% 0.24/0.50  thf(fact_1654_order__less__le__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst2
% 0.24/0.50  thf(fact_1655_order__less__le__subst2,axiom,
% 0.24/0.50      ! [A: num,B: num,F: num > real,C: real] :
% 0.24/0.50        ( ( ord_less_num @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst2
% 0.24/0.50  thf(fact_1656_order__less__le__subst2,axiom,
% 0.24/0.50      ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.50                ( ( ord_less_nat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst2
% 0.24/0.50  thf(fact_1657_order__less__le__subst2,axiom,
% 0.24/0.50      ! [A: int,B: int,F: int > real,C: real] :
% 0.24/0.50        ( ( ord_less_int @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: int,Y4: int] :
% 0.24/0.50                ( ( ord_less_int @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst2
% 0.24/0.50  thf(fact_1658_order__less__le__subst2,axiom,
% 0.24/0.50      ! [A: real,B: real,F: real > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_real @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: real,Y4: real] :
% 0.24/0.50                ( ( ord_less_real @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst2
% 0.24/0.50  thf(fact_1659_order__less__le__subst2,axiom,
% 0.24/0.50      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_rat @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.50                ( ( ord_less_rat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst2
% 0.24/0.50  thf(fact_1660_order__less__le__subst2,axiom,
% 0.24/0.50      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_num @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: num,Y4: num] :
% 0.24/0.50                ( ( ord_less_num @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst2
% 0.24/0.50  thf(fact_1661_order__less__le__subst2,axiom,
% 0.24/0.50      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_nat @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.50                ( ( ord_less_nat @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst2
% 0.24/0.50  thf(fact_1662_order__less__le__subst2,axiom,
% 0.24/0.50      ! [A: int,B: int,F: int > rat,C: rat] :
% 0.24/0.50        ( ( ord_less_int @ A @ B )
% 0.24/0.50       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.24/0.50         => ( ! [X3: int,Y4: int] :
% 0.24/0.50                ( ( ord_less_int @ X3 @ Y4 )
% 0.24/0.50               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_less_le_subst2
% 0.24/0.50  thf(fact_1663_linorder__le__less__linear,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ X @ Y )
% 0.24/0.50        | ( ord_less_real @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_le_less_linear
% 0.24/0.50  thf(fact_1664_linorder__le__less__linear,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.50        | ( ord_less_rat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_le_less_linear
% 0.24/0.50  thf(fact_1665_linorder__le__less__linear,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_eq_num @ X @ Y )
% 0.24/0.50        | ( ord_less_num @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_le_less_linear
% 0.24/0.50  thf(fact_1666_linorder__le__less__linear,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ X @ Y )
% 0.24/0.50        | ( ord_less_nat @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_le_less_linear
% 0.24/0.50  thf(fact_1667_linorder__le__less__linear,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ X @ Y )
% 0.24/0.50        | ( ord_less_int @ Y @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % linorder_le_less_linear
% 0.24/0.50  thf(fact_1668_order__le__imp__less__or__eq,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_eq_real @ X @ Y )
% 0.24/0.50       => ( ( ord_less_real @ X @ Y )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_imp_less_or_eq
% 0.24/0.50  thf(fact_1669_order__le__imp__less__or__eq,axiom,
% 0.24/0.50      ! [X: set_int,Y: set_int] :
% 0.24/0.50        ( ( ord_less_eq_set_int @ X @ Y )
% 0.24/0.50       => ( ( ord_less_set_int @ X @ Y )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_imp_less_or_eq
% 0.24/0.50  thf(fact_1670_order__le__imp__less__or__eq,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_rat @ X @ Y )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_imp_less_or_eq
% 0.24/0.50  thf(fact_1671_order__le__imp__less__or__eq,axiom,
% 0.24/0.50      ! [X: num,Y: num] :
% 0.24/0.50        ( ( ord_less_eq_num @ X @ Y )
% 0.24/0.50       => ( ( ord_less_num @ X @ Y )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_imp_less_or_eq
% 0.24/0.50  thf(fact_1672_order__le__imp__less__or__eq,axiom,
% 0.24/0.50      ! [X: nat,Y: nat] :
% 0.24/0.50        ( ( ord_less_eq_nat @ X @ Y )
% 0.24/0.50       => ( ( ord_less_nat @ X @ Y )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_imp_less_or_eq
% 0.24/0.50  thf(fact_1673_order__le__imp__less__or__eq,axiom,
% 0.24/0.50      ! [X: int,Y: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ X @ Y )
% 0.24/0.50       => ( ( ord_less_int @ X @ Y )
% 0.24/0.50          | ( X = Y ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % order_le_imp_less_or_eq
% 0.24/0.50  thf(fact_1674_ceiling__less__cancel,axiom,
% 0.24/0.50      ! [X: real,Y: real] :
% 0.24/0.50        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
% 0.24/0.50       => ( ord_less_real @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ceiling_less_cancel
% 0.24/0.50  thf(fact_1675_ceiling__less__cancel,axiom,
% 0.24/0.50      ! [X: rat,Y: rat] :
% 0.24/0.50        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
% 0.24/0.50       => ( ord_less_rat @ X @ Y ) ) ).
% 0.24/0.50  
% 0.24/0.50  % ceiling_less_cancel
% 0.24/0.50  thf(fact_1676_nat__power__less__imp__less,axiom,
% 0.24/0.50      ! [I: nat,M: nat,N: nat] :
% 0.24/0.50        ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.24/0.50       => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.24/0.50         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % nat_power_less_imp_less
% 0.24/0.50  thf(fact_1677_of__nat__less__of__int__iff,axiom,
% 0.24/0.50      ! [N: nat,X: int] :
% 0.24/0.50        ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( ring_18347121197199848620nteger @ X ) )
% 0.24/0.50        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_of_int_iff
% 0.24/0.50  thf(fact_1678_of__nat__less__of__int__iff,axiom,
% 0.24/0.50      ! [N: nat,X: int] :
% 0.24/0.50        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
% 0.24/0.50        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_of_int_iff
% 0.24/0.50  thf(fact_1679_of__nat__less__of__int__iff,axiom,
% 0.24/0.50      ! [N: nat,X: int] :
% 0.24/0.50        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
% 0.24/0.50        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_of_int_iff
% 0.24/0.50  thf(fact_1680_of__nat__less__of__int__iff,axiom,
% 0.24/0.50      ! [N: nat,X: int] :
% 0.24/0.50        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
% 0.24/0.50        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_nat_less_of_int_iff
% 0.24/0.50  thf(fact_1681_power__strict__decreasing,axiom,
% 0.24/0.50      ! [N: nat,N4: nat,A: real] :
% 0.24/0.50        ( ( ord_less_nat @ N @ N4 )
% 0.24/0.50       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.50         => ( ( ord_less_real @ A @ one_one_real )
% 0.24/0.50           => ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_decreasing
% 0.24/0.50  thf(fact_1682_power__strict__decreasing,axiom,
% 0.24/0.50      ! [N: nat,N4: nat,A: rat] :
% 0.24/0.50        ( ( ord_less_nat @ N @ N4 )
% 0.24/0.50       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.50         => ( ( ord_less_rat @ A @ one_one_rat )
% 0.24/0.50           => ( ord_less_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_decreasing
% 0.24/0.50  thf(fact_1683_power__strict__decreasing,axiom,
% 0.24/0.50      ! [N: nat,N4: nat,A: nat] :
% 0.24/0.50        ( ( ord_less_nat @ N @ N4 )
% 0.24/0.50       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.24/0.50         => ( ( ord_less_nat @ A @ one_one_nat )
% 0.24/0.50           => ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_decreasing
% 0.24/0.50  thf(fact_1684_power__strict__decreasing,axiom,
% 0.24/0.50      ! [N: nat,N4: nat,A: int] :
% 0.24/0.50        ( ( ord_less_nat @ N @ N4 )
% 0.24/0.50       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.24/0.50         => ( ( ord_less_int @ A @ one_one_int )
% 0.24/0.50           => ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % power_strict_decreasing
% 0.24/0.50  thf(fact_1685_length__induct,axiom,
% 0.24/0.50      ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.24/0.50        ( ! [Xs2: list_VEBT_VEBT] :
% 0.24/0.50            ( ! [Ys: list_VEBT_VEBT] :
% 0.24/0.50                ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.24/0.50               => ( P @ Ys ) )
% 0.24/0.50           => ( P @ Xs2 ) )
% 0.24/0.50       => ( P @ Xs ) ) ).
% 0.24/0.50  
% 0.24/0.50  % length_induct
% 0.24/0.50  thf(fact_1686_length__induct,axiom,
% 0.24/0.50      ! [P: list_o > $o,Xs: list_o] :
% 0.24/0.50        ( ! [Xs2: list_o] :
% 0.24/0.50            ( ! [Ys: list_o] :
% 0.24/0.50                ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs2 ) )
% 0.24/0.50               => ( P @ Ys ) )
% 0.24/0.50           => ( P @ Xs2 ) )
% 0.24/0.50       => ( P @ Xs ) ) ).
% 0.24/0.50  
% 0.24/0.50  % length_induct
% 0.24/0.50  thf(fact_1687_length__induct,axiom,
% 0.24/0.50      ! [P: list_int > $o,Xs: list_int] :
% 0.24/0.50        ( ! [Xs2: list_int] :
% 0.24/0.50            ( ! [Ys: list_int] :
% 0.24/0.50                ( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs2 ) )
% 0.24/0.50               => ( P @ Ys ) )
% 0.24/0.50           => ( P @ Xs2 ) )
% 0.24/0.50       => ( P @ Xs ) ) ).
% 0.24/0.50  
% 0.24/0.50  % length_induct
% 0.24/0.50  thf(fact_1688_of__int__nonneg,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.24/0.50       => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_nonneg
% 0.24/0.50  thf(fact_1689_of__int__nonneg,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.24/0.50       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_nonneg
% 0.24/0.50  thf(fact_1690_of__int__nonneg,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.24/0.50       => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_nonneg
% 0.24/0.50  thf(fact_1691_of__int__nonneg,axiom,
% 0.24/0.50      ! [Z: int] :
% 0.24/0.50        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.24/0.50       => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % of_int_nonneg
% 0.24/0.50  thf(fact_1692_one__less__power,axiom,
% 0.24/0.50      ! [A: real,N: nat] :
% 0.24/0.50        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.50       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50         => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.24/0.50  
% 0.24/0.50  % one_less_power
% 0.24/0.50  thf(fact_1693_one__less__power,axiom,
% 0.24/0.50      ! [A: rat,N: nat] :
% 0.24/0.50        ( ( ord_less_rat @ one_one_rat @ A )
% 0.24/0.50       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.50         => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_less_power
% 0.24/0.51  thf(fact_1694_one__less__power,axiom,
% 0.24/0.51      ! [A: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ one_one_nat @ A )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51         => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_less_power
% 0.24/0.51  thf(fact_1695_one__less__power,axiom,
% 0.24/0.51      ! [A: int,N: nat] :
% 0.24/0.51        ( ( ord_less_int @ one_one_int @ A )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51         => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_less_power
% 0.24/0.51  thf(fact_1696_power__eq__iff__eq__base,axiom,
% 0.24/0.51      ! [N: nat,A: real,B: real] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.51           => ( ( ( power_power_real @ A @ N )
% 0.24/0.51                = ( power_power_real @ B @ N ) )
% 0.24/0.51              = ( A = B ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_eq_iff_eq_base
% 0.24/0.51  thf(fact_1697_power__eq__iff__eq__base,axiom,
% 0.24/0.51      ! [N: nat,A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.51           => ( ( ( power_power_rat @ A @ N )
% 0.24/0.51                = ( power_power_rat @ B @ N ) )
% 0.24/0.51              = ( A = B ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_eq_iff_eq_base
% 0.24/0.51  thf(fact_1698_power__eq__iff__eq__base,axiom,
% 0.24/0.51      ! [N: nat,A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.24/0.51           => ( ( ( power_power_nat @ A @ N )
% 0.24/0.51                = ( power_power_nat @ B @ N ) )
% 0.24/0.51              = ( A = B ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_eq_iff_eq_base
% 0.24/0.51  thf(fact_1699_power__eq__iff__eq__base,axiom,
% 0.24/0.51      ! [N: nat,A: int,B: int] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.24/0.51           => ( ( ( power_power_int @ A @ N )
% 0.24/0.51                = ( power_power_int @ B @ N ) )
% 0.24/0.51              = ( A = B ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_eq_iff_eq_base
% 0.24/0.51  thf(fact_1700_power__eq__imp__eq__base,axiom,
% 0.24/0.51      ! [A: real,N: nat,B: real] :
% 0.24/0.51        ( ( ( power_power_real @ A @ N )
% 0.24/0.51          = ( power_power_real @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.51           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51             => ( A = B ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_eq_imp_eq_base
% 0.24/0.51  thf(fact_1701_power__eq__imp__eq__base,axiom,
% 0.24/0.51      ! [A: rat,N: nat,B: rat] :
% 0.24/0.51        ( ( ( power_power_rat @ A @ N )
% 0.24/0.51          = ( power_power_rat @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.51           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51             => ( A = B ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_eq_imp_eq_base
% 0.24/0.51  thf(fact_1702_power__eq__imp__eq__base,axiom,
% 0.24/0.51      ! [A: nat,N: nat,B: nat] :
% 0.24/0.51        ( ( ( power_power_nat @ A @ N )
% 0.24/0.51          = ( power_power_nat @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.24/0.51           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51             => ( A = B ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_eq_imp_eq_base
% 0.24/0.51  thf(fact_1703_power__eq__imp__eq__base,axiom,
% 0.24/0.51      ! [A: int,N: nat,B: int] :
% 0.24/0.51        ( ( ( power_power_int @ A @ N )
% 0.24/0.51          = ( power_power_int @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.24/0.51           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51             => ( A = B ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_eq_imp_eq_base
% 0.24/0.51  thf(fact_1704_length__pos__if__in__set,axiom,
% 0.24/0.51      ! [X: complex,Xs: list_complex] :
% 0.24/0.51        ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 0.24/0.51       => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % length_pos_if_in_set
% 0.24/0.51  thf(fact_1705_length__pos__if__in__set,axiom,
% 0.24/0.51      ! [X: real,Xs: list_real] :
% 0.24/0.51        ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.24/0.51       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % length_pos_if_in_set
% 0.24/0.51  thf(fact_1706_length__pos__if__in__set,axiom,
% 0.24/0.51      ! [X: set_nat,Xs: list_set_nat] :
% 0.24/0.51        ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.24/0.51       => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % length_pos_if_in_set
% 0.24/0.51  thf(fact_1707_length__pos__if__in__set,axiom,
% 0.24/0.51      ! [X: nat,Xs: list_nat] :
% 0.24/0.51        ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.24/0.51       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % length_pos_if_in_set
% 0.24/0.51  thf(fact_1708_length__pos__if__in__set,axiom,
% 0.24/0.51      ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.24/0.51        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.24/0.51       => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % length_pos_if_in_set
% 0.24/0.51  thf(fact_1709_length__pos__if__in__set,axiom,
% 0.24/0.51      ! [X: $o,Xs: list_o] :
% 0.24/0.51        ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.24/0.51       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % length_pos_if_in_set
% 0.24/0.51  thf(fact_1710_length__pos__if__in__set,axiom,
% 0.24/0.51      ! [X: int,Xs: list_int] :
% 0.24/0.51        ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.24/0.51       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % length_pos_if_in_set
% 0.24/0.51  thf(fact_1711_half__gt__zero__iff,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.24/0.51        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.24/0.51  
% 0.24/0.51  % half_gt_zero_iff
% 0.24/0.51  thf(fact_1712_half__gt__zero__iff,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.24/0.51        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.24/0.51  
% 0.24/0.51  % half_gt_zero_iff
% 0.24/0.51  thf(fact_1713_half__gt__zero,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % half_gt_zero
% 0.24/0.51  thf(fact_1714_half__gt__zero,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % half_gt_zero
% 0.24/0.51  thf(fact_1715_nat__approx__posE,axiom,
% 0.24/0.51      ! [E2: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.24/0.51       => ~ ! [N2: nat] :
% 0.24/0.51              ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat_approx_posE
% 0.24/0.51  thf(fact_1716_nat__approx__posE,axiom,
% 0.24/0.51      ! [E2: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.24/0.51       => ~ ! [N2: nat] :
% 0.24/0.51              ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat_approx_posE
% 0.24/0.51  thf(fact_1717_power__strict__increasing,axiom,
% 0.24/0.51      ! [N: nat,N4: nat,A: real] :
% 0.24/0.51        ( ( ord_less_nat @ N @ N4 )
% 0.24/0.51       => ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_strict_increasing
% 0.24/0.51  thf(fact_1718_power__strict__increasing,axiom,
% 0.24/0.51      ! [N: nat,N4: nat,A: rat] :
% 0.24/0.51        ( ( ord_less_nat @ N @ N4 )
% 0.24/0.51       => ( ( ord_less_rat @ one_one_rat @ A )
% 0.24/0.51         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_strict_increasing
% 0.24/0.51  thf(fact_1719_power__strict__increasing,axiom,
% 0.24/0.51      ! [N: nat,N4: nat,A: nat] :
% 0.24/0.51        ( ( ord_less_nat @ N @ N4 )
% 0.24/0.51       => ( ( ord_less_nat @ one_one_nat @ A )
% 0.24/0.51         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_strict_increasing
% 0.24/0.51  thf(fact_1720_power__strict__increasing,axiom,
% 0.24/0.51      ! [N: nat,N4: nat,A: int] :
% 0.24/0.51        ( ( ord_less_nat @ N @ N4 )
% 0.24/0.51       => ( ( ord_less_int @ one_one_int @ A )
% 0.24/0.51         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_strict_increasing
% 0.24/0.51  thf(fact_1721_power__less__imp__less__exp,axiom,
% 0.24/0.51      ! [A: real,M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.24/0.51         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_less_imp_less_exp
% 0.24/0.51  thf(fact_1722_power__less__imp__less__exp,axiom,
% 0.24/0.51      ! [A: rat,M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_rat @ one_one_rat @ A )
% 0.24/0.51       => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.24/0.51         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_less_imp_less_exp
% 0.24/0.51  thf(fact_1723_power__less__imp__less__exp,axiom,
% 0.24/0.51      ! [A: nat,M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ one_one_nat @ A )
% 0.24/0.51       => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.24/0.51         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_less_imp_less_exp
% 0.24/0.51  thf(fact_1724_power__less__imp__less__exp,axiom,
% 0.24/0.51      ! [A: int,M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_int @ one_one_int @ A )
% 0.24/0.51       => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.24/0.51         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_less_imp_less_exp
% 0.24/0.51  thf(fact_1725_add__strict__increasing2,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ B @ C )
% 0.24/0.51         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_increasing2
% 0.24/0.51  thf(fact_1726_add__strict__increasing2,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_rat @ B @ C )
% 0.24/0.51         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_increasing2
% 0.24/0.51  thf(fact_1727_add__strict__increasing2,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51       => ( ( ord_less_nat @ B @ C )
% 0.24/0.51         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_increasing2
% 0.24/0.51  thf(fact_1728_add__strict__increasing2,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51       => ( ( ord_less_int @ B @ C )
% 0.24/0.51         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_increasing2
% 0.24/0.51  thf(fact_1729_add__strict__increasing,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_eq_real @ B @ C )
% 0.24/0.51         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_increasing
% 0.24/0.51  thf(fact_1730_add__strict__increasing,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_increasing
% 0.24/0.51  thf(fact_1731_add__strict__increasing,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ C )
% 0.24/0.51         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_increasing
% 0.24/0.51  thf(fact_1732_add__strict__increasing,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int] :
% 0.24/0.51        ( ( ord_less_int @ zero_zero_int @ A )
% 0.24/0.51       => ( ( ord_less_eq_int @ B @ C )
% 0.24/0.51         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_increasing
% 0.24/0.51  thf(fact_1733_add__pos__nonneg,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.51         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_pos_nonneg
% 0.24/0.51  thf(fact_1734_add__pos__nonneg,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.51         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_pos_nonneg
% 0.24/0.51  thf(fact_1735_add__pos__nonneg,axiom,
% 0.24/0.51      ! [A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.24/0.51         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_pos_nonneg
% 0.24/0.51  thf(fact_1736_add__pos__nonneg,axiom,
% 0.24/0.51      ! [A: int,B: int] :
% 0.24/0.51        ( ( ord_less_int @ zero_zero_int @ A )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.24/0.51         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_pos_nonneg
% 0.24/0.51  thf(fact_1737_add__nonpos__neg,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.24/0.51         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_neg
% 0.24/0.51  thf(fact_1738_add__nonpos__neg,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_neg
% 0.24/0.51  thf(fact_1739_add__nonpos__neg,axiom,
% 0.24/0.51      ! [A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.24/0.51       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.24/0.51         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_neg
% 0.24/0.51  thf(fact_1740_add__nonpos__neg,axiom,
% 0.24/0.51      ! [A: int,B: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.24/0.51       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.24/0.51         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_neg
% 0.24/0.51  thf(fact_1741_add__nonneg__pos,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.24/0.51         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_pos
% 0.24/0.51  thf(fact_1742_add__nonneg__pos,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.24/0.51         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_pos
% 0.24/0.51  thf(fact_1743_add__nonneg__pos,axiom,
% 0.24/0.51      ! [A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.24/0.51         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_pos
% 0.24/0.51  thf(fact_1744_add__nonneg__pos,axiom,
% 0.24/0.51      ! [A: int,B: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.24/0.51         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_pos
% 0.24/0.51  thf(fact_1745_add__neg__nonpos,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.24/0.51         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_neg_nonpos
% 0.24/0.51  thf(fact_1746_add__neg__nonpos,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_neg_nonpos
% 0.24/0.51  thf(fact_1747_add__neg__nonpos,axiom,
% 0.24/0.51      ! [A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.24/0.51         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_neg_nonpos
% 0.24/0.51  thf(fact_1748_add__neg__nonpos,axiom,
% 0.24/0.51      ! [A: int,B: int] :
% 0.24/0.51        ( ( ord_less_int @ A @ zero_zero_int )
% 0.24/0.51       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.24/0.51         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_neg_nonpos
% 0.24/0.51  thf(fact_1749_power__less__imp__less__base,axiom,
% 0.24/0.51      ! [A: real,N: nat,B: real] :
% 0.24/0.51        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.51         => ( ord_less_real @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_less_imp_less_base
% 0.24/0.51  thf(fact_1750_power__less__imp__less__base,axiom,
% 0.24/0.51      ! [A: rat,N: nat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.51         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_less_imp_less_base
% 0.24/0.51  thf(fact_1751_power__less__imp__less__base,axiom,
% 0.24/0.51      ! [A: nat,N: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.24/0.51         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_less_imp_less_base
% 0.24/0.51  thf(fact_1752_power__less__imp__less__base,axiom,
% 0.24/0.51      ! [A: int,N: nat,B: int] :
% 0.24/0.51        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.24/0.51         => ( ord_less_int @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_less_imp_less_base
% 0.24/0.51  thf(fact_1753_ex__least__nat__less,axiom,
% 0.24/0.51      ! [P: nat > $o,N: nat] :
% 0.24/0.51        ( ( P @ N )
% 0.24/0.51       => ( ~ ( P @ zero_zero_nat )
% 0.24/0.51         => ? [K3: nat] :
% 0.24/0.51              ( ( ord_less_nat @ K3 @ N )
% 0.24/0.51              & ! [I4: nat] :
% 0.24/0.51                  ( ( ord_less_eq_nat @ I4 @ K3 )
% 0.24/0.51                 => ~ ( P @ I4 ) )
% 0.24/0.51              & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ex_least_nat_less
% 0.24/0.51  thf(fact_1754_power__0__left,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ( N = zero_zero_nat )
% 0.24/0.51         => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.24/0.51            = one_one_rat ) )
% 0.24/0.51        & ( ( N != zero_zero_nat )
% 0.24/0.51         => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.24/0.51            = zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_0_left
% 0.24/0.51  thf(fact_1755_power__0__left,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ( N = zero_zero_nat )
% 0.24/0.51         => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.24/0.51            = one_one_nat ) )
% 0.24/0.51        & ( ( N != zero_zero_nat )
% 0.24/0.51         => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.24/0.51            = zero_zero_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_0_left
% 0.24/0.51  thf(fact_1756_power__0__left,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ( N = zero_zero_nat )
% 0.24/0.51         => ( ( power_power_real @ zero_zero_real @ N )
% 0.24/0.51            = one_one_real ) )
% 0.24/0.51        & ( ( N != zero_zero_nat )
% 0.24/0.51         => ( ( power_power_real @ zero_zero_real @ N )
% 0.24/0.51            = zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_0_left
% 0.24/0.51  thf(fact_1757_power__0__left,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ( N = zero_zero_nat )
% 0.24/0.51         => ( ( power_power_int @ zero_zero_int @ N )
% 0.24/0.51            = one_one_int ) )
% 0.24/0.51        & ( ( N != zero_zero_nat )
% 0.24/0.51         => ( ( power_power_int @ zero_zero_int @ N )
% 0.24/0.51            = zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_0_left
% 0.24/0.51  thf(fact_1758_power__0__left,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ( N = zero_zero_nat )
% 0.24/0.51         => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.24/0.51            = one_one_complex ) )
% 0.24/0.51        & ( ( N != zero_zero_nat )
% 0.24/0.51         => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.24/0.51            = zero_zero_complex ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_0_left
% 0.24/0.51  thf(fact_1759_ex__le__of__int,axiom,
% 0.24/0.51      ! [X: real] :
% 0.24/0.51      ? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ex_le_of_int
% 0.24/0.51  thf(fact_1760_ex__le__of__int,axiom,
% 0.24/0.51      ! [X: rat] :
% 0.24/0.51      ? [Z3: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ex_le_of_int
% 0.24/0.51  thf(fact_1761_nat__induct__non__zero,axiom,
% 0.24/0.51      ! [N: nat,P: nat > $o] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51       => ( ( P @ one_one_nat )
% 0.24/0.51         => ( ! [N2: nat] :
% 0.24/0.51                ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.24/0.51               => ( ( P @ N2 )
% 0.24/0.51                 => ( P @ ( suc @ N2 ) ) ) )
% 0.24/0.51           => ( P @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat_induct_non_zero
% 0.24/0.51  thf(fact_1762_power__gt__expt,axiom,
% 0.24/0.51      ! [N: nat,K: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.24/0.51       => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_gt_expt
% 0.24/0.51  thf(fact_1763_verit__comp__simplify1_I3_J,axiom,
% 0.24/0.51      ! [B7: real,A6: real] :
% 0.24/0.51        ( ( ~ ( ord_less_eq_real @ B7 @ A6 ) )
% 0.24/0.51        = ( ord_less_real @ A6 @ B7 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % verit_comp_simplify1(3)
% 0.24/0.51  thf(fact_1764_verit__comp__simplify1_I3_J,axiom,
% 0.24/0.51      ! [B7: rat,A6: rat] :
% 0.24/0.51        ( ( ~ ( ord_less_eq_rat @ B7 @ A6 ) )
% 0.24/0.51        = ( ord_less_rat @ A6 @ B7 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % verit_comp_simplify1(3)
% 0.24/0.51  thf(fact_1765_verit__comp__simplify1_I3_J,axiom,
% 0.24/0.51      ! [B7: num,A6: num] :
% 0.24/0.51        ( ( ~ ( ord_less_eq_num @ B7 @ A6 ) )
% 0.24/0.51        = ( ord_less_num @ A6 @ B7 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % verit_comp_simplify1(3)
% 0.24/0.51  thf(fact_1766_verit__comp__simplify1_I3_J,axiom,
% 0.24/0.51      ! [B7: nat,A6: nat] :
% 0.24/0.51        ( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
% 0.24/0.51        = ( ord_less_nat @ A6 @ B7 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % verit_comp_simplify1(3)
% 0.24/0.51  thf(fact_1767_verit__comp__simplify1_I3_J,axiom,
% 0.24/0.51      ! [B7: int,A6: int] :
% 0.24/0.51        ( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
% 0.24/0.51        = ( ord_less_int @ A6 @ B7 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % verit_comp_simplify1(3)
% 0.24/0.51  thf(fact_1768_power__divide,axiom,
% 0.24/0.51      ! [A: complex,B: complex,N: nat] :
% 0.24/0.51        ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
% 0.24/0.51        = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_divide
% 0.24/0.51  thf(fact_1769_power__divide,axiom,
% 0.24/0.51      ! [A: real,B: real,N: nat] :
% 0.24/0.51        ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 0.24/0.51        = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_divide
% 0.24/0.51  thf(fact_1770_power__divide,axiom,
% 0.24/0.51      ! [A: rat,B: rat,N: nat] :
% 0.24/0.51        ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
% 0.24/0.51        = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_divide
% 0.24/0.51  thf(fact_1771_less__numeral__extra_I4_J,axiom,
% 0.24/0.51      ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.24/0.51  
% 0.24/0.51  % less_numeral_extra(4)
% 0.24/0.51  thf(fact_1772_less__numeral__extra_I4_J,axiom,
% 0.24/0.51      ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % less_numeral_extra(4)
% 0.24/0.51  thf(fact_1773_less__numeral__extra_I4_J,axiom,
% 0.24/0.51      ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % less_numeral_extra(4)
% 0.24/0.51  thf(fact_1774_less__numeral__extra_I4_J,axiom,
% 0.24/0.51      ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.24/0.51  
% 0.24/0.51  % less_numeral_extra(4)
% 0.24/0.51  thf(fact_1775_add__less__imp__less__right,axiom,
% 0.24/0.51      ! [A: real,C: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.24/0.51       => ( ord_less_real @ A @ B ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_imp_less_right
% 0.24/0.51  thf(fact_1776_add__less__imp__less__right,axiom,
% 0.24/0.51      ! [A: rat,C: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.24/0.51       => ( ord_less_rat @ A @ B ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_imp_less_right
% 0.24/0.51  thf(fact_1777_add__less__imp__less__right,axiom,
% 0.24/0.51      ! [A: nat,C: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.24/0.51       => ( ord_less_nat @ A @ B ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_imp_less_right
% 0.24/0.51  thf(fact_1778_add__less__imp__less__right,axiom,
% 0.24/0.51      ! [A: int,C: int,B: int] :
% 0.24/0.51        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.24/0.51       => ( ord_less_int @ A @ B ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_imp_less_right
% 0.24/0.51  thf(fact_1779_add__less__imp__less__left,axiom,
% 0.24/0.51      ! [C: real,A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.24/0.51       => ( ord_less_real @ A @ B ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_imp_less_left
% 0.24/0.51  thf(fact_1780_add__less__imp__less__left,axiom,
% 0.24/0.51      ! [C: rat,A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.24/0.51       => ( ord_less_rat @ A @ B ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_imp_less_left
% 0.24/0.51  thf(fact_1781_add__less__imp__less__left,axiom,
% 0.24/0.51      ! [C: nat,A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.24/0.51       => ( ord_less_nat @ A @ B ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_imp_less_left
% 0.24/0.51  thf(fact_1782_add__less__imp__less__left,axiom,
% 0.24/0.51      ! [C: int,A: int,B: int] :
% 0.24/0.51        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.24/0.51       => ( ord_less_int @ A @ B ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_imp_less_left
% 0.24/0.51  thf(fact_1783_add__strict__right__mono,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ B )
% 0.24/0.51       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_right_mono
% 0.24/0.51  thf(fact_1784_add__strict__right__mono,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ B )
% 0.24/0.51       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_right_mono
% 0.24/0.51  thf(fact_1785_add__strict__right__mono,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat] :
% 0.24/0.51        ( ( ord_less_nat @ A @ B )
% 0.24/0.51       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_right_mono
% 0.24/0.51  thf(fact_1786_add__strict__right__mono,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int] :
% 0.24/0.51        ( ( ord_less_int @ A @ B )
% 0.24/0.51       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_right_mono
% 0.24/0.51  thf(fact_1787_add__strict__left__mono,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ B )
% 0.24/0.51       => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_left_mono
% 0.24/0.51  thf(fact_1788_add__strict__left__mono,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ B )
% 0.24/0.51       => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_left_mono
% 0.24/0.51  thf(fact_1789_add__strict__left__mono,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat] :
% 0.24/0.51        ( ( ord_less_nat @ A @ B )
% 0.24/0.51       => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_left_mono
% 0.24/0.51  thf(fact_1790_add__strict__left__mono,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int] :
% 0.24/0.51        ( ( ord_less_int @ A @ B )
% 0.24/0.51       => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_left_mono
% 0.24/0.51  thf(fact_1791_add__strict__mono,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real,D: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ B )
% 0.24/0.51       => ( ( ord_less_real @ C @ D )
% 0.24/0.51         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_mono
% 0.24/0.51  thf(fact_1792_add__strict__mono,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_rat @ C @ D )
% 0.24/0.51         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_mono
% 0.24/0.51  thf(fact_1793_add__strict__mono,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.24/0.51        ( ( ord_less_nat @ A @ B )
% 0.24/0.51       => ( ( ord_less_nat @ C @ D )
% 0.24/0.51         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_mono
% 0.24/0.51  thf(fact_1794_add__strict__mono,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int,D: int] :
% 0.24/0.51        ( ( ord_less_int @ A @ B )
% 0.24/0.51       => ( ( ord_less_int @ C @ D )
% 0.24/0.51         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_strict_mono
% 0.24/0.51  thf(fact_1795_add__mono__thms__linordered__field_I1_J,axiom,
% 0.24/0.51      ! [I: real,J: real,K: real,L: real] :
% 0.24/0.51        ( ( ( ord_less_real @ I @ J )
% 0.24/0.51          & ( K = L ) )
% 0.24/0.51       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(1)
% 0.24/0.51  thf(fact_1796_add__mono__thms__linordered__field_I1_J,axiom,
% 0.24/0.51      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.24/0.51        ( ( ( ord_less_rat @ I @ J )
% 0.24/0.51          & ( K = L ) )
% 0.24/0.51       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(1)
% 0.24/0.51  thf(fact_1797_add__mono__thms__linordered__field_I1_J,axiom,
% 0.24/0.51      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.51        ( ( ( ord_less_nat @ I @ J )
% 0.24/0.51          & ( K = L ) )
% 0.24/0.51       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(1)
% 0.24/0.51  thf(fact_1798_add__mono__thms__linordered__field_I1_J,axiom,
% 0.24/0.51      ! [I: int,J: int,K: int,L: int] :
% 0.24/0.51        ( ( ( ord_less_int @ I @ J )
% 0.24/0.51          & ( K = L ) )
% 0.24/0.51       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(1)
% 0.24/0.51  thf(fact_1799_add__mono__thms__linordered__field_I2_J,axiom,
% 0.24/0.51      ! [I: real,J: real,K: real,L: real] :
% 0.24/0.51        ( ( ( I = J )
% 0.24/0.51          & ( ord_less_real @ K @ L ) )
% 0.24/0.51       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(2)
% 0.24/0.51  thf(fact_1800_add__mono__thms__linordered__field_I2_J,axiom,
% 0.24/0.51      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.24/0.51        ( ( ( I = J )
% 0.24/0.51          & ( ord_less_rat @ K @ L ) )
% 0.24/0.51       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(2)
% 0.24/0.51  thf(fact_1801_add__mono__thms__linordered__field_I2_J,axiom,
% 0.24/0.51      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.51        ( ( ( I = J )
% 0.24/0.51          & ( ord_less_nat @ K @ L ) )
% 0.24/0.51       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(2)
% 0.24/0.51  thf(fact_1802_add__mono__thms__linordered__field_I2_J,axiom,
% 0.24/0.51      ! [I: int,J: int,K: int,L: int] :
% 0.24/0.51        ( ( ( I = J )
% 0.24/0.51          & ( ord_less_int @ K @ L ) )
% 0.24/0.51       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(2)
% 0.24/0.51  thf(fact_1803_add__mono__thms__linordered__field_I5_J,axiom,
% 0.24/0.51      ! [I: real,J: real,K: real,L: real] :
% 0.24/0.51        ( ( ( ord_less_real @ I @ J )
% 0.24/0.51          & ( ord_less_real @ K @ L ) )
% 0.24/0.51       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(5)
% 0.24/0.51  thf(fact_1804_add__mono__thms__linordered__field_I5_J,axiom,
% 0.24/0.51      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.24/0.51        ( ( ( ord_less_rat @ I @ J )
% 0.24/0.51          & ( ord_less_rat @ K @ L ) )
% 0.24/0.51       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(5)
% 0.24/0.51  thf(fact_1805_add__mono__thms__linordered__field_I5_J,axiom,
% 0.24/0.51      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.51        ( ( ( ord_less_nat @ I @ J )
% 0.24/0.51          & ( ord_less_nat @ K @ L ) )
% 0.24/0.51       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(5)
% 0.24/0.51  thf(fact_1806_add__mono__thms__linordered__field_I5_J,axiom,
% 0.24/0.51      ! [I: int,J: int,K: int,L: int] :
% 0.24/0.51        ( ( ( ord_less_int @ I @ J )
% 0.24/0.51          & ( ord_less_int @ K @ L ) )
% 0.24/0.51       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(5)
% 0.24/0.51  thf(fact_1807_reals__Archimedean2,axiom,
% 0.24/0.51      ! [X: real] :
% 0.24/0.51      ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % reals_Archimedean2
% 0.24/0.51  thf(fact_1808_reals__Archimedean2,axiom,
% 0.24/0.51      ! [X: rat] :
% 0.24/0.51      ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % reals_Archimedean2
% 0.24/0.51  thf(fact_1809_not__less__less__Suc__eq,axiom,
% 0.24/0.51      ! [N: nat,M: nat] :
% 0.24/0.51        ( ~ ( ord_less_nat @ N @ M )
% 0.24/0.51       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.24/0.51          = ( N = M ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % not_less_less_Suc_eq
% 0.24/0.51  thf(fact_1810_strict__inc__induct,axiom,
% 0.24/0.51      ! [I: nat,J: nat,P: nat > $o] :
% 0.24/0.51        ( ( ord_less_nat @ I @ J )
% 0.24/0.51       => ( ! [I2: nat] :
% 0.24/0.51              ( ( J
% 0.24/0.51                = ( suc @ I2 ) )
% 0.24/0.51             => ( P @ I2 ) )
% 0.24/0.51         => ( ! [I2: nat] :
% 0.24/0.51                ( ( ord_less_nat @ I2 @ J )
% 0.24/0.51               => ( ( P @ ( suc @ I2 ) )
% 0.24/0.51                 => ( P @ I2 ) ) )
% 0.24/0.51           => ( P @ I ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % strict_inc_induct
% 0.24/0.51  thf(fact_1811_less__Suc__induct,axiom,
% 0.24/0.51      ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.24/0.51        ( ( ord_less_nat @ I @ J )
% 0.24/0.51       => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 0.24/0.51         => ( ! [I2: nat,J3: nat,K3: nat] :
% 0.24/0.51                ( ( ord_less_nat @ I2 @ J3 )
% 0.24/0.51               => ( ( ord_less_nat @ J3 @ K3 )
% 0.24/0.51                 => ( ( P @ I2 @ J3 )
% 0.24/0.51                   => ( ( P @ J3 @ K3 )
% 0.24/0.51                     => ( P @ I2 @ K3 ) ) ) ) )
% 0.24/0.51           => ( P @ I @ J ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_Suc_induct
% 0.24/0.51  thf(fact_1812_less__trans__Suc,axiom,
% 0.24/0.51      ! [I: nat,J: nat,K: nat] :
% 0.24/0.51        ( ( ord_less_nat @ I @ J )
% 0.24/0.51       => ( ( ord_less_nat @ J @ K )
% 0.24/0.51         => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_trans_Suc
% 0.24/0.51  thf(fact_1813_Suc__less__SucD,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.24/0.51       => ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_less_SucD
% 0.24/0.51  thf(fact_1814_less__antisym,axiom,
% 0.24/0.51      ! [N: nat,M: nat] :
% 0.24/0.51        ( ~ ( ord_less_nat @ N @ M )
% 0.24/0.51       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.24/0.51         => ( M = N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_antisym
% 0.24/0.51  thf(fact_1815_Suc__less__eq2,axiom,
% 0.24/0.51      ! [N: nat,M: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.24/0.51        = ( ? [M7: nat] :
% 0.24/0.51              ( ( M
% 0.24/0.51                = ( suc @ M7 ) )
% 0.24/0.51              & ( ord_less_nat @ N @ M7 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_less_eq2
% 0.24/0.51  thf(fact_1816_All__less__Suc,axiom,
% 0.24/0.51      ! [N: nat,P: nat > $o] :
% 0.24/0.51        ( ( ! [I3: nat] :
% 0.24/0.51              ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.24/0.51             => ( P @ I3 ) ) )
% 0.24/0.51        = ( ( P @ N )
% 0.24/0.51          & ! [I3: nat] :
% 0.24/0.51              ( ( ord_less_nat @ I3 @ N )
% 0.24/0.51             => ( P @ I3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % All_less_Suc
% 0.24/0.51  thf(fact_1817_not__less__eq,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.24/0.51        = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % not_less_eq
% 0.24/0.51  thf(fact_1818_less__Suc__eq,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.24/0.51        = ( ( ord_less_nat @ M @ N )
% 0.24/0.51          | ( M = N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_Suc_eq
% 0.24/0.51  thf(fact_1819_Ex__less__Suc,axiom,
% 0.24/0.51      ! [N: nat,P: nat > $o] :
% 0.24/0.51        ( ( ? [I3: nat] :
% 0.24/0.51              ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.24/0.51              & ( P @ I3 ) ) )
% 0.24/0.51        = ( ( P @ N )
% 0.24/0.51          | ? [I3: nat] :
% 0.24/0.51              ( ( ord_less_nat @ I3 @ N )
% 0.24/0.51              & ( P @ I3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Ex_less_Suc
% 0.24/0.51  thf(fact_1820_less__SucI,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ N )
% 0.24/0.51       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_SucI
% 0.24/0.51  thf(fact_1821_less__SucE,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.24/0.51       => ( ~ ( ord_less_nat @ M @ N )
% 0.24/0.51         => ( M = N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_SucE
% 0.24/0.51  thf(fact_1822_Suc__lessI,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ N )
% 0.24/0.51       => ( ( ( suc @ M )
% 0.24/0.51           != N )
% 0.24/0.51         => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_lessI
% 0.24/0.51  thf(fact_1823_Suc__lessE,axiom,
% 0.24/0.51      ! [I: nat,K: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.24/0.51       => ~ ! [J3: nat] :
% 0.24/0.51              ( ( ord_less_nat @ I @ J3 )
% 0.24/0.51             => ( K
% 0.24/0.51               != ( suc @ J3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_lessE
% 0.24/0.51  thf(fact_1824_Suc__lessD,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.24/0.51       => ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_lessD
% 0.24/0.51  thf(fact_1825_Nat_OlessE,axiom,
% 0.24/0.51      ! [I: nat,K: nat] :
% 0.24/0.51        ( ( ord_less_nat @ I @ K )
% 0.24/0.51       => ( ( K
% 0.24/0.51           != ( suc @ I ) )
% 0.24/0.51         => ~ ! [J3: nat] :
% 0.24/0.51                ( ( ord_less_nat @ I @ J3 )
% 0.24/0.51               => ( K
% 0.24/0.51                 != ( suc @ J3 ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Nat.lessE
% 0.24/0.51  thf(fact_1826_less__mono__imp__le__mono,axiom,
% 0.24/0.51      ! [F: nat > nat,I: nat,J: nat] :
% 0.24/0.51        ( ! [I2: nat,J3: nat] :
% 0.24/0.51            ( ( ord_less_nat @ I2 @ J3 )
% 0.24/0.51           => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.51         => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_mono_imp_le_mono
% 0.24/0.51  thf(fact_1827_le__neq__implies__less,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.51       => ( ( M != N )
% 0.24/0.51         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_neq_implies_less
% 0.24/0.51  thf(fact_1828_less__or__eq__imp__le,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ( ord_less_nat @ M @ N )
% 0.24/0.51          | ( M = N ) )
% 0.24/0.51       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_or_eq_imp_le
% 0.24/0.51  thf(fact_1829_le__eq__less__or__eq,axiom,
% 0.24/0.51      ( ord_less_eq_nat
% 0.24/0.51      = ( ^ [M6: nat,N3: nat] :
% 0.24/0.51            ( ( ord_less_nat @ M6 @ N3 )
% 0.24/0.51            | ( M6 = N3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_eq_less_or_eq
% 0.24/0.51  thf(fact_1830_less__imp__le__nat,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ N )
% 0.24/0.51       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_imp_le_nat
% 0.24/0.51  thf(fact_1831_nat__less__le,axiom,
% 0.24/0.51      ( ord_less_nat
% 0.24/0.51      = ( ^ [M6: nat,N3: nat] :
% 0.24/0.51            ( ( ord_less_eq_nat @ M6 @ N3 )
% 0.24/0.51            & ( M6 != N3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat_less_le
% 0.24/0.51  thf(fact_1832_zero__le,axiom,
% 0.24/0.51      ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le
% 0.24/0.51  thf(fact_1833_le__numeral__extra_I3_J,axiom,
% 0.24/0.51      ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.24/0.51  
% 0.24/0.51  % le_numeral_extra(3)
% 0.24/0.51  thf(fact_1834_le__numeral__extra_I3_J,axiom,
% 0.24/0.51      ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.24/0.51  
% 0.24/0.51  % le_numeral_extra(3)
% 0.24/0.51  thf(fact_1835_le__numeral__extra_I3_J,axiom,
% 0.24/0.51      ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.24/0.51  
% 0.24/0.51  % le_numeral_extra(3)
% 0.24/0.51  thf(fact_1836_le__numeral__extra_I3_J,axiom,
% 0.24/0.51      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.24/0.51  
% 0.24/0.51  % le_numeral_extra(3)
% 0.24/0.51  thf(fact_1837_add__lessD1,axiom,
% 0.24/0.51      ! [I: nat,J: nat,K: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.24/0.51       => ( ord_less_nat @ I @ K ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_lessD1
% 0.24/0.51  thf(fact_1838_add__less__mono,axiom,
% 0.24/0.51      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.51        ( ( ord_less_nat @ I @ J )
% 0.24/0.51       => ( ( ord_less_nat @ K @ L )
% 0.24/0.51         => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_mono
% 0.24/0.51  thf(fact_1839_not__add__less1,axiom,
% 0.24/0.51      ! [I: nat,J: nat] :
% 0.24/0.51        ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.24/0.51  
% 0.24/0.51  % not_add_less1
% 0.24/0.51  thf(fact_1840_not__add__less2,axiom,
% 0.24/0.51      ! [J: nat,I: nat] :
% 0.24/0.51        ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.24/0.51  
% 0.24/0.51  % not_add_less2
% 0.24/0.51  thf(fact_1841_add__less__mono1,axiom,
% 0.24/0.51      ! [I: nat,J: nat,K: nat] :
% 0.24/0.51        ( ( ord_less_nat @ I @ J )
% 0.24/0.51       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_mono1
% 0.24/0.51  thf(fact_1842_trans__less__add1,axiom,
% 0.24/0.51      ! [I: nat,J: nat,M: nat] :
% 0.24/0.51        ( ( ord_less_nat @ I @ J )
% 0.24/0.51       => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % trans_less_add1
% 0.24/0.51  thf(fact_1843_trans__less__add2,axiom,
% 0.24/0.51      ! [I: nat,J: nat,M: nat] :
% 0.24/0.51        ( ( ord_less_nat @ I @ J )
% 0.24/0.51       => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % trans_less_add2
% 0.24/0.51  thf(fact_1844_less__add__eq__less,axiom,
% 0.24/0.51      ! [K: nat,L: nat,M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ K @ L )
% 0.24/0.51       => ( ( ( plus_plus_nat @ M @ L )
% 0.24/0.51            = ( plus_plus_nat @ K @ N ) )
% 0.24/0.51         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_add_eq_less
% 0.24/0.51  thf(fact_1845_zero__neq__numeral,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ( zero_zero_complex
% 0.24/0.51       != ( numera6690914467698888265omplex @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_neq_numeral
% 0.24/0.51  thf(fact_1846_zero__neq__numeral,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ( zero_zero_real
% 0.24/0.51       != ( numeral_numeral_real @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_neq_numeral
% 0.24/0.51  thf(fact_1847_zero__neq__numeral,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ( zero_zero_rat
% 0.24/0.51       != ( numeral_numeral_rat @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_neq_numeral
% 0.24/0.51  thf(fact_1848_zero__neq__numeral,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ( zero_zero_nat
% 0.24/0.51       != ( numeral_numeral_nat @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_neq_numeral
% 0.24/0.51  thf(fact_1849_zero__neq__numeral,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ( zero_zero_int
% 0.24/0.51       != ( numeral_numeral_int @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_neq_numeral
% 0.24/0.51  thf(fact_1850_add_Ogroup__left__neutral,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % add.group_left_neutral
% 0.24/0.51  thf(fact_1851_add_Ogroup__left__neutral,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % add.group_left_neutral
% 0.24/0.51  thf(fact_1852_add_Ogroup__left__neutral,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % add.group_left_neutral
% 0.24/0.51  thf(fact_1853_add_Ogroup__left__neutral,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % add.group_left_neutral
% 0.24/0.51  thf(fact_1854_add_Ocomm__neutral,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % add.comm_neutral
% 0.24/0.51  thf(fact_1855_add_Ocomm__neutral,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % add.comm_neutral
% 0.24/0.51  thf(fact_1856_add_Ocomm__neutral,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % add.comm_neutral
% 0.24/0.51  thf(fact_1857_add_Ocomm__neutral,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % add.comm_neutral
% 0.24/0.51  thf(fact_1858_add_Ocomm__neutral,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % add.comm_neutral
% 0.24/0.51  thf(fact_1859_comm__monoid__add__class_Oadd__0,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % comm_monoid_add_class.add_0
% 0.24/0.51  thf(fact_1860_comm__monoid__add__class_Oadd__0,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % comm_monoid_add_class.add_0
% 0.24/0.51  thf(fact_1861_comm__monoid__add__class_Oadd__0,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % comm_monoid_add_class.add_0
% 0.24/0.51  thf(fact_1862_comm__monoid__add__class_Oadd__0,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % comm_monoid_add_class.add_0
% 0.24/0.51  thf(fact_1863_comm__monoid__add__class_Oadd__0,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % comm_monoid_add_class.add_0
% 0.24/0.51  thf(fact_1864_verit__sum__simplify,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % verit_sum_simplify
% 0.24/0.51  thf(fact_1865_verit__sum__simplify,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % verit_sum_simplify
% 0.24/0.51  thf(fact_1866_verit__sum__simplify,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % verit_sum_simplify
% 0.24/0.51  thf(fact_1867_verit__sum__simplify,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % verit_sum_simplify
% 0.24/0.51  thf(fact_1868_verit__sum__simplify,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % verit_sum_simplify
% 0.24/0.51  thf(fact_1869_power__not__zero,axiom,
% 0.24/0.51      ! [A: rat,N: nat] :
% 0.24/0.51        ( ( A != zero_zero_rat )
% 0.24/0.51       => ( ( power_power_rat @ A @ N )
% 0.24/0.51         != zero_zero_rat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_not_zero
% 0.24/0.51  thf(fact_1870_power__not__zero,axiom,
% 0.24/0.51      ! [A: nat,N: nat] :
% 0.24/0.51        ( ( A != zero_zero_nat )
% 0.24/0.51       => ( ( power_power_nat @ A @ N )
% 0.24/0.51         != zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_not_zero
% 0.24/0.51  thf(fact_1871_power__not__zero,axiom,
% 0.24/0.51      ! [A: real,N: nat] :
% 0.24/0.51        ( ( A != zero_zero_real )
% 0.24/0.51       => ( ( power_power_real @ A @ N )
% 0.24/0.51         != zero_zero_real ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_not_zero
% 0.24/0.51  thf(fact_1872_power__not__zero,axiom,
% 0.24/0.51      ! [A: int,N: nat] :
% 0.24/0.51        ( ( A != zero_zero_int )
% 0.24/0.51       => ( ( power_power_int @ A @ N )
% 0.24/0.51         != zero_zero_int ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_not_zero
% 0.24/0.51  thf(fact_1873_power__not__zero,axiom,
% 0.24/0.51      ! [A: complex,N: nat] :
% 0.24/0.51        ( ( A != zero_zero_complex )
% 0.24/0.51       => ( ( power_power_complex @ A @ N )
% 0.24/0.51         != zero_zero_complex ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_not_zero
% 0.24/0.51  thf(fact_1874_num_Osize_I4_J,axiom,
% 0.24/0.51      ( ( size_size_num @ one )
% 0.24/0.51      = zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % num.size(4)
% 0.24/0.51  thf(fact_1875_vebt__buildup_Ocases,axiom,
% 0.24/0.51      ! [X: nat] :
% 0.24/0.51        ( ( X != zero_zero_nat )
% 0.24/0.51       => ( ( X
% 0.24/0.51           != ( suc @ zero_zero_nat ) )
% 0.24/0.51         => ~ ! [Va: nat] :
% 0.24/0.51                ( X
% 0.24/0.51               != ( suc @ ( suc @ Va ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % vebt_buildup.cases
% 0.24/0.51  thf(fact_1876_exists__least__lemma,axiom,
% 0.24/0.51      ! [P: nat > $o] :
% 0.24/0.51        ( ~ ( P @ zero_zero_nat )
% 0.24/0.51       => ( ? [X_1: nat] : ( P @ X_1 )
% 0.24/0.51         => ? [N2: nat] :
% 0.24/0.51              ( ~ ( P @ N2 )
% 0.24/0.51              & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % exists_least_lemma
% 0.24/0.51  thf(fact_1877_not0__implies__Suc,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( N != zero_zero_nat )
% 0.24/0.51       => ? [M5: nat] :
% 0.24/0.51            ( N
% 0.24/0.51            = ( suc @ M5 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % not0_implies_Suc
% 0.24/0.51  thf(fact_1878_Zero__not__Suc,axiom,
% 0.24/0.51      ! [M: nat] :
% 0.24/0.51        ( zero_zero_nat
% 0.24/0.51       != ( suc @ M ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Zero_not_Suc
% 0.24/0.51  thf(fact_1879_Zero__neq__Suc,axiom,
% 0.24/0.51      ! [M: nat] :
% 0.24/0.51        ( zero_zero_nat
% 0.24/0.51       != ( suc @ M ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Zero_neq_Suc
% 0.24/0.51  thf(fact_1880_Suc__neq__Zero,axiom,
% 0.24/0.51      ! [M: nat] :
% 0.24/0.51        ( ( suc @ M )
% 0.24/0.51       != zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_neq_Zero
% 0.24/0.51  thf(fact_1881_zero__induct,axiom,
% 0.24/0.51      ! [P: nat > $o,K: nat] :
% 0.24/0.51        ( ( P @ K )
% 0.24/0.51       => ( ! [N2: nat] :
% 0.24/0.51              ( ( P @ ( suc @ N2 ) )
% 0.24/0.51             => ( P @ N2 ) )
% 0.24/0.51         => ( P @ zero_zero_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_induct
% 0.24/0.51  thf(fact_1882_diff__induct,axiom,
% 0.24/0.51      ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.24/0.51        ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 0.24/0.51       => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 0.24/0.51         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.51                ( ( P @ X3 @ Y4 )
% 0.24/0.51               => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 0.24/0.51           => ( P @ M @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % diff_induct
% 0.24/0.51  thf(fact_1883_nat__induct,axiom,
% 0.24/0.51      ! [P: nat > $o,N: nat] :
% 0.24/0.51        ( ( P @ zero_zero_nat )
% 0.24/0.51       => ( ! [N2: nat] :
% 0.24/0.51              ( ( P @ N2 )
% 0.24/0.51             => ( P @ ( suc @ N2 ) ) )
% 0.24/0.51         => ( P @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat_induct
% 0.24/0.51  thf(fact_1884_old_Onat_Oexhaust,axiom,
% 0.24/0.51      ! [Y: nat] :
% 0.24/0.51        ( ( Y != zero_zero_nat )
% 0.24/0.51       => ~ ! [Nat3: nat] :
% 0.24/0.51              ( Y
% 0.24/0.51             != ( suc @ Nat3 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % old.nat.exhaust
% 0.24/0.51  thf(fact_1885_nat_OdiscI,axiom,
% 0.24/0.51      ! [Nat: nat,X22: nat] :
% 0.24/0.51        ( ( Nat
% 0.24/0.51          = ( suc @ X22 ) )
% 0.24/0.51       => ( Nat != zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat.discI
% 0.24/0.51  thf(fact_1886_old_Onat_Odistinct_I1_J,axiom,
% 0.24/0.51      ! [Nat2: nat] :
% 0.24/0.51        ( zero_zero_nat
% 0.24/0.51       != ( suc @ Nat2 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % old.nat.distinct(1)
% 0.24/0.51  thf(fact_1887_old_Onat_Odistinct_I2_J,axiom,
% 0.24/0.51      ! [Nat2: nat] :
% 0.24/0.51        ( ( suc @ Nat2 )
% 0.24/0.51       != zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % old.nat.distinct(2)
% 0.24/0.51  thf(fact_1888_nat_Odistinct_I1_J,axiom,
% 0.24/0.51      ! [X22: nat] :
% 0.24/0.51        ( zero_zero_nat
% 0.24/0.51       != ( suc @ X22 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat.distinct(1)
% 0.24/0.51  thf(fact_1889_le__0__eq,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.24/0.51        = ( N = zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_0_eq
% 0.24/0.51  thf(fact_1890_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.24/0.51       => ( A = zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % bot_nat_0.extremum_uniqueI
% 0.24/0.51  thf(fact_1891_bot__nat__0_Oextremum__unique,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.24/0.51        = ( A = zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % bot_nat_0.extremum_unique
% 0.24/0.51  thf(fact_1892_less__eq__nat_Osimps_I1_J,axiom,
% 0.24/0.51      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.24/0.51  
% 0.24/0.51  % less_eq_nat.simps(1)
% 0.24/0.51  thf(fact_1893_plus__nat_Oadd__0,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.24/0.51        = N ) ).
% 0.24/0.51  
% 0.24/0.51  % plus_nat.add_0
% 0.24/0.51  thf(fact_1894_add__eq__self__zero,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ( plus_plus_nat @ M @ N )
% 0.24/0.51          = M )
% 0.24/0.51       => ( N = zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_eq_self_zero
% 0.24/0.51  thf(fact_1895_floor__exists1,axiom,
% 0.24/0.51      ! [X: real] :
% 0.24/0.51      ? [X3: int] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
% 0.24/0.51        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 0.24/0.51        & ! [Y5: int] :
% 0.24/0.51            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
% 0.24/0.51              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.24/0.51           => ( Y5 = X3 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % floor_exists1
% 0.24/0.51  thf(fact_1896_floor__exists1,axiom,
% 0.24/0.51      ! [X: rat] :
% 0.24/0.51      ? [X3: int] :
% 0.24/0.51        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
% 0.24/0.51        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 0.24/0.51        & ! [Y5: int] :
% 0.24/0.51            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
% 0.24/0.51              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.24/0.51           => ( Y5 = X3 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % floor_exists1
% 0.24/0.51  thf(fact_1897_floor__exists,axiom,
% 0.24/0.51      ! [X: real] :
% 0.24/0.51      ? [Z3: int] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
% 0.24/0.51        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % floor_exists
% 0.24/0.51  thf(fact_1898_floor__exists,axiom,
% 0.24/0.51      ! [X: rat] :
% 0.24/0.51      ? [Z3: int] :
% 0.24/0.51        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
% 0.24/0.51        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % floor_exists
% 0.24/0.51  thf(fact_1899_power__Suc__less__one,axiom,
% 0.24/0.51      ! [A: real,N: nat] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ A @ one_one_real )
% 0.24/0.51         => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_Suc_less_one
% 0.24/0.51  thf(fact_1900_power__Suc__less__one,axiom,
% 0.24/0.51      ! [A: rat,N: nat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_rat @ A @ one_one_rat )
% 0.24/0.51         => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_Suc_less_one
% 0.24/0.51  thf(fact_1901_power__Suc__less__one,axiom,
% 0.24/0.51      ! [A: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.24/0.51       => ( ( ord_less_nat @ A @ one_one_nat )
% 0.24/0.51         => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_Suc_less_one
% 0.24/0.51  thf(fact_1902_power__Suc__less__one,axiom,
% 0.24/0.51      ! [A: int,N: nat] :
% 0.24/0.51        ( ( ord_less_int @ zero_zero_int @ A )
% 0.24/0.51       => ( ( ord_less_int @ A @ one_one_int )
% 0.24/0.51         => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_Suc_less_one
% 0.24/0.51  thf(fact_1903_self__le__power,axiom,
% 0.24/0.51      ! [A: real,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51         => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % self_le_power
% 0.24/0.51  thf(fact_1904_self__le__power,axiom,
% 0.24/0.51      ! [A: rat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51         => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % self_le_power
% 0.24/0.51  thf(fact_1905_self__le__power,axiom,
% 0.24/0.51      ! [A: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51         => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % self_le_power
% 0.24/0.51  thf(fact_1906_self__le__power,axiom,
% 0.24/0.51      ! [A: int,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_int @ one_one_int @ A )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51         => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % self_le_power
% 0.24/0.51  thf(fact_1907_field__less__half__sum,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ X @ Y )
% 0.24/0.51       => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % field_less_half_sum
% 0.24/0.51  thf(fact_1908_field__less__half__sum,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_rat @ X @ Y )
% 0.24/0.51       => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % field_less_half_sum
% 0.24/0.51  thf(fact_1909_inverse__of__nat__le,axiom,
% 0.24/0.51      ! [N: nat,M: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ N @ M )
% 0.24/0.51       => ( ( N != zero_zero_nat )
% 0.24/0.51         => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % inverse_of_nat_le
% 0.24/0.51  thf(fact_1910_inverse__of__nat__le,axiom,
% 0.24/0.51      ! [N: nat,M: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ N @ M )
% 0.24/0.51       => ( ( N != zero_zero_nat )
% 0.24/0.51         => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % inverse_of_nat_le
% 0.24/0.51  thf(fact_1911_power2__less__0,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_less_0
% 0.24/0.51  thf(fact_1912_power2__less__0,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_less_0
% 0.24/0.51  thf(fact_1913_power2__less__0,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_less_0
% 0.24/0.51  thf(fact_1914_less__2__cases,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51       => ( ( N = zero_zero_nat )
% 0.24/0.51          | ( N
% 0.24/0.51            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_2_cases
% 0.24/0.51  thf(fact_1915_less__2__cases__iff,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51        = ( ( N = zero_zero_nat )
% 0.24/0.51          | ( N
% 0.24/0.51            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_2_cases_iff
% 0.24/0.51  thf(fact_1916_divide__numeral__1,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_numeral_1
% 0.24/0.51  thf(fact_1917_divide__numeral__1,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_numeral_1
% 0.24/0.51  thf(fact_1918_divide__numeral__1,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_numeral_1
% 0.24/0.51  thf(fact_1919_le__of__int__ceiling,axiom,
% 0.24/0.51      ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_of_int_ceiling
% 0.24/0.51  thf(fact_1920_le__of__int__ceiling,axiom,
% 0.24/0.51      ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_of_int_ceiling
% 0.24/0.51  thf(fact_1921_power__one__over,axiom,
% 0.24/0.51      ! [A: complex,N: nat] :
% 0.24/0.51        ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 0.24/0.51        = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_one_over
% 0.24/0.51  thf(fact_1922_power__one__over,axiom,
% 0.24/0.51      ! [A: real,N: nat] :
% 0.24/0.51        ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 0.24/0.51        = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_one_over
% 0.24/0.51  thf(fact_1923_power__one__over,axiom,
% 0.24/0.51      ! [A: rat,N: nat] :
% 0.24/0.51        ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 0.24/0.51        = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_one_over
% 0.24/0.51  thf(fact_1924_add__less__le__mono,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real,D: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_real @ C @ D )
% 0.24/0.51         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_le_mono
% 0.24/0.51  thf(fact_1925_add__less__le__mono,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_rat @ C @ D )
% 0.24/0.51         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_le_mono
% 0.24/0.51  thf(fact_1926_add__less__le__mono,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.24/0.51        ( ( ord_less_nat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_nat @ C @ D )
% 0.24/0.51         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_le_mono
% 0.24/0.51  thf(fact_1927_add__less__le__mono,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int,D: int] :
% 0.24/0.51        ( ( ord_less_int @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_int @ C @ D )
% 0.24/0.51         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_le_mono
% 0.24/0.51  thf(fact_1928_add__le__less__mono,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real,D: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.51       => ( ( ord_less_real @ C @ D )
% 0.24/0.51         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_le_less_mono
% 0.24/0.51  thf(fact_1929_add__le__less__mono,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_rat @ C @ D )
% 0.24/0.51         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_le_less_mono
% 0.24/0.51  thf(fact_1930_add__le__less__mono,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51       => ( ( ord_less_nat @ C @ D )
% 0.24/0.51         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_le_less_mono
% 0.24/0.51  thf(fact_1931_add__le__less__mono,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int,D: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.51       => ( ( ord_less_int @ C @ D )
% 0.24/0.51         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_le_less_mono
% 0.24/0.51  thf(fact_1932_add__mono__thms__linordered__field_I3_J,axiom,
% 0.24/0.51      ! [I: real,J: real,K: real,L: real] :
% 0.24/0.51        ( ( ( ord_less_real @ I @ J )
% 0.24/0.51          & ( ord_less_eq_real @ K @ L ) )
% 0.24/0.51       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(3)
% 0.24/0.51  thf(fact_1933_add__mono__thms__linordered__field_I3_J,axiom,
% 0.24/0.51      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.24/0.51        ( ( ( ord_less_rat @ I @ J )
% 0.24/0.51          & ( ord_less_eq_rat @ K @ L ) )
% 0.24/0.51       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(3)
% 0.24/0.51  thf(fact_1934_add__mono__thms__linordered__field_I3_J,axiom,
% 0.24/0.51      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.51        ( ( ( ord_less_nat @ I @ J )
% 0.24/0.51          & ( ord_less_eq_nat @ K @ L ) )
% 0.24/0.51       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(3)
% 0.24/0.51  thf(fact_1935_add__mono__thms__linordered__field_I3_J,axiom,
% 0.24/0.51      ! [I: int,J: int,K: int,L: int] :
% 0.24/0.51        ( ( ( ord_less_int @ I @ J )
% 0.24/0.51          & ( ord_less_eq_int @ K @ L ) )
% 0.24/0.51       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(3)
% 0.24/0.51  thf(fact_1936_add__mono__thms__linordered__field_I4_J,axiom,
% 0.24/0.51      ! [I: real,J: real,K: real,L: real] :
% 0.24/0.51        ( ( ( ord_less_eq_real @ I @ J )
% 0.24/0.51          & ( ord_less_real @ K @ L ) )
% 0.24/0.51       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(4)
% 0.24/0.51  thf(fact_1937_add__mono__thms__linordered__field_I4_J,axiom,
% 0.24/0.51      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.24/0.51        ( ( ( ord_less_eq_rat @ I @ J )
% 0.24/0.51          & ( ord_less_rat @ K @ L ) )
% 0.24/0.51       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(4)
% 0.24/0.51  thf(fact_1938_add__mono__thms__linordered__field_I4_J,axiom,
% 0.24/0.51      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.51        ( ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.51          & ( ord_less_nat @ K @ L ) )
% 0.24/0.51       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(4)
% 0.24/0.51  thf(fact_1939_add__mono__thms__linordered__field_I4_J,axiom,
% 0.24/0.51      ! [I: int,J: int,K: int,L: int] :
% 0.24/0.51        ( ( ( ord_less_eq_int @ I @ J )
% 0.24/0.51          & ( ord_less_int @ K @ L ) )
% 0.24/0.51       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono_thms_linordered_field(4)
% 0.24/0.51  thf(fact_1940_not__numeral__less__one,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 0.24/0.51  
% 0.24/0.51  % not_numeral_less_one
% 0.24/0.51  thf(fact_1941_not__numeral__less__one,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % not_numeral_less_one
% 0.24/0.51  thf(fact_1942_not__numeral__less__one,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % not_numeral_less_one
% 0.24/0.51  thf(fact_1943_not__numeral__less__one,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 0.24/0.51  
% 0.24/0.51  % not_numeral_less_one
% 0.24/0.51  thf(fact_1944_le__imp__less__Suc,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.51       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_imp_less_Suc
% 0.24/0.51  thf(fact_1945_less__eq__Suc__le,axiom,
% 0.24/0.51      ( ord_less_nat
% 0.24/0.51      = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_eq_Suc_le
% 0.24/0.51  thf(fact_1946_less__Suc__eq__le,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.24/0.51        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_Suc_eq_le
% 0.24/0.51  thf(fact_1947_le__less__Suc__eq,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.51       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.24/0.51          = ( N = M ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_less_Suc_eq
% 0.24/0.51  thf(fact_1948_Suc__le__lessD,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.24/0.51       => ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_le_lessD
% 0.24/0.51  thf(fact_1949_inc__induct,axiom,
% 0.24/0.51      ! [I: nat,J: nat,P: nat > $o] :
% 0.24/0.51        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.51       => ( ( P @ J )
% 0.24/0.51         => ( ! [N2: nat] :
% 0.24/0.51                ( ( ord_less_eq_nat @ I @ N2 )
% 0.24/0.51               => ( ( ord_less_nat @ N2 @ J )
% 0.24/0.51                 => ( ( P @ ( suc @ N2 ) )
% 0.24/0.51                   => ( P @ N2 ) ) ) )
% 0.24/0.51           => ( P @ I ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % inc_induct
% 0.24/0.51  thf(fact_1950_dec__induct,axiom,
% 0.24/0.51      ! [I: nat,J: nat,P: nat > $o] :
% 0.24/0.51        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.51       => ( ( P @ I )
% 0.24/0.51         => ( ! [N2: nat] :
% 0.24/0.51                ( ( ord_less_eq_nat @ I @ N2 )
% 0.24/0.51               => ( ( ord_less_nat @ N2 @ J )
% 0.24/0.51                 => ( ( P @ N2 )
% 0.24/0.51                   => ( P @ ( suc @ N2 ) ) ) ) )
% 0.24/0.51           => ( P @ J ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dec_induct
% 0.24/0.51  thf(fact_1951_Suc__le__eq,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.24/0.51        = ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_le_eq
% 0.24/0.51  thf(fact_1952_Suc__leI,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ N )
% 0.24/0.51       => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_leI
% 0.24/0.51  thf(fact_1953_less__natE,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ N )
% 0.24/0.51       => ~ ! [Q2: nat] :
% 0.24/0.51              ( N
% 0.24/0.51             != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_natE
% 0.24/0.51  thf(fact_1954_less__add__Suc1,axiom,
% 0.24/0.51      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_add_Suc1
% 0.24/0.51  thf(fact_1955_less__add__Suc2,axiom,
% 0.24/0.51      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_add_Suc2
% 0.24/0.51  thf(fact_1956_less__iff__Suc__add,axiom,
% 0.24/0.51      ( ord_less_nat
% 0.24/0.51      = ( ^ [M6: nat,N3: nat] :
% 0.24/0.51          ? [K2: nat] :
% 0.24/0.51            ( N3
% 0.24/0.51            = ( suc @ ( plus_plus_nat @ M6 @ K2 ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_iff_Suc_add
% 0.24/0.51  thf(fact_1957_less__imp__Suc__add,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ N )
% 0.24/0.51       => ? [K3: nat] :
% 0.24/0.51            ( N
% 0.24/0.51            = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_imp_Suc_add
% 0.24/0.51  thf(fact_1958_mono__nat__linear__lb,axiom,
% 0.24/0.51      ! [F: nat > nat,M: nat,K: nat] :
% 0.24/0.51        ( ! [M5: nat,N2: nat] :
% 0.24/0.51            ( ( ord_less_nat @ M5 @ N2 )
% 0.24/0.51           => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
% 0.24/0.51       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % mono_nat_linear_lb
% 0.24/0.51  thf(fact_1959_zero__le__numeral,axiom,
% 0.24/0.51      ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_numeral
% 0.24/0.51  thf(fact_1960_zero__le__numeral,axiom,
% 0.24/0.51      ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_numeral
% 0.24/0.51  thf(fact_1961_zero__le__numeral,axiom,
% 0.24/0.51      ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_numeral
% 0.24/0.51  thf(fact_1962_zero__le__numeral,axiom,
% 0.24/0.51      ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_numeral
% 0.24/0.51  thf(fact_1963_not__numeral__le__zero,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % not_numeral_le_zero
% 0.24/0.51  thf(fact_1964_not__numeral__le__zero,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % not_numeral_le_zero
% 0.24/0.51  thf(fact_1965_not__numeral__le__zero,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % not_numeral_le_zero
% 0.24/0.51  thf(fact_1966_not__numeral__le__zero,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % not_numeral_le_zero
% 0.24/0.51  thf(fact_1967_add__nonpos__eq__0__iff,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.24/0.51         => ( ( ( plus_plus_real @ X @ Y )
% 0.24/0.51              = zero_zero_real )
% 0.24/0.51            = ( ( X = zero_zero_real )
% 0.24/0.51              & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_eq_0_iff
% 0.24/0.51  thf(fact_1968_add__nonpos__eq__0__iff,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.24/0.51         => ( ( ( plus_plus_rat @ X @ Y )
% 0.24/0.51              = zero_zero_rat )
% 0.24/0.51            = ( ( X = zero_zero_rat )
% 0.24/0.51              & ( Y = zero_zero_rat ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_eq_0_iff
% 0.24/0.51  thf(fact_1969_add__nonpos__eq__0__iff,axiom,
% 0.24/0.51      ! [X: nat,Y: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 0.24/0.51       => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 0.24/0.51         => ( ( ( plus_plus_nat @ X @ Y )
% 0.24/0.51              = zero_zero_nat )
% 0.24/0.51            = ( ( X = zero_zero_nat )
% 0.24/0.51              & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_eq_0_iff
% 0.24/0.51  thf(fact_1970_add__nonpos__eq__0__iff,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.24/0.51       => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.24/0.51         => ( ( ( plus_plus_int @ X @ Y )
% 0.24/0.51              = zero_zero_int )
% 0.24/0.51            = ( ( X = zero_zero_int )
% 0.24/0.51              & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_eq_0_iff
% 0.24/0.51  thf(fact_1971_add__nonneg__eq__0__iff,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.24/0.51         => ( ( ( plus_plus_real @ X @ Y )
% 0.24/0.51              = zero_zero_real )
% 0.24/0.51            = ( ( X = zero_zero_real )
% 0.24/0.51              & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_eq_0_iff
% 0.24/0.51  thf(fact_1972_add__nonneg__eq__0__iff,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.24/0.51         => ( ( ( plus_plus_rat @ X @ Y )
% 0.24/0.51              = zero_zero_rat )
% 0.24/0.51            = ( ( X = zero_zero_rat )
% 0.24/0.51              & ( Y = zero_zero_rat ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_eq_0_iff
% 0.24/0.51  thf(fact_1973_add__nonneg__eq__0__iff,axiom,
% 0.24/0.51      ! [X: nat,Y: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.24/0.51         => ( ( ( plus_plus_nat @ X @ Y )
% 0.24/0.51              = zero_zero_nat )
% 0.24/0.51            = ( ( X = zero_zero_nat )
% 0.24/0.51              & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_eq_0_iff
% 0.24/0.51  thf(fact_1974_add__nonneg__eq__0__iff,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.24/0.51         => ( ( ( plus_plus_int @ X @ Y )
% 0.24/0.51              = zero_zero_int )
% 0.24/0.51            = ( ( X = zero_zero_int )
% 0.24/0.51              & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_eq_0_iff
% 0.24/0.51  thf(fact_1975_add__nonpos__nonpos,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.24/0.51         => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_nonpos
% 0.24/0.51  thf(fact_1976_add__nonpos__nonpos,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_nonpos
% 0.24/0.51  thf(fact_1977_add__nonpos__nonpos,axiom,
% 0.24/0.51      ! [A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.24/0.51         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_nonpos
% 0.24/0.51  thf(fact_1978_add__nonpos__nonpos,axiom,
% 0.24/0.51      ! [A: int,B: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.24/0.51       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.24/0.51         => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonpos_nonpos
% 0.24/0.51  thf(fact_1979_add__nonneg__nonneg,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.51         => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_nonneg
% 0.24/0.51  thf(fact_1980_add__nonneg__nonneg,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.51         => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_nonneg
% 0.24/0.51  thf(fact_1981_add__nonneg__nonneg,axiom,
% 0.24/0.51      ! [A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.24/0.51         => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_nonneg
% 0.24/0.51  thf(fact_1982_add__nonneg__nonneg,axiom,
% 0.24/0.51      ! [A: int,B: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.24/0.51         => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_nonneg_nonneg
% 0.24/0.51  thf(fact_1983_add__increasing2,axiom,
% 0.24/0.51      ! [C: real,B: real,A: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.24/0.51       => ( ( ord_less_eq_real @ B @ A )
% 0.24/0.51         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_increasing2
% 0.24/0.51  thf(fact_1984_add__increasing2,axiom,
% 0.24/0.51      ! [C: rat,B: rat,A: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ A )
% 0.24/0.51         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_increasing2
% 0.24/0.51  thf(fact_1985_add__increasing2,axiom,
% 0.24/0.51      ! [C: nat,B: nat,A: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ A )
% 0.24/0.51         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_increasing2
% 0.24/0.51  thf(fact_1986_add__increasing2,axiom,
% 0.24/0.51      ! [C: int,B: int,A: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.24/0.51       => ( ( ord_less_eq_int @ B @ A )
% 0.24/0.51         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_increasing2
% 0.24/0.51  thf(fact_1987_add__decreasing2,axiom,
% 0.24/0.51      ! [C: real,A: real,B: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_eq_real @ A @ B )
% 0.24/0.51         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_decreasing2
% 0.24/0.51  thf(fact_1988_add__decreasing2,axiom,
% 0.24/0.51      ! [C: rat,A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_decreasing2
% 0.24/0.51  thf(fact_1989_add__decreasing2,axiom,
% 0.24/0.51      ! [C: nat,A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.24/0.51       => ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_decreasing2
% 0.24/0.51  thf(fact_1990_add__decreasing2,axiom,
% 0.24/0.51      ! [C: int,A: int,B: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.24/0.51       => ( ( ord_less_eq_int @ A @ B )
% 0.24/0.51         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_decreasing2
% 0.24/0.51  thf(fact_1991_add__increasing,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_eq_real @ B @ C )
% 0.24/0.51         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_increasing
% 0.24/0.51  thf(fact_1992_add__increasing,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_increasing
% 0.24/0.51  thf(fact_1993_add__increasing,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ C )
% 0.24/0.51         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_increasing
% 0.24/0.51  thf(fact_1994_add__increasing,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51       => ( ( ord_less_eq_int @ B @ C )
% 0.24/0.51         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_increasing
% 0.24/0.51  thf(fact_1995_add__decreasing,axiom,
% 0.24/0.51      ! [A: real,C: real,B: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_eq_real @ C @ B )
% 0.24/0.51         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_decreasing
% 0.24/0.51  thf(fact_1996_add__decreasing,axiom,
% 0.24/0.51      ! [A: rat,C: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_eq_rat @ C @ B )
% 0.24/0.51         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_decreasing
% 0.24/0.51  thf(fact_1997_add__decreasing,axiom,
% 0.24/0.51      ! [A: nat,C: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.24/0.51       => ( ( ord_less_eq_nat @ C @ B )
% 0.24/0.51         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_decreasing
% 0.24/0.51  thf(fact_1998_add__decreasing,axiom,
% 0.24/0.51      ! [A: int,C: int,B: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.24/0.51       => ( ( ord_less_eq_int @ C @ B )
% 0.24/0.51         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_decreasing
% 0.24/0.51  thf(fact_1999_power__mono,axiom,
% 0.24/0.51      ! [A: real,B: real,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_mono
% 0.24/0.51  thf(fact_2000_power__mono,axiom,
% 0.24/0.51      ! [A: rat,B: rat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_mono
% 0.24/0.51  thf(fact_2001_power__mono,axiom,
% 0.24/0.51      ! [A: nat,B: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_mono
% 0.24/0.51  thf(fact_2002_power__mono,axiom,
% 0.24/0.51      ! [A: int,B: int,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_mono
% 0.24/0.51  thf(fact_2003_zero__le__power,axiom,
% 0.24/0.51      ! [A: real,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_power
% 0.24/0.51  thf(fact_2004_zero__le__power,axiom,
% 0.24/0.51      ! [A: rat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_power
% 0.24/0.51  thf(fact_2005_zero__le__power,axiom,
% 0.24/0.51      ! [A: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51       => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_power
% 0.24/0.51  thf(fact_2006_zero__le__power,axiom,
% 0.24/0.51      ! [A: int,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_power
% 0.24/0.51  thf(fact_2007_of__nat__0__le__iff,axiom,
% 0.24/0.51      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_0_le_iff
% 0.24/0.51  thf(fact_2008_of__nat__0__le__iff,axiom,
% 0.24/0.51      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_0_le_iff
% 0.24/0.51  thf(fact_2009_of__nat__0__le__iff,axiom,
% 0.24/0.51      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_0_le_iff
% 0.24/0.51  thf(fact_2010_of__nat__0__le__iff,axiom,
% 0.24/0.51      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_0_le_iff
% 0.24/0.51  thf(fact_2011_nat__less__real__le,axiom,
% 0.24/0.51      ( ord_less_nat
% 0.24/0.51      = ( ^ [N3: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat_less_real_le
% 0.24/0.51  thf(fact_2012_power__0,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( power_power_rat @ A @ zero_zero_nat )
% 0.24/0.51        = one_one_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % power_0
% 0.24/0.51  thf(fact_2013_power__0,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( power_power_nat @ A @ zero_zero_nat )
% 0.24/0.51        = one_one_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % power_0
% 0.24/0.51  thf(fact_2014_power__0,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( power_power_real @ A @ zero_zero_nat )
% 0.24/0.51        = one_one_real ) ).
% 0.24/0.51  
% 0.24/0.51  % power_0
% 0.24/0.51  thf(fact_2015_power__0,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( power_power_int @ A @ zero_zero_nat )
% 0.24/0.51        = one_one_int ) ).
% 0.24/0.51  
% 0.24/0.51  % power_0
% 0.24/0.51  thf(fact_2016_power__0,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( power_power_complex @ A @ zero_zero_nat )
% 0.24/0.51        = one_one_complex ) ).
% 0.24/0.51  
% 0.24/0.51  % power_0
% 0.24/0.51  thf(fact_2017_of__nat__neq__0,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 0.24/0.51       != zero_zero_complex ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_neq_0
% 0.24/0.51  thf(fact_2018_of__nat__neq__0,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 0.24/0.51       != zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_neq_0
% 0.24/0.51  thf(fact_2019_of__nat__neq__0,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 0.24/0.51       != zero_zero_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_neq_0
% 0.24/0.51  thf(fact_2020_of__nat__neq__0,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 0.24/0.51       != zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_neq_0
% 0.24/0.51  thf(fact_2021_of__nat__neq__0,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.24/0.51       != zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_neq_0
% 0.24/0.51  thf(fact_2022_add__is__1,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ( plus_plus_nat @ M @ N )
% 0.24/0.51          = ( suc @ zero_zero_nat ) )
% 0.24/0.51        = ( ( ( M
% 0.24/0.51              = ( suc @ zero_zero_nat ) )
% 0.24/0.51            & ( N = zero_zero_nat ) )
% 0.24/0.51          | ( ( M = zero_zero_nat )
% 0.24/0.51            & ( N
% 0.24/0.51              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_is_1
% 0.24/0.51  thf(fact_2023_one__is__add,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ( suc @ zero_zero_nat )
% 0.24/0.51          = ( plus_plus_nat @ M @ N ) )
% 0.24/0.51        = ( ( ( M
% 0.24/0.51              = ( suc @ zero_zero_nat ) )
% 0.24/0.51            & ( N = zero_zero_nat ) )
% 0.24/0.51          | ( ( M = zero_zero_nat )
% 0.24/0.51            & ( N
% 0.24/0.51              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_is_add
% 0.24/0.51  thf(fact_2024_One__nat__def,axiom,
% 0.24/0.51      ( one_one_nat
% 0.24/0.51      = ( suc @ zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % One_nat_def
% 0.24/0.51  thf(fact_2025_pow_Osimps_I1_J,axiom,
% 0.24/0.51      ! [X: num] :
% 0.24/0.51        ( ( pow @ X @ one )
% 0.24/0.51        = X ) ).
% 0.24/0.51  
% 0.24/0.51  % pow.simps(1)
% 0.24/0.51  thf(fact_2026_power2__less__imp__less,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.24/0.51         => ( ord_less_real @ X @ Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_less_imp_less
% 0.24/0.51  thf(fact_2027_power2__less__imp__less,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.24/0.51         => ( ord_less_rat @ X @ Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_less_imp_less
% 0.24/0.51  thf(fact_2028_power2__less__imp__less,axiom,
% 0.24/0.51      ! [X: nat,Y: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.24/0.51         => ( ord_less_nat @ X @ Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_less_imp_less
% 0.24/0.51  thf(fact_2029_power2__less__imp__less,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.24/0.51         => ( ord_less_int @ X @ Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_less_imp_less
% 0.24/0.51  thf(fact_2030_sum__power2__gt__zero__iff,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.24/0.51        = ( ( X != zero_zero_real )
% 0.24/0.51          | ( Y != zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % sum_power2_gt_zero_iff
% 0.24/0.51  thf(fact_2031_sum__power2__gt__zero__iff,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.24/0.51        = ( ( X != zero_zero_rat )
% 0.24/0.51          | ( Y != zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % sum_power2_gt_zero_iff
% 0.24/0.51  thf(fact_2032_sum__power2__gt__zero__iff,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.24/0.51        = ( ( X != zero_zero_int )
% 0.24/0.51          | ( Y != zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % sum_power2_gt_zero_iff
% 0.24/0.51  thf(fact_2033_not__sum__power2__lt__zero,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % not_sum_power2_lt_zero
% 0.24/0.51  thf(fact_2034_not__sum__power2__lt__zero,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % not_sum_power2_lt_zero
% 0.24/0.51  thf(fact_2035_not__sum__power2__lt__zero,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % not_sum_power2_lt_zero
% 0.24/0.51  thf(fact_2036_ceiling__eq,axiom,
% 0.24/0.51      ! [N: int,X: real] :
% 0.24/0.51        ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.24/0.51       => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.24/0.51         => ( ( archim7802044766580827645g_real @ X )
% 0.24/0.51            = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ceiling_eq
% 0.24/0.51  thf(fact_2037_ceiling__eq,axiom,
% 0.24/0.51      ! [N: int,X: rat] :
% 0.24/0.51        ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
% 0.24/0.51       => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 0.24/0.51         => ( ( archim2889992004027027881ng_rat @ X )
% 0.24/0.51            = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ceiling_eq
% 0.24/0.51  thf(fact_2038_ceiling__le__iff,axiom,
% 0.24/0.51      ! [X: real,Z: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
% 0.24/0.51        = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ceiling_le_iff
% 0.24/0.51  thf(fact_2039_ceiling__le__iff,axiom,
% 0.24/0.51      ! [X: rat,Z: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
% 0.24/0.51        = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ceiling_le_iff
% 0.24/0.51  thf(fact_2040_ceiling__le,axiom,
% 0.24/0.51      ! [X: real,A: int] :
% 0.24/0.51        ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 0.24/0.51       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ceiling_le
% 0.24/0.51  thf(fact_2041_ceiling__le,axiom,
% 0.24/0.51      ! [X: rat,A: int] :
% 0.24/0.51        ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
% 0.24/0.51       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ceiling_le
% 0.24/0.51  thf(fact_2042_power__gt1,axiom,
% 0.24/0.51      ! [A: real,N: nat] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_gt1
% 0.24/0.51  thf(fact_2043_power__gt1,axiom,
% 0.24/0.51      ! [A: rat,N: nat] :
% 0.24/0.51        ( ( ord_less_rat @ one_one_rat @ A )
% 0.24/0.51       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_gt1
% 0.24/0.51  thf(fact_2044_power__gt1,axiom,
% 0.24/0.51      ! [A: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ one_one_nat @ A )
% 0.24/0.51       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_gt1
% 0.24/0.51  thf(fact_2045_power__gt1,axiom,
% 0.24/0.51      ! [A: int,N: nat] :
% 0.24/0.51        ( ( ord_less_int @ one_one_int @ A )
% 0.24/0.51       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_gt1
% 0.24/0.51  thf(fact_2046_nle__le,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 0.24/0.51        = ( ( ord_less_eq_rat @ B @ A )
% 0.24/0.51          & ( B != A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nle_le
% 0.24/0.51  thf(fact_2047_nle__le,axiom,
% 0.24/0.51      ! [A: num,B: num] :
% 0.24/0.51        ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 0.24/0.51        = ( ( ord_less_eq_num @ B @ A )
% 0.24/0.51          & ( B != A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nle_le
% 0.24/0.51  thf(fact_2048_nle__le,axiom,
% 0.24/0.51      ! [A: nat,B: nat] :
% 0.24/0.51        ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 0.24/0.51        = ( ( ord_less_eq_nat @ B @ A )
% 0.24/0.51          & ( B != A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nle_le
% 0.24/0.51  thf(fact_2049_nle__le,axiom,
% 0.24/0.51      ! [A: int,B: int] :
% 0.24/0.51        ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 0.24/0.51        = ( ( ord_less_eq_int @ B @ A )
% 0.24/0.51          & ( B != A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nle_le
% 0.24/0.51  thf(fact_2050_le__cases3,axiom,
% 0.24/0.51      ! [X: rat,Y: rat,Z: rat] :
% 0.24/0.51        ( ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.51         => ~ ( ord_less_eq_rat @ Y @ Z ) )
% 0.24/0.51       => ( ( ( ord_less_eq_rat @ Y @ X )
% 0.24/0.51           => ~ ( ord_less_eq_rat @ X @ Z ) )
% 0.24/0.51         => ( ( ( ord_less_eq_rat @ X @ Z )
% 0.24/0.51             => ~ ( ord_less_eq_rat @ Z @ Y ) )
% 0.24/0.51           => ( ( ( ord_less_eq_rat @ Z @ Y )
% 0.24/0.51               => ~ ( ord_less_eq_rat @ Y @ X ) )
% 0.24/0.51             => ( ( ( ord_less_eq_rat @ Y @ Z )
% 0.24/0.51                 => ~ ( ord_less_eq_rat @ Z @ X ) )
% 0.24/0.51               => ~ ( ( ord_less_eq_rat @ Z @ X )
% 0.24/0.51                   => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_cases3
% 0.24/0.51  thf(fact_2051_le__cases3,axiom,
% 0.24/0.51      ! [X: num,Y: num,Z: num] :
% 0.24/0.51        ( ( ( ord_less_eq_num @ X @ Y )
% 0.24/0.51         => ~ ( ord_less_eq_num @ Y @ Z ) )
% 0.24/0.51       => ( ( ( ord_less_eq_num @ Y @ X )
% 0.24/0.51           => ~ ( ord_less_eq_num @ X @ Z ) )
% 0.24/0.51         => ( ( ( ord_less_eq_num @ X @ Z )
% 0.24/0.51             => ~ ( ord_less_eq_num @ Z @ Y ) )
% 0.24/0.51           => ( ( ( ord_less_eq_num @ Z @ Y )
% 0.24/0.51               => ~ ( ord_less_eq_num @ Y @ X ) )
% 0.24/0.51             => ( ( ( ord_less_eq_num @ Y @ Z )
% 0.24/0.51                 => ~ ( ord_less_eq_num @ Z @ X ) )
% 0.24/0.51               => ~ ( ( ord_less_eq_num @ Z @ X )
% 0.24/0.51                   => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_cases3
% 0.24/0.51  thf(fact_2052_le__cases3,axiom,
% 0.24/0.51      ! [X: nat,Y: nat,Z: nat] :
% 0.24/0.51        ( ( ( ord_less_eq_nat @ X @ Y )
% 0.24/0.51         => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 0.24/0.51       => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.24/0.51           => ~ ( ord_less_eq_nat @ X @ Z ) )
% 0.24/0.51         => ( ( ( ord_less_eq_nat @ X @ Z )
% 0.24/0.51             => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 0.24/0.51           => ( ( ( ord_less_eq_nat @ Z @ Y )
% 0.24/0.51               => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.24/0.51             => ( ( ( ord_less_eq_nat @ Y @ Z )
% 0.24/0.51                 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 0.24/0.51               => ~ ( ( ord_less_eq_nat @ Z @ X )
% 0.24/0.51                   => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_cases3
% 0.24/0.51  thf(fact_2053_le__cases3,axiom,
% 0.24/0.51      ! [X: int,Y: int,Z: int] :
% 0.24/0.51        ( ( ( ord_less_eq_int @ X @ Y )
% 0.24/0.51         => ~ ( ord_less_eq_int @ Y @ Z ) )
% 0.24/0.51       => ( ( ( ord_less_eq_int @ Y @ X )
% 0.24/0.51           => ~ ( ord_less_eq_int @ X @ Z ) )
% 0.24/0.51         => ( ( ( ord_less_eq_int @ X @ Z )
% 0.24/0.51             => ~ ( ord_less_eq_int @ Z @ Y ) )
% 0.24/0.51           => ( ( ( ord_less_eq_int @ Z @ Y )
% 0.24/0.51               => ~ ( ord_less_eq_int @ Y @ X ) )
% 0.24/0.51             => ( ( ( ord_less_eq_int @ Y @ Z )
% 0.24/0.51                 => ~ ( ord_less_eq_int @ Z @ X ) )
% 0.24/0.51               => ~ ( ( ord_less_eq_int @ Z @ X )
% 0.24/0.51                   => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_cases3
% 0.24/0.51  thf(fact_2054_order__class_Oorder__eq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: set_int,Z2: set_int] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [X4: set_int,Y6: set_int] :
% 0.24/0.51            ( ( ord_less_eq_set_int @ X4 @ Y6 )
% 0.24/0.51            & ( ord_less_eq_set_int @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_class.order_eq_iff
% 0.24/0.51  thf(fact_2055_order__class_Oorder__eq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [X4: rat,Y6: rat] :
% 0.24/0.51            ( ( ord_less_eq_rat @ X4 @ Y6 )
% 0.24/0.51            & ( ord_less_eq_rat @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_class.order_eq_iff
% 0.24/0.51  thf(fact_2056_order__class_Oorder__eq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [X4: num,Y6: num] :
% 0.24/0.51            ( ( ord_less_eq_num @ X4 @ Y6 )
% 0.24/0.51            & ( ord_less_eq_num @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_class.order_eq_iff
% 0.24/0.51  thf(fact_2057_order__class_Oorder__eq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [X4: nat,Y6: nat] :
% 0.24/0.51            ( ( ord_less_eq_nat @ X4 @ Y6 )
% 0.24/0.51            & ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_class.order_eq_iff
% 0.24/0.51  thf(fact_2058_order__class_Oorder__eq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [X4: int,Y6: int] :
% 0.24/0.51            ( ( ord_less_eq_int @ X4 @ Y6 )
% 0.24/0.51            & ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_class.order_eq_iff
% 0.24/0.51  thf(fact_2059_ord__eq__le__trans,axiom,
% 0.24/0.51      ! [A: set_int,B: set_int,C: set_int] :
% 0.24/0.51        ( ( A = B )
% 0.24/0.51       => ( ( ord_less_eq_set_int @ B @ C )
% 0.24/0.51         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_trans
% 0.24/0.51  thf(fact_2060_ord__eq__le__trans,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( A = B )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_trans
% 0.24/0.51  thf(fact_2061_ord__eq__le__trans,axiom,
% 0.24/0.51      ! [A: num,B: num,C: num] :
% 0.24/0.51        ( ( A = B )
% 0.24/0.51       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.51         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_trans
% 0.24/0.51  thf(fact_2062_ord__eq__le__trans,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat] :
% 0.24/0.51        ( ( A = B )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ C )
% 0.24/0.51         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_trans
% 0.24/0.51  thf(fact_2063_ord__eq__le__trans,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int] :
% 0.24/0.51        ( ( A = B )
% 0.24/0.51       => ( ( ord_less_eq_int @ B @ C )
% 0.24/0.51         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_trans
% 0.24/0.51  thf(fact_2064_ord__le__eq__trans,axiom,
% 0.24/0.51      ! [A: set_int,B: set_int,C: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ A @ B )
% 0.24/0.51       => ( ( B = C )
% 0.24/0.51         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_trans
% 0.24/0.51  thf(fact_2065_ord__le__eq__trans,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( B = C )
% 0.24/0.51         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_trans
% 0.24/0.51  thf(fact_2066_ord__le__eq__trans,axiom,
% 0.24/0.51      ! [A: num,B: num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( B = C )
% 0.24/0.51         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_trans
% 0.24/0.51  thf(fact_2067_ord__le__eq__trans,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51       => ( ( B = C )
% 0.24/0.51         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_trans
% 0.24/0.51  thf(fact_2068_ord__le__eq__trans,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.51       => ( ( B = C )
% 0.24/0.51         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_trans
% 0.24/0.51  thf(fact_2069_order__antisym,axiom,
% 0.24/0.51      ! [X: set_int,Y: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ X @ Y )
% 0.24/0.51       => ( ( ord_less_eq_set_int @ Y @ X )
% 0.24/0.51         => ( X = Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_antisym
% 0.24/0.51  thf(fact_2070_order__antisym,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.51       => ( ( ord_less_eq_rat @ Y @ X )
% 0.24/0.51         => ( X = Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_antisym
% 0.24/0.51  thf(fact_2071_order__antisym,axiom,
% 0.24/0.51      ! [X: num,Y: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ X @ Y )
% 0.24/0.51       => ( ( ord_less_eq_num @ Y @ X )
% 0.24/0.51         => ( X = Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_antisym
% 0.24/0.51  thf(fact_2072_order__antisym,axiom,
% 0.24/0.51      ! [X: nat,Y: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ X @ Y )
% 0.24/0.51       => ( ( ord_less_eq_nat @ Y @ X )
% 0.24/0.51         => ( X = Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_antisym
% 0.24/0.51  thf(fact_2073_order__antisym,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ X @ Y )
% 0.24/0.51       => ( ( ord_less_eq_int @ Y @ X )
% 0.24/0.51         => ( X = Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_antisym
% 0.24/0.51  thf(fact_2074_order_Otrans,axiom,
% 0.24/0.51      ! [A: set_int,B: set_int,C: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_set_int @ B @ C )
% 0.24/0.51         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order.trans
% 0.24/0.51  thf(fact_2075_order_Otrans,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order.trans
% 0.24/0.51  thf(fact_2076_order_Otrans,axiom,
% 0.24/0.51      ! [A: num,B: num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.51         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order.trans
% 0.24/0.51  thf(fact_2077_order_Otrans,axiom,
% 0.24/0.51      ! [A: nat,B: nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ C )
% 0.24/0.51         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order.trans
% 0.24/0.51  thf(fact_2078_order_Otrans,axiom,
% 0.24/0.51      ! [A: int,B: int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_int @ B @ C )
% 0.24/0.51         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order.trans
% 0.24/0.51  thf(fact_2079_order__trans,axiom,
% 0.24/0.51      ! [X: set_int,Y: set_int,Z: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ X @ Y )
% 0.24/0.51       => ( ( ord_less_eq_set_int @ Y @ Z )
% 0.24/0.51         => ( ord_less_eq_set_int @ X @ Z ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_trans
% 0.24/0.51  thf(fact_2080_order__trans,axiom,
% 0.24/0.51      ! [X: rat,Y: rat,Z: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.51       => ( ( ord_less_eq_rat @ Y @ Z )
% 0.24/0.51         => ( ord_less_eq_rat @ X @ Z ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_trans
% 0.24/0.51  thf(fact_2081_order__trans,axiom,
% 0.24/0.51      ! [X: num,Y: num,Z: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ X @ Y )
% 0.24/0.51       => ( ( ord_less_eq_num @ Y @ Z )
% 0.24/0.51         => ( ord_less_eq_num @ X @ Z ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_trans
% 0.24/0.51  thf(fact_2082_order__trans,axiom,
% 0.24/0.51      ! [X: nat,Y: nat,Z: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ X @ Y )
% 0.24/0.51       => ( ( ord_less_eq_nat @ Y @ Z )
% 0.24/0.51         => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_trans
% 0.24/0.51  thf(fact_2083_order__trans,axiom,
% 0.24/0.51      ! [X: int,Y: int,Z: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ X @ Y )
% 0.24/0.51       => ( ( ord_less_eq_int @ Y @ Z )
% 0.24/0.51         => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_trans
% 0.24/0.51  thf(fact_2084_linorder__wlog,axiom,
% 0.24/0.51      ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.24/0.51        ( ! [A5: rat,B6: rat] :
% 0.24/0.51            ( ( ord_less_eq_rat @ A5 @ B6 )
% 0.24/0.51           => ( P @ A5 @ B6 ) )
% 0.24/0.51       => ( ! [A5: rat,B6: rat] :
% 0.24/0.51              ( ( P @ B6 @ A5 )
% 0.24/0.51             => ( P @ A5 @ B6 ) )
% 0.24/0.51         => ( P @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_wlog
% 0.24/0.51  thf(fact_2085_linorder__wlog,axiom,
% 0.24/0.51      ! [P: num > num > $o,A: num,B: num] :
% 0.24/0.51        ( ! [A5: num,B6: num] :
% 0.24/0.51            ( ( ord_less_eq_num @ A5 @ B6 )
% 0.24/0.51           => ( P @ A5 @ B6 ) )
% 0.24/0.51       => ( ! [A5: num,B6: num] :
% 0.24/0.51              ( ( P @ B6 @ A5 )
% 0.24/0.51             => ( P @ A5 @ B6 ) )
% 0.24/0.51         => ( P @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_wlog
% 0.24/0.51  thf(fact_2086_linorder__wlog,axiom,
% 0.24/0.51      ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.24/0.51        ( ! [A5: nat,B6: nat] :
% 0.24/0.51            ( ( ord_less_eq_nat @ A5 @ B6 )
% 0.24/0.51           => ( P @ A5 @ B6 ) )
% 0.24/0.51       => ( ! [A5: nat,B6: nat] :
% 0.24/0.51              ( ( P @ B6 @ A5 )
% 0.24/0.51             => ( P @ A5 @ B6 ) )
% 0.24/0.51         => ( P @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_wlog
% 0.24/0.51  thf(fact_2087_linorder__wlog,axiom,
% 0.24/0.51      ! [P: int > int > $o,A: int,B: int] :
% 0.24/0.51        ( ! [A5: int,B6: int] :
% 0.24/0.51            ( ( ord_less_eq_int @ A5 @ B6 )
% 0.24/0.51           => ( P @ A5 @ B6 ) )
% 0.24/0.51       => ( ! [A5: int,B6: int] :
% 0.24/0.51              ( ( P @ B6 @ A5 )
% 0.24/0.51             => ( P @ A5 @ B6 ) )
% 0.24/0.51         => ( P @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_wlog
% 0.24/0.51  thf(fact_2088_dual__order_Oeq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: set_int,Z2: set_int] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [A3: set_int,B4: set_int] :
% 0.24/0.51            ( ( ord_less_eq_set_int @ B4 @ A3 )
% 0.24/0.51            & ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.eq_iff
% 0.24/0.51  thf(fact_2089_dual__order_Oeq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [A3: rat,B4: rat] :
% 0.24/0.51            ( ( ord_less_eq_rat @ B4 @ A3 )
% 0.24/0.51            & ( ord_less_eq_rat @ A3 @ B4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.eq_iff
% 0.24/0.51  thf(fact_2090_dual__order_Oeq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [A3: num,B4: num] :
% 0.24/0.51            ( ( ord_less_eq_num @ B4 @ A3 )
% 0.24/0.51            & ( ord_less_eq_num @ A3 @ B4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.eq_iff
% 0.24/0.51  thf(fact_2091_dual__order_Oeq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [A3: nat,B4: nat] :
% 0.24/0.51            ( ( ord_less_eq_nat @ B4 @ A3 )
% 0.24/0.51            & ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.eq_iff
% 0.24/0.51  thf(fact_2092_dual__order_Oeq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [A3: int,B4: int] :
% 0.24/0.51            ( ( ord_less_eq_int @ B4 @ A3 )
% 0.24/0.51            & ( ord_less_eq_int @ A3 @ B4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.eq_iff
% 0.24/0.51  thf(fact_2093_dual__order_Oantisym,axiom,
% 0.24/0.51      ! [B: set_int,A: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ B @ A )
% 0.24/0.51       => ( ( ord_less_eq_set_int @ A @ B )
% 0.24/0.51         => ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.antisym
% 0.24/0.51  thf(fact_2094_dual__order_Oantisym,axiom,
% 0.24/0.51      ! [B: rat,A: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ B @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51         => ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.antisym
% 0.24/0.51  thf(fact_2095_dual__order_Oantisym,axiom,
% 0.24/0.51      ! [B: num,A: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ B @ A )
% 0.24/0.51       => ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51         => ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.antisym
% 0.24/0.51  thf(fact_2096_dual__order_Oantisym,axiom,
% 0.24/0.51      ! [B: nat,A: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ B @ A )
% 0.24/0.51       => ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51         => ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.antisym
% 0.24/0.51  thf(fact_2097_dual__order_Oantisym,axiom,
% 0.24/0.51      ! [B: int,A: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ B @ A )
% 0.24/0.51       => ( ( ord_less_eq_int @ A @ B )
% 0.24/0.51         => ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.antisym
% 0.24/0.51  thf(fact_2098_dual__order_Otrans,axiom,
% 0.24/0.51      ! [B: set_int,A: set_int,C: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ B @ A )
% 0.24/0.51       => ( ( ord_less_eq_set_int @ C @ B )
% 0.24/0.51         => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.trans
% 0.24/0.51  thf(fact_2099_dual__order_Otrans,axiom,
% 0.24/0.51      ! [B: rat,A: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ B @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ C @ B )
% 0.24/0.51         => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.trans
% 0.24/0.51  thf(fact_2100_dual__order_Otrans,axiom,
% 0.24/0.51      ! [B: num,A: num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ B @ A )
% 0.24/0.51       => ( ( ord_less_eq_num @ C @ B )
% 0.24/0.51         => ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.trans
% 0.24/0.51  thf(fact_2101_dual__order_Otrans,axiom,
% 0.24/0.51      ! [B: nat,A: nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ B @ A )
% 0.24/0.51       => ( ( ord_less_eq_nat @ C @ B )
% 0.24/0.51         => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.trans
% 0.24/0.51  thf(fact_2102_dual__order_Otrans,axiom,
% 0.24/0.51      ! [B: int,A: int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ B @ A )
% 0.24/0.51       => ( ( ord_less_eq_int @ C @ B )
% 0.24/0.51         => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % dual_order.trans
% 0.24/0.51  thf(fact_2103_antisym,axiom,
% 0.24/0.51      ! [A: set_int,B: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_set_int @ B @ A )
% 0.24/0.51         => ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % antisym
% 0.24/0.51  thf(fact_2104_antisym,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ A )
% 0.24/0.51         => ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % antisym
% 0.24/0.51  thf(fact_2105_antisym,axiom,
% 0.24/0.51      ! [A: num,B: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_num @ B @ A )
% 0.24/0.51         => ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % antisym
% 0.24/0.51  thf(fact_2106_antisym,axiom,
% 0.24/0.51      ! [A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ A )
% 0.24/0.51         => ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % antisym
% 0.24/0.51  thf(fact_2107_antisym,axiom,
% 0.24/0.51      ! [A: int,B: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_int @ B @ A )
% 0.24/0.51         => ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % antisym
% 0.24/0.51  thf(fact_2108_Orderings_Oorder__eq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: set_int,Z2: set_int] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [A3: set_int,B4: set_int] :
% 0.24/0.51            ( ( ord_less_eq_set_int @ A3 @ B4 )
% 0.24/0.51            & ( ord_less_eq_set_int @ B4 @ A3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Orderings.order_eq_iff
% 0.24/0.51  thf(fact_2109_Orderings_Oorder__eq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [A3: rat,B4: rat] :
% 0.24/0.51            ( ( ord_less_eq_rat @ A3 @ B4 )
% 0.24/0.51            & ( ord_less_eq_rat @ B4 @ A3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Orderings.order_eq_iff
% 0.24/0.51  thf(fact_2110_Orderings_Oorder__eq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [A3: num,B4: num] :
% 0.24/0.51            ( ( ord_less_eq_num @ A3 @ B4 )
% 0.24/0.51            & ( ord_less_eq_num @ B4 @ A3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Orderings.order_eq_iff
% 0.24/0.51  thf(fact_2111_Orderings_Oorder__eq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [A3: nat,B4: nat] :
% 0.24/0.51            ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.24/0.51            & ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Orderings.order_eq_iff
% 0.24/0.51  thf(fact_2112_Orderings_Oorder__eq__iff,axiom,
% 0.24/0.51      ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 0.24/0.51      = ( ^ [A3: int,B4: int] :
% 0.24/0.51            ( ( ord_less_eq_int @ A3 @ B4 )
% 0.24/0.51            & ( ord_less_eq_int @ B4 @ A3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Orderings.order_eq_iff
% 0.24/0.51  thf(fact_2113_order__subst1,axiom,
% 0.24/0.51      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst1
% 0.24/0.51  thf(fact_2114_order__subst1,axiom,
% 0.24/0.51      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst1
% 0.24/0.51  thf(fact_2115_order__subst1,axiom,
% 0.24/0.51      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ C )
% 0.24/0.51         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.51                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst1
% 0.24/0.51  thf(fact_2116_order__subst1,axiom,
% 0.24/0.51      ! [A: rat,F: int > rat,B: int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_int @ B @ C )
% 0.24/0.51         => ( ! [X3: int,Y4: int] :
% 0.24/0.51                ( ( ord_less_eq_int @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst1
% 0.24/0.51  thf(fact_2117_order__subst1,axiom,
% 0.24/0.51      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst1
% 0.24/0.51  thf(fact_2118_order__subst1,axiom,
% 0.24/0.51      ! [A: num,F: num > num,B: num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst1
% 0.24/0.51  thf(fact_2119_order__subst1,axiom,
% 0.24/0.51      ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ C )
% 0.24/0.51         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.51                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst1
% 0.24/0.51  thf(fact_2120_order__subst1,axiom,
% 0.24/0.51      ! [A: num,F: int > num,B: int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_int @ B @ C )
% 0.24/0.51         => ( ! [X3: int,Y4: int] :
% 0.24/0.51                ( ( ord_less_eq_int @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst1
% 0.24/0.51  thf(fact_2121_order__subst1,axiom,
% 0.24/0.51      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst1
% 0.24/0.51  thf(fact_2122_order__subst1,axiom,
% 0.24/0.51      ! [A: nat,F: num > nat,B: num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst1
% 0.24/0.51  thf(fact_2123_order__subst2,axiom,
% 0.24/0.51      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst2
% 0.24/0.51  thf(fact_2124_order__subst2,axiom,
% 0.24/0.51      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst2
% 0.24/0.51  thf(fact_2125_order__subst2,axiom,
% 0.24/0.51      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst2
% 0.24/0.51  thf(fact_2126_order__subst2,axiom,
% 0.24/0.51      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst2
% 0.24/0.51  thf(fact_2127_order__subst2,axiom,
% 0.24/0.51      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst2
% 0.24/0.51  thf(fact_2128_order__subst2,axiom,
% 0.24/0.51      ! [A: num,B: num,F: num > num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst2
% 0.24/0.51  thf(fact_2129_order__subst2,axiom,
% 0.24/0.51      ! [A: num,B: num,F: num > nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst2
% 0.24/0.51  thf(fact_2130_order__subst2,axiom,
% 0.24/0.51      ! [A: num,B: num,F: num > int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst2
% 0.24/0.51  thf(fact_2131_order__subst2,axiom,
% 0.24/0.51      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.24/0.51         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.51                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst2
% 0.24/0.51  thf(fact_2132_order__subst2,axiom,
% 0.24/0.51      ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.24/0.51         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.51                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_subst2
% 0.24/0.51  thf(fact_2133_order__eq__refl,axiom,
% 0.24/0.51      ! [X: set_int,Y: set_int] :
% 0.24/0.51        ( ( X = Y )
% 0.24/0.51       => ( ord_less_eq_set_int @ X @ Y ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_eq_refl
% 0.24/0.51  thf(fact_2134_order__eq__refl,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( X = Y )
% 0.24/0.51       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_eq_refl
% 0.24/0.51  thf(fact_2135_order__eq__refl,axiom,
% 0.24/0.51      ! [X: num,Y: num] :
% 0.24/0.51        ( ( X = Y )
% 0.24/0.51       => ( ord_less_eq_num @ X @ Y ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_eq_refl
% 0.24/0.51  thf(fact_2136_order__eq__refl,axiom,
% 0.24/0.51      ! [X: nat,Y: nat] :
% 0.24/0.51        ( ( X = Y )
% 0.24/0.51       => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_eq_refl
% 0.24/0.51  thf(fact_2137_order__eq__refl,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( X = Y )
% 0.24/0.51       => ( ord_less_eq_int @ X @ Y ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_eq_refl
% 0.24/0.51  thf(fact_2138_linorder__linear,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.51        | ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_linear
% 0.24/0.51  thf(fact_2139_linorder__linear,axiom,
% 0.24/0.51      ! [X: num,Y: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ X @ Y )
% 0.24/0.51        | ( ord_less_eq_num @ Y @ X ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_linear
% 0.24/0.51  thf(fact_2140_linorder__linear,axiom,
% 0.24/0.51      ! [X: nat,Y: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ X @ Y )
% 0.24/0.51        | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_linear
% 0.24/0.51  thf(fact_2141_linorder__linear,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ X @ Y )
% 0.24/0.51        | ( ord_less_eq_int @ Y @ X ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_linear
% 0.24/0.51  thf(fact_2142_ord__eq__le__subst,axiom,
% 0.24/0.51      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.24/0.51        ( ( A
% 0.24/0.51          = ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_subst
% 0.24/0.51  thf(fact_2143_ord__eq__le__subst,axiom,
% 0.24/0.51      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.24/0.51        ( ( A
% 0.24/0.51          = ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_subst
% 0.24/0.51  thf(fact_2144_ord__eq__le__subst,axiom,
% 0.24/0.51      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.24/0.51        ( ( A
% 0.24/0.51          = ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_subst
% 0.24/0.51  thf(fact_2145_ord__eq__le__subst,axiom,
% 0.24/0.51      ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.24/0.51        ( ( A
% 0.24/0.51          = ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ B @ C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_subst
% 0.24/0.51  thf(fact_2146_ord__eq__le__subst,axiom,
% 0.24/0.51      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.24/0.51        ( ( A
% 0.24/0.51          = ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_subst
% 0.24/0.51  thf(fact_2147_ord__eq__le__subst,axiom,
% 0.24/0.51      ! [A: num,F: num > num,B: num,C: num] :
% 0.24/0.51        ( ( A
% 0.24/0.51          = ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_subst
% 0.24/0.51  thf(fact_2148_ord__eq__le__subst,axiom,
% 0.24/0.51      ! [A: nat,F: num > nat,B: num,C: num] :
% 0.24/0.51        ( ( A
% 0.24/0.51          = ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_subst
% 0.24/0.51  thf(fact_2149_ord__eq__le__subst,axiom,
% 0.24/0.51      ! [A: int,F: num > int,B: num,C: num] :
% 0.24/0.51        ( ( A
% 0.24/0.51          = ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_num @ B @ C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_subst
% 0.24/0.51  thf(fact_2150_ord__eq__le__subst,axiom,
% 0.24/0.51      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.24/0.51        ( ( A
% 0.24/0.51          = ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ C )
% 0.24/0.51         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.51                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_subst
% 0.24/0.51  thf(fact_2151_ord__eq__le__subst,axiom,
% 0.24/0.51      ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.24/0.51        ( ( A
% 0.24/0.51          = ( F @ B ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ B @ C )
% 0.24/0.51         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.51                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_eq_le_subst
% 0.24/0.51  thf(fact_2152_ord__le__eq__subst,axiom,
% 0.24/0.51      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ( F @ B )
% 0.24/0.51            = C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_subst
% 0.24/0.51  thf(fact_2153_ord__le__eq__subst,axiom,
% 0.24/0.51      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ( F @ B )
% 0.24/0.51            = C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_subst
% 0.24/0.51  thf(fact_2154_ord__le__eq__subst,axiom,
% 0.24/0.51      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ( F @ B )
% 0.24/0.51            = C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_subst
% 0.24/0.51  thf(fact_2155_ord__le__eq__subst,axiom,
% 0.24/0.51      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ( F @ B )
% 0.24/0.51            = C )
% 0.24/0.51         => ( ! [X3: rat,Y4: rat] :
% 0.24/0.51                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_subst
% 0.24/0.51  thf(fact_2156_ord__le__eq__subst,axiom,
% 0.24/0.51      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( ( F @ B )
% 0.24/0.51            = C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_subst
% 0.24/0.51  thf(fact_2157_ord__le__eq__subst,axiom,
% 0.24/0.51      ! [A: num,B: num,F: num > num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( ( F @ B )
% 0.24/0.51            = C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_subst
% 0.24/0.51  thf(fact_2158_ord__le__eq__subst,axiom,
% 0.24/0.51      ! [A: num,B: num,F: num > nat,C: nat] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( ( F @ B )
% 0.24/0.51            = C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_subst
% 0.24/0.51  thf(fact_2159_ord__le__eq__subst,axiom,
% 0.24/0.51      ! [A: num,B: num,F: num > int,C: int] :
% 0.24/0.51        ( ( ord_less_eq_num @ A @ B )
% 0.24/0.51       => ( ( ( F @ B )
% 0.24/0.51            = C )
% 0.24/0.51         => ( ! [X3: num,Y4: num] :
% 0.24/0.51                ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_subst
% 0.24/0.51  thf(fact_2160_ord__le__eq__subst,axiom,
% 0.24/0.51      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51       => ( ( ( F @ B )
% 0.24/0.51            = C )
% 0.24/0.51         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.51                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_subst
% 0.24/0.51  thf(fact_2161_ord__le__eq__subst,axiom,
% 0.24/0.51      ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.24/0.51        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.51       => ( ( ( F @ B )
% 0.24/0.51            = C )
% 0.24/0.51         => ( ! [X3: nat,Y4: nat] :
% 0.24/0.51                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.24/0.51               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.24/0.51           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ord_le_eq_subst
% 0.24/0.51  thf(fact_2162_linorder__le__cases,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ~ ( ord_less_eq_rat @ X @ Y )
% 0.24/0.51       => ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_le_cases
% 0.24/0.51  thf(fact_2163_linorder__le__cases,axiom,
% 0.24/0.51      ! [X: num,Y: num] :
% 0.24/0.51        ( ~ ( ord_less_eq_num @ X @ Y )
% 0.24/0.51       => ( ord_less_eq_num @ Y @ X ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_le_cases
% 0.24/0.51  thf(fact_2164_linorder__le__cases,axiom,
% 0.24/0.51      ! [X: nat,Y: nat] :
% 0.24/0.51        ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.24/0.51       => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_le_cases
% 0.24/0.51  thf(fact_2165_linorder__le__cases,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ~ ( ord_less_eq_int @ X @ Y )
% 0.24/0.51       => ( ord_less_eq_int @ Y @ X ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_le_cases
% 0.24/0.51  thf(fact_2166_order__antisym__conv,axiom,
% 0.24/0.51      ! [Y: set_int,X: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ Y @ X )
% 0.24/0.51       => ( ( ord_less_eq_set_int @ X @ Y )
% 0.24/0.51          = ( X = Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_antisym_conv
% 0.24/0.51  thf(fact_2167_order__antisym__conv,axiom,
% 0.24/0.51      ! [Y: rat,X: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ Y @ X )
% 0.24/0.51       => ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.51          = ( X = Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_antisym_conv
% 0.24/0.51  thf(fact_2168_order__antisym__conv,axiom,
% 0.24/0.51      ! [Y: num,X: num] :
% 0.24/0.51        ( ( ord_less_eq_num @ Y @ X )
% 0.24/0.51       => ( ( ord_less_eq_num @ X @ Y )
% 0.24/0.51          = ( X = Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_antisym_conv
% 0.24/0.51  thf(fact_2169_order__antisym__conv,axiom,
% 0.24/0.51      ! [Y: nat,X: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ Y @ X )
% 0.24/0.51       => ( ( ord_less_eq_nat @ X @ Y )
% 0.24/0.51          = ( X = Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_antisym_conv
% 0.24/0.51  thf(fact_2170_order__antisym__conv,axiom,
% 0.24/0.51      ! [Y: int,X: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ Y @ X )
% 0.24/0.51       => ( ( ord_less_eq_int @ X @ Y )
% 0.24/0.51          = ( X = Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % order_antisym_conv
% 0.24/0.51  thf(fact_2171_power__le__one,axiom,
% 0.24/0.51      ! [A: real,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.24/0.51         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_one
% 0.24/0.51  thf(fact_2172_power__le__one,axiom,
% 0.24/0.51      ! [A: rat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.24/0.51         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_one
% 0.24/0.51  thf(fact_2173_power__le__one,axiom,
% 0.24/0.51      ! [A: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.24/0.51         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_one
% 0.24/0.51  thf(fact_2174_power__le__one,axiom,
% 0.24/0.51      ! [A: int,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51       => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.24/0.51         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_one
% 0.24/0.51  thf(fact_2175_power__le__imp__le__base,axiom,
% 0.24/0.51      ! [A: real,N: nat,B: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.51         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_imp_le_base
% 0.24/0.51  thf(fact_2176_power__le__imp__le__base,axiom,
% 0.24/0.51      ! [A: rat,N: nat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.51         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_imp_le_base
% 0.24/0.51  thf(fact_2177_power__le__imp__le__base,axiom,
% 0.24/0.51      ! [A: nat,N: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.24/0.51         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_imp_le_base
% 0.24/0.51  thf(fact_2178_power__le__imp__le__base,axiom,
% 0.24/0.51      ! [A: int,N: nat,B: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.24/0.51         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_imp_le_base
% 0.24/0.51  thf(fact_2179_power__inject__base,axiom,
% 0.24/0.51      ! [A: real,N: nat,B: real] :
% 0.24/0.51        ( ( ( power_power_real @ A @ ( suc @ N ) )
% 0.24/0.51          = ( power_power_real @ B @ ( suc @ N ) ) )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.51           => ( A = B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_inject_base
% 0.24/0.51  thf(fact_2180_power__inject__base,axiom,
% 0.24/0.51      ! [A: rat,N: nat,B: rat] :
% 0.24/0.51        ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.24/0.51          = ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.51           => ( A = B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_inject_base
% 0.24/0.51  thf(fact_2181_power__inject__base,axiom,
% 0.24/0.51      ! [A: nat,N: nat,B: nat] :
% 0.24/0.51        ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.24/0.51          = ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.24/0.51           => ( A = B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_inject_base
% 0.24/0.51  thf(fact_2182_power__inject__base,axiom,
% 0.24/0.51      ! [A: int,N: nat,B: int] :
% 0.24/0.51        ( ( ( power_power_int @ A @ ( suc @ N ) )
% 0.24/0.51          = ( power_power_int @ B @ ( suc @ N ) ) )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.24/0.51           => ( A = B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_inject_base
% 0.24/0.51  thf(fact_2183_numeral__1__eq__Suc__0,axiom,
% 0.24/0.51      ( ( numeral_numeral_nat @ one )
% 0.24/0.51      = ( suc @ zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % numeral_1_eq_Suc_0
% 0.24/0.51  thf(fact_2184_subset__code_I1_J,axiom,
% 0.24/0.51      ! [Xs: list_complex,B2: set_complex] :
% 0.24/0.51        ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B2 )
% 0.24/0.51        = ( ! [X4: complex] :
% 0.24/0.51              ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
% 0.24/0.51             => ( member_complex @ X4 @ B2 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % subset_code(1)
% 0.24/0.51  thf(fact_2185_subset__code_I1_J,axiom,
% 0.24/0.51      ! [Xs: list_real,B2: set_real] :
% 0.24/0.51        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
% 0.24/0.51        = ( ! [X4: real] :
% 0.24/0.51              ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 0.24/0.51             => ( member_real @ X4 @ B2 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % subset_code(1)
% 0.24/0.51  thf(fact_2186_subset__code_I1_J,axiom,
% 0.24/0.51      ! [Xs: list_set_nat,B2: set_set_nat] :
% 0.24/0.51        ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
% 0.24/0.51        = ( ! [X4: set_nat] :
% 0.24/0.51              ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 0.24/0.51             => ( member_set_nat @ X4 @ B2 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % subset_code(1)
% 0.24/0.51  thf(fact_2187_subset__code_I1_J,axiom,
% 0.24/0.51      ! [Xs: list_nat,B2: set_nat] :
% 0.24/0.51        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
% 0.24/0.51        = ( ! [X4: nat] :
% 0.24/0.51              ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.24/0.51             => ( member_nat @ X4 @ B2 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % subset_code(1)
% 0.24/0.51  thf(fact_2188_subset__code_I1_J,axiom,
% 0.24/0.51      ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.24/0.51        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
% 0.24/0.51        = ( ! [X4: vEBT_VEBT] :
% 0.24/0.51              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.24/0.51             => ( member_VEBT_VEBT @ X4 @ B2 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % subset_code(1)
% 0.24/0.51  thf(fact_2189_subset__code_I1_J,axiom,
% 0.24/0.51      ! [Xs: list_int,B2: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
% 0.24/0.51        = ( ! [X4: int] :
% 0.24/0.51              ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.24/0.51             => ( member_int @ X4 @ B2 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % subset_code(1)
% 0.24/0.51  thf(fact_2190_num_Osize_I5_J,axiom,
% 0.24/0.51      ! [X22: num] :
% 0.24/0.51        ( ( size_size_num @ ( bit0 @ X22 ) )
% 0.24/0.51        = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % num.size(5)
% 0.24/0.51  thf(fact_2191_Ex__list__of__length,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51      ? [Xs2: list_VEBT_VEBT] :
% 0.24/0.51        ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.24/0.51        = N ) ).
% 0.24/0.51  
% 0.24/0.51  % Ex_list_of_length
% 0.24/0.51  thf(fact_2192_Ex__list__of__length,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51      ? [Xs2: list_o] :
% 0.24/0.51        ( ( size_size_list_o @ Xs2 )
% 0.24/0.51        = N ) ).
% 0.24/0.51  
% 0.24/0.51  % Ex_list_of_length
% 0.24/0.51  thf(fact_2193_Ex__list__of__length,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51      ? [Xs2: list_int] :
% 0.24/0.51        ( ( size_size_list_int @ Xs2 )
% 0.24/0.51        = N ) ).
% 0.24/0.51  
% 0.24/0.51  % Ex_list_of_length
% 0.24/0.51  thf(fact_2194_neq__if__length__neq,axiom,
% 0.24/0.51      ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.24/0.51        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.24/0.51         != ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 0.24/0.51       => ( Xs != Ys2 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % neq_if_length_neq
% 0.24/0.51  thf(fact_2195_neq__if__length__neq,axiom,
% 0.24/0.51      ! [Xs: list_o,Ys2: list_o] :
% 0.24/0.51        ( ( ( size_size_list_o @ Xs )
% 0.24/0.51         != ( size_size_list_o @ Ys2 ) )
% 0.24/0.51       => ( Xs != Ys2 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % neq_if_length_neq
% 0.24/0.51  thf(fact_2196_neq__if__length__neq,axiom,
% 0.24/0.51      ! [Xs: list_int,Ys2: list_int] :
% 0.24/0.51        ( ( ( size_size_list_int @ Xs )
% 0.24/0.51         != ( size_size_list_int @ Ys2 ) )
% 0.24/0.51       => ( Xs != Ys2 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % neq_if_length_neq
% 0.24/0.51  thf(fact_2197_nat__one__le__power,axiom,
% 0.24/0.51      ! [I: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.24/0.51       => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat_one_le_power
% 0.24/0.51  thf(fact_2198_of__int__ceiling__le__add__one,axiom,
% 0.24/0.51      ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 0.24/0.51  
% 0.24/0.51  % of_int_ceiling_le_add_one
% 0.24/0.51  thf(fact_2199_of__int__ceiling__le__add__one,axiom,
% 0.24/0.51      ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % of_int_ceiling_le_add_one
% 0.24/0.51  thf(fact_2200_field__sum__of__halves,axiom,
% 0.24/0.51      ! [X: real] :
% 0.24/0.51        ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.24/0.51        = X ) ).
% 0.24/0.51  
% 0.24/0.51  % field_sum_of_halves
% 0.24/0.51  thf(fact_2201_field__sum__of__halves,axiom,
% 0.24/0.51      ! [X: rat] :
% 0.24/0.51        ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.24/0.51        = X ) ).
% 0.24/0.51  
% 0.24/0.51  % field_sum_of_halves
% 0.24/0.51  thf(fact_2202_power__le__imp__le__exp,axiom,
% 0.24/0.51      ! [A: real,M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.24/0.51         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_imp_le_exp
% 0.24/0.51  thf(fact_2203_power__le__imp__le__exp,axiom,
% 0.24/0.51      ! [A: rat,M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_rat @ one_one_rat @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.24/0.51         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_imp_le_exp
% 0.24/0.51  thf(fact_2204_power__le__imp__le__exp,axiom,
% 0.24/0.51      ! [A: nat,M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ one_one_nat @ A )
% 0.24/0.51       => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.24/0.51         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_imp_le_exp
% 0.24/0.51  thf(fact_2205_power__le__imp__le__exp,axiom,
% 0.24/0.51      ! [A: int,M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_int @ one_one_int @ A )
% 0.24/0.51       => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.24/0.51         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_le_imp_le_exp
% 0.24/0.51  thf(fact_2206_less__exp,axiom,
% 0.24/0.51      ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_exp
% 0.24/0.51  thf(fact_2207_power__Suc__le__self,axiom,
% 0.24/0.51      ! [A: real,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.24/0.51         => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_Suc_le_self
% 0.24/0.51  thf(fact_2208_power__Suc__le__self,axiom,
% 0.24/0.51      ! [A: rat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.24/0.51         => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_Suc_le_self
% 0.24/0.51  thf(fact_2209_power__Suc__le__self,axiom,
% 0.24/0.51      ! [A: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.24/0.51         => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_Suc_le_self
% 0.24/0.51  thf(fact_2210_power__Suc__le__self,axiom,
% 0.24/0.51      ! [A: int,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51       => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.24/0.51         => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_Suc_le_self
% 0.24/0.51  thf(fact_2211_power__decreasing,axiom,
% 0.24/0.51      ! [N: nat,N4: nat,A: real] :
% 0.24/0.51        ( ( ord_less_eq_nat @ N @ N4 )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51         => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.24/0.51           => ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_decreasing
% 0.24/0.51  thf(fact_2212_power__decreasing,axiom,
% 0.24/0.51      ! [N: nat,N4: nat,A: rat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ N @ N4 )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51         => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.24/0.51           => ( ord_less_eq_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_decreasing
% 0.24/0.51  thf(fact_2213_power__decreasing,axiom,
% 0.24/0.51      ! [N: nat,N4: nat,A: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ N @ N4 )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.51         => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.24/0.51           => ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_decreasing
% 0.24/0.51  thf(fact_2214_power__decreasing,axiom,
% 0.24/0.51      ! [N: nat,N4: nat,A: int] :
% 0.24/0.51        ( ( ord_less_eq_nat @ N @ N4 )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.51         => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.24/0.51           => ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power_decreasing
% 0.24/0.51  thf(fact_2215_zero__power2,axiom,
% 0.24/0.51      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51      = zero_zero_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_power2
% 0.24/0.51  thf(fact_2216_zero__power2,axiom,
% 0.24/0.51      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51      = zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_power2
% 0.24/0.51  thf(fact_2217_zero__power2,axiom,
% 0.24/0.51      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51      = zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_power2
% 0.24/0.51  thf(fact_2218_zero__power2,axiom,
% 0.24/0.51      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51      = zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_power2
% 0.24/0.51  thf(fact_2219_zero__power2,axiom,
% 0.24/0.51      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51      = zero_zero_complex ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_power2
% 0.24/0.51  thf(fact_2220_numeral__2__eq__2,axiom,
% 0.24/0.51      ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.24/0.51      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % numeral_2_eq_2
% 0.24/0.51  thf(fact_2221_log2__of__power__le,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.24/0.51         => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log2_of_power_le
% 0.24/0.51  thf(fact_2222_of__nat__less__two__power,axiom,
% 0.24/0.51      ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_less_two_power
% 0.24/0.51  thf(fact_2223_of__nat__less__two__power,axiom,
% 0.24/0.51      ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_less_two_power
% 0.24/0.51  thf(fact_2224_of__nat__less__two__power,axiom,
% 0.24/0.51      ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % of_nat_less_two_power
% 0.24/0.51  thf(fact_2225_zero__le__power2,axiom,
% 0.24/0.51      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_power2
% 0.24/0.51  thf(fact_2226_zero__le__power2,axiom,
% 0.24/0.51      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_power2
% 0.24/0.51  thf(fact_2227_zero__le__power2,axiom,
% 0.24/0.51      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_power2
% 0.24/0.51  thf(fact_2228_power2__eq__imp__eq,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51          = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.24/0.51           => ( X = Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_eq_imp_eq
% 0.24/0.51  thf(fact_2229_power2__eq__imp__eq,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51          = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.24/0.51         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.24/0.51           => ( X = Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_eq_imp_eq
% 0.24/0.51  thf(fact_2230_power2__eq__imp__eq,axiom,
% 0.24/0.51      ! [X: nat,Y: nat] :
% 0.24/0.51        ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51          = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.24/0.51         => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.24/0.51           => ( X = Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_eq_imp_eq
% 0.24/0.51  thf(fact_2231_power2__eq__imp__eq,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51          = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.24/0.51         => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.24/0.51           => ( X = Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_eq_imp_eq
% 0.24/0.51  thf(fact_2232_power2__le__imp__le,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.24/0.51         => ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_le_imp_le
% 0.24/0.51  thf(fact_2233_power2__le__imp__le,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.24/0.51         => ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_le_imp_le
% 0.24/0.51  thf(fact_2234_power2__le__imp__le,axiom,
% 0.24/0.51      ! [X: nat,Y: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.24/0.51         => ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_le_imp_le
% 0.24/0.51  thf(fact_2235_power2__le__imp__le,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.24/0.51       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.24/0.51         => ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % power2_le_imp_le
% 0.24/0.51  thf(fact_2236_sum__power2__le__zero__iff,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 0.24/0.51        = ( ( X = zero_zero_real )
% 0.24/0.51          & ( Y = zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % sum_power2_le_zero_iff
% 0.24/0.51  thf(fact_2237_sum__power2__le__zero__iff,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 0.24/0.51        = ( ( X = zero_zero_rat )
% 0.24/0.51          & ( Y = zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % sum_power2_le_zero_iff
% 0.24/0.51  thf(fact_2238_sum__power2__le__zero__iff,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 0.24/0.51        = ( ( X = zero_zero_int )
% 0.24/0.51          & ( Y = zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % sum_power2_le_zero_iff
% 0.24/0.51  thf(fact_2239_sum__power2__ge__zero,axiom,
% 0.24/0.51      ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % sum_power2_ge_zero
% 0.24/0.51  thf(fact_2240_sum__power2__ge__zero,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % sum_power2_ge_zero
% 0.24/0.51  thf(fact_2241_sum__power2__ge__zero,axiom,
% 0.24/0.51      ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % sum_power2_ge_zero
% 0.24/0.51  thf(fact_2242_one__div__two__eq__zero,axiom,
% 0.24/0.51      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51      = zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % one_div_two_eq_zero
% 0.24/0.51  thf(fact_2243_one__div__two__eq__zero,axiom,
% 0.24/0.51      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.24/0.51      = zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % one_div_two_eq_zero
% 0.24/0.51  thf(fact_2244_bits__1__div__2,axiom,
% 0.24/0.51      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51      = zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % bits_1_div_2
% 0.24/0.51  thf(fact_2245_bits__1__div__2,axiom,
% 0.24/0.51      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.24/0.51      = zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % bits_1_div_2
% 0.24/0.51  thf(fact_2246_add__self__div__2,axiom,
% 0.24/0.51      ! [M: nat] :
% 0.24/0.51        ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51        = M ) ).
% 0.24/0.51  
% 0.24/0.51  % add_self_div_2
% 0.24/0.51  thf(fact_2247_div2__Suc__Suc,axiom,
% 0.24/0.51      ! [M: nat] :
% 0.24/0.51        ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51        = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div2_Suc_Suc
% 0.24/0.51  thf(fact_2248_le__divide__eq__1__pos,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.24/0.51          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_divide_eq_1_pos
% 0.24/0.51  thf(fact_2249_le__divide__eq__1__pos,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.24/0.51          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_divide_eq_1_pos
% 0.24/0.51  thf(fact_2250_le__divide__eq__1__neg,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.24/0.51          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_divide_eq_1_neg
% 0.24/0.51  thf(fact_2251_le__divide__eq__1__neg,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.24/0.51          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_divide_eq_1_neg
% 0.24/0.51  thf(fact_2252_divide__le__eq__1__pos,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.24/0.51          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_eq_1_pos
% 0.24/0.51  thf(fact_2253_divide__le__eq__1__pos,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.24/0.51          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_eq_1_pos
% 0.24/0.51  thf(fact_2254_divide__le__eq__1__neg,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.24/0.51          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_eq_1_neg
% 0.24/0.51  thf(fact_2255_divide__le__eq__1__neg,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.24/0.51          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_eq_1_neg
% 0.24/0.51  thf(fact_2256_zero__less__divide__1__iff,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.24/0.51        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_divide_1_iff
% 0.24/0.51  thf(fact_2257_zero__less__divide__1__iff,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.24/0.51        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_divide_1_iff
% 0.24/0.51  thf(fact_2258_less__divide__eq__1__pos,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.24/0.51          = ( ord_less_real @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_divide_eq_1_pos
% 0.24/0.51  thf(fact_2259_less__divide__eq__1__pos,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.24/0.51          = ( ord_less_rat @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_divide_eq_1_pos
% 0.24/0.51  thf(fact_2260_less__divide__eq__1__neg,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.24/0.51          = ( ord_less_real @ B @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_divide_eq_1_neg
% 0.24/0.51  thf(fact_2261_less__divide__eq__1__neg,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.24/0.51          = ( ord_less_rat @ B @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_divide_eq_1_neg
% 0.24/0.51  thf(fact_2262_psubsetI,axiom,
% 0.24/0.51      ! [A2: set_int,B2: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.51       => ( ( A2 != B2 )
% 0.24/0.51         => ( ord_less_set_int @ A2 @ B2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % psubsetI
% 0.24/0.51  thf(fact_2263_division__ring__divide__zero,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.24/0.51        = zero_zero_complex ) ).
% 0.24/0.51  
% 0.24/0.51  % division_ring_divide_zero
% 0.24/0.51  thf(fact_2264_division__ring__divide__zero,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( divide_divide_real @ A @ zero_zero_real )
% 0.24/0.51        = zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % division_ring_divide_zero
% 0.24/0.51  thf(fact_2265_division__ring__divide__zero,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.24/0.51        = zero_zero_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % division_ring_divide_zero
% 0.24/0.51  thf(fact_2266_bits__div__by__0,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.24/0.51        = zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % bits_div_by_0
% 0.24/0.51  thf(fact_2267_bits__div__by__0,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( divide_divide_int @ A @ zero_zero_int )
% 0.24/0.51        = zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % bits_div_by_0
% 0.24/0.51  thf(fact_2268_bits__div__0,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.24/0.51        = zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % bits_div_0
% 0.24/0.51  thf(fact_2269_bits__div__0,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( divide_divide_int @ zero_zero_int @ A )
% 0.24/0.51        = zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % bits_div_0
% 0.24/0.51  thf(fact_2270_divide__cancel__right,axiom,
% 0.24/0.51      ! [A: complex,C: complex,B: complex] :
% 0.24/0.51        ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.24/0.51          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.24/0.51        = ( ( C = zero_zero_complex )
% 0.24/0.51          | ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_cancel_right
% 0.24/0.51  thf(fact_2271_divide__cancel__right,axiom,
% 0.24/0.51      ! [A: real,C: real,B: real] :
% 0.24/0.51        ( ( ( divide_divide_real @ A @ C )
% 0.24/0.51          = ( divide_divide_real @ B @ C ) )
% 0.24/0.51        = ( ( C = zero_zero_real )
% 0.24/0.51          | ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_cancel_right
% 0.24/0.51  thf(fact_2272_divide__cancel__right,axiom,
% 0.24/0.51      ! [A: rat,C: rat,B: rat] :
% 0.24/0.51        ( ( ( divide_divide_rat @ A @ C )
% 0.24/0.51          = ( divide_divide_rat @ B @ C ) )
% 0.24/0.51        = ( ( C = zero_zero_rat )
% 0.24/0.51          | ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_cancel_right
% 0.24/0.51  thf(fact_2273_divide__cancel__left,axiom,
% 0.24/0.51      ! [C: complex,A: complex,B: complex] :
% 0.24/0.51        ( ( ( divide1717551699836669952omplex @ C @ A )
% 0.24/0.51          = ( divide1717551699836669952omplex @ C @ B ) )
% 0.24/0.51        = ( ( C = zero_zero_complex )
% 0.24/0.51          | ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_cancel_left
% 0.24/0.51  thf(fact_2274_divide__cancel__left,axiom,
% 0.24/0.51      ! [C: real,A: real,B: real] :
% 0.24/0.51        ( ( ( divide_divide_real @ C @ A )
% 0.24/0.51          = ( divide_divide_real @ C @ B ) )
% 0.24/0.51        = ( ( C = zero_zero_real )
% 0.24/0.51          | ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_cancel_left
% 0.24/0.51  thf(fact_2275_divide__cancel__left,axiom,
% 0.24/0.51      ! [C: rat,A: rat,B: rat] :
% 0.24/0.51        ( ( ( divide_divide_rat @ C @ A )
% 0.24/0.51          = ( divide_divide_rat @ C @ B ) )
% 0.24/0.51        = ( ( C = zero_zero_rat )
% 0.24/0.51          | ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_cancel_left
% 0.24/0.51  thf(fact_2276_divide__eq__0__iff,axiom,
% 0.24/0.51      ! [A: complex,B: complex] :
% 0.24/0.51        ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.24/0.51          = zero_zero_complex )
% 0.24/0.51        = ( ( A = zero_zero_complex )
% 0.24/0.51          | ( B = zero_zero_complex ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_eq_0_iff
% 0.24/0.51  thf(fact_2277_divide__eq__0__iff,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ( divide_divide_real @ A @ B )
% 0.24/0.51          = zero_zero_real )
% 0.24/0.51        = ( ( A = zero_zero_real )
% 0.24/0.51          | ( B = zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_eq_0_iff
% 0.24/0.51  thf(fact_2278_divide__eq__0__iff,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ( divide_divide_rat @ A @ B )
% 0.24/0.51          = zero_zero_rat )
% 0.24/0.51        = ( ( A = zero_zero_rat )
% 0.24/0.51          | ( B = zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_eq_0_iff
% 0.24/0.51  thf(fact_2279_bits__div__by__1,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( divide_divide_nat @ A @ one_one_nat )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % bits_div_by_1
% 0.24/0.51  thf(fact_2280_bits__div__by__1,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( divide_divide_int @ A @ one_one_int )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % bits_div_by_1
% 0.24/0.51  thf(fact_2281_log__eq__one,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( A != one_one_real )
% 0.24/0.51         => ( ( log @ A @ A )
% 0.24/0.51            = one_one_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_eq_one
% 0.24/0.51  thf(fact_2282_log__less__cancel__iff,axiom,
% 0.24/0.51      ! [A: real,X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.24/0.51           => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 0.24/0.51              = ( ord_less_real @ X @ Y ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_less_cancel_iff
% 0.24/0.51  thf(fact_2283_log__less__one__cancel__iff,axiom,
% 0.24/0.51      ! [A: real,X: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
% 0.24/0.51            = ( ord_less_real @ X @ A ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_less_one_cancel_iff
% 0.24/0.51  thf(fact_2284_one__less__log__cancel__iff,axiom,
% 0.24/0.51      ! [A: real,X: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
% 0.24/0.51            = ( ord_less_real @ A @ X ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_less_log_cancel_iff
% 0.24/0.51  thf(fact_2285_log__less__zero__cancel__iff,axiom,
% 0.24/0.51      ! [A: real,X: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
% 0.24/0.51            = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_less_zero_cancel_iff
% 0.24/0.51  thf(fact_2286_zero__less__log__cancel__iff,axiom,
% 0.24/0.51      ! [A: real,X: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
% 0.24/0.51            = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_log_cancel_iff
% 0.24/0.51  thf(fact_2287_log__one,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( log @ A @ one_one_real )
% 0.24/0.51        = zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % log_one
% 0.24/0.51  thf(fact_2288_semiring__norm_I78_J,axiom,
% 0.24/0.51      ! [M: num,N: num] :
% 0.24/0.51        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.24/0.51        = ( ord_less_num @ M @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % semiring_norm(78)
% 0.24/0.51  thf(fact_2289_semiring__norm_I75_J,axiom,
% 0.24/0.51      ! [M: num] :
% 0.24/0.51        ~ ( ord_less_num @ M @ one ) ).
% 0.24/0.51  
% 0.24/0.51  % semiring_norm(75)
% 0.24/0.51  thf(fact_2290_divide__eq__1__iff,axiom,
% 0.24/0.51      ! [A: complex,B: complex] :
% 0.24/0.51        ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.24/0.51          = one_one_complex )
% 0.24/0.51        = ( ( B != zero_zero_complex )
% 0.24/0.51          & ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_eq_1_iff
% 0.24/0.51  thf(fact_2291_divide__eq__1__iff,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ( divide_divide_real @ A @ B )
% 0.24/0.51          = one_one_real )
% 0.24/0.51        = ( ( B != zero_zero_real )
% 0.24/0.51          & ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_eq_1_iff
% 0.24/0.51  thf(fact_2292_divide__eq__1__iff,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ( divide_divide_rat @ A @ B )
% 0.24/0.51          = one_one_rat )
% 0.24/0.51        = ( ( B != zero_zero_rat )
% 0.24/0.51          & ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_eq_1_iff
% 0.24/0.51  thf(fact_2293_one__eq__divide__iff,axiom,
% 0.24/0.51      ! [A: complex,B: complex] :
% 0.24/0.51        ( ( one_one_complex
% 0.24/0.51          = ( divide1717551699836669952omplex @ A @ B ) )
% 0.24/0.51        = ( ( B != zero_zero_complex )
% 0.24/0.51          & ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_eq_divide_iff
% 0.24/0.51  thf(fact_2294_one__eq__divide__iff,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( one_one_real
% 0.24/0.51          = ( divide_divide_real @ A @ B ) )
% 0.24/0.51        = ( ( B != zero_zero_real )
% 0.24/0.51          & ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_eq_divide_iff
% 0.24/0.51  thf(fact_2295_one__eq__divide__iff,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( one_one_rat
% 0.24/0.51          = ( divide_divide_rat @ A @ B ) )
% 0.24/0.51        = ( ( B != zero_zero_rat )
% 0.24/0.51          & ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_eq_divide_iff
% 0.24/0.51  thf(fact_2296_divide__self,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( A != zero_zero_complex )
% 0.24/0.51       => ( ( divide1717551699836669952omplex @ A @ A )
% 0.24/0.51          = one_one_complex ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_self
% 0.24/0.51  thf(fact_2297_divide__self,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( A != zero_zero_real )
% 0.24/0.51       => ( ( divide_divide_real @ A @ A )
% 0.24/0.51          = one_one_real ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_self
% 0.24/0.51  thf(fact_2298_divide__self,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( A != zero_zero_rat )
% 0.24/0.51       => ( ( divide_divide_rat @ A @ A )
% 0.24/0.51          = one_one_rat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_self
% 0.24/0.51  thf(fact_2299_divide__self__if,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( ( A = zero_zero_complex )
% 0.24/0.51         => ( ( divide1717551699836669952omplex @ A @ A )
% 0.24/0.51            = zero_zero_complex ) )
% 0.24/0.51        & ( ( A != zero_zero_complex )
% 0.24/0.51         => ( ( divide1717551699836669952omplex @ A @ A )
% 0.24/0.51            = one_one_complex ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_self_if
% 0.24/0.51  thf(fact_2300_divide__self__if,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( ( A = zero_zero_real )
% 0.24/0.51         => ( ( divide_divide_real @ A @ A )
% 0.24/0.51            = zero_zero_real ) )
% 0.24/0.51        & ( ( A != zero_zero_real )
% 0.24/0.51         => ( ( divide_divide_real @ A @ A )
% 0.24/0.51            = one_one_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_self_if
% 0.24/0.51  thf(fact_2301_divide__self__if,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( ( A = zero_zero_rat )
% 0.24/0.51         => ( ( divide_divide_rat @ A @ A )
% 0.24/0.51            = zero_zero_rat ) )
% 0.24/0.51        & ( ( A != zero_zero_rat )
% 0.24/0.51         => ( ( divide_divide_rat @ A @ A )
% 0.24/0.51            = one_one_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_self_if
% 0.24/0.51  thf(fact_2302_divide__eq__eq__1,axiom,
% 0.24/0.51      ! [B: real,A: real] :
% 0.24/0.51        ( ( ( divide_divide_real @ B @ A )
% 0.24/0.51          = one_one_real )
% 0.24/0.51        = ( ( A != zero_zero_real )
% 0.24/0.51          & ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_eq_eq_1
% 0.24/0.51  thf(fact_2303_divide__eq__eq__1,axiom,
% 0.24/0.51      ! [B: rat,A: rat] :
% 0.24/0.51        ( ( ( divide_divide_rat @ B @ A )
% 0.24/0.51          = one_one_rat )
% 0.24/0.51        = ( ( A != zero_zero_rat )
% 0.24/0.51          & ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_eq_eq_1
% 0.24/0.51  thf(fact_2304_eq__divide__eq__1,axiom,
% 0.24/0.51      ! [B: real,A: real] :
% 0.24/0.51        ( ( one_one_real
% 0.24/0.51          = ( divide_divide_real @ B @ A ) )
% 0.24/0.51        = ( ( A != zero_zero_real )
% 0.24/0.51          & ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % eq_divide_eq_1
% 0.24/0.51  thf(fact_2305_eq__divide__eq__1,axiom,
% 0.24/0.51      ! [B: rat,A: rat] :
% 0.24/0.51        ( ( one_one_rat
% 0.24/0.51          = ( divide_divide_rat @ B @ A ) )
% 0.24/0.51        = ( ( A != zero_zero_rat )
% 0.24/0.51          & ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % eq_divide_eq_1
% 0.24/0.51  thf(fact_2306_one__divide__eq__0__iff,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( ( divide_divide_real @ one_one_real @ A )
% 0.24/0.51          = zero_zero_real )
% 0.24/0.51        = ( A = zero_zero_real ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_divide_eq_0_iff
% 0.24/0.51  thf(fact_2307_one__divide__eq__0__iff,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( ( divide_divide_rat @ one_one_rat @ A )
% 0.24/0.51          = zero_zero_rat )
% 0.24/0.51        = ( A = zero_zero_rat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_divide_eq_0_iff
% 0.24/0.51  thf(fact_2308_zero__eq__1__divide__iff,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( zero_zero_real
% 0.24/0.51          = ( divide_divide_real @ one_one_real @ A ) )
% 0.24/0.51        = ( A = zero_zero_real ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_eq_1_divide_iff
% 0.24/0.51  thf(fact_2309_zero__eq__1__divide__iff,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( zero_zero_rat
% 0.24/0.51          = ( divide_divide_rat @ one_one_rat @ A ) )
% 0.24/0.51        = ( A = zero_zero_rat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_eq_1_divide_iff
% 0.24/0.51  thf(fact_2310_div__by__Suc__0,axiom,
% 0.24/0.51      ! [M: nat] :
% 0.24/0.51        ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.24/0.51        = M ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_Suc_0
% 0.24/0.51  thf(fact_2311_div__less,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ N )
% 0.24/0.51       => ( ( divide_divide_nat @ M @ N )
% 0.24/0.51          = zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_less
% 0.24/0.51  thf(fact_2312_div__neg__neg__trivial,axiom,
% 0.24/0.51      ! [K: int,L: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.24/0.51       => ( ( ord_less_int @ L @ K )
% 0.24/0.51         => ( ( divide_divide_int @ K @ L )
% 0.24/0.51            = zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_neg_neg_trivial
% 0.24/0.51  thf(fact_2313_div__pos__pos__trivial,axiom,
% 0.24/0.51      ! [K: int,L: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.24/0.51       => ( ( ord_less_int @ K @ L )
% 0.24/0.51         => ( ( divide_divide_int @ K @ L )
% 0.24/0.51            = zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_pos_pos_trivial
% 0.24/0.51  thf(fact_2314_log__le__cancel__iff,axiom,
% 0.24/0.51      ! [A: real,X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.24/0.51           => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 0.24/0.51              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_le_cancel_iff
% 0.24/0.51  thf(fact_2315_log__le__one__cancel__iff,axiom,
% 0.24/0.51      ! [A: real,X: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
% 0.24/0.51            = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_le_one_cancel_iff
% 0.24/0.51  thf(fact_2316_one__le__log__cancel__iff,axiom,
% 0.24/0.51      ! [A: real,X: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
% 0.24/0.51            = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % one_le_log_cancel_iff
% 0.24/0.51  thf(fact_2317_log__le__zero__cancel__iff,axiom,
% 0.24/0.51      ! [A: real,X: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
% 0.24/0.51            = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_le_zero_cancel_iff
% 0.24/0.51  thf(fact_2318_zero__le__log__cancel__iff,axiom,
% 0.24/0.51      ! [A: real,X: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51         => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
% 0.24/0.51            = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_log_cancel_iff
% 0.24/0.51  thf(fact_2319_semiring__norm_I76_J,axiom,
% 0.24/0.51      ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % semiring_norm(76)
% 0.24/0.51  thf(fact_2320_enat__ord__number_I2_J,axiom,
% 0.24/0.51      ! [M: num,N: num] :
% 0.24/0.51        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.24/0.51        = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % enat_ord_number(2)
% 0.24/0.51  thf(fact_2321_divide__le__0__1__iff,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.24/0.51        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_0_1_iff
% 0.24/0.51  thf(fact_2322_divide__le__0__1__iff,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.24/0.51        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_0_1_iff
% 0.24/0.51  thf(fact_2323_zero__le__divide__1__iff,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.24/0.51        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_divide_1_iff
% 0.24/0.51  thf(fact_2324_zero__le__divide__1__iff,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.24/0.51        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_divide_1_iff
% 0.24/0.51  thf(fact_2325_divide__less__0__1__iff,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.24/0.51        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_0_1_iff
% 0.24/0.51  thf(fact_2326_divide__less__0__1__iff,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.24/0.51        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_0_1_iff
% 0.24/0.51  thf(fact_2327_divide__less__eq__1__neg,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.24/0.51          = ( ord_less_real @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_eq_1_neg
% 0.24/0.51  thf(fact_2328_divide__less__eq__1__neg,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.24/0.51          = ( ord_less_rat @ A @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_eq_1_neg
% 0.24/0.51  thf(fact_2329_divide__less__eq__1__pos,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.24/0.51          = ( ord_less_real @ B @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_eq_1_pos
% 0.24/0.51  thf(fact_2330_divide__less__eq__1__pos,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.24/0.51          = ( ord_less_rat @ B @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_eq_1_pos
% 0.24/0.51  thf(fact_2331_numeral__less__real__of__nat__iff,axiom,
% 0.24/0.51      ! [W: num,N: nat] :
% 0.24/0.51        ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.51        = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % numeral_less_real_of_nat_iff
% 0.24/0.51  thf(fact_2332_real__of__nat__less__numeral__iff,axiom,
% 0.24/0.51      ! [N: nat,W: num] :
% 0.24/0.51        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 0.24/0.51        = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % real_of_nat_less_numeral_iff
% 0.24/0.51  thf(fact_2333_zle__add1__eq__le,axiom,
% 0.24/0.51      ! [W: int,Z: int] :
% 0.24/0.51        ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.24/0.51        = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zle_add1_eq_le
% 0.24/0.51  thf(fact_2334_log__pow__cancel,axiom,
% 0.24/0.51      ! [A: real,B: nat] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( A != one_one_real )
% 0.24/0.51         => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 0.24/0.51            = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_pow_cancel
% 0.24/0.51  thf(fact_2335_half__nonnegative__int__iff,axiom,
% 0.24/0.51      ! [K: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.24/0.51        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.24/0.51  
% 0.24/0.51  % half_nonnegative_int_iff
% 0.24/0.51  thf(fact_2336_half__negative__int__iff,axiom,
% 0.24/0.51      ! [K: int] :
% 0.24/0.51        ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.24/0.51        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.24/0.51  
% 0.24/0.51  % half_negative_int_iff
% 0.24/0.51  thf(fact_2337_ceiling__less__one,axiom,
% 0.24/0.51      ! [X: real] :
% 0.24/0.51        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 0.24/0.51        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ceiling_less_one
% 0.24/0.51  thf(fact_2338_ceiling__less__one,axiom,
% 0.24/0.51      ! [X: rat] :
% 0.24/0.51        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 0.24/0.51        = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ceiling_less_one
% 0.24/0.51  thf(fact_2339_less__int__code_I1_J,axiom,
% 0.24/0.51      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % less_int_code(1)
% 0.24/0.51  thf(fact_2340_log__base__change,axiom,
% 0.24/0.51      ! [A: real,B: real,X: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( A != one_one_real )
% 0.24/0.51         => ( ( log @ B @ X )
% 0.24/0.51            = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_base_change
% 0.24/0.51  thf(fact_2341_log__base__pow,axiom,
% 0.24/0.51      ! [A: real,N: nat,X: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ( ( log @ ( power_power_real @ A @ N ) @ X )
% 0.24/0.51          = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_base_pow
% 0.24/0.51  thf(fact_2342_real__of__int__div4,axiom,
% 0.24/0.51      ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % real_of_int_div4
% 0.24/0.51  thf(fact_2343_int__one__le__iff__zero__less,axiom,
% 0.24/0.51      ! [Z: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ one_one_int @ Z )
% 0.24/0.51        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.24/0.51  
% 0.24/0.51  % int_one_le_iff_zero_less
% 0.24/0.51  thf(fact_2344_real__arch__pow__inv,axiom,
% 0.24/0.51      ! [Y: real,X: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.24/0.51       => ( ( ord_less_real @ X @ one_one_real )
% 0.24/0.51         => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % real_arch_pow_inv
% 0.24/0.51  thf(fact_2345_odd__less__0__iff,axiom,
% 0.24/0.51      ! [Z: int] :
% 0.24/0.51        ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 0.24/0.51        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.24/0.51  
% 0.24/0.51  % odd_less_0_iff
% 0.24/0.51  thf(fact_2346_psubsetE,axiom,
% 0.24/0.51      ! [A2: set_int,B2: set_int] :
% 0.24/0.51        ( ( ord_less_set_int @ A2 @ B2 )
% 0.24/0.51       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.51           => ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % psubsetE
% 0.24/0.51  thf(fact_2347_psubset__eq,axiom,
% 0.24/0.51      ( ord_less_set_int
% 0.24/0.51      = ( ^ [A4: set_int,B5: set_int] :
% 0.24/0.51            ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.24/0.51            & ( A4 != B5 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % psubset_eq
% 0.24/0.51  thf(fact_2348_psubset__imp__subset,axiom,
% 0.24/0.51      ! [A2: set_int,B2: set_int] :
% 0.24/0.51        ( ( ord_less_set_int @ A2 @ B2 )
% 0.24/0.51       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.24/0.51  
% 0.24/0.51  % psubset_imp_subset
% 0.24/0.51  thf(fact_2349_psubset__subset__trans,axiom,
% 0.24/0.51      ! [A2: set_int,B2: set_int,C6: set_int] :
% 0.24/0.51        ( ( ord_less_set_int @ A2 @ B2 )
% 0.24/0.51       => ( ( ord_less_eq_set_int @ B2 @ C6 )
% 0.24/0.51         => ( ord_less_set_int @ A2 @ C6 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % psubset_subset_trans
% 0.24/0.51  thf(fact_2350_subset__not__subset__eq,axiom,
% 0.24/0.51      ( ord_less_set_int
% 0.24/0.51      = ( ^ [A4: set_int,B5: set_int] :
% 0.24/0.51            ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.24/0.51            & ~ ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % subset_not_subset_eq
% 0.24/0.51  thf(fact_2351_subset__psubset__trans,axiom,
% 0.24/0.51      ! [A2: set_int,B2: set_int,C6: set_int] :
% 0.24/0.51        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.24/0.51       => ( ( ord_less_set_int @ B2 @ C6 )
% 0.24/0.51         => ( ord_less_set_int @ A2 @ C6 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % subset_psubset_trans
% 0.24/0.51  thf(fact_2352_subset__iff__psubset__eq,axiom,
% 0.24/0.51      ( ord_less_eq_set_int
% 0.24/0.51      = ( ^ [A4: set_int,B5: set_int] :
% 0.24/0.51            ( ( ord_less_set_int @ A4 @ B5 )
% 0.24/0.51            | ( A4 = B5 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % subset_iff_psubset_eq
% 0.24/0.51  thf(fact_2353_less__eq__real__def,axiom,
% 0.24/0.51      ( ord_less_eq_real
% 0.24/0.51      = ( ^ [X4: real,Y6: real] :
% 0.24/0.51            ( ( ord_less_real @ X4 @ Y6 )
% 0.24/0.51            | ( X4 = Y6 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_eq_real_def
% 0.24/0.51  thf(fact_2354_less__eq__int__code_I1_J,axiom,
% 0.24/0.51      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.24/0.51  
% 0.24/0.51  % less_eq_int_code(1)
% 0.24/0.51  thf(fact_2355_plus__int__code_I2_J,axiom,
% 0.24/0.51      ! [L: int] :
% 0.24/0.51        ( ( plus_plus_int @ zero_zero_int @ L )
% 0.24/0.51        = L ) ).
% 0.24/0.51  
% 0.24/0.51  % plus_int_code(2)
% 0.24/0.51  thf(fact_2356_plus__int__code_I1_J,axiom,
% 0.24/0.51      ! [K: int] :
% 0.24/0.51        ( ( plus_plus_int @ K @ zero_zero_int )
% 0.24/0.51        = K ) ).
% 0.24/0.51  
% 0.24/0.51  % plus_int_code(1)
% 0.24/0.51  thf(fact_2357_ile0__eq,axiom,
% 0.24/0.51      ! [N: extended_enat] :
% 0.24/0.51        ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.24/0.51        = ( N = zero_z5237406670263579293d_enat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % ile0_eq
% 0.24/0.51  thf(fact_2358_i0__lb,axiom,
% 0.24/0.51      ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 0.24/0.51  
% 0.24/0.51  % i0_lb
% 0.24/0.51  thf(fact_2359_zero__less__imp__eq__int,axiom,
% 0.24/0.51      ! [K: int] :
% 0.24/0.51        ( ( ord_less_int @ zero_zero_int @ K )
% 0.24/0.51       => ? [N2: nat] :
% 0.24/0.51            ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.24/0.51            & ( K
% 0.24/0.51              = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_imp_eq_int
% 0.24/0.51  thf(fact_2360_pos__int__cases,axiom,
% 0.24/0.51      ! [K: int] :
% 0.24/0.51        ( ( ord_less_int @ zero_zero_int @ K )
% 0.24/0.51       => ~ ! [N2: nat] :
% 0.24/0.51              ( ( K
% 0.24/0.51                = ( semiri1314217659103216013at_int @ N2 ) )
% 0.24/0.51             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % pos_int_cases
% 0.24/0.51  thf(fact_2361_le__imp__0__less,axiom,
% 0.24/0.51      ! [Z: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.24/0.51       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_imp_0_less
% 0.24/0.51  thf(fact_2362_nat__int__comparison_I2_J,axiom,
% 0.24/0.51      ( ord_less_nat
% 0.24/0.51      = ( ^ [A3: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat_int_comparison(2)
% 0.24/0.51  thf(fact_2363_real__arch__pow,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ X )
% 0.24/0.51       => ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % real_arch_pow
% 0.24/0.51  thf(fact_2364_int__gr__induct,axiom,
% 0.24/0.51      ! [K: int,I: int,P: int > $o] :
% 0.24/0.51        ( ( ord_less_int @ K @ I )
% 0.24/0.51       => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.24/0.51         => ( ! [I2: int] :
% 0.24/0.51                ( ( ord_less_int @ K @ I2 )
% 0.24/0.51               => ( ( P @ I2 )
% 0.24/0.51                 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.24/0.51           => ( P @ I ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % int_gr_induct
% 0.24/0.51  thf(fact_2365_zless__add1__eq,axiom,
% 0.24/0.51      ! [W: int,Z: int] :
% 0.24/0.51        ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.24/0.51        = ( ( ord_less_int @ W @ Z )
% 0.24/0.51          | ( W = Z ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zless_add1_eq
% 0.24/0.51  thf(fact_2366_int__ops_I1_J,axiom,
% 0.24/0.51      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.24/0.51      = zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % int_ops(1)
% 0.24/0.51  thf(fact_2367_nonneg__int__cases,axiom,
% 0.24/0.51      ! [K: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.24/0.51       => ~ ! [N2: nat] :
% 0.24/0.51              ( K
% 0.24/0.51             != ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nonneg_int_cases
% 0.24/0.51  thf(fact_2368_zero__le__imp__eq__int,axiom,
% 0.24/0.51      ! [K: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.24/0.51       => ? [N2: nat] :
% 0.24/0.51            ( K
% 0.24/0.51            = ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_imp_eq_int
% 0.24/0.51  thf(fact_2369_odd__nonzero,axiom,
% 0.24/0.51      ! [Z: int] :
% 0.24/0.51        ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 0.24/0.51       != zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % odd_nonzero
% 0.24/0.51  thf(fact_2370_linordered__field__no__ub,axiom,
% 0.24/0.51      ! [X2: real] :
% 0.24/0.51      ? [X_12: real] : ( ord_less_real @ X2 @ X_12 ) ).
% 0.24/0.51  
% 0.24/0.51  % linordered_field_no_ub
% 0.24/0.51  thf(fact_2371_linordered__field__no__ub,axiom,
% 0.24/0.51      ! [X2: rat] :
% 0.24/0.51      ? [X_12: rat] : ( ord_less_rat @ X2 @ X_12 ) ).
% 0.24/0.51  
% 0.24/0.51  % linordered_field_no_ub
% 0.24/0.51  thf(fact_2372_linordered__field__no__lb,axiom,
% 0.24/0.51      ! [X2: real] :
% 0.24/0.51      ? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% 0.24/0.51  
% 0.24/0.51  % linordered_field_no_lb
% 0.24/0.51  thf(fact_2373_linordered__field__no__lb,axiom,
% 0.24/0.51      ! [X2: rat] :
% 0.24/0.51      ? [Y4: rat] : ( ord_less_rat @ Y4 @ X2 ) ).
% 0.24/0.51  
% 0.24/0.51  % linordered_field_no_lb
% 0.24/0.51  thf(fact_2374_real__of__nat__div4,axiom,
% 0.24/0.51      ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % real_of_nat_div4
% 0.24/0.51  thf(fact_2375_zless__iff__Suc__zadd,axiom,
% 0.24/0.51      ( ord_less_int
% 0.24/0.51      = ( ^ [W2: int,Z4: int] :
% 0.24/0.51          ? [N3: nat] :
% 0.24/0.51            ( Z4
% 0.24/0.51            = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zless_iff_Suc_zadd
% 0.24/0.51  thf(fact_2376_add1__zle__eq,axiom,
% 0.24/0.51      ! [W: int,Z: int] :
% 0.24/0.51        ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 0.24/0.51        = ( ord_less_int @ W @ Z ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add1_zle_eq
% 0.24/0.51  thf(fact_2377_zless__imp__add1__zle,axiom,
% 0.24/0.51      ! [W: int,Z: int] :
% 0.24/0.51        ( ( ord_less_int @ W @ Z )
% 0.24/0.51       => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zless_imp_add1_zle
% 0.24/0.51  thf(fact_2378_nat__le__real__less,axiom,
% 0.24/0.51      ( ord_less_eq_nat
% 0.24/0.51      = ( ^ [N3: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat_le_real_less
% 0.24/0.51  thf(fact_2379_int__less__real__le,axiom,
% 0.24/0.51      ( ord_less_int
% 0.24/0.51      = ( ^ [N3: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % int_less_real_le
% 0.24/0.51  thf(fact_2380_int__le__real__less,axiom,
% 0.24/0.51      ( ord_less_eq_int
% 0.24/0.51      = ( ^ [N3: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % int_le_real_less
% 0.24/0.51  thf(fact_2381_not__exp__less__eq__0__int,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % not_exp_less_eq_0_int
% 0.24/0.51  thf(fact_2382_log__of__power__eq,axiom,
% 0.24/0.51      ! [M: nat,B: real,N: nat] :
% 0.24/0.51        ( ( ( semiri5074537144036343181t_real @ M )
% 0.24/0.51          = ( power_power_real @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_real @ one_one_real @ B )
% 0.24/0.51         => ( ( semiri5074537144036343181t_real @ N )
% 0.24/0.51            = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_of_power_eq
% 0.24/0.51  thf(fact_2383_less__log__of__power,axiom,
% 0.24/0.51      ! [B: real,N: nat,M: real] :
% 0.24/0.51        ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 0.24/0.51       => ( ( ord_less_real @ one_one_real @ B )
% 0.24/0.51         => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_log_of_power
% 0.24/0.51  thf(fact_2384_le__log__of__power,axiom,
% 0.24/0.51      ! [B: real,N: nat,M: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 0.24/0.51       => ( ( ord_less_real @ one_one_real @ B )
% 0.24/0.51         => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_log_of_power
% 0.24/0.51  thf(fact_2385_log__of__power__less,axiom,
% 0.24/0.51      ! [M: nat,B: real,N: nat] :
% 0.24/0.51        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_real @ one_one_real @ B )
% 0.24/0.51         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.24/0.51           => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_of_power_less
% 0.24/0.51  thf(fact_2386_add__divide__distrib,axiom,
% 0.24/0.51      ! [A: complex,B: complex,C: complex] :
% 0.24/0.51        ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.24/0.51        = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_divide_distrib
% 0.24/0.51  thf(fact_2387_add__divide__distrib,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real] :
% 0.24/0.51        ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.24/0.51        = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_divide_distrib
% 0.24/0.51  thf(fact_2388_add__divide__distrib,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.24/0.51        = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_divide_distrib
% 0.24/0.51  thf(fact_2389_div__le__dividend,axiom,
% 0.24/0.51      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.24/0.51  
% 0.24/0.51  % div_le_dividend
% 0.24/0.51  thf(fact_2390_div__le__mono,axiom,
% 0.24/0.51      ! [M: nat,N: nat,K: nat] :
% 0.24/0.51        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.51       => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_le_mono
% 0.24/0.51  thf(fact_2391_log__of__power__le,axiom,
% 0.24/0.51      ! [M: nat,B: real,N: nat] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.24/0.51       => ( ( ord_less_real @ one_one_real @ B )
% 0.24/0.51         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.24/0.51           => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log_of_power_le
% 0.24/0.51  thf(fact_2392_less__log2__of__power,axiom,
% 0.24/0.51      ! [N: nat,M: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.24/0.51       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_log2_of_power
% 0.24/0.51  thf(fact_2393_log2__of__power__less,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.24/0.51         => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % log2_of_power_less
% 0.24/0.51  thf(fact_2394_divide__right__mono__neg,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.24/0.51         => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_right_mono_neg
% 0.24/0.51  thf(fact_2395_divide__right__mono__neg,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_right_mono_neg
% 0.24/0.51  thf(fact_2396_divide__nonpos__nonpos,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.24/0.51         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonpos_nonpos
% 0.24/0.51  thf(fact_2397_divide__nonpos__nonpos,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonpos_nonpos
% 0.24/0.51  thf(fact_2398_divide__nonpos__nonneg,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.24/0.51         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonpos_nonneg
% 0.24/0.51  thf(fact_2399_divide__nonpos__nonneg,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.24/0.51         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonpos_nonneg
% 0.24/0.51  thf(fact_2400_divide__nonneg__nonpos,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.51       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.24/0.51         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonneg_nonpos
% 0.24/0.51  thf(fact_2401_divide__nonneg__nonpos,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.24/0.51       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonneg_nonpos
% 0.24/0.51  thf(fact_2402_divide__nonneg__nonneg,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.24/0.51         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonneg_nonneg
% 0.24/0.51  thf(fact_2403_divide__nonneg__nonneg,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.24/0.51         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonneg_nonneg
% 0.24/0.51  thf(fact_2404_zero__le__divide__iff,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.24/0.51        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.24/0.51          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.24/0.51            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_divide_iff
% 0.24/0.51  thf(fact_2405_zero__le__divide__iff,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.24/0.51        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.24/0.51          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.24/0.51            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_le_divide_iff
% 0.24/0.51  thf(fact_2406_divide__right__mono,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.24/0.51         => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_right_mono
% 0.24/0.51  thf(fact_2407_divide__right__mono,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.24/0.51         => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_right_mono
% 0.24/0.51  thf(fact_2408_divide__le__0__iff,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.24/0.51        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.51            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.24/0.51          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.24/0.51            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_0_iff
% 0.24/0.51  thf(fact_2409_divide__le__0__iff,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.24/0.51        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.51            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.24/0.51          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.24/0.51            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_0_iff
% 0.24/0.51  thf(fact_2410_divide__strict__right__mono__neg,axiom,
% 0.24/0.51      ! [B: real,A: real,C: real] :
% 0.24/0.51        ( ( ord_less_real @ B @ A )
% 0.24/0.51       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.24/0.51         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_strict_right_mono_neg
% 0.24/0.51  thf(fact_2411_divide__strict__right__mono__neg,axiom,
% 0.24/0.51      ! [B: rat,A: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_rat @ B @ A )
% 0.24/0.51       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_strict_right_mono_neg
% 0.24/0.51  thf(fact_2412_divide__strict__right__mono,axiom,
% 0.24/0.51      ! [A: real,B: real,C: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ B )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.24/0.51         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_strict_right_mono
% 0.24/0.51  thf(fact_2413_divide__strict__right__mono,axiom,
% 0.24/0.51      ! [A: rat,B: rat,C: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ B )
% 0.24/0.51       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.24/0.51         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_strict_right_mono
% 0.24/0.51  thf(fact_2414_zero__less__divide__iff,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.24/0.51        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51            & ( ord_less_real @ zero_zero_real @ B ) )
% 0.24/0.51          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_divide_iff
% 0.24/0.51  thf(fact_2415_zero__less__divide__iff,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.24/0.51        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.24/0.51          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_divide_iff
% 0.24/0.51  thf(fact_2416_divide__less__cancel,axiom,
% 0.24/0.51      ! [A: real,C: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.24/0.51        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.24/0.51           => ( ord_less_real @ A @ B ) )
% 0.24/0.51          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.24/0.51           => ( ord_less_real @ B @ A ) )
% 0.24/0.51          & ( C != zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_cancel
% 0.24/0.51  thf(fact_2417_divide__less__cancel,axiom,
% 0.24/0.51      ! [A: rat,C: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.24/0.51        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.24/0.51           => ( ord_less_rat @ A @ B ) )
% 0.24/0.51          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.24/0.51           => ( ord_less_rat @ B @ A ) )
% 0.24/0.51          & ( C != zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_cancel
% 0.24/0.51  thf(fact_2418_divide__less__0__iff,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.24/0.51        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51            & ( ord_less_real @ B @ zero_zero_real ) )
% 0.24/0.51          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_0_iff
% 0.24/0.51  thf(fact_2419_divide__less__0__iff,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.24/0.51        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.24/0.51          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_0_iff
% 0.24/0.51  thf(fact_2420_divide__pos__pos,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.24/0.51         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_pos_pos
% 0.24/0.51  thf(fact_2421_divide__pos__pos,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.24/0.51       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.24/0.51         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_pos_pos
% 0.24/0.51  thf(fact_2422_divide__pos__neg,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51       => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.24/0.51         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_pos_neg
% 0.24/0.51  thf(fact_2423_divide__pos__neg,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.24/0.51       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_pos_neg
% 0.24/0.51  thf(fact_2424_divide__neg__pos,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ X @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.24/0.51         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_neg_pos
% 0.24/0.51  thf(fact_2425_divide__neg__pos,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.24/0.51         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_neg_pos
% 0.24/0.51  thf(fact_2426_divide__neg__neg,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ X @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.24/0.51         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_neg_neg
% 0.24/0.51  thf(fact_2427_divide__neg__neg,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_neg_neg
% 0.24/0.51  thf(fact_2428_right__inverse__eq,axiom,
% 0.24/0.51      ! [B: complex,A: complex] :
% 0.24/0.51        ( ( B != zero_zero_complex )
% 0.24/0.51       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.24/0.51            = one_one_complex )
% 0.24/0.51          = ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % right_inverse_eq
% 0.24/0.51  thf(fact_2429_right__inverse__eq,axiom,
% 0.24/0.51      ! [B: real,A: real] :
% 0.24/0.51        ( ( B != zero_zero_real )
% 0.24/0.51       => ( ( ( divide_divide_real @ A @ B )
% 0.24/0.51            = one_one_real )
% 0.24/0.51          = ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % right_inverse_eq
% 0.24/0.51  thf(fact_2430_right__inverse__eq,axiom,
% 0.24/0.51      ! [B: rat,A: rat] :
% 0.24/0.51        ( ( B != zero_zero_rat )
% 0.24/0.51       => ( ( ( divide_divide_rat @ A @ B )
% 0.24/0.51            = one_one_rat )
% 0.24/0.51          = ( A = B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % right_inverse_eq
% 0.24/0.51  thf(fact_2431_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 0.24/0.51        = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.24/0.51  thf(fact_2432_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 0.24/0.51        = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.24/0.51  thf(fact_2433_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ( divide_divide_nat @ M @ N )
% 0.24/0.51          = zero_zero_nat )
% 0.24/0.51        = ( ( ord_less_nat @ M @ N )
% 0.24/0.51          | ( N = zero_zero_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Euclidean_Division.div_eq_0_iff
% 0.24/0.51  thf(fact_2434_Suc__div__le__mono,axiom,
% 0.24/0.51      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_div_le_mono
% 0.24/0.51  thf(fact_2435_field__le__epsilon,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ! [E: real] :
% 0.24/0.51            ( ( ord_less_real @ zero_zero_real @ E )
% 0.24/0.51           => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
% 0.24/0.51       => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.24/0.51  
% 0.24/0.51  % field_le_epsilon
% 0.24/0.51  thf(fact_2436_field__le__epsilon,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ! [E: rat] :
% 0.24/0.51            ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.24/0.51           => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
% 0.24/0.51       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.24/0.51  
% 0.24/0.51  % field_le_epsilon
% 0.24/0.51  thf(fact_2437_divide__nonpos__pos,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.24/0.51         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonpos_pos
% 0.24/0.51  thf(fact_2438_divide__nonpos__pos,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.24/0.51         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonpos_pos
% 0.24/0.51  thf(fact_2439_divide__nonpos__neg,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.24/0.51         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonpos_neg
% 0.24/0.51  thf(fact_2440_divide__nonpos__neg,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonpos_neg
% 0.24/0.51  thf(fact_2441_divide__nonneg__pos,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.51       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.24/0.51         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonneg_pos
% 0.24/0.51  thf(fact_2442_divide__nonneg__pos,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.24/0.51       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.24/0.51         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonneg_pos
% 0.24/0.51  thf(fact_2443_divide__nonneg__neg,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.51       => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.24/0.51         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonneg_neg
% 0.24/0.51  thf(fact_2444_divide__nonneg__neg,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.24/0.51       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.24/0.51         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_nonneg_neg
% 0.24/0.51  thf(fact_2445_divide__le__cancel,axiom,
% 0.24/0.51      ! [A: real,C: real,B: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.24/0.51        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.24/0.51           => ( ord_less_eq_real @ A @ B ) )
% 0.24/0.51          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.24/0.51           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_cancel
% 0.24/0.51  thf(fact_2446_divide__le__cancel,axiom,
% 0.24/0.51      ! [A: rat,C: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.24/0.51        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.24/0.51           => ( ord_less_eq_rat @ A @ B ) )
% 0.24/0.51          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.24/0.51           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_cancel
% 0.24/0.51  thf(fact_2447_frac__less2,axiom,
% 0.24/0.51      ! [X: real,Y: real,W: real,Z: real] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.51       => ( ( ord_less_eq_real @ X @ Y )
% 0.24/0.51         => ( ( ord_less_real @ zero_zero_real @ W )
% 0.24/0.51           => ( ( ord_less_real @ W @ Z )
% 0.24/0.51             => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % frac_less2
% 0.24/0.51  thf(fact_2448_frac__less2,axiom,
% 0.24/0.51      ! [X: rat,Y: rat,W: rat,Z: rat] :
% 0.24/0.51        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.24/0.51       => ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.51         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.24/0.51           => ( ( ord_less_rat @ W @ Z )
% 0.24/0.51             => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % frac_less2
% 0.24/0.51  thf(fact_2449_frac__less,axiom,
% 0.24/0.51      ! [X: real,Y: real,W: real,Z: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.51       => ( ( ord_less_real @ X @ Y )
% 0.24/0.51         => ( ( ord_less_real @ zero_zero_real @ W )
% 0.24/0.51           => ( ( ord_less_eq_real @ W @ Z )
% 0.24/0.51             => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % frac_less
% 0.24/0.51  thf(fact_2450_frac__less,axiom,
% 0.24/0.51      ! [X: rat,Y: rat,W: rat,Z: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.24/0.51       => ( ( ord_less_rat @ X @ Y )
% 0.24/0.51         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.24/0.51           => ( ( ord_less_eq_rat @ W @ Z )
% 0.24/0.51             => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % frac_less
% 0.24/0.51  thf(fact_2451_frac__le,axiom,
% 0.24/0.51      ! [Y: real,X: real,W: real,Z: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.24/0.51       => ( ( ord_less_eq_real @ X @ Y )
% 0.24/0.51         => ( ( ord_less_real @ zero_zero_real @ W )
% 0.24/0.51           => ( ( ord_less_eq_real @ W @ Z )
% 0.24/0.51             => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % frac_le
% 0.24/0.51  thf(fact_2452_frac__le,axiom,
% 0.24/0.51      ! [Y: rat,X: rat,W: rat,Z: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.24/0.51       => ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.51         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.24/0.51           => ( ( ord_less_eq_rat @ W @ Z )
% 0.24/0.51             => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % frac_le
% 0.24/0.51  thf(fact_2453_divide__less__eq__1,axiom,
% 0.24/0.51      ! [B: real,A: real] :
% 0.24/0.51        ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.24/0.51        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51            & ( ord_less_real @ B @ A ) )
% 0.24/0.51          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51            & ( ord_less_real @ A @ B ) )
% 0.24/0.51          | ( A = zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_eq_1
% 0.24/0.51  thf(fact_2454_divide__less__eq__1,axiom,
% 0.24/0.51      ! [B: rat,A: rat] :
% 0.24/0.51        ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.24/0.51        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51            & ( ord_less_rat @ B @ A ) )
% 0.24/0.51          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51            & ( ord_less_rat @ A @ B ) )
% 0.24/0.51          | ( A = zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_less_eq_1
% 0.24/0.51  thf(fact_2455_less__divide__eq__1,axiom,
% 0.24/0.51      ! [B: real,A: real] :
% 0.24/0.51        ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.24/0.51        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51            & ( ord_less_real @ A @ B ) )
% 0.24/0.51          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51            & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_divide_eq_1
% 0.24/0.51  thf(fact_2456_less__divide__eq__1,axiom,
% 0.24/0.51      ! [B: rat,A: rat] :
% 0.24/0.51        ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.24/0.51        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51            & ( ord_less_rat @ A @ B ) )
% 0.24/0.51          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_divide_eq_1
% 0.24/0.51  thf(fact_2457_div__add__self1,axiom,
% 0.24/0.51      ! [B: nat,A: nat] :
% 0.24/0.51        ( ( B != zero_zero_nat )
% 0.24/0.51       => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.24/0.51          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_add_self1
% 0.24/0.51  thf(fact_2458_div__add__self1,axiom,
% 0.24/0.51      ! [B: int,A: int] :
% 0.24/0.51        ( ( B != zero_zero_int )
% 0.24/0.51       => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.24/0.51          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_add_self1
% 0.24/0.51  thf(fact_2459_div__add__self2,axiom,
% 0.24/0.51      ! [B: nat,A: nat] :
% 0.24/0.51        ( ( B != zero_zero_nat )
% 0.24/0.51       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.24/0.51          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_add_self2
% 0.24/0.51  thf(fact_2460_div__add__self2,axiom,
% 0.24/0.51      ! [B: int,A: int] :
% 0.24/0.51        ( ( B != zero_zero_int )
% 0.24/0.51       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.24/0.51          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_add_self2
% 0.24/0.51  thf(fact_2461_gt__half__sum,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ B )
% 0.24/0.51       => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 0.24/0.51  
% 0.24/0.51  % gt_half_sum
% 0.24/0.51  thf(fact_2462_gt__half__sum,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ B )
% 0.24/0.51       => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 0.24/0.51  
% 0.24/0.51  % gt_half_sum
% 0.24/0.51  thf(fact_2463_less__half__sum,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ B )
% 0.24/0.51       => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_half_sum
% 0.24/0.51  thf(fact_2464_less__half__sum,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ B )
% 0.24/0.51       => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_half_sum
% 0.24/0.51  thf(fact_2465_numeral__Bit0__div__2,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.24/0.51        = ( numeral_numeral_nat @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % numeral_Bit0_div_2
% 0.24/0.51  thf(fact_2466_numeral__Bit0__div__2,axiom,
% 0.24/0.51      ! [N: num] :
% 0.24/0.51        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.24/0.51        = ( numeral_numeral_int @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % numeral_Bit0_div_2
% 0.24/0.51  thf(fact_2467_div__greater__zero__iff,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 0.24/0.51        = ( ( ord_less_eq_nat @ N @ M )
% 0.24/0.51          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_greater_zero_iff
% 0.24/0.51  thf(fact_2468_div__le__mono2,axiom,
% 0.24/0.51      ! [M: nat,N: nat,K: nat] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.24/0.51       => ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.51         => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_le_mono2
% 0.24/0.51  thf(fact_2469_div__less__dividend,axiom,
% 0.24/0.51      ! [N: nat,M: nat] :
% 0.24/0.51        ( ( ord_less_nat @ one_one_nat @ N )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.24/0.51         => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_less_dividend
% 0.24/0.51  thf(fact_2470_div__eq__dividend__iff,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.24/0.51       => ( ( ( divide_divide_nat @ M @ N )
% 0.24/0.51            = M )
% 0.24/0.51          = ( N = one_one_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_eq_dividend_iff
% 0.24/0.51  thf(fact_2471_le__divide__eq__1,axiom,
% 0.24/0.51      ! [B: real,A: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.24/0.51        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51            & ( ord_less_eq_real @ A @ B ) )
% 0.24/0.51          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51            & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_divide_eq_1
% 0.24/0.51  thf(fact_2472_le__divide__eq__1,axiom,
% 0.24/0.51      ! [B: rat,A: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.24/0.51        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51            & ( ord_less_eq_rat @ A @ B ) )
% 0.24/0.51          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51            & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % le_divide_eq_1
% 0.24/0.51  thf(fact_2473_divide__le__eq__1,axiom,
% 0.24/0.51      ! [B: real,A: real] :
% 0.24/0.51        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.24/0.51        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51            & ( ord_less_eq_real @ B @ A ) )
% 0.24/0.51          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.51            & ( ord_less_eq_real @ A @ B ) )
% 0.24/0.51          | ( A = zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_eq_1
% 0.24/0.51  thf(fact_2474_divide__le__eq__1,axiom,
% 0.24/0.51      ! [B: rat,A: rat] :
% 0.24/0.51        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.24/0.51        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.24/0.51            & ( ord_less_eq_rat @ B @ A ) )
% 0.24/0.51          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.24/0.51            & ( ord_less_eq_rat @ A @ B ) )
% 0.24/0.51          | ( A = zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % divide_le_eq_1
% 0.24/0.51  thf(fact_2475_exp__add__not__zero__imp__left,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.51         != zero_zero_nat )
% 0.24/0.51       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.24/0.51         != zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % exp_add_not_zero_imp_left
% 0.24/0.51  thf(fact_2476_exp__add__not__zero__imp__left,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.51         != zero_zero_int )
% 0.24/0.51       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.24/0.51         != zero_zero_int ) ) ).
% 0.24/0.51  
% 0.24/0.51  % exp_add_not_zero_imp_left
% 0.24/0.51  thf(fact_2477_exp__add__not__zero__imp__right,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.51         != zero_zero_nat )
% 0.24/0.51       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.24/0.51         != zero_zero_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % exp_add_not_zero_imp_right
% 0.24/0.51  thf(fact_2478_exp__add__not__zero__imp__right,axiom,
% 0.24/0.51      ! [M: nat,N: nat] :
% 0.24/0.51        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.51         != zero_zero_int )
% 0.24/0.51       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.24/0.51         != zero_zero_int ) ) ).
% 0.24/0.51  
% 0.24/0.51  % exp_add_not_zero_imp_right
% 0.24/0.51  thf(fact_2479_div__exp__eq,axiom,
% 0.24/0.51      ! [A: nat,M: nat,N: nat] :
% 0.24/0.51        ( ( 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 ) )
% 0.24/0.51        = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_exp_eq
% 0.24/0.51  thf(fact_2480_div__exp__eq,axiom,
% 0.24/0.51      ! [A: int,M: nat,N: nat] :
% 0.24/0.51        ( ( 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 ) )
% 0.24/0.51        = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_exp_eq
% 0.24/0.51  thf(fact_2481_div__2__gt__zero,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.24/0.51       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_2_gt_zero
% 0.24/0.51  thf(fact_2482_Suc__n__div__2__gt__zero,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % Suc_n_div_2_gt_zero
% 0.24/0.51  thf(fact_2483_buildup__gives__valid,axiom,
% 0.24/0.51      ! [N: nat] :
% 0.24/0.51        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51       => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % buildup_gives_valid
% 0.24/0.51  thf(fact_2484_div__self,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( A != zero_zero_complex )
% 0.24/0.51       => ( ( divide1717551699836669952omplex @ A @ A )
% 0.24/0.51          = one_one_complex ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_self
% 0.24/0.51  thf(fact_2485_div__self,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( A != zero_zero_real )
% 0.24/0.51       => ( ( divide_divide_real @ A @ A )
% 0.24/0.51          = one_one_real ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_self
% 0.24/0.51  thf(fact_2486_div__self,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( A != zero_zero_rat )
% 0.24/0.51       => ( ( divide_divide_rat @ A @ A )
% 0.24/0.51          = one_one_rat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_self
% 0.24/0.51  thf(fact_2487_div__self,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( A != zero_zero_nat )
% 0.24/0.51       => ( ( divide_divide_nat @ A @ A )
% 0.24/0.51          = one_one_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_self
% 0.24/0.51  thf(fact_2488_div__self,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( A != zero_zero_int )
% 0.24/0.51       => ( ( divide_divide_int @ A @ A )
% 0.24/0.51          = one_one_int ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_self
% 0.24/0.51  thf(fact_2489_arcosh__1,axiom,
% 0.24/0.51      ( ( arcosh_real @ one_one_real )
% 0.24/0.51      = zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % arcosh_1
% 0.24/0.51  thf(fact_2490_nat__induct2,axiom,
% 0.24/0.51      ! [P: nat > $o,N: nat] :
% 0.24/0.51        ( ( P @ zero_zero_nat )
% 0.24/0.51       => ( ( P @ one_one_nat )
% 0.24/0.51         => ( ! [N2: nat] :
% 0.24/0.51                ( ( P @ N2 )
% 0.24/0.51               => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.24/0.51           => ( P @ N ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % nat_induct2
% 0.24/0.51  thf(fact_2491_high__bound__aux,axiom,
% 0.24/0.51      ! [Ma: nat,N: nat,M: nat] :
% 0.24/0.51        ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.24/0.51       => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % high_bound_aux
% 0.24/0.51  thf(fact_2492_zdiv__numeral__Bit0,axiom,
% 0.24/0.51      ! [V: num,W: num] :
% 0.24/0.51        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.24/0.51        = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zdiv_numeral_Bit0
% 0.24/0.51  thf(fact_2493_pos2,axiom,
% 0.24/0.51      ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 0.24/0.51  
% 0.24/0.51  % pos2
% 0.24/0.51  thf(fact_2494_div__by__1,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_1
% 0.24/0.51  thf(fact_2495_div__by__1,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( divide_divide_real @ A @ one_one_real )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_1
% 0.24/0.51  thf(fact_2496_div__by__1,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( divide_divide_rat @ A @ one_one_rat )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_1
% 0.24/0.51  thf(fact_2497_div__by__1,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( divide_divide_nat @ A @ one_one_nat )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_1
% 0.24/0.51  thf(fact_2498_div__by__1,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( divide_divide_int @ A @ one_one_int )
% 0.24/0.51        = A ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_1
% 0.24/0.51  thf(fact_2499_div__by__0,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.24/0.51        = zero_zero_complex ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_0
% 0.24/0.51  thf(fact_2500_div__by__0,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( divide_divide_real @ A @ zero_zero_real )
% 0.24/0.51        = zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_0
% 0.24/0.51  thf(fact_2501_div__by__0,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.24/0.51        = zero_zero_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_0
% 0.24/0.51  thf(fact_2502_div__by__0,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.24/0.51        = zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_0
% 0.24/0.51  thf(fact_2503_div__by__0,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( divide_divide_int @ A @ zero_zero_int )
% 0.24/0.51        = zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % div_by_0
% 0.24/0.51  thf(fact_2504_div__0,axiom,
% 0.24/0.51      ! [A: complex] :
% 0.24/0.51        ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 0.24/0.51        = zero_zero_complex ) ).
% 0.24/0.51  
% 0.24/0.51  % div_0
% 0.24/0.51  thf(fact_2505_div__0,axiom,
% 0.24/0.51      ! [A: real] :
% 0.24/0.51        ( ( divide_divide_real @ zero_zero_real @ A )
% 0.24/0.51        = zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % div_0
% 0.24/0.51  thf(fact_2506_div__0,axiom,
% 0.24/0.51      ! [A: rat] :
% 0.24/0.51        ( ( divide_divide_rat @ zero_zero_rat @ A )
% 0.24/0.51        = zero_zero_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % div_0
% 0.24/0.51  thf(fact_2507_div__0,axiom,
% 0.24/0.51      ! [A: nat] :
% 0.24/0.51        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.24/0.51        = zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % div_0
% 0.24/0.51  thf(fact_2508_div__0,axiom,
% 0.24/0.51      ! [A: int] :
% 0.24/0.51        ( ( divide_divide_int @ zero_zero_int @ A )
% 0.24/0.51        = zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % div_0
% 0.24/0.51  thf(fact_2509_high__def,axiom,
% 0.24/0.51      ( vEBT_VEBT_high
% 0.24/0.51      = ( ^ [X4: nat,N3: nat] : ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % high_def
% 0.24/0.51  thf(fact_2510_i0__less,axiom,
% 0.24/0.51      ! [N: extended_enat] :
% 0.24/0.51        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.24/0.51        = ( N != zero_z5237406670263579293d_enat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % i0_less
% 0.24/0.51  thf(fact_2511_psubsetD,axiom,
% 0.24/0.51      ! [A2: set_complex,B2: set_complex,C: complex] :
% 0.24/0.51        ( ( ord_less_set_complex @ A2 @ B2 )
% 0.24/0.51       => ( ( member_complex @ C @ A2 )
% 0.24/0.51         => ( member_complex @ C @ B2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % psubsetD
% 0.24/0.51  thf(fact_2512_psubsetD,axiom,
% 0.24/0.51      ! [A2: set_real,B2: set_real,C: real] :
% 0.24/0.51        ( ( ord_less_set_real @ A2 @ B2 )
% 0.24/0.51       => ( ( member_real @ C @ A2 )
% 0.24/0.51         => ( member_real @ C @ B2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % psubsetD
% 0.24/0.51  thf(fact_2513_psubsetD,axiom,
% 0.24/0.51      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 0.24/0.51        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 0.24/0.51       => ( ( member_set_nat @ C @ A2 )
% 0.24/0.51         => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % psubsetD
% 0.24/0.51  thf(fact_2514_psubsetD,axiom,
% 0.24/0.51      ! [A2: set_nat,B2: set_nat,C: nat] :
% 0.24/0.51        ( ( ord_less_set_nat @ A2 @ B2 )
% 0.24/0.51       => ( ( member_nat @ C @ A2 )
% 0.24/0.51         => ( member_nat @ C @ B2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % psubsetD
% 0.24/0.51  thf(fact_2515_psubsetD,axiom,
% 0.24/0.51      ! [A2: set_int,B2: set_int,C: int] :
% 0.24/0.51        ( ( ord_less_set_int @ A2 @ B2 )
% 0.24/0.51       => ( ( member_int @ C @ A2 )
% 0.24/0.51         => ( member_int @ C @ B2 ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % psubsetD
% 0.24/0.51  thf(fact_2516_not__iless0,axiom,
% 0.24/0.51      ! [N: extended_enat] :
% 0.24/0.51        ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 0.24/0.51  
% 0.24/0.51  % not_iless0
% 0.24/0.51  thf(fact_2517_enat__less__induct,axiom,
% 0.24/0.51      ! [P: extended_enat > $o,N: extended_enat] :
% 0.24/0.51        ( ! [N2: extended_enat] :
% 0.24/0.51            ( ! [M3: extended_enat] :
% 0.24/0.51                ( ( ord_le72135733267957522d_enat @ M3 @ N2 )
% 0.24/0.51               => ( P @ M3 ) )
% 0.24/0.51           => ( P @ N2 ) )
% 0.24/0.51       => ( P @ N ) ) ).
% 0.24/0.51  
% 0.24/0.51  % enat_less_induct
% 0.24/0.51  thf(fact_2518_iadd__is__0,axiom,
% 0.24/0.51      ! [M: extended_enat,N: extended_enat] :
% 0.24/0.51        ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 0.24/0.51          = zero_z5237406670263579293d_enat )
% 0.24/0.51        = ( ( M = zero_z5237406670263579293d_enat )
% 0.24/0.51          & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % iadd_is_0
% 0.24/0.51  thf(fact_2519_zero__one__enat__neq_I1_J,axiom,
% 0.24/0.51      zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 0.24/0.51  
% 0.24/0.51  % zero_one_enat_neq(1)
% 0.24/0.51  thf(fact_2520_linorder__neqE__linordered__idom,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( X != Y )
% 0.24/0.51       => ( ~ ( ord_less_real @ X @ Y )
% 0.24/0.51         => ( ord_less_real @ Y @ X ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_neqE_linordered_idom
% 0.24/0.51  thf(fact_2521_linorder__neqE__linordered__idom,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( X != Y )
% 0.24/0.51       => ( ~ ( ord_less_rat @ X @ Y )
% 0.24/0.51         => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_neqE_linordered_idom
% 0.24/0.51  thf(fact_2522_linorder__neqE__linordered__idom,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( X != Y )
% 0.24/0.51       => ( ~ ( ord_less_int @ X @ Y )
% 0.24/0.51         => ( ord_less_int @ Y @ X ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % linorder_neqE_linordered_idom
% 0.24/0.51  thf(fact_2523_zero__neq__one,axiom,
% 0.24/0.51      zero_zero_complex != one_one_complex ).
% 0.24/0.51  
% 0.24/0.51  % zero_neq_one
% 0.24/0.51  thf(fact_2524_zero__neq__one,axiom,
% 0.24/0.51      zero_zero_real != one_one_real ).
% 0.24/0.51  
% 0.24/0.51  % zero_neq_one
% 0.24/0.51  thf(fact_2525_zero__neq__one,axiom,
% 0.24/0.51      zero_zero_rat != one_one_rat ).
% 0.24/0.51  
% 0.24/0.51  % zero_neq_one
% 0.24/0.51  thf(fact_2526_zero__neq__one,axiom,
% 0.24/0.51      zero_zero_nat != one_one_nat ).
% 0.24/0.51  
% 0.24/0.51  % zero_neq_one
% 0.24/0.51  thf(fact_2527_zero__neq__one,axiom,
% 0.24/0.51      zero_zero_int != one_one_int ).
% 0.24/0.51  
% 0.24/0.51  % zero_neq_one
% 0.24/0.51  thf(fact_2528_realpow__pos__nth2,axiom,
% 0.24/0.51      ! [A: real,N: nat] :
% 0.24/0.51        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.51       => ? [R3: real] :
% 0.24/0.51            ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.24/0.51            & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 0.24/0.51              = A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % realpow_pos_nth2
% 0.24/0.51  thf(fact_2529_pos__imp__zdiv__neg__iff,axiom,
% 0.24/0.51      ! [B: int,A: int] :
% 0.24/0.51        ( ( ord_less_int @ zero_zero_int @ B )
% 0.24/0.51       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.24/0.51          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % pos_imp_zdiv_neg_iff
% 0.24/0.51  thf(fact_2530_neg__imp__zdiv__neg__iff,axiom,
% 0.24/0.51      ! [B: int,A: int] :
% 0.24/0.51        ( ( ord_less_int @ B @ zero_zero_int )
% 0.24/0.51       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.24/0.51          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % neg_imp_zdiv_neg_iff
% 0.24/0.51  thf(fact_2531_div__neg__pos__less0,axiom,
% 0.24/0.51      ! [A: int,B: int] :
% 0.24/0.51        ( ( ord_less_int @ A @ zero_zero_int )
% 0.24/0.51       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.24/0.51         => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % div_neg_pos_less0
% 0.24/0.51  thf(fact_2532_zdiv__int,axiom,
% 0.24/0.51      ! [A: nat,B: nat] :
% 0.24/0.51        ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 0.24/0.51        = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % zdiv_int
% 0.24/0.51  thf(fact_2533_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 0.24/0.51      ! [X: nat,N: nat,M: nat] :
% 0.24/0.51        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.24/0.51       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.51         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.24/0.51           => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % VEBT_internal.exp_split_high_low(1)
% 0.24/0.51  thf(fact_2534_not__one__le__zero,axiom,
% 0.24/0.51      ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % not_one_le_zero
% 0.24/0.51  thf(fact_2535_not__one__le__zero,axiom,
% 0.24/0.51      ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % not_one_le_zero
% 0.24/0.51  thf(fact_2536_not__one__le__zero,axiom,
% 0.24/0.51      ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % not_one_le_zero
% 0.24/0.51  thf(fact_2537_not__one__le__zero,axiom,
% 0.24/0.51      ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % not_one_le_zero
% 0.24/0.51  thf(fact_2538_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.24/0.51      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.24/0.51  
% 0.24/0.51  % linordered_nonzero_semiring_class.zero_le_one
% 0.24/0.51  thf(fact_2539_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.24/0.51      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.24/0.51  
% 0.24/0.51  % linordered_nonzero_semiring_class.zero_le_one
% 0.24/0.51  thf(fact_2540_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.24/0.51      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.24/0.51  
% 0.24/0.51  % linordered_nonzero_semiring_class.zero_le_one
% 0.24/0.51  thf(fact_2541_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.24/0.51      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.24/0.51  
% 0.24/0.51  % linordered_nonzero_semiring_class.zero_le_one
% 0.24/0.51  thf(fact_2542_zero__less__one__class_Ozero__le__one,axiom,
% 0.24/0.51      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_one_class.zero_le_one
% 0.24/0.51  thf(fact_2543_zero__less__one__class_Ozero__le__one,axiom,
% 0.24/0.51      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_one_class.zero_le_one
% 0.24/0.51  thf(fact_2544_zero__less__one__class_Ozero__le__one,axiom,
% 0.24/0.51      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_one_class.zero_le_one
% 0.24/0.51  thf(fact_2545_zero__less__one__class_Ozero__le__one,axiom,
% 0.24/0.51      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_one_class.zero_le_one
% 0.24/0.51  thf(fact_2546_not__one__less__zero,axiom,
% 0.24/0.51      ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.24/0.51  
% 0.24/0.51  % not_one_less_zero
% 0.24/0.51  thf(fact_2547_not__one__less__zero,axiom,
% 0.24/0.51      ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.24/0.51  
% 0.24/0.51  % not_one_less_zero
% 0.24/0.51  thf(fact_2548_not__one__less__zero,axiom,
% 0.24/0.51      ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.24/0.51  
% 0.24/0.51  % not_one_less_zero
% 0.24/0.51  thf(fact_2549_not__one__less__zero,axiom,
% 0.24/0.51      ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.24/0.51  
% 0.24/0.51  % not_one_less_zero
% 0.24/0.51  thf(fact_2550_zero__less__one,axiom,
% 0.24/0.51      ord_less_real @ zero_zero_real @ one_one_real ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_one
% 0.24/0.51  thf(fact_2551_zero__less__one,axiom,
% 0.24/0.51      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_one
% 0.24/0.51  thf(fact_2552_zero__less__one,axiom,
% 0.24/0.51      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_one
% 0.24/0.51  thf(fact_2553_zero__less__one,axiom,
% 0.24/0.51      ord_less_int @ zero_zero_int @ one_one_int ).
% 0.24/0.51  
% 0.24/0.51  % zero_less_one
% 0.24/0.51  thf(fact_2554_add__less__zeroD,axiom,
% 0.24/0.51      ! [X: real,Y: real] :
% 0.24/0.51        ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.24/0.51       => ( ( ord_less_real @ X @ zero_zero_real )
% 0.24/0.51          | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_zeroD
% 0.24/0.51  thf(fact_2555_add__less__zeroD,axiom,
% 0.24/0.51      ! [X: rat,Y: rat] :
% 0.24/0.51        ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
% 0.24/0.51       => ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.24/0.51          | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_zeroD
% 0.24/0.51  thf(fact_2556_add__less__zeroD,axiom,
% 0.24/0.51      ! [X: int,Y: int] :
% 0.24/0.51        ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
% 0.24/0.51       => ( ( ord_less_int @ X @ zero_zero_int )
% 0.24/0.51          | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_less_zeroD
% 0.24/0.51  thf(fact_2557_add__mono1,axiom,
% 0.24/0.51      ! [A: real,B: real] :
% 0.24/0.51        ( ( ord_less_real @ A @ B )
% 0.24/0.51       => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono1
% 0.24/0.51  thf(fact_2558_add__mono1,axiom,
% 0.24/0.51      ! [A: rat,B: rat] :
% 0.24/0.51        ( ( ord_less_rat @ A @ B )
% 0.24/0.51       => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono1
% 0.24/0.51  thf(fact_2559_add__mono1,axiom,
% 0.24/0.51      ! [A: nat,B: nat] :
% 0.24/0.51        ( ( ord_less_nat @ A @ B )
% 0.24/0.51       => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono1
% 0.24/0.51  thf(fact_2560_add__mono1,axiom,
% 0.24/0.51      ! [A: int,B: int] :
% 0.24/0.51        ( ( ord_less_int @ A @ B )
% 0.24/0.51       => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 0.24/0.51  
% 0.24/0.51  % add_mono1
% 0.24/0.51  thf(fact_2561_less__add__one,axiom,
% 0.24/0.51      ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_add_one
% 0.24/0.51  thf(fact_2562_less__add__one,axiom,
% 0.24/0.51      ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_add_one
% 0.24/0.51  thf(fact_2563_less__add__one,axiom,
% 0.24/0.51      ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 0.24/0.51  
% 0.24/0.51  % less_add_one
% 0.24/0.51  thf(fact_2564_less__add__one,axiom,
% 0.24/0.52      ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_add_one
% 0.24/0.52  thf(fact_2565_realpow__pos__nth,axiom,
% 0.24/0.52      ! [N: nat,A: real] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.52         => ? [R3: real] :
% 0.24/0.52              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.24/0.52              & ( ( power_power_real @ R3 @ N )
% 0.24/0.52                = A ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % realpow_pos_nth
% 0.24/0.52  thf(fact_2566_realpow__pos__nth__unique,axiom,
% 0.24/0.52      ! [N: nat,A: real] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.52         => ? [X3: real] :
% 0.24/0.52              ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.24/0.52              & ( ( power_power_real @ X3 @ N )
% 0.24/0.52                = A )
% 0.24/0.52              & ! [Y5: real] :
% 0.24/0.52                  ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 0.24/0.52                    & ( ( power_power_real @ Y5 @ N )
% 0.24/0.52                      = A ) )
% 0.24/0.52                 => ( Y5 = X3 ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % realpow_pos_nth_unique
% 0.24/0.52  thf(fact_2567_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.52       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.24/0.52          = ( ( ord_less_eq_int @ B @ A )
% 0.24/0.52            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonneg1_imp_zdiv_pos_iff
% 0.24/0.52  thf(fact_2568_pos__imp__zdiv__nonneg__iff,axiom,
% 0.24/0.52      ! [B: int,A: int] :
% 0.24/0.52        ( ( ord_less_int @ zero_zero_int @ B )
% 0.24/0.52       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.24/0.52          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % pos_imp_zdiv_nonneg_iff
% 0.24/0.52  thf(fact_2569_neg__imp__zdiv__nonneg__iff,axiom,
% 0.24/0.52      ! [B: int,A: int] :
% 0.24/0.52        ( ( ord_less_int @ B @ zero_zero_int )
% 0.24/0.52       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.24/0.52          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % neg_imp_zdiv_nonneg_iff
% 0.24/0.52  thf(fact_2570_pos__imp__zdiv__pos__iff,axiom,
% 0.24/0.52      ! [K: int,I: int] :
% 0.24/0.52        ( ( ord_less_int @ zero_zero_int @ K )
% 0.24/0.52       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 0.24/0.52          = ( ord_less_eq_int @ K @ I ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % pos_imp_zdiv_pos_iff
% 0.24/0.52  thf(fact_2571_div__nonpos__pos__le0,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.24/0.52       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.24/0.52         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_nonpos_pos_le0
% 0.24/0.52  thf(fact_2572_div__nonneg__neg__le0,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.52       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.24/0.52         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_nonneg_neg_le0
% 0.24/0.52  thf(fact_2573_div__positive__int,axiom,
% 0.24/0.52      ! [L: int,K: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ L @ K )
% 0.24/0.52       => ( ( ord_less_int @ zero_zero_int @ L )
% 0.24/0.52         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_positive_int
% 0.24/0.52  thf(fact_2574_div__int__pos__iff,axiom,
% 0.24/0.52      ! [K: int,L: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 0.24/0.52        = ( ( K = zero_zero_int )
% 0.24/0.52          | ( L = zero_zero_int )
% 0.24/0.52          | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.24/0.52            & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 0.24/0.52          | ( ( ord_less_int @ K @ zero_zero_int )
% 0.24/0.52            & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_int_pos_iff
% 0.24/0.52  thf(fact_2575_zdiv__mono2__neg,axiom,
% 0.24/0.52      ! [A: int,B7: int,B: int] :
% 0.24/0.52        ( ( ord_less_int @ A @ zero_zero_int )
% 0.24/0.52       => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.24/0.52         => ( ( ord_less_eq_int @ B7 @ B )
% 0.24/0.52           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B7 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zdiv_mono2_neg
% 0.24/0.52  thf(fact_2576_zdiv__mono1__neg,axiom,
% 0.24/0.52      ! [A: int,A6: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ A @ A6 )
% 0.24/0.52       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.24/0.52         => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zdiv_mono1_neg
% 0.24/0.52  thf(fact_2577_zdiv__eq__0__iff,axiom,
% 0.24/0.52      ! [I: int,K: int] :
% 0.24/0.52        ( ( ( divide_divide_int @ I @ K )
% 0.24/0.52          = zero_zero_int )
% 0.24/0.52        = ( ( K = zero_zero_int )
% 0.24/0.52          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.24/0.52            & ( ord_less_int @ I @ K ) )
% 0.24/0.52          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.24/0.52            & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zdiv_eq_0_iff
% 0.24/0.52  thf(fact_2578_zdiv__mono2,axiom,
% 0.24/0.52      ! [A: int,B7: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.52       => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.24/0.52         => ( ( ord_less_eq_int @ B7 @ B )
% 0.24/0.52           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B7 ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zdiv_mono2
% 0.24/0.52  thf(fact_2579_zdiv__mono1,axiom,
% 0.24/0.52      ! [A: int,A6: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ A @ A6 )
% 0.24/0.52       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.24/0.52         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zdiv_mono1
% 0.24/0.52  thf(fact_2580_int__div__less__self,axiom,
% 0.24/0.52      ! [X: int,K: int] :
% 0.24/0.52        ( ( ord_less_int @ zero_zero_int @ X )
% 0.24/0.52       => ( ( ord_less_int @ one_one_int @ K )
% 0.24/0.52         => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % int_div_less_self
% 0.24/0.52  thf(fact_2581_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.52       => ( ( ord_less_nat @ A @ B )
% 0.24/0.52         => ( ( divide_divide_nat @ A @ B )
% 0.24/0.52            = zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % unique_euclidean_semiring_numeral_class.div_less
% 0.24/0.52  thf(fact_2582_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.52       => ( ( ord_less_int @ A @ B )
% 0.24/0.52         => ( ( divide_divide_int @ A @ B )
% 0.24/0.52            = zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % unique_euclidean_semiring_numeral_class.div_less
% 0.24/0.52  thf(fact_2583_div__positive,axiom,
% 0.24/0.52      ! [B: nat,A: nat] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.24/0.52       => ( ( ord_less_eq_nat @ B @ A )
% 0.24/0.52         => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_positive
% 0.24/0.52  thf(fact_2584_div__positive,axiom,
% 0.24/0.52      ! [B: int,A: int] :
% 0.24/0.52        ( ( ord_less_int @ zero_zero_int @ B )
% 0.24/0.52       => ( ( ord_less_eq_int @ B @ A )
% 0.24/0.52         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_positive
% 0.24/0.52  thf(fact_2585_discrete,axiom,
% 0.24/0.52      ( ord_less_nat
% 0.24/0.52      = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % discrete
% 0.24/0.52  thf(fact_2586_discrete,axiom,
% 0.24/0.52      ( ord_less_int
% 0.24/0.52      = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % discrete
% 0.24/0.52  thf(fact_2587_zero__less__two,axiom,
% 0.24/0.52      ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_less_two
% 0.24/0.52  thf(fact_2588_zero__less__two,axiom,
% 0.24/0.52      ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_less_two
% 0.24/0.52  thf(fact_2589_zero__less__two,axiom,
% 0.24/0.52      ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_less_two
% 0.24/0.52  thf(fact_2590_zero__less__two,axiom,
% 0.24/0.52      ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_less_two
% 0.24/0.52  thf(fact_2591_high__inv,axiom,
% 0.24/0.52      ! [X: nat,N: nat,Y: nat] :
% 0.24/0.52        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.24/0.52       => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 0.24/0.52          = Y ) ) ).
% 0.24/0.52  
% 0.24/0.52  % high_inv
% 0.24/0.52  thf(fact_2592_buildup__nothing__in__leaf,axiom,
% 0.24/0.52      ! [N: nat,X: nat] :
% 0.24/0.52        ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 0.24/0.52  
% 0.24/0.52  % buildup_nothing_in_leaf
% 0.24/0.52  thf(fact_2593_ceiling__log2__div2,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.24/0.52       => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.24/0.52          = ( 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 ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ceiling_log2_div2
% 0.24/0.52  thf(fact_2594_artanh__0,axiom,
% 0.24/0.52      ( ( artanh_real @ zero_zero_real )
% 0.24/0.52      = zero_zero_real ) ).
% 0.24/0.52  
% 0.24/0.52  % artanh_0
% 0.24/0.52  thf(fact_2595_arsinh__0,axiom,
% 0.24/0.52      ( ( arsinh_real @ zero_zero_real )
% 0.24/0.52      = zero_zero_real ) ).
% 0.24/0.52  
% 0.24/0.52  % arsinh_0
% 0.24/0.52  thf(fact_2596_ceiling__log__eq__powr__iff,axiom,
% 0.24/0.52      ! [X: real,B: real,K: nat] :
% 0.24/0.52        ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.52       => ( ( ord_less_real @ one_one_real @ B )
% 0.24/0.52         => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
% 0.24/0.52              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.24/0.52            = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 0.24/0.52              & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ceiling_log_eq_powr_iff
% 0.24/0.52  thf(fact_2597_floor__log__nat__eq__powr__iff,axiom,
% 0.24/0.52      ! [B: nat,K: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.24/0.52       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.24/0.52         => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.24/0.52              = ( semiri1314217659103216013at_int @ N ) )
% 0.24/0.52            = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.24/0.52              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_log_nat_eq_powr_iff
% 0.24/0.52  thf(fact_2598_of__int__round__le,axiom,
% 0.24/0.52      ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_round_le
% 0.24/0.52  thf(fact_2599_of__int__round__le,axiom,
% 0.24/0.52      ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_round_le
% 0.24/0.52  thf(fact_2600_mult__cancel__right,axiom,
% 0.24/0.52      ! [A: complex,C: complex,B: complex] :
% 0.24/0.52        ( ( ( times_times_complex @ A @ C )
% 0.24/0.52          = ( times_times_complex @ B @ C ) )
% 0.24/0.52        = ( ( C = zero_zero_complex )
% 0.24/0.52          | ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right
% 0.24/0.52  thf(fact_2601_mult__cancel__right,axiom,
% 0.24/0.52      ! [A: real,C: real,B: real] :
% 0.24/0.52        ( ( ( times_times_real @ A @ C )
% 0.24/0.52          = ( times_times_real @ B @ C ) )
% 0.24/0.52        = ( ( C = zero_zero_real )
% 0.24/0.52          | ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right
% 0.24/0.52  thf(fact_2602_mult__cancel__right,axiom,
% 0.24/0.52      ! [A: rat,C: rat,B: rat] :
% 0.24/0.52        ( ( ( times_times_rat @ A @ C )
% 0.24/0.52          = ( times_times_rat @ B @ C ) )
% 0.24/0.52        = ( ( C = zero_zero_rat )
% 0.24/0.52          | ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right
% 0.24/0.52  thf(fact_2603_mult__cancel__right,axiom,
% 0.24/0.52      ! [A: nat,C: nat,B: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ A @ C )
% 0.24/0.52          = ( times_times_nat @ B @ C ) )
% 0.24/0.52        = ( ( C = zero_zero_nat )
% 0.24/0.52          | ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right
% 0.24/0.52  thf(fact_2604_mult__cancel__right,axiom,
% 0.24/0.52      ! [A: int,C: int,B: int] :
% 0.24/0.52        ( ( ( times_times_int @ A @ C )
% 0.24/0.52          = ( times_times_int @ B @ C ) )
% 0.24/0.52        = ( ( C = zero_zero_int )
% 0.24/0.52          | ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right
% 0.24/0.52  thf(fact_2605_mult__cancel__left,axiom,
% 0.24/0.52      ! [C: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( ( times_times_complex @ C @ A )
% 0.24/0.52          = ( times_times_complex @ C @ B ) )
% 0.24/0.52        = ( ( C = zero_zero_complex )
% 0.24/0.52          | ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left
% 0.24/0.52  thf(fact_2606_mult__cancel__left,axiom,
% 0.24/0.52      ! [C: real,A: real,B: real] :
% 0.24/0.52        ( ( ( times_times_real @ C @ A )
% 0.24/0.52          = ( times_times_real @ C @ B ) )
% 0.24/0.52        = ( ( C = zero_zero_real )
% 0.24/0.52          | ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left
% 0.24/0.52  thf(fact_2607_mult__cancel__left,axiom,
% 0.24/0.52      ! [C: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( ( times_times_rat @ C @ A )
% 0.24/0.52          = ( times_times_rat @ C @ B ) )
% 0.24/0.52        = ( ( C = zero_zero_rat )
% 0.24/0.52          | ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left
% 0.24/0.52  thf(fact_2608_mult__cancel__left,axiom,
% 0.24/0.52      ! [C: nat,A: nat,B: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ C @ A )
% 0.24/0.52          = ( times_times_nat @ C @ B ) )
% 0.24/0.52        = ( ( C = zero_zero_nat )
% 0.24/0.52          | ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left
% 0.24/0.52  thf(fact_2609_mult__cancel__left,axiom,
% 0.24/0.52      ! [C: int,A: int,B: int] :
% 0.24/0.52        ( ( ( times_times_int @ C @ A )
% 0.24/0.52          = ( times_times_int @ C @ B ) )
% 0.24/0.52        = ( ( C = zero_zero_int )
% 0.24/0.52          | ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left
% 0.24/0.52  thf(fact_2610_mult__eq__0__iff,axiom,
% 0.24/0.52      ! [A: complex,B: complex] :
% 0.24/0.52        ( ( ( times_times_complex @ A @ B )
% 0.24/0.52          = zero_zero_complex )
% 0.24/0.52        = ( ( A = zero_zero_complex )
% 0.24/0.52          | ( B = zero_zero_complex ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_eq_0_iff
% 0.24/0.52  thf(fact_2611_mult__eq__0__iff,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ( times_times_real @ A @ B )
% 0.24/0.52          = zero_zero_real )
% 0.24/0.52        = ( ( A = zero_zero_real )
% 0.24/0.52          | ( B = zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_eq_0_iff
% 0.24/0.52  thf(fact_2612_mult__eq__0__iff,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ( times_times_rat @ A @ B )
% 0.24/0.52          = zero_zero_rat )
% 0.24/0.52        = ( ( A = zero_zero_rat )
% 0.24/0.52          | ( B = zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_eq_0_iff
% 0.24/0.52  thf(fact_2613_mult__eq__0__iff,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ A @ B )
% 0.24/0.52          = zero_zero_nat )
% 0.24/0.52        = ( ( A = zero_zero_nat )
% 0.24/0.52          | ( B = zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_eq_0_iff
% 0.24/0.52  thf(fact_2614_mult__eq__0__iff,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ( times_times_int @ A @ B )
% 0.24/0.52          = zero_zero_int )
% 0.24/0.52        = ( ( A = zero_zero_int )
% 0.24/0.52          | ( B = zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_eq_0_iff
% 0.24/0.52  thf(fact_2615_mult__zero__right,axiom,
% 0.24/0.52      ! [A: complex] :
% 0.24/0.52        ( ( times_times_complex @ A @ zero_zero_complex )
% 0.24/0.52        = zero_zero_complex ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_zero_right
% 0.24/0.52  thf(fact_2616_mult__zero__right,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( times_times_real @ A @ zero_zero_real )
% 0.24/0.52        = zero_zero_real ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_zero_right
% 0.24/0.52  thf(fact_2617_mult__zero__right,axiom,
% 0.24/0.52      ! [A: rat] :
% 0.24/0.52        ( ( times_times_rat @ A @ zero_zero_rat )
% 0.24/0.52        = zero_zero_rat ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_zero_right
% 0.24/0.52  thf(fact_2618_mult__zero__right,axiom,
% 0.24/0.52      ! [A: nat] :
% 0.24/0.52        ( ( times_times_nat @ A @ zero_zero_nat )
% 0.24/0.52        = zero_zero_nat ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_zero_right
% 0.24/0.52  thf(fact_2619_mult__zero__right,axiom,
% 0.24/0.52      ! [A: int] :
% 0.24/0.52        ( ( times_times_int @ A @ zero_zero_int )
% 0.24/0.52        = zero_zero_int ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_zero_right
% 0.24/0.52  thf(fact_2620_mult__zero__left,axiom,
% 0.24/0.52      ! [A: complex] :
% 0.24/0.52        ( ( times_times_complex @ zero_zero_complex @ A )
% 0.24/0.52        = zero_zero_complex ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_zero_left
% 0.24/0.52  thf(fact_2621_mult__zero__left,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( times_times_real @ zero_zero_real @ A )
% 0.24/0.52        = zero_zero_real ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_zero_left
% 0.24/0.52  thf(fact_2622_mult__zero__left,axiom,
% 0.24/0.52      ! [A: rat] :
% 0.24/0.52        ( ( times_times_rat @ zero_zero_rat @ A )
% 0.24/0.52        = zero_zero_rat ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_zero_left
% 0.24/0.52  thf(fact_2623_mult__zero__left,axiom,
% 0.24/0.52      ! [A: nat] :
% 0.24/0.52        ( ( times_times_nat @ zero_zero_nat @ A )
% 0.24/0.52        = zero_zero_nat ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_zero_left
% 0.24/0.52  thf(fact_2624_mult__zero__left,axiom,
% 0.24/0.52      ! [A: int] :
% 0.24/0.52        ( ( times_times_int @ zero_zero_int @ A )
% 0.24/0.52        = zero_zero_int ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_zero_left
% 0.24/0.52  thf(fact_2625_diff__self,axiom,
% 0.24/0.52      ! [A: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ A @ A )
% 0.24/0.52        = zero_zero_complex ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_self
% 0.24/0.52  thf(fact_2626_diff__self,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( minus_minus_real @ A @ A )
% 0.24/0.52        = zero_zero_real ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_self
% 0.24/0.52  thf(fact_2627_diff__self,axiom,
% 0.24/0.52      ! [A: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ A @ A )
% 0.24/0.52        = zero_zero_rat ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_self
% 0.24/0.52  thf(fact_2628_diff__self,axiom,
% 0.24/0.52      ! [A: int] :
% 0.24/0.52        ( ( minus_minus_int @ A @ A )
% 0.24/0.52        = zero_zero_int ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_self
% 0.24/0.52  thf(fact_2629_diff__0__right,axiom,
% 0.24/0.52      ! [A: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_0_right
% 0.24/0.52  thf(fact_2630_diff__0__right,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( minus_minus_real @ A @ zero_zero_real )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_0_right
% 0.24/0.52  thf(fact_2631_diff__0__right,axiom,
% 0.24/0.52      ! [A: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_0_right
% 0.24/0.52  thf(fact_2632_diff__0__right,axiom,
% 0.24/0.52      ! [A: int] :
% 0.24/0.52        ( ( minus_minus_int @ A @ zero_zero_int )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_0_right
% 0.24/0.52  thf(fact_2633_zero__diff,axiom,
% 0.24/0.52      ! [A: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.24/0.52        = zero_zero_nat ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_diff
% 0.24/0.52  thf(fact_2634_diff__zero,axiom,
% 0.24/0.52      ! [A: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_zero
% 0.24/0.52  thf(fact_2635_diff__zero,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( minus_minus_real @ A @ zero_zero_real )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_zero
% 0.24/0.52  thf(fact_2636_diff__zero,axiom,
% 0.24/0.52      ! [A: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_zero
% 0.24/0.52  thf(fact_2637_diff__zero,axiom,
% 0.24/0.52      ! [A: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_zero
% 0.24/0.52  thf(fact_2638_diff__zero,axiom,
% 0.24/0.52      ! [A: int] :
% 0.24/0.52        ( ( minus_minus_int @ A @ zero_zero_int )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_zero
% 0.24/0.52  thf(fact_2639_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.24/0.52      ! [A: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ A @ A )
% 0.24/0.52        = zero_zero_complex ) ).
% 0.24/0.52  
% 0.24/0.52  % cancel_comm_monoid_add_class.diff_cancel
% 0.24/0.52  thf(fact_2640_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( minus_minus_real @ A @ A )
% 0.24/0.52        = zero_zero_real ) ).
% 0.24/0.52  
% 0.24/0.52  % cancel_comm_monoid_add_class.diff_cancel
% 0.24/0.52  thf(fact_2641_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.24/0.52      ! [A: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ A @ A )
% 0.24/0.52        = zero_zero_rat ) ).
% 0.24/0.52  
% 0.24/0.52  % cancel_comm_monoid_add_class.diff_cancel
% 0.24/0.52  thf(fact_2642_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.24/0.52      ! [A: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ A @ A )
% 0.24/0.52        = zero_zero_nat ) ).
% 0.24/0.52  
% 0.24/0.52  % cancel_comm_monoid_add_class.diff_cancel
% 0.24/0.52  thf(fact_2643_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.24/0.52      ! [A: int] :
% 0.24/0.52        ( ( minus_minus_int @ A @ A )
% 0.24/0.52        = zero_zero_int ) ).
% 0.24/0.52  
% 0.24/0.52  % cancel_comm_monoid_add_class.diff_cancel
% 0.24/0.52  thf(fact_2644_numeral__times__numeral,axiom,
% 0.24/0.52      ! [M: num,N: num] :
% 0.24/0.52        ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.24/0.52        = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_times_numeral
% 0.24/0.52  thf(fact_2645_numeral__times__numeral,axiom,
% 0.24/0.52      ! [M: num,N: num] :
% 0.24/0.52        ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.24/0.52        = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_times_numeral
% 0.24/0.52  thf(fact_2646_numeral__times__numeral,axiom,
% 0.24/0.52      ! [M: num,N: num] :
% 0.24/0.52        ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.24/0.52        = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_times_numeral
% 0.24/0.52  thf(fact_2647_numeral__times__numeral,axiom,
% 0.24/0.52      ! [M: num,N: num] :
% 0.24/0.52        ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.24/0.52        = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_times_numeral
% 0.24/0.52  thf(fact_2648_numeral__times__numeral,axiom,
% 0.24/0.52      ! [M: num,N: num] :
% 0.24/0.52        ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.24/0.52        = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_times_numeral
% 0.24/0.52  thf(fact_2649_mult__numeral__left__semiring__numeral,axiom,
% 0.24/0.52      ! [V: num,W: num,Z: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.24/0.52        = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_numeral_left_semiring_numeral
% 0.24/0.52  thf(fact_2650_mult__numeral__left__semiring__numeral,axiom,
% 0.24/0.52      ! [V: num,W: num,Z: real] :
% 0.24/0.52        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.24/0.52        = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_numeral_left_semiring_numeral
% 0.24/0.52  thf(fact_2651_mult__numeral__left__semiring__numeral,axiom,
% 0.24/0.52      ! [V: num,W: num,Z: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.24/0.52        = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_numeral_left_semiring_numeral
% 0.24/0.52  thf(fact_2652_mult__numeral__left__semiring__numeral,axiom,
% 0.24/0.52      ! [V: num,W: num,Z: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.24/0.52        = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_numeral_left_semiring_numeral
% 0.24/0.52  thf(fact_2653_mult__numeral__left__semiring__numeral,axiom,
% 0.24/0.52      ! [V: num,W: num,Z: int] :
% 0.24/0.52        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.24/0.52        = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_numeral_left_semiring_numeral
% 0.24/0.52  thf(fact_2654_mult_Oright__neutral,axiom,
% 0.24/0.52      ! [A: complex] :
% 0.24/0.52        ( ( times_times_complex @ A @ one_one_complex )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.right_neutral
% 0.24/0.52  thf(fact_2655_mult_Oright__neutral,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( times_times_real @ A @ one_one_real )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.right_neutral
% 0.24/0.52  thf(fact_2656_mult_Oright__neutral,axiom,
% 0.24/0.52      ! [A: rat] :
% 0.24/0.52        ( ( times_times_rat @ A @ one_one_rat )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.right_neutral
% 0.24/0.52  thf(fact_2657_mult_Oright__neutral,axiom,
% 0.24/0.52      ! [A: nat] :
% 0.24/0.52        ( ( times_times_nat @ A @ one_one_nat )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.right_neutral
% 0.24/0.52  thf(fact_2658_mult_Oright__neutral,axiom,
% 0.24/0.52      ! [A: int] :
% 0.24/0.52        ( ( times_times_int @ A @ one_one_int )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.right_neutral
% 0.24/0.52  thf(fact_2659_mult__1,axiom,
% 0.24/0.52      ! [A: complex] :
% 0.24/0.52        ( ( times_times_complex @ one_one_complex @ A )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_1
% 0.24/0.52  thf(fact_2660_mult__1,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( times_times_real @ one_one_real @ A )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_1
% 0.24/0.52  thf(fact_2661_mult__1,axiom,
% 0.24/0.52      ! [A: rat] :
% 0.24/0.52        ( ( times_times_rat @ one_one_rat @ A )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_1
% 0.24/0.52  thf(fact_2662_mult__1,axiom,
% 0.24/0.52      ! [A: nat] :
% 0.24/0.52        ( ( times_times_nat @ one_one_nat @ A )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_1
% 0.24/0.52  thf(fact_2663_mult__1,axiom,
% 0.24/0.52      ! [A: int] :
% 0.24/0.52        ( ( times_times_int @ one_one_int @ A )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_1
% 0.24/0.52  thf(fact_2664_add__diff__cancel,axiom,
% 0.24/0.52      ! [A: complex,B: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel
% 0.24/0.52  thf(fact_2665_add__diff__cancel,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel
% 0.24/0.52  thf(fact_2666_add__diff__cancel,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel
% 0.24/0.52  thf(fact_2667_add__diff__cancel,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel
% 0.24/0.52  thf(fact_2668_diff__add__cancel,axiom,
% 0.24/0.52      ! [A: complex,B: complex] :
% 0.24/0.52        ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_cancel
% 0.24/0.52  thf(fact_2669_diff__add__cancel,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_cancel
% 0.24/0.52  thf(fact_2670_diff__add__cancel,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_cancel
% 0.24/0.52  thf(fact_2671_diff__add__cancel,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_cancel
% 0.24/0.52  thf(fact_2672_add__diff__cancel__left,axiom,
% 0.24/0.52      ! [C: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
% 0.24/0.52        = ( minus_minus_complex @ A @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_left
% 0.24/0.52  thf(fact_2673_add__diff__cancel__left,axiom,
% 0.24/0.52      ! [C: real,A: real,B: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.24/0.52        = ( minus_minus_real @ A @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_left
% 0.24/0.52  thf(fact_2674_add__diff__cancel__left,axiom,
% 0.24/0.52      ! [C: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.24/0.52        = ( minus_minus_rat @ A @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_left
% 0.24/0.52  thf(fact_2675_add__diff__cancel__left,axiom,
% 0.24/0.52      ! [C: nat,A: nat,B: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.24/0.52        = ( minus_minus_nat @ A @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_left
% 0.24/0.52  thf(fact_2676_add__diff__cancel__left,axiom,
% 0.24/0.52      ! [C: int,A: int,B: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.24/0.52        = ( minus_minus_int @ A @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_left
% 0.24/0.52  thf(fact_2677_add__diff__cancel__left_H,axiom,
% 0.24/0.52      ! [A: complex,B: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ A )
% 0.24/0.52        = B ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_left'
% 0.24/0.52  thf(fact_2678_add__diff__cancel__left_H,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 0.24/0.52        = B ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_left'
% 0.24/0.52  thf(fact_2679_add__diff__cancel__left_H,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 0.24/0.52        = B ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_left'
% 0.24/0.52  thf(fact_2680_add__diff__cancel__left_H,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.24/0.52        = B ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_left'
% 0.24/0.52  thf(fact_2681_add__diff__cancel__left_H,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.24/0.52        = B ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_left'
% 0.24/0.52  thf(fact_2682_add__diff__cancel__right,axiom,
% 0.24/0.52      ! [A: complex,C: complex,B: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
% 0.24/0.52        = ( minus_minus_complex @ A @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_right
% 0.24/0.52  thf(fact_2683_add__diff__cancel__right,axiom,
% 0.24/0.52      ! [A: real,C: real,B: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.24/0.52        = ( minus_minus_real @ A @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_right
% 0.24/0.52  thf(fact_2684_add__diff__cancel__right,axiom,
% 0.24/0.52      ! [A: rat,C: rat,B: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.24/0.52        = ( minus_minus_rat @ A @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_right
% 0.24/0.52  thf(fact_2685_add__diff__cancel__right,axiom,
% 0.24/0.52      ! [A: nat,C: nat,B: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.24/0.52        = ( minus_minus_nat @ A @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_right
% 0.24/0.52  thf(fact_2686_add__diff__cancel__right,axiom,
% 0.24/0.52      ! [A: int,C: int,B: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.24/0.52        = ( minus_minus_int @ A @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_right
% 0.24/0.52  thf(fact_2687_add__diff__cancel__right_H,axiom,
% 0.24/0.52      ! [A: complex,B: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_right'
% 0.24/0.52  thf(fact_2688_add__diff__cancel__right_H,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_right'
% 0.24/0.52  thf(fact_2689_add__diff__cancel__right_H,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_right'
% 0.24/0.52  thf(fact_2690_add__diff__cancel__right_H,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_right'
% 0.24/0.52  thf(fact_2691_add__diff__cancel__right_H,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_cancel_right'
% 0.24/0.52  thf(fact_2692_times__divide__eq__right,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.24/0.52        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % times_divide_eq_right
% 0.24/0.52  thf(fact_2693_times__divide__eq__right,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.24/0.52        = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % times_divide_eq_right
% 0.24/0.52  thf(fact_2694_times__divide__eq__right,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.24/0.52        = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % times_divide_eq_right
% 0.24/0.52  thf(fact_2695_divide__divide__eq__right,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.24/0.52        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_eq_right
% 0.24/0.52  thf(fact_2696_divide__divide__eq__right,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.24/0.52        = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_eq_right
% 0.24/0.52  thf(fact_2697_divide__divide__eq__right,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.24/0.52        = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_eq_right
% 0.24/0.52  thf(fact_2698_divide__divide__eq__left,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.24/0.52        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_eq_left
% 0.24/0.52  thf(fact_2699_divide__divide__eq__left,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.24/0.52        = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_eq_left
% 0.24/0.52  thf(fact_2700_divide__divide__eq__left,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.24/0.52        = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_eq_left
% 0.24/0.52  thf(fact_2701_times__divide__eq__left,axiom,
% 0.24/0.52      ! [B: complex,C: complex,A: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 0.24/0.52        = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % times_divide_eq_left
% 0.24/0.52  thf(fact_2702_times__divide__eq__left,axiom,
% 0.24/0.52      ! [B: real,C: real,A: real] :
% 0.24/0.52        ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.24/0.52        = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % times_divide_eq_left
% 0.24/0.52  thf(fact_2703_times__divide__eq__left,axiom,
% 0.24/0.52      ! [B: rat,C: rat,A: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.24/0.52        = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % times_divide_eq_left
% 0.24/0.52  thf(fact_2704_of__int__mult,axiom,
% 0.24/0.52      ! [W: int,Z: int] :
% 0.24/0.52        ( ( ring_18347121197199848620nteger @ ( times_times_int @ W @ Z ) )
% 0.24/0.52        = ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_mult
% 0.24/0.52  thf(fact_2705_of__int__mult,axiom,
% 0.24/0.52      ! [W: int,Z: int] :
% 0.24/0.52        ( ( ring_17405671764205052669omplex @ ( times_times_int @ W @ Z ) )
% 0.24/0.52        = ( times_times_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_mult
% 0.24/0.52  thf(fact_2706_of__int__mult,axiom,
% 0.24/0.52      ! [W: int,Z: int] :
% 0.24/0.52        ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 0.24/0.52        = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_mult
% 0.24/0.52  thf(fact_2707_of__int__mult,axiom,
% 0.24/0.52      ! [W: int,Z: int] :
% 0.24/0.52        ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 0.24/0.52        = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_mult
% 0.24/0.52  thf(fact_2708_of__int__mult,axiom,
% 0.24/0.52      ! [W: int,Z: int] :
% 0.24/0.52        ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 0.24/0.52        = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_mult
% 0.24/0.52  thf(fact_2709_diff__Suc__Suc,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.24/0.52        = ( minus_minus_nat @ M @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_Suc_Suc
% 0.24/0.52  thf(fact_2710_Suc__diff__diff,axiom,
% 0.24/0.52      ! [M: nat,N: nat,K: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.24/0.52        = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.24/0.52  
% 0.24/0.52  % Suc_diff_diff
% 0.24/0.52  thf(fact_2711_diff__0__eq__0,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.24/0.52        = zero_zero_nat ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_0_eq_0
% 0.24/0.52  thf(fact_2712_diff__self__eq__0,axiom,
% 0.24/0.52      ! [M: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ M @ M )
% 0.24/0.52        = zero_zero_nat ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_self_eq_0
% 0.24/0.52  thf(fact_2713_of__int__diff,axiom,
% 0.24/0.52      ! [W: int,Z: int] :
% 0.24/0.52        ( ( ring_18347121197199848620nteger @ ( minus_minus_int @ W @ Z ) )
% 0.24/0.52        = ( minus_8373710615458151222nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_diff
% 0.24/0.52  thf(fact_2714_of__int__diff,axiom,
% 0.24/0.52      ! [W: int,Z: int] :
% 0.24/0.52        ( ( ring_17405671764205052669omplex @ ( minus_minus_int @ W @ Z ) )
% 0.24/0.52        = ( minus_minus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_diff
% 0.24/0.52  thf(fact_2715_of__int__diff,axiom,
% 0.24/0.52      ! [W: int,Z: int] :
% 0.24/0.52        ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 0.24/0.52        = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_diff
% 0.24/0.52  thf(fact_2716_of__int__diff,axiom,
% 0.24/0.52      ! [W: int,Z: int] :
% 0.24/0.52        ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 0.24/0.52        = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_diff
% 0.24/0.52  thf(fact_2717_of__int__diff,axiom,
% 0.24/0.52      ! [W: int,Z: int] :
% 0.24/0.52        ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 0.24/0.52        = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_diff
% 0.24/0.52  thf(fact_2718_mult__is__0,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ M @ N )
% 0.24/0.52          = zero_zero_nat )
% 0.24/0.52        = ( ( M = zero_zero_nat )
% 0.24/0.52          | ( N = zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_is_0
% 0.24/0.52  thf(fact_2719_mult__0__right,axiom,
% 0.24/0.52      ! [M: nat] :
% 0.24/0.52        ( ( times_times_nat @ M @ zero_zero_nat )
% 0.24/0.52        = zero_zero_nat ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_0_right
% 0.24/0.52  thf(fact_2720_mult__cancel1,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ K @ M )
% 0.24/0.52          = ( times_times_nat @ K @ N ) )
% 0.24/0.52        = ( ( M = N )
% 0.24/0.52          | ( K = zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel1
% 0.24/0.52  thf(fact_2721_mult__cancel2,axiom,
% 0.24/0.52      ! [M: nat,K: nat,N: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ M @ K )
% 0.24/0.52          = ( times_times_nat @ N @ K ) )
% 0.24/0.52        = ( ( M = N )
% 0.24/0.52          | ( K = zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel2
% 0.24/0.52  thf(fact_2722_diff__diff__cancel,axiom,
% 0.24/0.52      ! [I: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ I @ N )
% 0.24/0.52       => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.24/0.52          = I ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_cancel
% 0.24/0.52  thf(fact_2723_diff__diff__left,axiom,
% 0.24/0.52      ! [I: nat,J: nat,K: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.24/0.52        = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_left
% 0.24/0.52  thf(fact_2724_powr__0,axiom,
% 0.24/0.52      ! [Z: real] :
% 0.24/0.52        ( ( powr_real @ zero_zero_real @ Z )
% 0.24/0.52        = zero_zero_real ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_0
% 0.24/0.52  thf(fact_2725_powr__eq__0__iff,axiom,
% 0.24/0.52      ! [W: real,Z: real] :
% 0.24/0.52        ( ( ( powr_real @ W @ Z )
% 0.24/0.52          = zero_zero_real )
% 0.24/0.52        = ( W = zero_zero_real ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_eq_0_iff
% 0.24/0.52  thf(fact_2726_powr__one__eq__one,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( powr_real @ one_one_real @ A )
% 0.24/0.52        = one_one_real ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_one_eq_one
% 0.24/0.52  thf(fact_2727_nat__mult__eq__1__iff,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ M @ N )
% 0.24/0.52          = one_one_nat )
% 0.24/0.52        = ( ( M = one_one_nat )
% 0.24/0.52          & ( N = one_one_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_mult_eq_1_iff
% 0.24/0.52  thf(fact_2728_nat__1__eq__mult__iff,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( one_one_nat
% 0.24/0.52          = ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( ( M = one_one_nat )
% 0.24/0.52          & ( N = one_one_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_1_eq_mult_iff
% 0.24/0.52  thf(fact_2729_floor__of__int,axiom,
% 0.24/0.52      ! [Z: int] :
% 0.24/0.52        ( ( archim6058952711729229775r_real @ ( ring_1_of_int_real @ Z ) )
% 0.24/0.52        = Z ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_of_int
% 0.24/0.52  thf(fact_2730_floor__of__int,axiom,
% 0.24/0.52      ! [Z: int] :
% 0.24/0.52        ( ( archim3151403230148437115or_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.24/0.52        = Z ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_of_int
% 0.24/0.52  thf(fact_2731_of__int__floor__cancel,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
% 0.24/0.52          = X )
% 0.24/0.52        = ( ? [N3: int] :
% 0.24/0.52              ( X
% 0.24/0.52              = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_floor_cancel
% 0.24/0.52  thf(fact_2732_of__int__floor__cancel,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
% 0.24/0.52          = X )
% 0.24/0.52        = ( ? [N3: int] :
% 0.24/0.52              ( X
% 0.24/0.52              = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_floor_cancel
% 0.24/0.52  thf(fact_2733_round__of__int,axiom,
% 0.24/0.52      ! [N: int] :
% 0.24/0.52        ( ( archim8280529875227126926d_real @ ( ring_1_of_int_real @ N ) )
% 0.24/0.52        = N ) ).
% 0.24/0.52  
% 0.24/0.52  % round_of_int
% 0.24/0.52  thf(fact_2734_round__of__int,axiom,
% 0.24/0.52      ! [N: int] :
% 0.24/0.52        ( ( archim7778729529865785530nd_rat @ ( ring_1_of_int_rat @ N ) )
% 0.24/0.52        = N ) ).
% 0.24/0.52  
% 0.24/0.52  % round_of_int
% 0.24/0.52  thf(fact_2735_diff__ge__0__iff__ge,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.24/0.52        = ( ord_less_eq_real @ B @ A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_ge_0_iff_ge
% 0.24/0.52  thf(fact_2736_diff__ge__0__iff__ge,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.24/0.52        = ( ord_less_eq_rat @ B @ A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_ge_0_iff_ge
% 0.24/0.52  thf(fact_2737_diff__ge__0__iff__ge,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.24/0.52        = ( ord_less_eq_int @ B @ A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_ge_0_iff_ge
% 0.24/0.52  thf(fact_2738_diff__gt__0__iff__gt,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.24/0.52        = ( ord_less_real @ B @ A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_gt_0_iff_gt
% 0.24/0.52  thf(fact_2739_diff__gt__0__iff__gt,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.24/0.52        = ( ord_less_rat @ B @ A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_gt_0_iff_gt
% 0.24/0.52  thf(fact_2740_diff__gt__0__iff__gt,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.24/0.52        = ( ord_less_int @ B @ A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_gt_0_iff_gt
% 0.24/0.52  thf(fact_2741_mult__cancel__left1,axiom,
% 0.24/0.52      ! [C: complex,B: complex] :
% 0.24/0.52        ( ( C
% 0.24/0.52          = ( times_times_complex @ C @ B ) )
% 0.24/0.52        = ( ( C = zero_zero_complex )
% 0.24/0.52          | ( B = one_one_complex ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left1
% 0.24/0.52  thf(fact_2742_mult__cancel__left1,axiom,
% 0.24/0.52      ! [C: real,B: real] :
% 0.24/0.52        ( ( C
% 0.24/0.52          = ( times_times_real @ C @ B ) )
% 0.24/0.52        = ( ( C = zero_zero_real )
% 0.24/0.52          | ( B = one_one_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left1
% 0.24/0.52  thf(fact_2743_mult__cancel__left1,axiom,
% 0.24/0.52      ! [C: rat,B: rat] :
% 0.24/0.52        ( ( C
% 0.24/0.52          = ( times_times_rat @ C @ B ) )
% 0.24/0.52        = ( ( C = zero_zero_rat )
% 0.24/0.52          | ( B = one_one_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left1
% 0.24/0.52  thf(fact_2744_mult__cancel__left1,axiom,
% 0.24/0.52      ! [C: int,B: int] :
% 0.24/0.52        ( ( C
% 0.24/0.52          = ( times_times_int @ C @ B ) )
% 0.24/0.52        = ( ( C = zero_zero_int )
% 0.24/0.52          | ( B = one_one_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left1
% 0.24/0.52  thf(fact_2745_mult__cancel__left2,axiom,
% 0.24/0.52      ! [C: complex,A: complex] :
% 0.24/0.52        ( ( ( times_times_complex @ C @ A )
% 0.24/0.52          = C )
% 0.24/0.52        = ( ( C = zero_zero_complex )
% 0.24/0.52          | ( A = one_one_complex ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left2
% 0.24/0.52  thf(fact_2746_mult__cancel__left2,axiom,
% 0.24/0.52      ! [C: real,A: real] :
% 0.24/0.52        ( ( ( times_times_real @ C @ A )
% 0.24/0.52          = C )
% 0.24/0.52        = ( ( C = zero_zero_real )
% 0.24/0.52          | ( A = one_one_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left2
% 0.24/0.52  thf(fact_2747_mult__cancel__left2,axiom,
% 0.24/0.52      ! [C: rat,A: rat] :
% 0.24/0.52        ( ( ( times_times_rat @ C @ A )
% 0.24/0.52          = C )
% 0.24/0.52        = ( ( C = zero_zero_rat )
% 0.24/0.52          | ( A = one_one_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left2
% 0.24/0.52  thf(fact_2748_mult__cancel__left2,axiom,
% 0.24/0.52      ! [C: int,A: int] :
% 0.24/0.52        ( ( ( times_times_int @ C @ A )
% 0.24/0.52          = C )
% 0.24/0.52        = ( ( C = zero_zero_int )
% 0.24/0.52          | ( A = one_one_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_left2
% 0.24/0.52  thf(fact_2749_mult__cancel__right1,axiom,
% 0.24/0.52      ! [C: complex,B: complex] :
% 0.24/0.52        ( ( C
% 0.24/0.52          = ( times_times_complex @ B @ C ) )
% 0.24/0.52        = ( ( C = zero_zero_complex )
% 0.24/0.52          | ( B = one_one_complex ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right1
% 0.24/0.52  thf(fact_2750_mult__cancel__right1,axiom,
% 0.24/0.52      ! [C: real,B: real] :
% 0.24/0.52        ( ( C
% 0.24/0.52          = ( times_times_real @ B @ C ) )
% 0.24/0.52        = ( ( C = zero_zero_real )
% 0.24/0.52          | ( B = one_one_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right1
% 0.24/0.52  thf(fact_2751_mult__cancel__right1,axiom,
% 0.24/0.52      ! [C: rat,B: rat] :
% 0.24/0.52        ( ( C
% 0.24/0.52          = ( times_times_rat @ B @ C ) )
% 0.24/0.52        = ( ( C = zero_zero_rat )
% 0.24/0.52          | ( B = one_one_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right1
% 0.24/0.52  thf(fact_2752_mult__cancel__right1,axiom,
% 0.24/0.52      ! [C: int,B: int] :
% 0.24/0.52        ( ( C
% 0.24/0.52          = ( times_times_int @ B @ C ) )
% 0.24/0.52        = ( ( C = zero_zero_int )
% 0.24/0.52          | ( B = one_one_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right1
% 0.24/0.52  thf(fact_2753_mult__cancel__right2,axiom,
% 0.24/0.52      ! [A: complex,C: complex] :
% 0.24/0.52        ( ( ( times_times_complex @ A @ C )
% 0.24/0.52          = C )
% 0.24/0.52        = ( ( C = zero_zero_complex )
% 0.24/0.52          | ( A = one_one_complex ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right2
% 0.24/0.52  thf(fact_2754_mult__cancel__right2,axiom,
% 0.24/0.52      ! [A: real,C: real] :
% 0.24/0.52        ( ( ( times_times_real @ A @ C )
% 0.24/0.52          = C )
% 0.24/0.52        = ( ( C = zero_zero_real )
% 0.24/0.52          | ( A = one_one_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right2
% 0.24/0.52  thf(fact_2755_mult__cancel__right2,axiom,
% 0.24/0.52      ! [A: rat,C: rat] :
% 0.24/0.52        ( ( ( times_times_rat @ A @ C )
% 0.24/0.52          = C )
% 0.24/0.52        = ( ( C = zero_zero_rat )
% 0.24/0.52          | ( A = one_one_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right2
% 0.24/0.52  thf(fact_2756_mult__cancel__right2,axiom,
% 0.24/0.52      ! [A: int,C: int] :
% 0.24/0.52        ( ( ( times_times_int @ A @ C )
% 0.24/0.52          = C )
% 0.24/0.52        = ( ( C = zero_zero_int )
% 0.24/0.52          | ( A = one_one_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_cancel_right2
% 0.24/0.52  thf(fact_2757_sum__squares__eq__zero__iff,axiom,
% 0.24/0.52      ! [X: real,Y: real] :
% 0.24/0.52        ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 0.24/0.52          = zero_zero_real )
% 0.24/0.52        = ( ( X = zero_zero_real )
% 0.24/0.52          & ( Y = zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % sum_squares_eq_zero_iff
% 0.24/0.52  thf(fact_2758_sum__squares__eq__zero__iff,axiom,
% 0.24/0.52      ! [X: rat,Y: rat] :
% 0.24/0.52        ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 0.24/0.52          = zero_zero_rat )
% 0.24/0.52        = ( ( X = zero_zero_rat )
% 0.24/0.52          & ( Y = zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % sum_squares_eq_zero_iff
% 0.24/0.52  thf(fact_2759_sum__squares__eq__zero__iff,axiom,
% 0.24/0.52      ! [X: int,Y: int] :
% 0.24/0.52        ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 0.24/0.52          = zero_zero_int )
% 0.24/0.52        = ( ( X = zero_zero_int )
% 0.24/0.52          & ( Y = zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % sum_squares_eq_zero_iff
% 0.24/0.52  thf(fact_2760_le__add__diff__inverse,axiom,
% 0.24/0.52      ! [B: real,A: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ B @ A )
% 0.24/0.52       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_add_diff_inverse
% 0.24/0.52  thf(fact_2761_le__add__diff__inverse,axiom,
% 0.24/0.52      ! [B: rat,A: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ B @ A )
% 0.24/0.52       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_add_diff_inverse
% 0.24/0.52  thf(fact_2762_le__add__diff__inverse,axiom,
% 0.24/0.52      ! [B: nat,A: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ B @ A )
% 0.24/0.52       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_add_diff_inverse
% 0.24/0.52  thf(fact_2763_le__add__diff__inverse,axiom,
% 0.24/0.52      ! [B: int,A: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ B @ A )
% 0.24/0.52       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_add_diff_inverse
% 0.24/0.52  thf(fact_2764_le__add__diff__inverse2,axiom,
% 0.24/0.52      ! [B: real,A: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ B @ A )
% 0.24/0.52       => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_add_diff_inverse2
% 0.24/0.52  thf(fact_2765_le__add__diff__inverse2,axiom,
% 0.24/0.52      ! [B: rat,A: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ B @ A )
% 0.24/0.52       => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_add_diff_inverse2
% 0.24/0.52  thf(fact_2766_le__add__diff__inverse2,axiom,
% 0.24/0.52      ! [B: nat,A: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ B @ A )
% 0.24/0.52       => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_add_diff_inverse2
% 0.24/0.52  thf(fact_2767_le__add__diff__inverse2,axiom,
% 0.24/0.52      ! [B: int,A: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ B @ A )
% 0.24/0.52       => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_add_diff_inverse2
% 0.24/0.52  thf(fact_2768_diff__numeral__special_I9_J,axiom,
% 0.24/0.52      ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.24/0.52      = zero_zero_complex ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_numeral_special(9)
% 0.24/0.52  thf(fact_2769_diff__numeral__special_I9_J,axiom,
% 0.24/0.52      ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.24/0.52      = zero_zero_real ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_numeral_special(9)
% 0.24/0.52  thf(fact_2770_diff__numeral__special_I9_J,axiom,
% 0.24/0.52      ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.24/0.52      = zero_zero_rat ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_numeral_special(9)
% 0.24/0.52  thf(fact_2771_diff__numeral__special_I9_J,axiom,
% 0.24/0.52      ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.24/0.52      = zero_zero_int ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_numeral_special(9)
% 0.24/0.52  thf(fact_2772_diff__add__zero,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.24/0.52        = zero_zero_nat ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_zero
% 0.24/0.52  thf(fact_2773_div__mult__mult1__if,axiom,
% 0.24/0.52      ! [C: nat,A: nat,B: nat] :
% 0.24/0.52        ( ( ( C = zero_zero_nat )
% 0.24/0.52         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.24/0.52            = zero_zero_nat ) )
% 0.24/0.52        & ( ( C != zero_zero_nat )
% 0.24/0.52         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.24/0.52            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_mult1_if
% 0.24/0.52  thf(fact_2774_div__mult__mult1__if,axiom,
% 0.24/0.52      ! [C: int,A: int,B: int] :
% 0.24/0.52        ( ( ( C = zero_zero_int )
% 0.24/0.52         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.24/0.52            = zero_zero_int ) )
% 0.24/0.52        & ( ( C != zero_zero_int )
% 0.24/0.52         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.24/0.52            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_mult1_if
% 0.24/0.52  thf(fact_2775_div__mult__mult2,axiom,
% 0.24/0.52      ! [C: nat,A: nat,B: nat] :
% 0.24/0.52        ( ( C != zero_zero_nat )
% 0.24/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.24/0.52          = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_mult2
% 0.24/0.52  thf(fact_2776_div__mult__mult2,axiom,
% 0.24/0.52      ! [C: int,A: int,B: int] :
% 0.24/0.52        ( ( C != zero_zero_int )
% 0.24/0.52       => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.24/0.52          = ( divide_divide_int @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_mult2
% 0.24/0.52  thf(fact_2777_div__mult__mult1,axiom,
% 0.24/0.52      ! [C: nat,A: nat,B: nat] :
% 0.24/0.52        ( ( C != zero_zero_nat )
% 0.24/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.24/0.52          = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_mult1
% 0.24/0.52  thf(fact_2778_div__mult__mult1,axiom,
% 0.24/0.52      ! [C: int,A: int,B: int] :
% 0.24/0.52        ( ( C != zero_zero_int )
% 0.24/0.52       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.24/0.52          = ( divide_divide_int @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_mult1
% 0.24/0.52  thf(fact_2779_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.24/0.52      ! [C: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( C != zero_zero_complex )
% 0.24/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 0.24/0.52          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_right2
% 0.24/0.52  thf(fact_2780_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.24/0.52      ! [C: real,A: real,B: real] :
% 0.24/0.52        ( ( C != zero_zero_real )
% 0.24/0.52       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 0.24/0.52          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_right2
% 0.24/0.52  thf(fact_2781_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.24/0.52      ! [C: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( C != zero_zero_rat )
% 0.24/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 0.24/0.52          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_right2
% 0.24/0.52  thf(fact_2782_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.24/0.52      ! [C: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( C != zero_zero_complex )
% 0.24/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.24/0.52          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_right
% 0.24/0.52  thf(fact_2783_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.24/0.52      ! [C: real,A: real,B: real] :
% 0.24/0.52        ( ( C != zero_zero_real )
% 0.24/0.52       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.24/0.52          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_right
% 0.24/0.52  thf(fact_2784_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.24/0.52      ! [C: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( C != zero_zero_rat )
% 0.24/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.24/0.52          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_right
% 0.24/0.52  thf(fact_2785_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.24/0.52      ! [C: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( C != zero_zero_complex )
% 0.24/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 0.24/0.52          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_left2
% 0.24/0.52  thf(fact_2786_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.24/0.52      ! [C: real,A: real,B: real] :
% 0.24/0.52        ( ( C != zero_zero_real )
% 0.24/0.52       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 0.24/0.52          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_left2
% 0.24/0.52  thf(fact_2787_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.24/0.52      ! [C: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( C != zero_zero_rat )
% 0.24/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 0.24/0.52          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_left2
% 0.24/0.52  thf(fact_2788_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.24/0.52      ! [C: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( C != zero_zero_complex )
% 0.24/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.24/0.52          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_left
% 0.24/0.52  thf(fact_2789_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.24/0.52      ! [C: real,A: real,B: real] :
% 0.24/0.52        ( ( C != zero_zero_real )
% 0.24/0.52       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.24/0.52          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_left
% 0.24/0.52  thf(fact_2790_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.24/0.52      ! [C: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( C != zero_zero_rat )
% 0.24/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.24/0.52          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_divide_mult_cancel_left
% 0.24/0.52  thf(fact_2791_mult__divide__mult__cancel__left__if,axiom,
% 0.24/0.52      ! [C: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( ( C = zero_zero_complex )
% 0.24/0.52         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.24/0.52            = zero_zero_complex ) )
% 0.24/0.52        & ( ( C != zero_zero_complex )
% 0.24/0.52         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.24/0.52            = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_divide_mult_cancel_left_if
% 0.24/0.52  thf(fact_2792_mult__divide__mult__cancel__left__if,axiom,
% 0.24/0.52      ! [C: real,A: real,B: real] :
% 0.24/0.52        ( ( ( C = zero_zero_real )
% 0.24/0.52         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.24/0.52            = zero_zero_real ) )
% 0.24/0.52        & ( ( C != zero_zero_real )
% 0.24/0.52         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.24/0.52            = ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_divide_mult_cancel_left_if
% 0.24/0.52  thf(fact_2793_mult__divide__mult__cancel__left__if,axiom,
% 0.24/0.52      ! [C: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( ( C = zero_zero_rat )
% 0.24/0.52         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.24/0.52            = zero_zero_rat ) )
% 0.24/0.52        & ( ( C != zero_zero_rat )
% 0.24/0.52         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.24/0.52            = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_divide_mult_cancel_left_if
% 0.24/0.52  thf(fact_2794_nonzero__mult__div__cancel__right,axiom,
% 0.24/0.52      ! [B: complex,A: complex] :
% 0.24/0.52        ( ( B != zero_zero_complex )
% 0.24/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_div_cancel_right
% 0.24/0.52  thf(fact_2795_nonzero__mult__div__cancel__right,axiom,
% 0.24/0.52      ! [B: real,A: real] :
% 0.24/0.52        ( ( B != zero_zero_real )
% 0.24/0.52       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_div_cancel_right
% 0.24/0.52  thf(fact_2796_nonzero__mult__div__cancel__right,axiom,
% 0.24/0.52      ! [B: rat,A: rat] :
% 0.24/0.52        ( ( B != zero_zero_rat )
% 0.24/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_div_cancel_right
% 0.24/0.52  thf(fact_2797_nonzero__mult__div__cancel__right,axiom,
% 0.24/0.52      ! [B: nat,A: nat] :
% 0.24/0.52        ( ( B != zero_zero_nat )
% 0.24/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_div_cancel_right
% 0.24/0.52  thf(fact_2798_nonzero__mult__div__cancel__right,axiom,
% 0.24/0.52      ! [B: int,A: int] :
% 0.24/0.52        ( ( B != zero_zero_int )
% 0.24/0.52       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_div_cancel_right
% 0.24/0.52  thf(fact_2799_nonzero__mult__div__cancel__left,axiom,
% 0.24/0.52      ! [A: complex,B: complex] :
% 0.24/0.52        ( ( A != zero_zero_complex )
% 0.24/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 0.24/0.52          = B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_div_cancel_left
% 0.24/0.52  thf(fact_2800_nonzero__mult__div__cancel__left,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( A != zero_zero_real )
% 0.24/0.52       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 0.24/0.52          = B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_div_cancel_left
% 0.24/0.52  thf(fact_2801_nonzero__mult__div__cancel__left,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( A != zero_zero_rat )
% 0.24/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 0.24/0.52          = B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_div_cancel_left
% 0.24/0.52  thf(fact_2802_nonzero__mult__div__cancel__left,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( A != zero_zero_nat )
% 0.24/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 0.24/0.52          = B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_div_cancel_left
% 0.24/0.52  thf(fact_2803_nonzero__mult__div__cancel__left,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( A != zero_zero_int )
% 0.24/0.52       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 0.24/0.52          = B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_mult_div_cancel_left
% 0.24/0.52  thf(fact_2804_distrib__left__numeral,axiom,
% 0.24/0.52      ! [V: num,B: complex,C: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 0.24/0.52        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_left_numeral
% 0.24/0.52  thf(fact_2805_distrib__left__numeral,axiom,
% 0.24/0.52      ! [V: num,B: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 0.24/0.52        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_left_numeral
% 0.24/0.52  thf(fact_2806_distrib__left__numeral,axiom,
% 0.24/0.52      ! [V: num,B: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 0.24/0.52        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_left_numeral
% 0.24/0.52  thf(fact_2807_distrib__left__numeral,axiom,
% 0.24/0.52      ! [V: num,B: nat,C: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 0.24/0.52        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_left_numeral
% 0.24/0.52  thf(fact_2808_distrib__left__numeral,axiom,
% 0.24/0.52      ! [V: num,B: int,C: int] :
% 0.24/0.52        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 0.24/0.52        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_left_numeral
% 0.24/0.52  thf(fact_2809_distrib__right__numeral,axiom,
% 0.24/0.52      ! [A: complex,B: complex,V: num] :
% 0.24/0.52        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.24/0.52        = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_right_numeral
% 0.24/0.52  thf(fact_2810_distrib__right__numeral,axiom,
% 0.24/0.52      ! [A: real,B: real,V: num] :
% 0.24/0.52        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.24/0.52        = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_right_numeral
% 0.24/0.52  thf(fact_2811_distrib__right__numeral,axiom,
% 0.24/0.52      ! [A: rat,B: rat,V: num] :
% 0.24/0.52        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.24/0.52        = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_right_numeral
% 0.24/0.52  thf(fact_2812_distrib__right__numeral,axiom,
% 0.24/0.52      ! [A: nat,B: nat,V: num] :
% 0.24/0.52        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 0.24/0.52        = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_right_numeral
% 0.24/0.52  thf(fact_2813_distrib__right__numeral,axiom,
% 0.24/0.52      ! [A: int,B: int,V: num] :
% 0.24/0.52        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.24/0.52        = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_right_numeral
% 0.24/0.52  thf(fact_2814_left__diff__distrib__numeral,axiom,
% 0.24/0.52      ! [A: complex,B: complex,V: num] :
% 0.24/0.52        ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.24/0.52        = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % left_diff_distrib_numeral
% 0.24/0.52  thf(fact_2815_left__diff__distrib__numeral,axiom,
% 0.24/0.52      ! [A: real,B: real,V: num] :
% 0.24/0.52        ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.24/0.52        = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % left_diff_distrib_numeral
% 0.24/0.52  thf(fact_2816_left__diff__distrib__numeral,axiom,
% 0.24/0.52      ! [A: rat,B: rat,V: num] :
% 0.24/0.52        ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.24/0.52        = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % left_diff_distrib_numeral
% 0.24/0.52  thf(fact_2817_left__diff__distrib__numeral,axiom,
% 0.24/0.52      ! [A: int,B: int,V: num] :
% 0.24/0.52        ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.24/0.52        = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % left_diff_distrib_numeral
% 0.24/0.52  thf(fact_2818_right__diff__distrib__numeral,axiom,
% 0.24/0.52      ! [V: num,B: complex,C: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 0.24/0.52        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % right_diff_distrib_numeral
% 0.24/0.52  thf(fact_2819_right__diff__distrib__numeral,axiom,
% 0.24/0.52      ! [V: num,B: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 0.24/0.52        = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % right_diff_distrib_numeral
% 0.24/0.52  thf(fact_2820_right__diff__distrib__numeral,axiom,
% 0.24/0.52      ! [V: num,B: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 0.24/0.52        = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % right_diff_distrib_numeral
% 0.24/0.52  thf(fact_2821_right__diff__distrib__numeral,axiom,
% 0.24/0.52      ! [V: num,B: int,C: int] :
% 0.24/0.52        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 0.24/0.52        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % right_diff_distrib_numeral
% 0.24/0.52  thf(fact_2822_of__nat__mult,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_nat_mult
% 0.24/0.52  thf(fact_2823_of__nat__mult,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_nat_mult
% 0.24/0.52  thf(fact_2824_of__nat__mult,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_nat_mult
% 0.24/0.52  thf(fact_2825_of__nat__mult,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_nat_mult
% 0.24/0.52  thf(fact_2826_of__nat__mult,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_nat_mult
% 0.24/0.52  thf(fact_2827_mult__eq__1__iff,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ M @ N )
% 0.24/0.52          = ( suc @ zero_zero_nat ) )
% 0.24/0.52        = ( ( M
% 0.24/0.52            = ( suc @ zero_zero_nat ) )
% 0.24/0.52          & ( N
% 0.24/0.52            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_eq_1_iff
% 0.24/0.52  thf(fact_2828_one__eq__mult__iff,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( ( suc @ zero_zero_nat )
% 0.24/0.52          = ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( ( M
% 0.24/0.52            = ( suc @ zero_zero_nat ) )
% 0.24/0.52          & ( N
% 0.24/0.52            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % one_eq_mult_iff
% 0.24/0.52  thf(fact_2829_zero__less__diff,axiom,
% 0.24/0.52      ! [N: nat,M: nat] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.24/0.52        = ( ord_less_nat @ M @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_less_diff
% 0.24/0.52  thf(fact_2830_powr__zero__eq__one,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ( X = zero_zero_real )
% 0.24/0.52         => ( ( powr_real @ X @ zero_zero_real )
% 0.24/0.52            = zero_zero_real ) )
% 0.24/0.52        & ( ( X != zero_zero_real )
% 0.24/0.52         => ( ( powr_real @ X @ zero_zero_real )
% 0.24/0.52            = one_one_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_zero_eq_one
% 0.24/0.52  thf(fact_2831_mult__less__cancel2,axiom,
% 0.24/0.52      ! [M: nat,K: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.24/0.52        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.24/0.52          & ( ord_less_nat @ M @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_less_cancel2
% 0.24/0.52  thf(fact_2832_nat__0__less__mult__iff,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.24/0.52          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_0_less_mult_iff
% 0.24/0.52  thf(fact_2833_nat__mult__less__cancel__disj,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.24/0.52        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.24/0.52          & ( ord_less_nat @ M @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_mult_less_cancel_disj
% 0.24/0.52  thf(fact_2834_diff__is__0__eq,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( ( minus_minus_nat @ M @ N )
% 0.24/0.52          = zero_zero_nat )
% 0.24/0.52        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_is_0_eq
% 0.24/0.52  thf(fact_2835_diff__is__0__eq_H,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.52       => ( ( minus_minus_nat @ M @ N )
% 0.24/0.52          = zero_zero_nat ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_is_0_eq'
% 0.24/0.52  thf(fact_2836_mult__Suc__right,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( times_times_nat @ M @ ( suc @ N ) )
% 0.24/0.52        = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_Suc_right
% 0.24/0.52  thf(fact_2837_Nat_Odiff__diff__right,axiom,
% 0.24/0.52      ! [K: nat,J: nat,I: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ K @ J )
% 0.24/0.52       => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.24/0.52          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % Nat.diff_diff_right
% 0.24/0.52  thf(fact_2838_Nat_Oadd__diff__assoc2,axiom,
% 0.24/0.52      ! [K: nat,J: nat,I: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ K @ J )
% 0.24/0.52       => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.24/0.52          = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % Nat.add_diff_assoc2
% 0.24/0.52  thf(fact_2839_Nat_Oadd__diff__assoc,axiom,
% 0.24/0.52      ! [K: nat,J: nat,I: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ K @ J )
% 0.24/0.52       => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.24/0.52          = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % Nat.add_diff_assoc
% 0.24/0.52  thf(fact_2840_floor__zero,axiom,
% 0.24/0.52      ( ( archim6058952711729229775r_real @ zero_zero_real )
% 0.24/0.52      = zero_zero_int ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_zero
% 0.24/0.52  thf(fact_2841_floor__zero,axiom,
% 0.24/0.52      ( ( archim3151403230148437115or_rat @ zero_zero_rat )
% 0.24/0.52      = zero_zero_int ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_zero
% 0.24/0.52  thf(fact_2842_diff__Suc__1,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 0.24/0.52        = N ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_Suc_1
% 0.24/0.52  thf(fact_2843_floor__numeral,axiom,
% 0.24/0.52      ! [V: num] :
% 0.24/0.52        ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
% 0.24/0.52        = ( numeral_numeral_int @ V ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_numeral
% 0.24/0.52  thf(fact_2844_floor__numeral,axiom,
% 0.24/0.52      ! [V: num] :
% 0.24/0.52        ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
% 0.24/0.52        = ( numeral_numeral_int @ V ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_numeral
% 0.24/0.52  thf(fact_2845_nat__mult__div__cancel__disj,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ( K = zero_zero_nat )
% 0.24/0.52         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.24/0.52            = zero_zero_nat ) )
% 0.24/0.52        & ( ( K != zero_zero_nat )
% 0.24/0.52         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.24/0.52            = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_mult_div_cancel_disj
% 0.24/0.52  thf(fact_2846_powr__gt__zero,axiom,
% 0.24/0.52      ! [X: real,A: real] :
% 0.24/0.52        ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
% 0.24/0.52        = ( X != zero_zero_real ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_gt_zero
% 0.24/0.52  thf(fact_2847_floor__one,axiom,
% 0.24/0.52      ( ( archim6058952711729229775r_real @ one_one_real )
% 0.24/0.52      = one_one_int ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_one
% 0.24/0.52  thf(fact_2848_floor__one,axiom,
% 0.24/0.52      ( ( archim3151403230148437115or_rat @ one_one_rat )
% 0.24/0.52      = one_one_int ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_one
% 0.24/0.52  thf(fact_2849_powr__nonneg__iff,axiom,
% 0.24/0.52      ! [A: real,X: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
% 0.24/0.52        = ( A = zero_zero_real ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_nonneg_iff
% 0.24/0.52  thf(fact_2850_floor__diff__of__int,axiom,
% 0.24/0.52      ! [X: real,Z: int] :
% 0.24/0.52        ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_diff_of_int
% 0.24/0.52  thf(fact_2851_floor__diff__of__int,axiom,
% 0.24/0.52      ! [X: rat,Z: int] :
% 0.24/0.52        ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_diff_of_int
% 0.24/0.52  thf(fact_2852_powr__less__cancel__iff,axiom,
% 0.24/0.52      ! [X: real,A: real,B: real] :
% 0.24/0.52        ( ( ord_less_real @ one_one_real @ X )
% 0.24/0.52       => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.24/0.52          = ( ord_less_real @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_less_cancel_iff
% 0.24/0.52  thf(fact_2853_ceiling__diff__of__int,axiom,
% 0.24/0.52      ! [X: real,Z: int] :
% 0.24/0.52        ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ceiling_diff_of_int
% 0.24/0.52  thf(fact_2854_ceiling__diff__of__int,axiom,
% 0.24/0.52      ! [X: rat,Z: int] :
% 0.24/0.52        ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ceiling_diff_of_int
% 0.24/0.52  thf(fact_2855_floor__of__nat,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.52        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_of_nat
% 0.24/0.52  thf(fact_2856_floor__of__nat,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( archim3151403230148437115or_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.24/0.52        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_of_nat
% 0.24/0.52  thf(fact_2857_round__0,axiom,
% 0.24/0.52      ( ( archim8280529875227126926d_real @ zero_zero_real )
% 0.24/0.52      = zero_zero_int ) ).
% 0.24/0.52  
% 0.24/0.52  % round_0
% 0.24/0.52  thf(fact_2858_round__0,axiom,
% 0.24/0.52      ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 0.24/0.52      = zero_zero_int ) ).
% 0.24/0.52  
% 0.24/0.52  % round_0
% 0.24/0.52  thf(fact_2859_round__numeral,axiom,
% 0.24/0.52      ! [N: num] :
% 0.24/0.52        ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 0.24/0.52        = ( numeral_numeral_int @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % round_numeral
% 0.24/0.52  thf(fact_2860_round__numeral,axiom,
% 0.24/0.52      ! [N: num] :
% 0.24/0.52        ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 0.24/0.52        = ( numeral_numeral_int @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % round_numeral
% 0.24/0.52  thf(fact_2861_round__1,axiom,
% 0.24/0.52      ( ( archim8280529875227126926d_real @ one_one_real )
% 0.24/0.52      = one_one_int ) ).
% 0.24/0.52  
% 0.24/0.52  % round_1
% 0.24/0.52  thf(fact_2862_round__1,axiom,
% 0.24/0.52      ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 0.24/0.52      = one_one_int ) ).
% 0.24/0.52  
% 0.24/0.52  % round_1
% 0.24/0.52  thf(fact_2863_round__of__nat,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.24/0.52        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % round_of_nat
% 0.24/0.52  thf(fact_2864_round__of__nat,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( archim7778729529865785530nd_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.24/0.52        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % round_of_nat
% 0.24/0.52  thf(fact_2865_le__divide__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [A: real,B: real,W: num] :
% 0.24/0.52        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.24/0.52        = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_divide_eq_numeral1(1)
% 0.24/0.52  thf(fact_2866_le__divide__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [A: rat,B: rat,W: num] :
% 0.24/0.52        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.24/0.52        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_divide_eq_numeral1(1)
% 0.24/0.52  thf(fact_2867_divide__le__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [B: real,W: num,A: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.24/0.52        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_le_eq_numeral1(1)
% 0.24/0.52  thf(fact_2868_divide__le__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [B: rat,W: num,A: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.24/0.52        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_le_eq_numeral1(1)
% 0.24/0.52  thf(fact_2869_eq__divide__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [A: complex,B: complex,W: num] :
% 0.24/0.52        ( ( A
% 0.24/0.52          = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 0.24/0.52        = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.24/0.52             != zero_zero_complex )
% 0.24/0.52           => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 0.24/0.52              = B ) )
% 0.24/0.52          & ( ( ( numera6690914467698888265omplex @ W )
% 0.24/0.52              = zero_zero_complex )
% 0.24/0.52           => ( A = zero_zero_complex ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_divide_eq_numeral1(1)
% 0.24/0.52  thf(fact_2870_eq__divide__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [A: real,B: real,W: num] :
% 0.24/0.52        ( ( A
% 0.24/0.52          = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.24/0.52        = ( ( ( ( numeral_numeral_real @ W )
% 0.24/0.52             != zero_zero_real )
% 0.24/0.52           => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 0.24/0.52              = B ) )
% 0.24/0.52          & ( ( ( numeral_numeral_real @ W )
% 0.24/0.52              = zero_zero_real )
% 0.24/0.52           => ( A = zero_zero_real ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_divide_eq_numeral1(1)
% 0.24/0.52  thf(fact_2871_eq__divide__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [A: rat,B: rat,W: num] :
% 0.24/0.52        ( ( A
% 0.24/0.52          = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.24/0.52        = ( ( ( ( numeral_numeral_rat @ W )
% 0.24/0.52             != zero_zero_rat )
% 0.24/0.52           => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 0.24/0.52              = B ) )
% 0.24/0.52          & ( ( ( numeral_numeral_rat @ W )
% 0.24/0.52              = zero_zero_rat )
% 0.24/0.52           => ( A = zero_zero_rat ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_divide_eq_numeral1(1)
% 0.24/0.52  thf(fact_2872_divide__eq__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [B: complex,W: num,A: complex] :
% 0.24/0.52        ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 0.24/0.52          = A )
% 0.24/0.52        = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.24/0.52             != zero_zero_complex )
% 0.24/0.52           => ( B
% 0.24/0.52              = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.24/0.52          & ( ( ( numera6690914467698888265omplex @ W )
% 0.24/0.52              = zero_zero_complex )
% 0.24/0.52           => ( A = zero_zero_complex ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_eq_eq_numeral1(1)
% 0.24/0.52  thf(fact_2873_divide__eq__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [B: real,W: num,A: real] :
% 0.24/0.52        ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 0.24/0.52          = A )
% 0.24/0.52        = ( ( ( ( numeral_numeral_real @ W )
% 0.24/0.52             != zero_zero_real )
% 0.24/0.52           => ( B
% 0.24/0.52              = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 0.24/0.52          & ( ( ( numeral_numeral_real @ W )
% 0.24/0.52              = zero_zero_real )
% 0.24/0.52           => ( A = zero_zero_real ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_eq_eq_numeral1(1)
% 0.24/0.52  thf(fact_2874_divide__eq__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [B: rat,W: num,A: rat] :
% 0.24/0.52        ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 0.24/0.52          = A )
% 0.24/0.52        = ( ( ( ( numeral_numeral_rat @ W )
% 0.24/0.52             != zero_zero_rat )
% 0.24/0.52           => ( B
% 0.24/0.52              = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 0.24/0.52          & ( ( ( numeral_numeral_rat @ W )
% 0.24/0.52              = zero_zero_rat )
% 0.24/0.52           => ( A = zero_zero_rat ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_eq_eq_numeral1(1)
% 0.24/0.52  thf(fact_2875_less__divide__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [A: real,B: real,W: num] :
% 0.24/0.52        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.24/0.52        = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_divide_eq_numeral1(1)
% 0.24/0.52  thf(fact_2876_less__divide__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [A: rat,B: rat,W: num] :
% 0.24/0.52        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.24/0.52        = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_divide_eq_numeral1(1)
% 0.24/0.52  thf(fact_2877_divide__less__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [B: real,W: num,A: real] :
% 0.24/0.52        ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.24/0.52        = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_less_eq_numeral1(1)
% 0.24/0.52  thf(fact_2878_divide__less__eq__numeral1_I1_J,axiom,
% 0.24/0.52      ! [B: rat,W: num,A: rat] :
% 0.24/0.52        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.24/0.52        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_less_eq_numeral1(1)
% 0.24/0.52  thf(fact_2879_nonzero__divide__mult__cancel__left,axiom,
% 0.24/0.52      ! [A: complex,B: complex] :
% 0.24/0.52        ( ( A != zero_zero_complex )
% 0.24/0.52       => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 0.24/0.52          = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_divide_mult_cancel_left
% 0.24/0.52  thf(fact_2880_nonzero__divide__mult__cancel__left,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( A != zero_zero_real )
% 0.24/0.52       => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 0.24/0.52          = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_divide_mult_cancel_left
% 0.24/0.52  thf(fact_2881_nonzero__divide__mult__cancel__left,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( A != zero_zero_rat )
% 0.24/0.52       => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 0.24/0.52          = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_divide_mult_cancel_left
% 0.24/0.52  thf(fact_2882_nonzero__divide__mult__cancel__right,axiom,
% 0.24/0.52      ! [B: complex,A: complex] :
% 0.24/0.52        ( ( B != zero_zero_complex )
% 0.24/0.52       => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 0.24/0.52          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_divide_mult_cancel_right
% 0.24/0.52  thf(fact_2883_nonzero__divide__mult__cancel__right,axiom,
% 0.24/0.52      ! [B: real,A: real] :
% 0.24/0.52        ( ( B != zero_zero_real )
% 0.24/0.52       => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 0.24/0.52          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_divide_mult_cancel_right
% 0.24/0.52  thf(fact_2884_nonzero__divide__mult__cancel__right,axiom,
% 0.24/0.52      ! [B: rat,A: rat] :
% 0.24/0.52        ( ( B != zero_zero_rat )
% 0.24/0.52       => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 0.24/0.52          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nonzero_divide_mult_cancel_right
% 0.24/0.52  thf(fact_2885_div__mult__self1,axiom,
% 0.24/0.52      ! [B: nat,A: nat,C: nat] :
% 0.24/0.52        ( ( B != zero_zero_nat )
% 0.24/0.52       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.24/0.52          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_self1
% 0.24/0.52  thf(fact_2886_div__mult__self1,axiom,
% 0.24/0.52      ! [B: int,A: int,C: int] :
% 0.24/0.52        ( ( B != zero_zero_int )
% 0.24/0.52       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.24/0.52          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_self1
% 0.24/0.52  thf(fact_2887_div__mult__self2,axiom,
% 0.24/0.52      ! [B: nat,A: nat,C: nat] :
% 0.24/0.52        ( ( B != zero_zero_nat )
% 0.24/0.52       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.24/0.52          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_self2
% 0.24/0.52  thf(fact_2888_div__mult__self2,axiom,
% 0.24/0.52      ! [B: int,A: int,C: int] :
% 0.24/0.52        ( ( B != zero_zero_int )
% 0.24/0.52       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.24/0.52          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_self2
% 0.24/0.52  thf(fact_2889_div__mult__self3,axiom,
% 0.24/0.52      ! [B: nat,C: nat,A: nat] :
% 0.24/0.52        ( ( B != zero_zero_nat )
% 0.24/0.52       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.24/0.52          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_self3
% 0.24/0.52  thf(fact_2890_div__mult__self3,axiom,
% 0.24/0.52      ! [B: int,C: int,A: int] :
% 0.24/0.52        ( ( B != zero_zero_int )
% 0.24/0.52       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.24/0.52          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_self3
% 0.24/0.52  thf(fact_2891_div__mult__self4,axiom,
% 0.24/0.52      ! [B: nat,C: nat,A: nat] :
% 0.24/0.52        ( ( B != zero_zero_nat )
% 0.24/0.52       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.24/0.52          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_self4
% 0.24/0.52  thf(fact_2892_div__mult__self4,axiom,
% 0.24/0.52      ! [B: int,C: int,A: int] :
% 0.24/0.52        ( ( B != zero_zero_int )
% 0.24/0.52       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.24/0.52          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_self4
% 0.24/0.52  thf(fact_2893_Suc__pred,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.24/0.52          = N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % Suc_pred
% 0.24/0.52  thf(fact_2894_one__le__mult__iff,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.24/0.52          & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % one_le_mult_iff
% 0.24/0.52  thf(fact_2895_mult__le__cancel2,axiom,
% 0.24/0.52      ! [M: nat,K: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.24/0.52        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.24/0.52         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_le_cancel2
% 0.24/0.52  thf(fact_2896_nat__mult__le__cancel__disj,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.24/0.52        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.24/0.52         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_mult_le_cancel_disj
% 0.24/0.52  thf(fact_2897_diff__Suc__diff__eq1,axiom,
% 0.24/0.52      ! [K: nat,J: nat,I: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ K @ J )
% 0.24/0.52       => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.24/0.52          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_Suc_diff_eq1
% 0.24/0.52  thf(fact_2898_diff__Suc__diff__eq2,axiom,
% 0.24/0.52      ! [K: nat,J: nat,I: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ K @ J )
% 0.24/0.52       => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 0.24/0.52          = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_Suc_diff_eq2
% 0.24/0.52  thf(fact_2899_floor__diff__numeral,axiom,
% 0.24/0.52      ! [X: real,V: num] :
% 0.24/0.52        ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_diff_numeral
% 0.24/0.52  thf(fact_2900_floor__diff__numeral,axiom,
% 0.24/0.52      ! [X: rat,V: num] :
% 0.24/0.52        ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_diff_numeral
% 0.24/0.52  thf(fact_2901_ceiling__diff__numeral,axiom,
% 0.24/0.52      ! [X: real,V: num] :
% 0.24/0.52        ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ceiling_diff_numeral
% 0.24/0.52  thf(fact_2902_ceiling__diff__numeral,axiom,
% 0.24/0.52      ! [X: rat,V: num] :
% 0.24/0.52        ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ceiling_diff_numeral
% 0.24/0.52  thf(fact_2903_div__mult__self1__is__m,axiom,
% 0.24/0.52      ! [N: nat,M: nat] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 0.24/0.52          = M ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_self1_is_m
% 0.24/0.52  thf(fact_2904_div__mult__self__is__m,axiom,
% 0.24/0.52      ! [N: nat,M: nat] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 0.24/0.52          = M ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult_self_is_m
% 0.24/0.52  thf(fact_2905_floor__diff__one,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_diff_one
% 0.24/0.52  thf(fact_2906_floor__diff__one,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_diff_one
% 0.24/0.52  thf(fact_2907_ceiling__diff__one,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ceiling_diff_one
% 0.24/0.52  thf(fact_2908_ceiling__diff__one,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 0.24/0.52        = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ceiling_diff_one
% 0.24/0.52  thf(fact_2909_power__add__numeral,axiom,
% 0.24/0.52      ! [A: complex,M: num,N: num] :
% 0.24/0.52        ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.24/0.52        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_add_numeral
% 0.24/0.52  thf(fact_2910_power__add__numeral,axiom,
% 0.24/0.52      ! [A: real,M: num,N: num] :
% 0.24/0.52        ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.24/0.52        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_add_numeral
% 0.24/0.52  thf(fact_2911_power__add__numeral,axiom,
% 0.24/0.52      ! [A: rat,M: num,N: num] :
% 0.24/0.52        ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.24/0.52        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_add_numeral
% 0.24/0.52  thf(fact_2912_power__add__numeral,axiom,
% 0.24/0.52      ! [A: nat,M: num,N: num] :
% 0.24/0.52        ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.24/0.52        = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_add_numeral
% 0.24/0.52  thf(fact_2913_power__add__numeral,axiom,
% 0.24/0.52      ! [A: int,M: num,N: num] :
% 0.24/0.52        ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.24/0.52        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_add_numeral
% 0.24/0.52  thf(fact_2914_power__add__numeral2,axiom,
% 0.24/0.52      ! [A: complex,M: num,N: num,B: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.24/0.52        = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_add_numeral2
% 0.24/0.52  thf(fact_2915_power__add__numeral2,axiom,
% 0.24/0.52      ! [A: real,M: num,N: num,B: real] :
% 0.24/0.52        ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.24/0.52        = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_add_numeral2
% 0.24/0.52  thf(fact_2916_power__add__numeral2,axiom,
% 0.24/0.52      ! [A: rat,M: num,N: num,B: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.24/0.52        = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_add_numeral2
% 0.24/0.52  thf(fact_2917_power__add__numeral2,axiom,
% 0.24/0.52      ! [A: nat,M: num,N: num,B: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.24/0.52        = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_add_numeral2
% 0.24/0.52  thf(fact_2918_power__add__numeral2,axiom,
% 0.24/0.52      ! [A: int,M: num,N: num,B: int] :
% 0.24/0.52        ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.24/0.52        = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_add_numeral2
% 0.24/0.52  thf(fact_2919_powr__eq__one__iff,axiom,
% 0.24/0.52      ! [A: real,X: real] :
% 0.24/0.52        ( ( ord_less_real @ one_one_real @ A )
% 0.24/0.52       => ( ( ( powr_real @ A @ X )
% 0.24/0.52            = one_one_real )
% 0.24/0.52          = ( X = zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_eq_one_iff
% 0.24/0.52  thf(fact_2920_powr__one__gt__zero__iff,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ( powr_real @ X @ one_one_real )
% 0.24/0.52          = X )
% 0.24/0.52        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_one_gt_zero_iff
% 0.24/0.52  thf(fact_2921_powr__one,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.52       => ( ( powr_real @ X @ one_one_real )
% 0.24/0.52          = X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_one
% 0.24/0.52  thf(fact_2922_powr__le__cancel__iff,axiom,
% 0.24/0.52      ! [X: real,A: real,B: real] :
% 0.24/0.52        ( ( ord_less_real @ one_one_real @ X )
% 0.24/0.52       => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.24/0.52          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_le_cancel_iff
% 0.24/0.52  thf(fact_2923_numeral__powr__numeral__real,axiom,
% 0.24/0.52      ! [M: num,N: num] :
% 0.24/0.52        ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.24/0.52        = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_powr_numeral_real
% 0.24/0.52  thf(fact_2924_zero__le__floor,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 0.24/0.52        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_le_floor
% 0.24/0.52  thf(fact_2925_zero__le__floor,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.24/0.52        = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_le_floor
% 0.24/0.52  thf(fact_2926_Suc__diff__1,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.24/0.52          = N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % Suc_diff_1
% 0.24/0.52  thf(fact_2927_floor__less__zero,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 0.24/0.52        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_less_zero
% 0.24/0.52  thf(fact_2928_floor__less__zero,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 0.24/0.52        = ( ord_less_rat @ X @ zero_zero_rat ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_less_zero
% 0.24/0.52  thf(fact_2929_numeral__le__floor,axiom,
% 0.24/0.52      ! [V: num,X: real] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 0.24/0.52        = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_le_floor
% 0.24/0.52  thf(fact_2930_numeral__le__floor,axiom,
% 0.24/0.52      ! [V: num,X: rat] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.24/0.52        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_le_floor
% 0.24/0.52  thf(fact_2931_zero__less__floor,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 0.24/0.52        = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_less_floor
% 0.24/0.52  thf(fact_2932_zero__less__floor,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.24/0.52        = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_less_floor
% 0.24/0.52  thf(fact_2933_floor__le__zero,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 0.24/0.52        = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_le_zero
% 0.24/0.52  thf(fact_2934_floor__le__zero,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 0.24/0.52        = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_le_zero
% 0.24/0.52  thf(fact_2935_floor__less__numeral,axiom,
% 0.24/0.52      ! [X: real,V: num] :
% 0.24/0.52        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.24/0.52        = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_less_numeral
% 0.24/0.52  thf(fact_2936_floor__less__numeral,axiom,
% 0.24/0.52      ! [X: rat,V: num] :
% 0.24/0.52        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.24/0.52        = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_less_numeral
% 0.24/0.52  thf(fact_2937_one__le__floor,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 0.24/0.52        = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % one_le_floor
% 0.24/0.52  thf(fact_2938_one__le__floor,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.24/0.52        = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % one_le_floor
% 0.24/0.52  thf(fact_2939_floor__less__one,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 0.24/0.52        = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_less_one
% 0.24/0.52  thf(fact_2940_floor__less__one,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 0.24/0.52        = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_less_one
% 0.24/0.52  thf(fact_2941_floor__numeral__power,axiom,
% 0.24/0.52      ! [X: num,N: nat] :
% 0.24/0.52        ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.24/0.52        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_numeral_power
% 0.24/0.52  thf(fact_2942_floor__numeral__power,axiom,
% 0.24/0.52      ! [X: num,N: nat] :
% 0.24/0.52        ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.24/0.52        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_numeral_power
% 0.24/0.52  thf(fact_2943_powr__log__cancel,axiom,
% 0.24/0.52      ! [A: real,X: real] :
% 0.24/0.52        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.52       => ( ( A != one_one_real )
% 0.24/0.52         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.52           => ( ( powr_real @ A @ ( log @ A @ X ) )
% 0.24/0.52              = X ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_log_cancel
% 0.24/0.52  thf(fact_2944_log__powr__cancel,axiom,
% 0.24/0.52      ! [A: real,Y: real] :
% 0.24/0.52        ( ( ord_less_real @ zero_zero_real @ A )
% 0.24/0.52       => ( ( A != one_one_real )
% 0.24/0.52         => ( ( log @ A @ ( powr_real @ A @ Y ) )
% 0.24/0.52            = Y ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % log_powr_cancel
% 0.24/0.52  thf(fact_2945_floor__divide__eq__div__numeral,axiom,
% 0.24/0.52      ! [A: num,B: num] :
% 0.24/0.52        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.24/0.52        = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_divide_eq_div_numeral
% 0.24/0.52  thf(fact_2946_powr__numeral,axiom,
% 0.24/0.52      ! [X: real,N: num] :
% 0.24/0.52        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.52       => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
% 0.24/0.52          = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_numeral
% 0.24/0.52  thf(fact_2947_numeral__less__floor,axiom,
% 0.24/0.52      ! [V: num,X: real] :
% 0.24/0.52        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 0.24/0.52        = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_less_floor
% 0.24/0.52  thf(fact_2948_numeral__less__floor,axiom,
% 0.24/0.52      ! [V: num,X: rat] :
% 0.24/0.52        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.24/0.52        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_less_floor
% 0.24/0.52  thf(fact_2949_floor__le__numeral,axiom,
% 0.24/0.52      ! [X: real,V: num] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.24/0.52        = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_le_numeral
% 0.24/0.52  thf(fact_2950_floor__le__numeral,axiom,
% 0.24/0.52      ! [X: rat,V: num] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.24/0.52        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_le_numeral
% 0.24/0.52  thf(fact_2951_one__less__floor,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 0.24/0.52        = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % one_less_floor
% 0.24/0.52  thf(fact_2952_one__less__floor,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.24/0.52        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % one_less_floor
% 0.24/0.52  thf(fact_2953_floor__le__one,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 0.24/0.52        = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_le_one
% 0.24/0.52  thf(fact_2954_floor__le__one,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 0.24/0.52        = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_le_one
% 0.24/0.52  thf(fact_2955_ceiling__less__numeral,axiom,
% 0.24/0.52      ! [X: real,V: num] :
% 0.24/0.52        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.24/0.52        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ceiling_less_numeral
% 0.24/0.52  thf(fact_2956_ceiling__less__numeral,axiom,
% 0.24/0.52      ! [X: rat,V: num] :
% 0.24/0.52        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.24/0.52        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ceiling_less_numeral
% 0.24/0.52  thf(fact_2957_numeral__le__ceiling,axiom,
% 0.24/0.52      ! [V: num,X: real] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 0.24/0.52        = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_le_ceiling
% 0.24/0.52  thf(fact_2958_numeral__le__ceiling,axiom,
% 0.24/0.52      ! [V: num,X: rat] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.24/0.52        = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % numeral_le_ceiling
% 0.24/0.52  thf(fact_2959_floor__one__divide__eq__div__numeral,axiom,
% 0.24/0.52      ! [B: num] :
% 0.24/0.52        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 0.24/0.52        = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_one_divide_eq_div_numeral
% 0.24/0.52  thf(fact_2960_eq__add__iff1,axiom,
% 0.24/0.52      ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
% 0.24/0.52        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
% 0.24/0.52          = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E2 ) @ C )
% 0.24/0.52          = D ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_add_iff1
% 0.24/0.52  thf(fact_2961_eq__add__iff1,axiom,
% 0.24/0.52      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.24/0.52        ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 0.24/0.52          = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
% 0.24/0.52          = D ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_add_iff1
% 0.24/0.52  thf(fact_2962_eq__add__iff1,axiom,
% 0.24/0.52      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.24/0.52        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 0.24/0.52          = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
% 0.24/0.52          = D ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_add_iff1
% 0.24/0.52  thf(fact_2963_eq__add__iff1,axiom,
% 0.24/0.52      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.24/0.52        ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 0.24/0.52          = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
% 0.24/0.52          = D ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_add_iff1
% 0.24/0.52  thf(fact_2964_eq__add__iff2,axiom,
% 0.24/0.52      ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
% 0.24/0.52        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
% 0.24/0.52          = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( C
% 0.24/0.52          = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_add_iff2
% 0.24/0.52  thf(fact_2965_eq__add__iff2,axiom,
% 0.24/0.52      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.24/0.52        ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 0.24/0.52          = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( C
% 0.24/0.52          = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_add_iff2
% 0.24/0.52  thf(fact_2966_eq__add__iff2,axiom,
% 0.24/0.52      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.24/0.52        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 0.24/0.52          = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( C
% 0.24/0.52          = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_add_iff2
% 0.24/0.52  thf(fact_2967_eq__add__iff2,axiom,
% 0.24/0.52      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.24/0.52        ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 0.24/0.52          = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( C
% 0.24/0.52          = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_add_iff2
% 0.24/0.52  thf(fact_2968_square__diff__square__factored,axiom,
% 0.24/0.52      ! [X: complex,Y: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ ( times_times_complex @ Y @ Y ) )
% 0.24/0.52        = ( times_times_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_complex @ X @ Y ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % square_diff_square_factored
% 0.24/0.52  thf(fact_2969_square__diff__square__factored,axiom,
% 0.24/0.52      ! [X: real,Y: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 0.24/0.52        = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % square_diff_square_factored
% 0.24/0.52  thf(fact_2970_square__diff__square__factored,axiom,
% 0.24/0.52      ! [X: rat,Y: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 0.24/0.52        = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % square_diff_square_factored
% 0.24/0.52  thf(fact_2971_square__diff__square__factored,axiom,
% 0.24/0.52      ! [X: int,Y: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 0.24/0.52        = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % square_diff_square_factored
% 0.24/0.52  thf(fact_2972_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.24/0.52        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ab_semigroup_mult_class.mult_ac(1)
% 0.24/0.52  thf(fact_2973_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.24/0.52        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ab_semigroup_mult_class.mult_ac(1)
% 0.24/0.52  thf(fact_2974_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.24/0.52        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ab_semigroup_mult_class.mult_ac(1)
% 0.24/0.52  thf(fact_2975_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.24/0.52        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ab_semigroup_mult_class.mult_ac(1)
% 0.24/0.52  thf(fact_2976_diff__eq__diff__eq,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real,D: real] :
% 0.24/0.52        ( ( ( minus_minus_real @ A @ B )
% 0.24/0.52          = ( minus_minus_real @ C @ D ) )
% 0.24/0.52       => ( ( A = B )
% 0.24/0.52          = ( C = D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_diff_eq
% 0.24/0.52  thf(fact_2977_diff__eq__diff__eq,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.24/0.52        ( ( ( minus_minus_rat @ A @ B )
% 0.24/0.52          = ( minus_minus_rat @ C @ D ) )
% 0.24/0.52       => ( ( A = B )
% 0.24/0.52          = ( C = D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_diff_eq
% 0.24/0.52  thf(fact_2978_diff__eq__diff__eq,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int,D: int] :
% 0.24/0.52        ( ( ( minus_minus_int @ A @ B )
% 0.24/0.52          = ( minus_minus_int @ C @ D ) )
% 0.24/0.52       => ( ( A = B )
% 0.24/0.52          = ( C = D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_diff_eq
% 0.24/0.52  thf(fact_2979_mult_Oassoc,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.24/0.52        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.assoc
% 0.24/0.52  thf(fact_2980_mult_Oassoc,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.24/0.52        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.assoc
% 0.24/0.52  thf(fact_2981_mult_Oassoc,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.24/0.52        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.assoc
% 0.24/0.52  thf(fact_2982_mult_Oassoc,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.24/0.52        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.assoc
% 0.24/0.52  thf(fact_2983_mult_Ocommute,axiom,
% 0.24/0.52      ( times_times_real
% 0.24/0.52      = ( ^ [A3: real,B4: real] : ( times_times_real @ B4 @ A3 ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.commute
% 0.24/0.52  thf(fact_2984_mult_Ocommute,axiom,
% 0.24/0.52      ( times_times_rat
% 0.24/0.52      = ( ^ [A3: rat,B4: rat] : ( times_times_rat @ B4 @ A3 ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.commute
% 0.24/0.52  thf(fact_2985_mult_Ocommute,axiom,
% 0.24/0.52      ( times_times_nat
% 0.24/0.52      = ( ^ [A3: nat,B4: nat] : ( times_times_nat @ B4 @ A3 ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.commute
% 0.24/0.52  thf(fact_2986_mult_Ocommute,axiom,
% 0.24/0.52      ( times_times_int
% 0.24/0.52      = ( ^ [A3: int,B4: int] : ( times_times_int @ B4 @ A3 ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.commute
% 0.24/0.52  thf(fact_2987_mult_Oleft__commute,axiom,
% 0.24/0.52      ! [B: real,A: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 0.24/0.52        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.left_commute
% 0.24/0.52  thf(fact_2988_mult_Oleft__commute,axiom,
% 0.24/0.52      ! [B: rat,A: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 0.24/0.52        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.left_commute
% 0.24/0.52  thf(fact_2989_mult_Oleft__commute,axiom,
% 0.24/0.52      ! [B: nat,A: nat,C: nat] :
% 0.24/0.52        ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.24/0.52        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.left_commute
% 0.24/0.52  thf(fact_2990_mult_Oleft__commute,axiom,
% 0.24/0.52      ! [B: int,A: int,C: int] :
% 0.24/0.52        ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 0.24/0.52        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.left_commute
% 0.24/0.52  thf(fact_2991_diff__right__commute,axiom,
% 0.24/0.52      ! [A: real,C: real,B: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.24/0.52        = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_right_commute
% 0.24/0.52  thf(fact_2992_diff__right__commute,axiom,
% 0.24/0.52      ! [A: rat,C: rat,B: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 0.24/0.52        = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_right_commute
% 0.24/0.52  thf(fact_2993_diff__right__commute,axiom,
% 0.24/0.52      ! [A: nat,C: nat,B: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.24/0.52        = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_right_commute
% 0.24/0.52  thf(fact_2994_diff__right__commute,axiom,
% 0.24/0.52      ! [A: int,C: int,B: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.24/0.52        = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_right_commute
% 0.24/0.52  thf(fact_2995_diff__commute,axiom,
% 0.24/0.52      ! [I: nat,J: nat,K: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.24/0.52        = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_commute
% 0.24/0.52  thf(fact_2996_diff__mult__distrib,axiom,
% 0.24/0.52      ! [M: nat,N: nat,K: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.24/0.52        = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_mult_distrib
% 0.24/0.52  thf(fact_2997_diff__mult__distrib2,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.24/0.52        = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_mult_distrib2
% 0.24/0.52  thf(fact_2998_powr__powr__swap,axiom,
% 0.24/0.52      ! [X: real,A: real,B: real] :
% 0.24/0.52        ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 0.24/0.52        = ( powr_real @ ( powr_real @ X @ B ) @ A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_powr_swap
% 0.24/0.52  thf(fact_2999_mult__diff__mult,axiom,
% 0.24/0.52      ! [X: complex,Y: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ ( times_times_complex @ X @ Y ) @ ( times_times_complex @ A @ B ) )
% 0.24/0.52        = ( plus_plus_complex @ ( times_times_complex @ X @ ( minus_minus_complex @ Y @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X @ A ) @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_diff_mult
% 0.24/0.52  thf(fact_3000_mult__diff__mult,axiom,
% 0.24/0.52      ! [X: real,Y: real,A: real,B: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
% 0.24/0.52        = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_diff_mult
% 0.24/0.52  thf(fact_3001_mult__diff__mult,axiom,
% 0.24/0.52      ! [X: rat,Y: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
% 0.24/0.52        = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_diff_mult
% 0.24/0.52  thf(fact_3002_mult__diff__mult,axiom,
% 0.24/0.52      ! [X: int,Y: int,A: int,B: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
% 0.24/0.52        = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_diff_mult
% 0.24/0.52  thf(fact_3003_floor__le__round,axiom,
% 0.24/0.52      ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_le_round
% 0.24/0.52  thf(fact_3004_floor__le__round,axiom,
% 0.24/0.52      ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_le_round
% 0.24/0.52  thf(fact_3005_powr__diff,axiom,
% 0.24/0.52      ! [W: real,Z1: real,Z22: real] :
% 0.24/0.52        ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
% 0.24/0.52        = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_diff
% 0.24/0.52  thf(fact_3006_powr__add,axiom,
% 0.24/0.52      ! [X: real,A: real,B: real] :
% 0.24/0.52        ( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
% 0.24/0.52        = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_add
% 0.24/0.52  thf(fact_3007_le__mult__floor,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.52         => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_mult_floor
% 0.24/0.52  thf(fact_3008_le__mult__floor,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.52         => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_mult_floor
% 0.24/0.52  thf(fact_3009_ordered__ring__class_Ole__add__iff1,axiom,
% 0.24/0.52      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_ring_class.le_add_iff1
% 0.24/0.52  thf(fact_3010_ordered__ring__class_Ole__add__iff1,axiom,
% 0.24/0.52      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_ring_class.le_add_iff1
% 0.24/0.52  thf(fact_3011_ordered__ring__class_Ole__add__iff1,axiom,
% 0.24/0.52      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_ring_class.le_add_iff1
% 0.24/0.52  thf(fact_3012_ordered__ring__class_Ole__add__iff2,axiom,
% 0.24/0.52      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_ring_class.le_add_iff2
% 0.24/0.52  thf(fact_3013_ordered__ring__class_Ole__add__iff2,axiom,
% 0.24/0.52      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_ring_class.le_add_iff2
% 0.24/0.52  thf(fact_3014_ordered__ring__class_Ole__add__iff2,axiom,
% 0.24/0.52      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_ring_class.le_add_iff2
% 0.24/0.52  thf(fact_3015_less__add__iff1,axiom,
% 0.24/0.52      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.24/0.52        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_add_iff1
% 0.24/0.52  thf(fact_3016_less__add__iff1,axiom,
% 0.24/0.52      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.24/0.52        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_add_iff1
% 0.24/0.52  thf(fact_3017_less__add__iff1,axiom,
% 0.24/0.52      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.24/0.52        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_add_iff1
% 0.24/0.52  thf(fact_3018_less__add__iff2,axiom,
% 0.24/0.52      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.24/0.52        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_add_iff2
% 0.24/0.52  thf(fact_3019_less__add__iff2,axiom,
% 0.24/0.52      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.24/0.52        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_add_iff2
% 0.24/0.52  thf(fact_3020_less__add__iff2,axiom,
% 0.24/0.52      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.24/0.52        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.24/0.52        = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_add_iff2
% 0.24/0.52  thf(fact_3021_divide__diff__eq__iff,axiom,
% 0.24/0.52      ! [Z: complex,X: complex,Y: complex] :
% 0.24/0.52        ( ( Z != zero_zero_complex )
% 0.24/0.52       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
% 0.24/0.52          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_diff_eq_iff
% 0.24/0.52  thf(fact_3022_divide__diff__eq__iff,axiom,
% 0.24/0.52      ! [Z: real,X: real,Y: real] :
% 0.24/0.52        ( ( Z != zero_zero_real )
% 0.24/0.52       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
% 0.24/0.52          = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_diff_eq_iff
% 0.24/0.52  thf(fact_3023_divide__diff__eq__iff,axiom,
% 0.24/0.52      ! [Z: rat,X: rat,Y: rat] :
% 0.24/0.52        ( ( Z != zero_zero_rat )
% 0.24/0.52       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
% 0.24/0.52          = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_diff_eq_iff
% 0.24/0.52  thf(fact_3024_diff__divide__eq__iff,axiom,
% 0.24/0.52      ! [Z: complex,X: complex,Y: complex] :
% 0.24/0.52        ( ( Z != zero_zero_complex )
% 0.24/0.52       => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
% 0.24/0.52          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_divide_eq_iff
% 0.24/0.52  thf(fact_3025_diff__divide__eq__iff,axiom,
% 0.24/0.52      ! [Z: real,X: real,Y: real] :
% 0.24/0.52        ( ( Z != zero_zero_real )
% 0.24/0.52       => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
% 0.24/0.52          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_divide_eq_iff
% 0.24/0.52  thf(fact_3026_diff__divide__eq__iff,axiom,
% 0.24/0.52      ! [Z: rat,X: rat,Y: rat] :
% 0.24/0.52        ( ( Z != zero_zero_rat )
% 0.24/0.52       => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
% 0.24/0.52          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_divide_eq_iff
% 0.24/0.52  thf(fact_3027_diff__frac__eq,axiom,
% 0.24/0.52      ! [Y: complex,Z: complex,X: complex,W: complex] :
% 0.24/0.52        ( ( Y != zero_zero_complex )
% 0.24/0.52       => ( ( Z != zero_zero_complex )
% 0.24/0.52         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.24/0.52            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_frac_eq
% 0.24/0.52  thf(fact_3028_diff__frac__eq,axiom,
% 0.24/0.52      ! [Y: real,Z: real,X: real,W: real] :
% 0.24/0.52        ( ( Y != zero_zero_real )
% 0.24/0.52       => ( ( Z != zero_zero_real )
% 0.24/0.52         => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 0.24/0.52            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_frac_eq
% 0.24/0.52  thf(fact_3029_diff__frac__eq,axiom,
% 0.24/0.52      ! [Y: rat,Z: rat,X: rat,W: rat] :
% 0.24/0.52        ( ( Y != zero_zero_rat )
% 0.24/0.52       => ( ( Z != zero_zero_rat )
% 0.24/0.52         => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 0.24/0.52            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_frac_eq
% 0.24/0.52  thf(fact_3030_add__divide__eq__if__simps_I4_J,axiom,
% 0.24/0.52      ! [Z: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( ( Z = zero_zero_complex )
% 0.24/0.52         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.24/0.52            = A ) )
% 0.24/0.52        & ( ( Z != zero_zero_complex )
% 0.24/0.52         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.24/0.52            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_divide_eq_if_simps(4)
% 0.24/0.52  thf(fact_3031_add__divide__eq__if__simps_I4_J,axiom,
% 0.24/0.52      ! [Z: real,A: real,B: real] :
% 0.24/0.52        ( ( ( Z = zero_zero_real )
% 0.24/0.52         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.24/0.52            = A ) )
% 0.24/0.52        & ( ( Z != zero_zero_real )
% 0.24/0.52         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.24/0.52            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_divide_eq_if_simps(4)
% 0.24/0.52  thf(fact_3032_add__divide__eq__if__simps_I4_J,axiom,
% 0.24/0.52      ! [Z: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( ( Z = zero_zero_rat )
% 0.24/0.52         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.24/0.52            = A ) )
% 0.24/0.52        & ( ( Z != zero_zero_rat )
% 0.24/0.52         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.24/0.52            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_divide_eq_if_simps(4)
% 0.24/0.52  thf(fact_3033_square__diff__one__factored,axiom,
% 0.24/0.52      ! [X: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
% 0.24/0.52        = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % square_diff_one_factored
% 0.24/0.52  thf(fact_3034_square__diff__one__factored,axiom,
% 0.24/0.52      ! [X: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
% 0.24/0.52        = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % square_diff_one_factored
% 0.24/0.52  thf(fact_3035_square__diff__one__factored,axiom,
% 0.24/0.52      ! [X: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 0.24/0.52        = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % square_diff_one_factored
% 0.24/0.52  thf(fact_3036_square__diff__one__factored,axiom,
% 0.24/0.52      ! [X: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 0.24/0.52        = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % square_diff_one_factored
% 0.24/0.52  thf(fact_3037_of__nat__diff,axiom,
% 0.24/0.52      ! [N: nat,M: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ N @ M )
% 0.24/0.52       => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 0.24/0.52          = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_nat_diff
% 0.24/0.52  thf(fact_3038_of__nat__diff,axiom,
% 0.24/0.52      ! [N: nat,M: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ N @ M )
% 0.24/0.52       => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 0.24/0.52          = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_nat_diff
% 0.24/0.52  thf(fact_3039_of__nat__diff,axiom,
% 0.24/0.52      ! [N: nat,M: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ N @ M )
% 0.24/0.52       => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 0.24/0.52          = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_nat_diff
% 0.24/0.52  thf(fact_3040_of__nat__diff,axiom,
% 0.24/0.52      ! [N: nat,M: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ N @ M )
% 0.24/0.52       => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.24/0.52          = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_nat_diff
% 0.24/0.52  thf(fact_3041_of__nat__diff,axiom,
% 0.24/0.52      ! [N: nat,M: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ N @ M )
% 0.24/0.52       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 0.24/0.52          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % of_nat_diff
% 0.24/0.52  thf(fact_3042_nat__diff__add__eq2,axiom,
% 0.24/0.52      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.52       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.24/0.52          = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_diff_add_eq2
% 0.24/0.52  thf(fact_3043_nat__diff__add__eq1,axiom,
% 0.24/0.52      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ J @ I )
% 0.24/0.52       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.24/0.52          = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_diff_add_eq1
% 0.24/0.52  thf(fact_3044_nat__le__add__iff2,axiom,
% 0.24/0.52      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.52       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.24/0.52          = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_le_add_iff2
% 0.24/0.52  thf(fact_3045_nat__le__add__iff1,axiom,
% 0.24/0.52      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ J @ I )
% 0.24/0.52       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.24/0.52          = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_le_add_iff1
% 0.24/0.52  thf(fact_3046_nat__eq__add__iff2,axiom,
% 0.24/0.52      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.52       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.24/0.52            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.24/0.52          = ( M
% 0.24/0.52            = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_eq_add_iff2
% 0.24/0.52  thf(fact_3047_nat__eq__add__iff1,axiom,
% 0.24/0.52      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ J @ I )
% 0.24/0.52       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.24/0.52            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.24/0.52          = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 0.24/0.52            = N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_eq_add_iff1
% 0.24/0.52  thf(fact_3048_diff__eq__diff__less__eq,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real,D: real] :
% 0.24/0.52        ( ( ( minus_minus_real @ A @ B )
% 0.24/0.52          = ( minus_minus_real @ C @ D ) )
% 0.24/0.52       => ( ( ord_less_eq_real @ A @ B )
% 0.24/0.52          = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_diff_less_eq
% 0.24/0.52  thf(fact_3049_diff__eq__diff__less__eq,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.24/0.52        ( ( ( minus_minus_rat @ A @ B )
% 0.24/0.52          = ( minus_minus_rat @ C @ D ) )
% 0.24/0.52       => ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.52          = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_diff_less_eq
% 0.24/0.52  thf(fact_3050_diff__eq__diff__less__eq,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int,D: int] :
% 0.24/0.52        ( ( ( minus_minus_int @ A @ B )
% 0.24/0.52          = ( minus_minus_int @ C @ D ) )
% 0.24/0.52       => ( ( ord_less_eq_int @ A @ B )
% 0.24/0.52          = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_diff_less_eq
% 0.24/0.52  thf(fact_3051_diff__right__mono,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.52       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_right_mono
% 0.24/0.52  thf(fact_3052_diff__right__mono,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.52       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_right_mono
% 0.24/0.52  thf(fact_3053_diff__right__mono,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.52       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_right_mono
% 0.24/0.52  thf(fact_3054_diff__left__mono,axiom,
% 0.24/0.52      ! [B: real,A: real,C: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ B @ A )
% 0.24/0.52       => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_left_mono
% 0.24/0.52  thf(fact_3055_diff__left__mono,axiom,
% 0.24/0.52      ! [B: rat,A: rat,C: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ B @ A )
% 0.24/0.52       => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_left_mono
% 0.24/0.52  thf(fact_3056_diff__left__mono,axiom,
% 0.24/0.52      ! [B: int,A: int,C: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ B @ A )
% 0.24/0.52       => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_left_mono
% 0.24/0.52  thf(fact_3057_diff__mono,axiom,
% 0.24/0.52      ! [A: real,B: real,D: real,C: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_real @ D @ C )
% 0.24/0.52         => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_mono
% 0.24/0.52  thf(fact_3058_diff__mono,axiom,
% 0.24/0.52      ! [A: rat,B: rat,D: rat,C: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_rat @ D @ C )
% 0.24/0.52         => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_mono
% 0.24/0.52  thf(fact_3059_diff__mono,axiom,
% 0.24/0.52      ! [A: int,B: int,D: int,C: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_int @ D @ C )
% 0.24/0.52         => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_mono
% 0.24/0.52  thf(fact_3060_eq__iff__diff__eq__0,axiom,
% 0.24/0.52      ( ( ^ [Y3: complex,Z2: complex] : ( Y3 = Z2 ) )
% 0.24/0.52      = ( ^ [A3: complex,B4: complex] :
% 0.24/0.52            ( ( minus_minus_complex @ A3 @ B4 )
% 0.24/0.52            = zero_zero_complex ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_iff_diff_eq_0
% 0.24/0.52  thf(fact_3061_eq__iff__diff__eq__0,axiom,
% 0.24/0.52      ( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
% 0.24/0.52      = ( ^ [A3: real,B4: real] :
% 0.24/0.52            ( ( minus_minus_real @ A3 @ B4 )
% 0.24/0.52            = zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_iff_diff_eq_0
% 0.24/0.52  thf(fact_3062_eq__iff__diff__eq__0,axiom,
% 0.24/0.52      ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 0.24/0.52      = ( ^ [A3: rat,B4: rat] :
% 0.24/0.52            ( ( minus_minus_rat @ A3 @ B4 )
% 0.24/0.52            = zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_iff_diff_eq_0
% 0.24/0.52  thf(fact_3063_eq__iff__diff__eq__0,axiom,
% 0.24/0.52      ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 0.24/0.52      = ( ^ [A3: int,B4: int] :
% 0.24/0.52            ( ( minus_minus_int @ A3 @ B4 )
% 0.24/0.52            = zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_iff_diff_eq_0
% 0.24/0.52  thf(fact_3064_diff__strict__mono,axiom,
% 0.24/0.52      ! [A: real,B: real,D: real,C: real] :
% 0.24/0.52        ( ( ord_less_real @ A @ B )
% 0.24/0.52       => ( ( ord_less_real @ D @ C )
% 0.24/0.52         => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_strict_mono
% 0.24/0.52  thf(fact_3065_diff__strict__mono,axiom,
% 0.24/0.52      ! [A: rat,B: rat,D: rat,C: rat] :
% 0.24/0.52        ( ( ord_less_rat @ A @ B )
% 0.24/0.52       => ( ( ord_less_rat @ D @ C )
% 0.24/0.52         => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_strict_mono
% 0.24/0.52  thf(fact_3066_diff__strict__mono,axiom,
% 0.24/0.52      ! [A: int,B: int,D: int,C: int] :
% 0.24/0.52        ( ( ord_less_int @ A @ B )
% 0.24/0.52       => ( ( ord_less_int @ D @ C )
% 0.24/0.52         => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_strict_mono
% 0.24/0.52  thf(fact_3067_diff__eq__diff__less,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real,D: real] :
% 0.24/0.52        ( ( ( minus_minus_real @ A @ B )
% 0.24/0.52          = ( minus_minus_real @ C @ D ) )
% 0.24/0.52       => ( ( ord_less_real @ A @ B )
% 0.24/0.52          = ( ord_less_real @ C @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_diff_less
% 0.24/0.52  thf(fact_3068_diff__eq__diff__less,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.24/0.52        ( ( ( minus_minus_rat @ A @ B )
% 0.24/0.52          = ( minus_minus_rat @ C @ D ) )
% 0.24/0.52       => ( ( ord_less_rat @ A @ B )
% 0.24/0.52          = ( ord_less_rat @ C @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_diff_less
% 0.24/0.52  thf(fact_3069_diff__eq__diff__less,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int,D: int] :
% 0.24/0.52        ( ( ( minus_minus_int @ A @ B )
% 0.24/0.52          = ( minus_minus_int @ C @ D ) )
% 0.24/0.52       => ( ( ord_less_int @ A @ B )
% 0.24/0.52          = ( ord_less_int @ C @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_diff_less
% 0.24/0.52  thf(fact_3070_diff__strict__left__mono,axiom,
% 0.24/0.52      ! [B: real,A: real,C: real] :
% 0.24/0.52        ( ( ord_less_real @ B @ A )
% 0.24/0.52       => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_strict_left_mono
% 0.24/0.52  thf(fact_3071_diff__strict__left__mono,axiom,
% 0.24/0.52      ! [B: rat,A: rat,C: rat] :
% 0.24/0.52        ( ( ord_less_rat @ B @ A )
% 0.24/0.52       => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_strict_left_mono
% 0.24/0.52  thf(fact_3072_diff__strict__left__mono,axiom,
% 0.24/0.52      ! [B: int,A: int,C: int] :
% 0.24/0.52        ( ( ord_less_int @ B @ A )
% 0.24/0.52       => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_strict_left_mono
% 0.24/0.52  thf(fact_3073_diff__strict__right__mono,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( ord_less_real @ A @ B )
% 0.24/0.52       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_strict_right_mono
% 0.24/0.52  thf(fact_3074_diff__strict__right__mono,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( ord_less_rat @ A @ B )
% 0.24/0.52       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_strict_right_mono
% 0.24/0.52  thf(fact_3075_diff__strict__right__mono,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( ord_less_int @ A @ B )
% 0.24/0.52       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_strict_right_mono
% 0.24/0.52  thf(fact_3076_add__diff__add,axiom,
% 0.24/0.52      ! [A: complex,C: complex,B: complex,D: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) )
% 0.24/0.52        = ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ ( minus_minus_complex @ C @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_add
% 0.24/0.52  thf(fact_3077_add__diff__add,axiom,
% 0.24/0.52      ! [A: real,C: real,B: real,D: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 0.24/0.52        = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_add
% 0.24/0.52  thf(fact_3078_add__diff__add,axiom,
% 0.24/0.52      ! [A: rat,C: rat,B: rat,D: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 0.24/0.52        = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_add
% 0.24/0.52  thf(fact_3079_add__diff__add,axiom,
% 0.24/0.52      ! [A: int,C: int,B: int,D: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 0.24/0.52        = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_add
% 0.24/0.52  thf(fact_3080_group__cancel_Osub1,axiom,
% 0.24/0.52      ! [A2: complex,K: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( A2
% 0.24/0.52          = ( plus_plus_complex @ K @ A ) )
% 0.24/0.52       => ( ( minus_minus_complex @ A2 @ B )
% 0.24/0.52          = ( plus_plus_complex @ K @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % group_cancel.sub1
% 0.24/0.52  thf(fact_3081_group__cancel_Osub1,axiom,
% 0.24/0.52      ! [A2: real,K: real,A: real,B: real] :
% 0.24/0.52        ( ( A2
% 0.24/0.52          = ( plus_plus_real @ K @ A ) )
% 0.24/0.52       => ( ( minus_minus_real @ A2 @ B )
% 0.24/0.52          = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % group_cancel.sub1
% 0.24/0.52  thf(fact_3082_group__cancel_Osub1,axiom,
% 0.24/0.52      ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( A2
% 0.24/0.52          = ( plus_plus_rat @ K @ A ) )
% 0.24/0.52       => ( ( minus_minus_rat @ A2 @ B )
% 0.24/0.52          = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % group_cancel.sub1
% 0.24/0.52  thf(fact_3083_group__cancel_Osub1,axiom,
% 0.24/0.52      ! [A2: int,K: int,A: int,B: int] :
% 0.24/0.52        ( ( A2
% 0.24/0.52          = ( plus_plus_int @ K @ A ) )
% 0.24/0.52       => ( ( minus_minus_int @ A2 @ B )
% 0.24/0.52          = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % group_cancel.sub1
% 0.24/0.52  thf(fact_3084_diff__eq__eq,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( ( minus_minus_complex @ A @ B )
% 0.24/0.52          = C )
% 0.24/0.52        = ( A
% 0.24/0.52          = ( plus_plus_complex @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_eq
% 0.24/0.52  thf(fact_3085_diff__eq__eq,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( ( minus_minus_real @ A @ B )
% 0.24/0.52          = C )
% 0.24/0.52        = ( A
% 0.24/0.52          = ( plus_plus_real @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_eq
% 0.24/0.52  thf(fact_3086_diff__eq__eq,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( ( minus_minus_rat @ A @ B )
% 0.24/0.52          = C )
% 0.24/0.52        = ( A
% 0.24/0.52          = ( plus_plus_rat @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_eq
% 0.24/0.52  thf(fact_3087_diff__eq__eq,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( ( minus_minus_int @ A @ B )
% 0.24/0.52          = C )
% 0.24/0.52        = ( A
% 0.24/0.52          = ( plus_plus_int @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_eq_eq
% 0.24/0.52  thf(fact_3088_eq__diff__eq,axiom,
% 0.24/0.52      ! [A: complex,C: complex,B: complex] :
% 0.24/0.52        ( ( A
% 0.24/0.52          = ( minus_minus_complex @ C @ B ) )
% 0.24/0.52        = ( ( plus_plus_complex @ A @ B )
% 0.24/0.52          = C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_diff_eq
% 0.24/0.52  thf(fact_3089_eq__diff__eq,axiom,
% 0.24/0.52      ! [A: real,C: real,B: real] :
% 0.24/0.52        ( ( A
% 0.24/0.52          = ( minus_minus_real @ C @ B ) )
% 0.24/0.52        = ( ( plus_plus_real @ A @ B )
% 0.24/0.52          = C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_diff_eq
% 0.24/0.52  thf(fact_3090_eq__diff__eq,axiom,
% 0.24/0.52      ! [A: rat,C: rat,B: rat] :
% 0.24/0.52        ( ( A
% 0.24/0.52          = ( minus_minus_rat @ C @ B ) )
% 0.24/0.52        = ( ( plus_plus_rat @ A @ B )
% 0.24/0.52          = C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_diff_eq
% 0.24/0.52  thf(fact_3091_eq__diff__eq,axiom,
% 0.24/0.52      ! [A: int,C: int,B: int] :
% 0.24/0.52        ( ( A
% 0.24/0.52          = ( minus_minus_int @ C @ B ) )
% 0.24/0.52        = ( ( plus_plus_int @ A @ B )
% 0.24/0.52          = C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_diff_eq
% 0.24/0.52  thf(fact_3092_add__diff__eq,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( plus_plus_complex @ A @ ( minus_minus_complex @ B @ C ) )
% 0.24/0.52        = ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_eq
% 0.24/0.52  thf(fact_3093_add__diff__eq,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.24/0.52        = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_eq
% 0.24/0.52  thf(fact_3094_add__diff__eq,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.24/0.52        = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_eq
% 0.24/0.52  thf(fact_3095_add__diff__eq,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.24/0.52        = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_diff_eq
% 0.24/0.52  thf(fact_3096_diff__diff__eq2,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ A @ ( minus_minus_complex @ B @ C ) )
% 0.24/0.52        = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_eq2
% 0.24/0.52  thf(fact_3097_diff__diff__eq2,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.24/0.52        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_eq2
% 0.24/0.52  thf(fact_3098_diff__diff__eq2,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.24/0.52        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_eq2
% 0.24/0.52  thf(fact_3099_diff__diff__eq2,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.24/0.52        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_eq2
% 0.24/0.52  thf(fact_3100_diff__add__eq,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_eq
% 0.24/0.52  thf(fact_3101_diff__add__eq,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_eq
% 0.24/0.52  thf(fact_3102_diff__add__eq,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_eq
% 0.24/0.52  thf(fact_3103_diff__add__eq,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_eq
% 0.24/0.52  thf(fact_3104_diff__add__eq__diff__diff__swap,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 0.24/0.52        = ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_eq_diff_diff_swap
% 0.24/0.52  thf(fact_3105_diff__add__eq__diff__diff__swap,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.24/0.52        = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_eq_diff_diff_swap
% 0.24/0.52  thf(fact_3106_diff__add__eq__diff__diff__swap,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.24/0.52        = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_eq_diff_diff_swap
% 0.24/0.52  thf(fact_3107_diff__add__eq__diff__diff__swap,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.24/0.52        = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_eq_diff_diff_swap
% 0.24/0.52  thf(fact_3108_add__implies__diff,axiom,
% 0.24/0.52      ! [C: complex,B: complex,A: complex] :
% 0.24/0.52        ( ( ( plus_plus_complex @ C @ B )
% 0.24/0.52          = A )
% 0.24/0.52       => ( C
% 0.24/0.52          = ( minus_minus_complex @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_implies_diff
% 0.24/0.52  thf(fact_3109_add__implies__diff,axiom,
% 0.24/0.52      ! [C: real,B: real,A: real] :
% 0.24/0.52        ( ( ( plus_plus_real @ C @ B )
% 0.24/0.52          = A )
% 0.24/0.52       => ( C
% 0.24/0.52          = ( minus_minus_real @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_implies_diff
% 0.24/0.52  thf(fact_3110_add__implies__diff,axiom,
% 0.24/0.52      ! [C: rat,B: rat,A: rat] :
% 0.24/0.52        ( ( ( plus_plus_rat @ C @ B )
% 0.24/0.52          = A )
% 0.24/0.52       => ( C
% 0.24/0.52          = ( minus_minus_rat @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_implies_diff
% 0.24/0.52  thf(fact_3111_add__implies__diff,axiom,
% 0.24/0.52      ! [C: nat,B: nat,A: nat] :
% 0.24/0.52        ( ( ( plus_plus_nat @ C @ B )
% 0.24/0.52          = A )
% 0.24/0.52       => ( C
% 0.24/0.52          = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_implies_diff
% 0.24/0.52  thf(fact_3112_add__implies__diff,axiom,
% 0.24/0.52      ! [C: int,B: int,A: int] :
% 0.24/0.52        ( ( ( plus_plus_int @ C @ B )
% 0.24/0.52          = A )
% 0.24/0.52       => ( C
% 0.24/0.52          = ( minus_minus_int @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_implies_diff
% 0.24/0.52  thf(fact_3113_diff__diff__eq,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( minus_minus_complex @ ( minus_minus_complex @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_eq
% 0.24/0.52  thf(fact_3114_diff__diff__eq,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_eq
% 0.24/0.52  thf(fact_3115_diff__diff__eq,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_eq
% 0.24/0.52  thf(fact_3116_diff__diff__eq,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_eq
% 0.24/0.52  thf(fact_3117_diff__diff__eq,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_diff_eq
% 0.24/0.52  thf(fact_3118_diff__divide__distrib,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_divide_distrib
% 0.24/0.52  thf(fact_3119_diff__divide__distrib,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_divide_distrib
% 0.24/0.52  thf(fact_3120_diff__divide__distrib,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.24/0.52        = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_divide_distrib
% 0.24/0.52  thf(fact_3121_mult__right__cancel,axiom,
% 0.24/0.52      ! [C: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( C != zero_zero_complex )
% 0.24/0.52       => ( ( ( times_times_complex @ A @ C )
% 0.24/0.52            = ( times_times_complex @ B @ C ) )
% 0.24/0.52          = ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_right_cancel
% 0.24/0.52  thf(fact_3122_mult__right__cancel,axiom,
% 0.24/0.52      ! [C: real,A: real,B: real] :
% 0.24/0.52        ( ( C != zero_zero_real )
% 0.24/0.52       => ( ( ( times_times_real @ A @ C )
% 0.24/0.52            = ( times_times_real @ B @ C ) )
% 0.24/0.52          = ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_right_cancel
% 0.24/0.52  thf(fact_3123_mult__right__cancel,axiom,
% 0.24/0.52      ! [C: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( C != zero_zero_rat )
% 0.24/0.52       => ( ( ( times_times_rat @ A @ C )
% 0.24/0.52            = ( times_times_rat @ B @ C ) )
% 0.24/0.52          = ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_right_cancel
% 0.24/0.52  thf(fact_3124_mult__right__cancel,axiom,
% 0.24/0.52      ! [C: nat,A: nat,B: nat] :
% 0.24/0.52        ( ( C != zero_zero_nat )
% 0.24/0.52       => ( ( ( times_times_nat @ A @ C )
% 0.24/0.52            = ( times_times_nat @ B @ C ) )
% 0.24/0.52          = ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_right_cancel
% 0.24/0.52  thf(fact_3125_mult__right__cancel,axiom,
% 0.24/0.52      ! [C: int,A: int,B: int] :
% 0.24/0.52        ( ( C != zero_zero_int )
% 0.24/0.52       => ( ( ( times_times_int @ A @ C )
% 0.24/0.52            = ( times_times_int @ B @ C ) )
% 0.24/0.52          = ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_right_cancel
% 0.24/0.52  thf(fact_3126_mult__left__cancel,axiom,
% 0.24/0.52      ! [C: complex,A: complex,B: complex] :
% 0.24/0.52        ( ( C != zero_zero_complex )
% 0.24/0.52       => ( ( ( times_times_complex @ C @ A )
% 0.24/0.52            = ( times_times_complex @ C @ B ) )
% 0.24/0.52          = ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_left_cancel
% 0.24/0.52  thf(fact_3127_mult__left__cancel,axiom,
% 0.24/0.52      ! [C: real,A: real,B: real] :
% 0.24/0.52        ( ( C != zero_zero_real )
% 0.24/0.52       => ( ( ( times_times_real @ C @ A )
% 0.24/0.52            = ( times_times_real @ C @ B ) )
% 0.24/0.52          = ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_left_cancel
% 0.24/0.52  thf(fact_3128_mult__left__cancel,axiom,
% 0.24/0.52      ! [C: rat,A: rat,B: rat] :
% 0.24/0.52        ( ( C != zero_zero_rat )
% 0.24/0.52       => ( ( ( times_times_rat @ C @ A )
% 0.24/0.52            = ( times_times_rat @ C @ B ) )
% 0.24/0.52          = ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_left_cancel
% 0.24/0.52  thf(fact_3129_mult__left__cancel,axiom,
% 0.24/0.52      ! [C: nat,A: nat,B: nat] :
% 0.24/0.52        ( ( C != zero_zero_nat )
% 0.24/0.52       => ( ( ( times_times_nat @ C @ A )
% 0.24/0.52            = ( times_times_nat @ C @ B ) )
% 0.24/0.52          = ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_left_cancel
% 0.24/0.52  thf(fact_3130_mult__left__cancel,axiom,
% 0.24/0.52      ! [C: int,A: int,B: int] :
% 0.24/0.52        ( ( C != zero_zero_int )
% 0.24/0.52       => ( ( ( times_times_int @ C @ A )
% 0.24/0.52            = ( times_times_int @ C @ B ) )
% 0.24/0.52          = ( A = B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_left_cancel
% 0.24/0.52  thf(fact_3131_no__zero__divisors,axiom,
% 0.24/0.52      ! [A: complex,B: complex] :
% 0.24/0.52        ( ( A != zero_zero_complex )
% 0.24/0.52       => ( ( B != zero_zero_complex )
% 0.24/0.52         => ( ( times_times_complex @ A @ B )
% 0.24/0.52           != zero_zero_complex ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % no_zero_divisors
% 0.24/0.52  thf(fact_3132_no__zero__divisors,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( A != zero_zero_real )
% 0.24/0.52       => ( ( B != zero_zero_real )
% 0.24/0.52         => ( ( times_times_real @ A @ B )
% 0.24/0.52           != zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % no_zero_divisors
% 0.24/0.52  thf(fact_3133_no__zero__divisors,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( A != zero_zero_rat )
% 0.24/0.52       => ( ( B != zero_zero_rat )
% 0.24/0.52         => ( ( times_times_rat @ A @ B )
% 0.24/0.52           != zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % no_zero_divisors
% 0.24/0.52  thf(fact_3134_no__zero__divisors,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( A != zero_zero_nat )
% 0.24/0.52       => ( ( B != zero_zero_nat )
% 0.24/0.52         => ( ( times_times_nat @ A @ B )
% 0.24/0.52           != zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % no_zero_divisors
% 0.24/0.52  thf(fact_3135_no__zero__divisors,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( A != zero_zero_int )
% 0.24/0.52       => ( ( B != zero_zero_int )
% 0.24/0.52         => ( ( times_times_int @ A @ B )
% 0.24/0.52           != zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % no_zero_divisors
% 0.24/0.52  thf(fact_3136_divisors__zero,axiom,
% 0.24/0.52      ! [A: complex,B: complex] :
% 0.24/0.52        ( ( ( times_times_complex @ A @ B )
% 0.24/0.52          = zero_zero_complex )
% 0.24/0.52       => ( ( A = zero_zero_complex )
% 0.24/0.52          | ( B = zero_zero_complex ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divisors_zero
% 0.24/0.52  thf(fact_3137_divisors__zero,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ( times_times_real @ A @ B )
% 0.24/0.52          = zero_zero_real )
% 0.24/0.52       => ( ( A = zero_zero_real )
% 0.24/0.52          | ( B = zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divisors_zero
% 0.24/0.52  thf(fact_3138_divisors__zero,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ( times_times_rat @ A @ B )
% 0.24/0.52          = zero_zero_rat )
% 0.24/0.52       => ( ( A = zero_zero_rat )
% 0.24/0.52          | ( B = zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divisors_zero
% 0.24/0.52  thf(fact_3139_divisors__zero,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ A @ B )
% 0.24/0.52          = zero_zero_nat )
% 0.24/0.52       => ( ( A = zero_zero_nat )
% 0.24/0.52          | ( B = zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divisors_zero
% 0.24/0.52  thf(fact_3140_divisors__zero,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ( times_times_int @ A @ B )
% 0.24/0.52          = zero_zero_int )
% 0.24/0.52       => ( ( A = zero_zero_int )
% 0.24/0.52          | ( B = zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divisors_zero
% 0.24/0.52  thf(fact_3141_mult__not__zero,axiom,
% 0.24/0.52      ! [A: complex,B: complex] :
% 0.24/0.52        ( ( ( times_times_complex @ A @ B )
% 0.24/0.52         != zero_zero_complex )
% 0.24/0.52       => ( ( A != zero_zero_complex )
% 0.24/0.52          & ( B != zero_zero_complex ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_not_zero
% 0.24/0.52  thf(fact_3142_mult__not__zero,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ( times_times_real @ A @ B )
% 0.24/0.52         != zero_zero_real )
% 0.24/0.52       => ( ( A != zero_zero_real )
% 0.24/0.52          & ( B != zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_not_zero
% 0.24/0.52  thf(fact_3143_mult__not__zero,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ( times_times_rat @ A @ B )
% 0.24/0.52         != zero_zero_rat )
% 0.24/0.52       => ( ( A != zero_zero_rat )
% 0.24/0.52          & ( B != zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_not_zero
% 0.24/0.52  thf(fact_3144_mult__not__zero,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ A @ B )
% 0.24/0.52         != zero_zero_nat )
% 0.24/0.52       => ( ( A != zero_zero_nat )
% 0.24/0.52          & ( B != zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_not_zero
% 0.24/0.52  thf(fact_3145_mult__not__zero,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ( times_times_int @ A @ B )
% 0.24/0.52         != zero_zero_int )
% 0.24/0.52       => ( ( A != zero_zero_int )
% 0.24/0.52          & ( B != zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_not_zero
% 0.24/0.52  thf(fact_3146_comm__monoid__mult__class_Omult__1,axiom,
% 0.24/0.52      ! [A: complex] :
% 0.24/0.52        ( ( times_times_complex @ one_one_complex @ A )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % comm_monoid_mult_class.mult_1
% 0.24/0.52  thf(fact_3147_comm__monoid__mult__class_Omult__1,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( times_times_real @ one_one_real @ A )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % comm_monoid_mult_class.mult_1
% 0.24/0.52  thf(fact_3148_comm__monoid__mult__class_Omult__1,axiom,
% 0.24/0.52      ! [A: rat] :
% 0.24/0.52        ( ( times_times_rat @ one_one_rat @ A )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % comm_monoid_mult_class.mult_1
% 0.24/0.52  thf(fact_3149_comm__monoid__mult__class_Omult__1,axiom,
% 0.24/0.52      ! [A: nat] :
% 0.24/0.52        ( ( times_times_nat @ one_one_nat @ A )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % comm_monoid_mult_class.mult_1
% 0.24/0.52  thf(fact_3150_comm__monoid__mult__class_Omult__1,axiom,
% 0.24/0.52      ! [A: int] :
% 0.24/0.52        ( ( times_times_int @ one_one_int @ A )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % comm_monoid_mult_class.mult_1
% 0.24/0.52  thf(fact_3151_mult_Ocomm__neutral,axiom,
% 0.24/0.52      ! [A: complex] :
% 0.24/0.52        ( ( times_times_complex @ A @ one_one_complex )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.comm_neutral
% 0.24/0.52  thf(fact_3152_mult_Ocomm__neutral,axiom,
% 0.24/0.52      ! [A: real] :
% 0.24/0.52        ( ( times_times_real @ A @ one_one_real )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.comm_neutral
% 0.24/0.52  thf(fact_3153_mult_Ocomm__neutral,axiom,
% 0.24/0.52      ! [A: rat] :
% 0.24/0.52        ( ( times_times_rat @ A @ one_one_rat )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.comm_neutral
% 0.24/0.52  thf(fact_3154_mult_Ocomm__neutral,axiom,
% 0.24/0.52      ! [A: nat] :
% 0.24/0.52        ( ( times_times_nat @ A @ one_one_nat )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.comm_neutral
% 0.24/0.52  thf(fact_3155_mult_Ocomm__neutral,axiom,
% 0.24/0.52      ! [A: int] :
% 0.24/0.52        ( ( times_times_int @ A @ one_one_int )
% 0.24/0.52        = A ) ).
% 0.24/0.52  
% 0.24/0.52  % mult.comm_neutral
% 0.24/0.52  thf(fact_3156_ring__class_Oring__distribs_I2_J,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ring_class.ring_distribs(2)
% 0.24/0.52  thf(fact_3157_ring__class_Oring__distribs_I2_J,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ring_class.ring_distribs(2)
% 0.24/0.52  thf(fact_3158_ring__class_Oring__distribs_I2_J,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ring_class.ring_distribs(2)
% 0.24/0.52  thf(fact_3159_ring__class_Oring__distribs_I2_J,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ring_class.ring_distribs(2)
% 0.24/0.52  thf(fact_3160_ring__class_Oring__distribs_I1_J,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 0.24/0.52        = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ring_class.ring_distribs(1)
% 0.24/0.52  thf(fact_3161_ring__class_Oring__distribs_I1_J,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.24/0.52        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ring_class.ring_distribs(1)
% 0.24/0.52  thf(fact_3162_ring__class_Oring__distribs_I1_J,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.24/0.52        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ring_class.ring_distribs(1)
% 0.24/0.52  thf(fact_3163_ring__class_Oring__distribs_I1_J,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.24/0.52        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ring_class.ring_distribs(1)
% 0.24/0.52  thf(fact_3164_comm__semiring__class_Odistrib,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % comm_semiring_class.distrib
% 0.24/0.52  thf(fact_3165_comm__semiring__class_Odistrib,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % comm_semiring_class.distrib
% 0.24/0.52  thf(fact_3166_comm__semiring__class_Odistrib,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % comm_semiring_class.distrib
% 0.24/0.52  thf(fact_3167_comm__semiring__class_Odistrib,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % comm_semiring_class.distrib
% 0.24/0.52  thf(fact_3168_comm__semiring__class_Odistrib,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % comm_semiring_class.distrib
% 0.24/0.52  thf(fact_3169_distrib__left,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 0.24/0.52        = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_left
% 0.24/0.52  thf(fact_3170_distrib__left,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.24/0.52        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_left
% 0.24/0.52  thf(fact_3171_distrib__left,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.24/0.52        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_left
% 0.24/0.52  thf(fact_3172_distrib__left,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.24/0.52        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_left
% 0.24/0.52  thf(fact_3173_distrib__left,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.24/0.52        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_left
% 0.24/0.52  thf(fact_3174_distrib__right,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_right
% 0.24/0.52  thf(fact_3175_distrib__right,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_right
% 0.24/0.52  thf(fact_3176_distrib__right,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_right
% 0.24/0.52  thf(fact_3177_distrib__right,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_right
% 0.24/0.52  thf(fact_3178_distrib__right,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.24/0.52        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % distrib_right
% 0.24/0.52  thf(fact_3179_combine__common__factor,axiom,
% 0.24/0.52      ! [A: complex,E2: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ C ) )
% 0.24/0.52        = ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E2 ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % combine_common_factor
% 0.24/0.52  thf(fact_3180_combine__common__factor,axiom,
% 0.24/0.52      ! [A: real,E2: real,B: real,C: real] :
% 0.24/0.52        ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
% 0.24/0.52        = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % combine_common_factor
% 0.24/0.52  thf(fact_3181_combine__common__factor,axiom,
% 0.24/0.52      ! [A: rat,E2: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
% 0.24/0.52        = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % combine_common_factor
% 0.24/0.52  thf(fact_3182_combine__common__factor,axiom,
% 0.24/0.52      ! [A: nat,E2: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
% 0.24/0.52        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % combine_common_factor
% 0.24/0.52  thf(fact_3183_combine__common__factor,axiom,
% 0.24/0.52      ! [A: int,E2: int,B: int,C: int] :
% 0.24/0.52        ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
% 0.24/0.52        = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % combine_common_factor
% 0.24/0.52  thf(fact_3184_zero__induct__lemma,axiom,
% 0.24/0.52      ! [P: nat > $o,K: nat,I: nat] :
% 0.24/0.52        ( ( P @ K )
% 0.24/0.52       => ( ! [N2: nat] :
% 0.24/0.52              ( ( P @ ( suc @ N2 ) )
% 0.24/0.52             => ( P @ N2 ) )
% 0.24/0.52         => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_induct_lemma
% 0.24/0.52  thf(fact_3185_minus__nat_Odiff__0,axiom,
% 0.24/0.52      ! [M: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.24/0.52        = M ) ).
% 0.24/0.52  
% 0.24/0.52  % minus_nat.diff_0
% 0.24/0.52  thf(fact_3186_diffs0__imp__equal,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( ( minus_minus_nat @ M @ N )
% 0.24/0.52          = zero_zero_nat )
% 0.24/0.52       => ( ( ( minus_minus_nat @ N @ M )
% 0.24/0.52            = zero_zero_nat )
% 0.24/0.52         => ( M = N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diffs0_imp_equal
% 0.24/0.52  thf(fact_3187_divide__divide__eq__left_H,axiom,
% 0.24/0.52      ! [A: complex,B: complex,C: complex] :
% 0.24/0.52        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.24/0.52        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_eq_left'
% 0.24/0.52  thf(fact_3188_divide__divide__eq__left_H,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.24/0.52        = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_eq_left'
% 0.24/0.52  thf(fact_3189_divide__divide__eq__left_H,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.24/0.52        = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_eq_left'
% 0.24/0.52  thf(fact_3190_divide__divide__times__eq,axiom,
% 0.24/0.52      ! [X: complex,Y: complex,Z: complex,W: complex] :
% 0.24/0.52        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.24/0.52        = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_times_eq
% 0.24/0.52  thf(fact_3191_divide__divide__times__eq,axiom,
% 0.24/0.52      ! [X: real,Y: real,Z: real,W: real] :
% 0.24/0.52        ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
% 0.24/0.52        = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_times_eq
% 0.24/0.52  thf(fact_3192_divide__divide__times__eq,axiom,
% 0.24/0.52      ! [X: rat,Y: rat,Z: rat,W: rat] :
% 0.24/0.52        ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
% 0.24/0.52        = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % divide_divide_times_eq
% 0.24/0.52  thf(fact_3193_times__divide__times__eq,axiom,
% 0.24/0.52      ! [X: complex,Y: complex,Z: complex,W: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.24/0.52        = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % times_divide_times_eq
% 0.24/0.52  thf(fact_3194_times__divide__times__eq,axiom,
% 0.24/0.52      ! [X: real,Y: real,Z: real,W: real] :
% 0.24/0.52        ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
% 0.24/0.52        = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % times_divide_times_eq
% 0.24/0.52  thf(fact_3195_times__divide__times__eq,axiom,
% 0.24/0.52      ! [X: rat,Y: rat,Z: rat,W: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
% 0.24/0.52        = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % times_divide_times_eq
% 0.24/0.52  thf(fact_3196_diff__less__mono2,axiom,
% 0.24/0.52      ! [M: nat,N: nat,L: nat] :
% 0.24/0.52        ( ( ord_less_nat @ M @ N )
% 0.24/0.52       => ( ( ord_less_nat @ M @ L )
% 0.24/0.52         => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_less_mono2
% 0.24/0.52  thf(fact_3197_less__imp__diff__less,axiom,
% 0.24/0.52      ! [J: nat,K: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_nat @ J @ K )
% 0.24/0.52       => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_imp_diff_less
% 0.24/0.52  thf(fact_3198_power__commuting__commutes,axiom,
% 0.24/0.52      ! [X: complex,Y: complex,N: nat] :
% 0.24/0.52        ( ( ( times_times_complex @ X @ Y )
% 0.24/0.52          = ( times_times_complex @ Y @ X ) )
% 0.24/0.52       => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
% 0.24/0.52          = ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_commuting_commutes
% 0.24/0.52  thf(fact_3199_power__commuting__commutes,axiom,
% 0.24/0.52      ! [X: real,Y: real,N: nat] :
% 0.24/0.52        ( ( ( times_times_real @ X @ Y )
% 0.24/0.52          = ( times_times_real @ Y @ X ) )
% 0.24/0.52       => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
% 0.24/0.52          = ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_commuting_commutes
% 0.24/0.52  thf(fact_3200_power__commuting__commutes,axiom,
% 0.24/0.52      ! [X: rat,Y: rat,N: nat] :
% 0.24/0.52        ( ( ( times_times_rat @ X @ Y )
% 0.24/0.52          = ( times_times_rat @ Y @ X ) )
% 0.24/0.52       => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
% 0.24/0.52          = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_commuting_commutes
% 0.24/0.52  thf(fact_3201_power__commuting__commutes,axiom,
% 0.24/0.52      ! [X: nat,Y: nat,N: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ X @ Y )
% 0.24/0.52          = ( times_times_nat @ Y @ X ) )
% 0.24/0.52       => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
% 0.24/0.52          = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_commuting_commutes
% 0.24/0.52  thf(fact_3202_power__commuting__commutes,axiom,
% 0.24/0.52      ! [X: int,Y: int,N: nat] :
% 0.24/0.52        ( ( ( times_times_int @ X @ Y )
% 0.24/0.52          = ( times_times_int @ Y @ X ) )
% 0.24/0.52       => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
% 0.24/0.52          = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_commuting_commutes
% 0.24/0.52  thf(fact_3203_power__mult__distrib,axiom,
% 0.24/0.52      ! [A: complex,B: complex,N: nat] :
% 0.24/0.52        ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
% 0.24/0.52        = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_mult_distrib
% 0.24/0.52  thf(fact_3204_power__mult__distrib,axiom,
% 0.24/0.52      ! [A: real,B: real,N: nat] :
% 0.24/0.52        ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
% 0.24/0.52        = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_mult_distrib
% 0.24/0.52  thf(fact_3205_power__mult__distrib,axiom,
% 0.24/0.52      ! [A: rat,B: rat,N: nat] :
% 0.24/0.52        ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
% 0.24/0.52        = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_mult_distrib
% 0.24/0.52  thf(fact_3206_power__mult__distrib,axiom,
% 0.24/0.52      ! [A: nat,B: nat,N: nat] :
% 0.24/0.52        ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
% 0.24/0.52        = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_mult_distrib
% 0.24/0.52  thf(fact_3207_power__mult__distrib,axiom,
% 0.24/0.52      ! [A: int,B: int,N: nat] :
% 0.24/0.52        ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
% 0.24/0.52        = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_mult_distrib
% 0.24/0.52  thf(fact_3208_power__commutes,axiom,
% 0.24/0.52      ! [A: complex,N: nat] :
% 0.24/0.52        ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 0.24/0.52        = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_commutes
% 0.24/0.52  thf(fact_3209_power__commutes,axiom,
% 0.24/0.52      ! [A: real,N: nat] :
% 0.24/0.52        ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 0.24/0.52        = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_commutes
% 0.24/0.52  thf(fact_3210_power__commutes,axiom,
% 0.24/0.52      ! [A: rat,N: nat] :
% 0.24/0.52        ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 0.24/0.52        = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_commutes
% 0.24/0.52  thf(fact_3211_power__commutes,axiom,
% 0.24/0.52      ! [A: nat,N: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 0.24/0.52        = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_commutes
% 0.24/0.52  thf(fact_3212_power__commutes,axiom,
% 0.24/0.52      ! [A: int,N: nat] :
% 0.24/0.52        ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 0.24/0.52        = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_commutes
% 0.24/0.52  thf(fact_3213_eq__diff__iff,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ K @ M )
% 0.24/0.52       => ( ( ord_less_eq_nat @ K @ N )
% 0.24/0.52         => ( ( ( minus_minus_nat @ M @ K )
% 0.24/0.52              = ( minus_minus_nat @ N @ K ) )
% 0.24/0.52            = ( M = N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % eq_diff_iff
% 0.24/0.52  thf(fact_3214_le__diff__iff,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ K @ M )
% 0.24/0.52       => ( ( ord_less_eq_nat @ K @ N )
% 0.24/0.52         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.24/0.52            = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_diff_iff
% 0.24/0.52  thf(fact_3215_Nat_Odiff__diff__eq,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ K @ M )
% 0.24/0.52       => ( ( ord_less_eq_nat @ K @ N )
% 0.24/0.52         => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.24/0.52            = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % Nat.diff_diff_eq
% 0.24/0.52  thf(fact_3216_diff__le__mono,axiom,
% 0.24/0.52      ! [M: nat,N: nat,L: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.52       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_le_mono
% 0.24/0.52  thf(fact_3217_diff__le__self,axiom,
% 0.24/0.52      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_le_self
% 0.24/0.52  thf(fact_3218_le__diff__iff_H,axiom,
% 0.24/0.52      ! [A: nat,C: nat,B: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ C )
% 0.24/0.52       => ( ( ord_less_eq_nat @ B @ C )
% 0.24/0.52         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.24/0.52            = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_diff_iff'
% 0.24/0.52  thf(fact_3219_diff__le__mono2,axiom,
% 0.24/0.52      ! [M: nat,N: nat,L: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ M @ N )
% 0.24/0.52       => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_le_mono2
% 0.24/0.52  thf(fact_3220_power__mult,axiom,
% 0.24/0.52      ! [A: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_mult
% 0.24/0.52  thf(fact_3221_power__mult,axiom,
% 0.24/0.52      ! [A: real,M: nat,N: nat] :
% 0.24/0.52        ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_mult
% 0.24/0.52  thf(fact_3222_power__mult,axiom,
% 0.24/0.52      ! [A: int,M: nat,N: nat] :
% 0.24/0.52        ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_mult
% 0.24/0.52  thf(fact_3223_power__mult,axiom,
% 0.24/0.52      ! [A: complex,M: nat,N: nat] :
% 0.24/0.52        ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 0.24/0.52        = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_mult
% 0.24/0.52  thf(fact_3224_Suc__mult__cancel1,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.24/0.52          = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.24/0.52        = ( M = N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % Suc_mult_cancel1
% 0.24/0.52  thf(fact_3225_mult__0,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( times_times_nat @ zero_zero_nat @ N )
% 0.24/0.52        = zero_zero_nat ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_0
% 0.24/0.52  thf(fact_3226_nat__mult__eq__cancel__disj,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ( times_times_nat @ K @ M )
% 0.24/0.52          = ( times_times_nat @ K @ N ) )
% 0.24/0.52        = ( ( K = zero_zero_nat )
% 0.24/0.52          | ( M = N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_mult_eq_cancel_disj
% 0.24/0.52  thf(fact_3227_diff__add__inverse2,axiom,
% 0.24/0.52      ! [M: nat,N: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.24/0.52        = M ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_inverse2
% 0.24/0.52  thf(fact_3228_diff__add__inverse,axiom,
% 0.24/0.52      ! [N: nat,M: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.24/0.52        = M ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add_inverse
% 0.24/0.52  thf(fact_3229_diff__cancel2,axiom,
% 0.24/0.52      ! [M: nat,K: nat,N: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.24/0.52        = ( minus_minus_nat @ M @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_cancel2
% 0.24/0.52  thf(fact_3230_Nat_Odiff__cancel,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.24/0.52        = ( minus_minus_nat @ M @ N ) ) ).
% 0.24/0.52  
% 0.24/0.52  % Nat.diff_cancel
% 0.24/0.52  thf(fact_3231_mult__of__nat__commute,axiom,
% 0.24/0.52      ! [X: nat,Y: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
% 0.24/0.52        = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_of_nat_commute
% 0.24/0.52  thf(fact_3232_mult__of__nat__commute,axiom,
% 0.24/0.52      ! [X: nat,Y: real] :
% 0.24/0.52        ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
% 0.24/0.52        = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_of_nat_commute
% 0.24/0.52  thf(fact_3233_mult__of__nat__commute,axiom,
% 0.24/0.52      ! [X: nat,Y: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
% 0.24/0.52        = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_of_nat_commute
% 0.24/0.52  thf(fact_3234_mult__of__nat__commute,axiom,
% 0.24/0.52      ! [X: nat,Y: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
% 0.24/0.52        = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_of_nat_commute
% 0.24/0.52  thf(fact_3235_mult__of__nat__commute,axiom,
% 0.24/0.52      ! [X: nat,Y: int] :
% 0.24/0.52        ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
% 0.24/0.52        = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_of_nat_commute
% 0.24/0.52  thf(fact_3236_le__cube,axiom,
% 0.24/0.52      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_cube
% 0.24/0.52  thf(fact_3237_le__square,axiom,
% 0.24/0.52      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_square
% 0.24/0.52  thf(fact_3238_mult__le__mono,axiom,
% 0.24/0.52      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.52       => ( ( ord_less_eq_nat @ K @ L )
% 0.24/0.52         => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_le_mono
% 0.24/0.52  thf(fact_3239_mult__le__mono1,axiom,
% 0.24/0.52      ! [I: nat,J: nat,K: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.52       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_le_mono1
% 0.24/0.52  thf(fact_3240_mult__le__mono2,axiom,
% 0.24/0.52      ! [I: nat,J: nat,K: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.52       => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_le_mono2
% 0.24/0.52  thf(fact_3241_left__add__mult__distrib,axiom,
% 0.24/0.52      ! [I: nat,U: nat,J: nat,K: nat] :
% 0.24/0.52        ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 0.24/0.52        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 0.24/0.52  
% 0.24/0.52  % left_add_mult_distrib
% 0.24/0.52  thf(fact_3242_add__mult__distrib2,axiom,
% 0.24/0.52      ! [K: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.52        = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_mult_distrib2
% 0.24/0.52  thf(fact_3243_add__mult__distrib,axiom,
% 0.24/0.52      ! [M: nat,N: nat,K: nat] :
% 0.24/0.52        ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.24/0.52        = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_mult_distrib
% 0.24/0.52  thf(fact_3244_mult__of__int__commute,axiom,
% 0.24/0.52      ! [X: int,Y: code_integer] :
% 0.24/0.52        ( ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ X ) @ Y )
% 0.24/0.52        = ( times_3573771949741848930nteger @ Y @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_of_int_commute
% 0.24/0.52  thf(fact_3245_mult__of__int__commute,axiom,
% 0.24/0.52      ! [X: int,Y: complex] :
% 0.24/0.52        ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y )
% 0.24/0.52        = ( times_times_complex @ Y @ ( ring_17405671764205052669omplex @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_of_int_commute
% 0.24/0.52  thf(fact_3246_mult__of__int__commute,axiom,
% 0.24/0.52      ! [X: int,Y: real] :
% 0.24/0.52        ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
% 0.24/0.52        = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_of_int_commute
% 0.24/0.52  thf(fact_3247_mult__of__int__commute,axiom,
% 0.24/0.52      ! [X: int,Y: rat] :
% 0.24/0.52        ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
% 0.24/0.52        = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_of_int_commute
% 0.24/0.52  thf(fact_3248_mult__of__int__commute,axiom,
% 0.24/0.52      ! [X: int,Y: int] :
% 0.24/0.52        ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
% 0.24/0.52        = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_of_int_commute
% 0.24/0.52  thf(fact_3249_nat__mult__1__right,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( times_times_nat @ N @ one_one_nat )
% 0.24/0.52        = N ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_mult_1_right
% 0.24/0.52  thf(fact_3250_nat__mult__1,axiom,
% 0.24/0.52      ! [N: nat] :
% 0.24/0.52        ( ( times_times_nat @ one_one_nat @ N )
% 0.24/0.52        = N ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_mult_1
% 0.24/0.52  thf(fact_3251_div__mult2__eq,axiom,
% 0.24/0.52      ! [M: nat,N: nat,Q3: nat] :
% 0.24/0.52        ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q3 ) )
% 0.24/0.52        = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) ).
% 0.24/0.52  
% 0.24/0.52  % div_mult2_eq
% 0.24/0.52  thf(fact_3252_frac__le__eq,axiom,
% 0.24/0.52      ! [Y: real,Z: real,X: real,W: real] :
% 0.24/0.52        ( ( Y != zero_zero_real )
% 0.24/0.52       => ( ( Z != zero_zero_real )
% 0.24/0.52         => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 0.24/0.52            = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % frac_le_eq
% 0.24/0.52  thf(fact_3253_frac__le__eq,axiom,
% 0.24/0.52      ! [Y: rat,Z: rat,X: rat,W: rat] :
% 0.24/0.52        ( ( Y != zero_zero_rat )
% 0.24/0.52       => ( ( Z != zero_zero_rat )
% 0.24/0.52         => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 0.24/0.52            = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % frac_le_eq
% 0.24/0.52  thf(fact_3254_frac__less__eq,axiom,
% 0.24/0.52      ! [Y: real,Z: real,X: real,W: real] :
% 0.24/0.52        ( ( Y != zero_zero_real )
% 0.24/0.52       => ( ( Z != zero_zero_real )
% 0.24/0.52         => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 0.24/0.52            = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % frac_less_eq
% 0.24/0.52  thf(fact_3255_frac__less__eq,axiom,
% 0.24/0.52      ! [Y: rat,Z: rat,X: rat,W: rat] :
% 0.24/0.52        ( ( Y != zero_zero_rat )
% 0.24/0.52       => ( ( Z != zero_zero_rat )
% 0.24/0.52         => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 0.24/0.52            = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % frac_less_eq
% 0.24/0.52  thf(fact_3256_nat__less__add__iff2,axiom,
% 0.24/0.52      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ I @ J )
% 0.24/0.52       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.24/0.52          = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_less_add_iff2
% 0.24/0.52  thf(fact_3257_nat__less__add__iff1,axiom,
% 0.24/0.52      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ J @ I )
% 0.24/0.52       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.24/0.52          = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % nat_less_add_iff1
% 0.24/0.52  thf(fact_3258_mult__eq__if,axiom,
% 0.24/0.52      ( times_times_nat
% 0.24/0.52      = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_eq_if
% 0.24/0.52  thf(fact_3259_scaling__mono,axiom,
% 0.24/0.52      ! [U: real,V: real,R2: real,S: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ U @ V )
% 0.24/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.24/0.52         => ( ( ord_less_eq_real @ R2 @ S )
% 0.24/0.52           => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % scaling_mono
% 0.24/0.52  thf(fact_3260_scaling__mono,axiom,
% 0.24/0.52      ! [U: rat,V: rat,R2: rat,S: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ U @ V )
% 0.24/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.24/0.52         => ( ( ord_less_eq_rat @ R2 @ S )
% 0.24/0.52           => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % scaling_mono
% 0.24/0.52  thf(fact_3261_floor__mono,axiom,
% 0.24/0.52      ! [X: real,Y: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ X @ Y )
% 0.24/0.52       => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_mono
% 0.24/0.52  thf(fact_3262_floor__mono,axiom,
% 0.24/0.52      ! [X: rat,Y: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ X @ Y )
% 0.24/0.52       => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_mono
% 0.24/0.52  thf(fact_3263_of__int__floor__le,axiom,
% 0.24/0.52      ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_floor_le
% 0.24/0.52  thf(fact_3264_of__int__floor__le,axiom,
% 0.24/0.52      ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).
% 0.24/0.52  
% 0.24/0.52  % of_int_floor_le
% 0.24/0.52  thf(fact_3265_floor__less__cancel,axiom,
% 0.24/0.52      ! [X: real,Y: real] :
% 0.24/0.52        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
% 0.24/0.52       => ( ord_less_real @ X @ Y ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_less_cancel
% 0.24/0.52  thf(fact_3266_floor__less__cancel,axiom,
% 0.24/0.52      ! [X: rat,Y: rat] :
% 0.24/0.52        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
% 0.24/0.52       => ( ord_less_rat @ X @ Y ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_less_cancel
% 0.24/0.52  thf(fact_3267_powr__non__neg,axiom,
% 0.24/0.52      ! [A: real,X: real] :
% 0.24/0.52        ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_non_neg
% 0.24/0.52  thf(fact_3268_powr__less__mono2__neg,axiom,
% 0.24/0.52      ! [A: real,X: real,Y: real] :
% 0.24/0.52        ( ( ord_less_real @ A @ zero_zero_real )
% 0.24/0.52       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.24/0.52         => ( ( ord_less_real @ X @ Y )
% 0.24/0.52           => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_less_mono2_neg
% 0.24/0.52  thf(fact_3269_powr__mono2,axiom,
% 0.24/0.52      ! [A: real,X: real,Y: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.24/0.52         => ( ( ord_less_eq_real @ X @ Y )
% 0.24/0.52           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_mono2
% 0.24/0.52  thf(fact_3270_powr__ge__pzero,axiom,
% 0.24/0.52      ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_ge_pzero
% 0.24/0.52  thf(fact_3271_powr__less__cancel,axiom,
% 0.24/0.52      ! [X: real,A: real,B: real] :
% 0.24/0.52        ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.24/0.52       => ( ( ord_less_real @ one_one_real @ X )
% 0.24/0.52         => ( ord_less_real @ A @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_less_cancel
% 0.24/0.52  thf(fact_3272_powr__less__mono,axiom,
% 0.24/0.52      ! [A: real,B: real,X: real] :
% 0.24/0.52        ( ( ord_less_real @ A @ B )
% 0.24/0.52       => ( ( ord_less_real @ one_one_real @ X )
% 0.24/0.52         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_less_mono
% 0.24/0.52  thf(fact_3273_powr__mono,axiom,
% 0.24/0.52      ! [A: real,B: real,X: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.24/0.52         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % powr_mono
% 0.24/0.52  thf(fact_3274_power__eq__if,axiom,
% 0.24/0.52      ( power_power_complex
% 0.24/0.52      = ( ^ [P4: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P4 @ ( power_power_complex @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_eq_if
% 0.24/0.52  thf(fact_3275_power__eq__if,axiom,
% 0.24/0.52      ( power_power_real
% 0.24/0.52      = ( ^ [P4: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P4 @ ( power_power_real @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_eq_if
% 0.24/0.52  thf(fact_3276_power__eq__if,axiom,
% 0.24/0.52      ( power_power_rat
% 0.24/0.52      = ( ^ [P4: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P4 @ ( power_power_rat @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_eq_if
% 0.24/0.52  thf(fact_3277_power__eq__if,axiom,
% 0.24/0.52      ( power_power_nat
% 0.24/0.52      = ( ^ [P4: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_eq_if
% 0.24/0.52  thf(fact_3278_power__eq__if,axiom,
% 0.24/0.52      ( power_power_int
% 0.24/0.52      = ( ^ [P4: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P4 @ ( power_power_int @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_eq_if
% 0.24/0.52  thf(fact_3279_power__minus__mult,axiom,
% 0.24/0.52      ! [N: nat,A: complex] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.24/0.52          = ( power_power_complex @ A @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_minus_mult
% 0.24/0.52  thf(fact_3280_power__minus__mult,axiom,
% 0.24/0.52      ! [N: nat,A: real] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.24/0.52          = ( power_power_real @ A @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_minus_mult
% 0.24/0.52  thf(fact_3281_power__minus__mult,axiom,
% 0.24/0.52      ! [N: nat,A: rat] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.24/0.52          = ( power_power_rat @ A @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_minus_mult
% 0.24/0.52  thf(fact_3282_power__minus__mult,axiom,
% 0.24/0.52      ! [N: nat,A: nat] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.24/0.52          = ( power_power_nat @ A @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_minus_mult
% 0.24/0.52  thf(fact_3283_power__minus__mult,axiom,
% 0.24/0.52      ! [N: nat,A: int] :
% 0.24/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.24/0.52       => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.24/0.52          = ( power_power_int @ A @ N ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % power_minus_mult
% 0.24/0.52  thf(fact_3284_mult__ceiling__le,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.52         => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_ceiling_le
% 0.24/0.52  thf(fact_3285_mult__ceiling__le,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.52         => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_ceiling_le
% 0.24/0.52  thf(fact_3286_le__iff__diff__le__0,axiom,
% 0.24/0.52      ( ord_less_eq_real
% 0.24/0.52      = ( ^ [A3: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_iff_diff_le_0
% 0.24/0.52  thf(fact_3287_le__iff__diff__le__0,axiom,
% 0.24/0.52      ( ord_less_eq_rat
% 0.24/0.52      = ( ^ [A3: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B4 ) @ zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_iff_diff_le_0
% 0.24/0.52  thf(fact_3288_le__iff__diff__le__0,axiom,
% 0.24/0.52      ( ord_less_eq_int
% 0.24/0.52      = ( ^ [A3: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_iff_diff_le_0
% 0.24/0.52  thf(fact_3289_less__iff__diff__less__0,axiom,
% 0.24/0.52      ( ord_less_real
% 0.24/0.52      = ( ^ [A3: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_iff_diff_less_0
% 0.24/0.52  thf(fact_3290_less__iff__diff__less__0,axiom,
% 0.24/0.52      ( ord_less_rat
% 0.24/0.52      = ( ^ [A3: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B4 ) @ zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_iff_diff_less_0
% 0.24/0.52  thf(fact_3291_less__iff__diff__less__0,axiom,
% 0.24/0.52      ( ord_less_int
% 0.24/0.52      = ( ^ [A3: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_iff_diff_less_0
% 0.24/0.52  thf(fact_3292_floor__le__ceiling,axiom,
% 0.24/0.52      ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_le_ceiling
% 0.24/0.52  thf(fact_3293_floor__le__ceiling,axiom,
% 0.24/0.52      ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).
% 0.24/0.52  
% 0.24/0.52  % floor_le_ceiling
% 0.24/0.52  thf(fact_3294_diff__le__eq,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.24/0.52        = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_le_eq
% 0.24/0.52  thf(fact_3295_diff__le__eq,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.24/0.52        = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_le_eq
% 0.24/0.52  thf(fact_3296_diff__le__eq,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.24/0.52        = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_le_eq
% 0.24/0.52  thf(fact_3297_le__diff__eq,axiom,
% 0.24/0.52      ! [A: real,C: real,B: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.24/0.52        = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_diff_eq
% 0.24/0.52  thf(fact_3298_le__diff__eq,axiom,
% 0.24/0.52      ! [A: rat,C: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.24/0.52        = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_diff_eq
% 0.24/0.52  thf(fact_3299_le__diff__eq,axiom,
% 0.24/0.52      ! [A: int,C: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.24/0.52        = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_diff_eq
% 0.24/0.52  thf(fact_3300_diff__add,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.24/0.52          = B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_add
% 0.24/0.52  thf(fact_3301_le__add__diff,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % le_add_diff
% 0.24/0.52  thf(fact_3302_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.24/0.52          = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.24/0.52  thf(fact_3303_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.24/0.52          = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.24/0.52  thf(fact_3304_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.24/0.52          = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.24/0.52  thf(fact_3305_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.24/0.52          = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.24/0.52  thf(fact_3306_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.24/0.52          = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.24/0.52  thf(fact_3307_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.24/0.52          = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.24/0.52  thf(fact_3308_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.24/0.52          = B ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.24/0.52  thf(fact_3309_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52         => ( ( ( minus_minus_nat @ B @ A )
% 0.24/0.52              = C )
% 0.24/0.52            = ( B
% 0.24/0.52              = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.24/0.52  thf(fact_3310_add__le__imp__le__diff,axiom,
% 0.24/0.52      ! [I: real,K: real,N: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.24/0.52       => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_le_imp_le_diff
% 0.24/0.52  thf(fact_3311_add__le__imp__le__diff,axiom,
% 0.24/0.52      ! [I: rat,K: rat,N: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.24/0.52       => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_le_imp_le_diff
% 0.24/0.52  thf(fact_3312_add__le__imp__le__diff,axiom,
% 0.24/0.52      ! [I: nat,K: nat,N: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.24/0.52       => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_le_imp_le_diff
% 0.24/0.52  thf(fact_3313_add__le__imp__le__diff,axiom,
% 0.24/0.52      ! [I: int,K: int,N: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.24/0.52       => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_le_imp_le_diff
% 0.24/0.52  thf(fact_3314_add__le__add__imp__diff__le,axiom,
% 0.24/0.52      ! [I: real,K: real,N: real,J: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.24/0.52       => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.24/0.52         => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.24/0.52           => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.24/0.52             => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_le_add_imp_diff_le
% 0.24/0.52  thf(fact_3315_add__le__add__imp__diff__le,axiom,
% 0.24/0.52      ! [I: rat,K: rat,N: rat,J: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.24/0.52       => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.24/0.52         => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.24/0.52           => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.24/0.52             => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_le_add_imp_diff_le
% 0.24/0.52  thf(fact_3316_add__le__add__imp__diff__le,axiom,
% 0.24/0.52      ! [I: nat,K: nat,N: nat,J: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.24/0.52       => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.24/0.52         => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.24/0.52           => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.24/0.52             => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_le_add_imp_diff_le
% 0.24/0.52  thf(fact_3317_add__le__add__imp__diff__le,axiom,
% 0.24/0.52      ! [I: int,K: int,N: int,J: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.24/0.52       => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.24/0.52         => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.24/0.52           => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.24/0.52             => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % add_le_add_imp_diff_le
% 0.24/0.52  thf(fact_3318_diff__less__eq,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.24/0.52        = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_less_eq
% 0.24/0.52  thf(fact_3319_diff__less__eq,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.24/0.52        = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_less_eq
% 0.24/0.52  thf(fact_3320_diff__less__eq,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.24/0.52        = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % diff_less_eq
% 0.24/0.52  thf(fact_3321_less__diff__eq,axiom,
% 0.24/0.52      ! [A: real,C: real,B: real] :
% 0.24/0.52        ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.24/0.52        = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_diff_eq
% 0.24/0.52  thf(fact_3322_less__diff__eq,axiom,
% 0.24/0.52      ! [A: rat,C: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.24/0.52        = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_diff_eq
% 0.24/0.52  thf(fact_3323_less__diff__eq,axiom,
% 0.24/0.52      ! [A: int,C: int,B: int] :
% 0.24/0.52        ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.24/0.52        = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.24/0.52  
% 0.24/0.52  % less_diff_eq
% 0.24/0.52  thf(fact_3324_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ~ ( ord_less_real @ A @ B )
% 0.24/0.52       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % linordered_semidom_class.add_diff_inverse
% 0.24/0.52  thf(fact_3325_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ~ ( ord_less_rat @ A @ B )
% 0.24/0.52       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % linordered_semidom_class.add_diff_inverse
% 0.24/0.52  thf(fact_3326_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ~ ( ord_less_nat @ A @ B )
% 0.24/0.52       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % linordered_semidom_class.add_diff_inverse
% 0.24/0.52  thf(fact_3327_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ~ ( ord_less_int @ A @ B )
% 0.24/0.52       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.24/0.52          = A ) ) ).
% 0.24/0.52  
% 0.24/0.52  % linordered_semidom_class.add_diff_inverse
% 0.24/0.52  thf(fact_3328_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.24/0.52         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.24/0.52  thf(fact_3329_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.24/0.52         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.24/0.52  thf(fact_3330_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.24/0.52      ! [A: nat,B: nat,C: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.24/0.52         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.24/0.52  thf(fact_3331_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.24/0.52      ! [A: int,B: int,C: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.24/0.52         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.24/0.52  thf(fact_3332_zero__le__mult__iff,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.24/0.52        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.52            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.24/0.52          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.24/0.52            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_le_mult_iff
% 0.24/0.52  thf(fact_3333_zero__le__mult__iff,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.24/0.52        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.52            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.24/0.52          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.24/0.52            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_le_mult_iff
% 0.24/0.52  thf(fact_3334_zero__le__mult__iff,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.24/0.52        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.52            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.24/0.52          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.24/0.52            & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % zero_le_mult_iff
% 0.24/0.52  thf(fact_3335_mult__nonneg__nonpos2,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.52       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.24/0.52         => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonpos2
% 0.24/0.52  thf(fact_3336_mult__nonneg__nonpos2,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.52       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.24/0.52         => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonpos2
% 0.24/0.52  thf(fact_3337_mult__nonneg__nonpos2,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.52       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.24/0.52         => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonpos2
% 0.24/0.52  thf(fact_3338_mult__nonneg__nonpos2,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.52       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.24/0.52         => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonpos2
% 0.24/0.52  thf(fact_3339_mult__nonpos__nonneg,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.24/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.52         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonpos_nonneg
% 0.24/0.52  thf(fact_3340_mult__nonpos__nonneg,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.24/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.52         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonpos_nonneg
% 0.24/0.52  thf(fact_3341_mult__nonpos__nonneg,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.24/0.52       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.24/0.52         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonpos_nonneg
% 0.24/0.52  thf(fact_3342_mult__nonpos__nonneg,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.24/0.52       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.24/0.52         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonpos_nonneg
% 0.24/0.52  thf(fact_3343_mult__nonneg__nonpos,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.52       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.24/0.52         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonpos
% 0.24/0.52  thf(fact_3344_mult__nonneg__nonpos,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.52       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.24/0.52         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonpos
% 0.24/0.52  thf(fact_3345_mult__nonneg__nonpos,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.52       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.24/0.52         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonpos
% 0.24/0.52  thf(fact_3346_mult__nonneg__nonpos,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.52       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.24/0.52         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonpos
% 0.24/0.52  thf(fact_3347_mult__nonneg__nonneg,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.24/0.52         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonneg
% 0.24/0.52  thf(fact_3348_mult__nonneg__nonneg,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.24/0.52         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonneg
% 0.24/0.52  thf(fact_3349_mult__nonneg__nonneg,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.52       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.24/0.52         => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonneg
% 0.24/0.52  thf(fact_3350_mult__nonneg__nonneg,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.52       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.24/0.52         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_nonneg_nonneg
% 0.24/0.52  thf(fact_3351_split__mult__neg__le,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.52            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.24/0.52          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.24/0.52            & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 0.24/0.52       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 0.24/0.52  
% 0.24/0.52  % split_mult_neg_le
% 0.24/0.52  thf(fact_3352_split__mult__neg__le,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.52            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.24/0.52          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.24/0.52            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 0.24/0.52       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 0.24/0.52  
% 0.24/0.52  % split_mult_neg_le
% 0.24/0.52  thf(fact_3353_split__mult__neg__le,axiom,
% 0.24/0.52      ! [A: nat,B: nat] :
% 0.24/0.52        ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.24/0.52            & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.24/0.52          | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.24/0.52            & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.24/0.52       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 0.24/0.52  
% 0.24/0.52  % split_mult_neg_le
% 0.24/0.52  thf(fact_3354_split__mult__neg__le,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.52            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.24/0.52          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.24/0.52            & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 0.24/0.52       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 0.24/0.52  
% 0.24/0.52  % split_mult_neg_le
% 0.24/0.52  thf(fact_3355_mult__le__0__iff,axiom,
% 0.24/0.52      ! [A: real,B: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.24/0.52        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.24/0.52            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.24/0.52          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.24/0.52            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_le_0_iff
% 0.24/0.52  thf(fact_3356_mult__le__0__iff,axiom,
% 0.24/0.52      ! [A: rat,B: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.24/0.52        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.24/0.52            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.24/0.52          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.24/0.52            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_le_0_iff
% 0.24/0.52  thf(fact_3357_mult__le__0__iff,axiom,
% 0.24/0.52      ! [A: int,B: int] :
% 0.24/0.52        ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.24/0.52        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.24/0.52            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.24/0.52          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.24/0.52            & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_le_0_iff
% 0.24/0.52  thf(fact_3358_mult__right__mono,axiom,
% 0.24/0.52      ! [A: real,B: real,C: real] :
% 0.24/0.52        ( ( ord_less_eq_real @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.24/0.52         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_right_mono
% 0.24/0.52  thf(fact_3359_mult__right__mono,axiom,
% 0.24/0.52      ! [A: rat,B: rat,C: rat] :
% 0.24/0.52        ( ( ord_less_eq_rat @ A @ B )
% 0.24/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.24/0.52         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.24/0.52  
% 0.24/0.52  % mult_right_mono
% 0.24/0.52  thf(fact_3360_mult__right__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono
% 0.29/0.54  thf(fact_3361_mult__right__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono
% 0.29/0.54  thf(fact_3362_mult__right__mono__neg,axiom,
% 0.29/0.54      ! [B: real,A: real,C: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono_neg
% 0.29/0.54  thf(fact_3363_mult__right__mono__neg,axiom,
% 0.29/0.54      ! [B: rat,A: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono_neg
% 0.29/0.54  thf(fact_3364_mult__right__mono__neg,axiom,
% 0.29/0.54      ! [B: int,A: int,C: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono_neg
% 0.29/0.54  thf(fact_3365_mult__left__mono,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono
% 0.29/0.54  thf(fact_3366_mult__left__mono,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono
% 0.29/0.54  thf(fact_3367_mult__left__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono
% 0.29/0.54  thf(fact_3368_mult__left__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono
% 0.29/0.54  thf(fact_3369_mult__nonpos__nonpos,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.29/0.54         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonpos_nonpos
% 0.29/0.54  thf(fact_3370_mult__nonpos__nonpos,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.54       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonpos_nonpos
% 0.29/0.54  thf(fact_3371_mult__nonpos__nonpos,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.54       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.29/0.54         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonpos_nonpos
% 0.29/0.54  thf(fact_3372_mult__left__mono__neg,axiom,
% 0.29/0.54      ! [B: real,A: real,C: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono_neg
% 0.29/0.54  thf(fact_3373_mult__left__mono__neg,axiom,
% 0.29/0.54      ! [B: rat,A: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono_neg
% 0.29/0.54  thf(fact_3374_mult__left__mono__neg,axiom,
% 0.29/0.54      ! [B: int,A: int,C: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono_neg
% 0.29/0.54  thf(fact_3375_split__mult__pos__le,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.29/0.54          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.54            & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.29/0.54       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % split_mult_pos_le
% 0.29/0.54  thf(fact_3376_split__mult__pos__le,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.29/0.54          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.54            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.29/0.54       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % split_mult_pos_le
% 0.29/0.54  thf(fact_3377_split__mult__pos__le,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.29/0.54          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.54            & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.29/0.54       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % split_mult_pos_le
% 0.29/0.54  thf(fact_3378_zero__le__square,axiom,
% 0.29/0.54      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_le_square
% 0.29/0.54  thf(fact_3379_zero__le__square,axiom,
% 0.29/0.54      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_le_square
% 0.29/0.54  thf(fact_3380_zero__le__square,axiom,
% 0.29/0.54      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_le_square
% 0.29/0.54  thf(fact_3381_mult__mono_H,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_real @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono'
% 0.29/0.54  thf(fact_3382_mult__mono_H,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_rat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono'
% 0.29/0.54  thf(fact_3383_mult__mono_H,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono'
% 0.29/0.54  thf(fact_3384_mult__mono_H,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_int @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono'
% 0.29/0.54  thf(fact_3385_mult__mono,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_real @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.54           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono
% 0.29/0.54  thf(fact_3386_mult__mono,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_rat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.54           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono
% 0.29/0.54  thf(fact_3387_mult__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.54           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono
% 0.29/0.54  thf(fact_3388_mult__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_int @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.54           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono
% 0.29/0.54  thf(fact_3389_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.29/0.54  thf(fact_3390_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.29/0.54  thf(fact_3391_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( ord_less_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.29/0.54  thf(fact_3392_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( ord_less_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.29/0.54  thf(fact_3393_mult__less__cancel__right__disj,axiom,
% 0.29/0.54      ! [A: real,C: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54            & ( ord_less_real @ A @ B ) )
% 0.29/0.54          | ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54            & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_right_disj
% 0.29/0.54  thf(fact_3394_mult__less__cancel__right__disj,axiom,
% 0.29/0.54      ! [A: rat,C: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54            & ( ord_less_rat @ A @ B ) )
% 0.29/0.54          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_right_disj
% 0.29/0.54  thf(fact_3395_mult__less__cancel__right__disj,axiom,
% 0.29/0.54      ! [A: int,C: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54            & ( ord_less_int @ A @ B ) )
% 0.29/0.54          | ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.54            & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_right_disj
% 0.29/0.54  thf(fact_3396_mult__strict__right__mono,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_right_mono
% 0.29/0.54  thf(fact_3397_mult__strict__right__mono,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_right_mono
% 0.29/0.54  thf(fact_3398_mult__strict__right__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( ord_less_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_right_mono
% 0.29/0.54  thf(fact_3399_mult__strict__right__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( ord_less_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_right_mono
% 0.29/0.54  thf(fact_3400_mult__strict__right__mono__neg,axiom,
% 0.29/0.54      ! [B: real,A: real,C: real] :
% 0.29/0.54        ( ( ord_less_real @ B @ A )
% 0.29/0.54       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_right_mono_neg
% 0.29/0.54  thf(fact_3401_mult__strict__right__mono__neg,axiom,
% 0.29/0.54      ! [B: rat,A: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_rat @ B @ A )
% 0.29/0.54       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_right_mono_neg
% 0.29/0.54  thf(fact_3402_mult__strict__right__mono__neg,axiom,
% 0.29/0.54      ! [B: int,A: int,C: int] :
% 0.29/0.54        ( ( ord_less_int @ B @ A )
% 0.29/0.54       => ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.54         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_right_mono_neg
% 0.29/0.54  thf(fact_3403_mult__less__cancel__left__disj,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54            & ( ord_less_real @ A @ B ) )
% 0.29/0.54          | ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54            & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left_disj
% 0.29/0.54  thf(fact_3404_mult__less__cancel__left__disj,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54            & ( ord_less_rat @ A @ B ) )
% 0.29/0.54          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left_disj
% 0.29/0.54  thf(fact_3405_mult__less__cancel__left__disj,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54            & ( ord_less_int @ A @ B ) )
% 0.29/0.54          | ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.54            & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left_disj
% 0.29/0.54  thf(fact_3406_mult__strict__left__mono,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_left_mono
% 0.29/0.54  thf(fact_3407_mult__strict__left__mono,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_left_mono
% 0.29/0.54  thf(fact_3408_mult__strict__left__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( ord_less_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_left_mono
% 0.29/0.54  thf(fact_3409_mult__strict__left__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( ord_less_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_left_mono
% 0.29/0.54  thf(fact_3410_mult__strict__left__mono__neg,axiom,
% 0.29/0.54      ! [B: real,A: real,C: real] :
% 0.29/0.54        ( ( ord_less_real @ B @ A )
% 0.29/0.54       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_left_mono_neg
% 0.29/0.54  thf(fact_3411_mult__strict__left__mono__neg,axiom,
% 0.29/0.54      ! [B: rat,A: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_rat @ B @ A )
% 0.29/0.54       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_left_mono_neg
% 0.29/0.54  thf(fact_3412_mult__strict__left__mono__neg,axiom,
% 0.29/0.54      ! [B: int,A: int,C: int] :
% 0.29/0.54        ( ( ord_less_int @ B @ A )
% 0.29/0.54       => ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.54         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_left_mono_neg
% 0.29/0.54  thf(fact_3413_mult__less__cancel__left__pos,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54          = ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left_pos
% 0.29/0.54  thf(fact_3414_mult__less__cancel__left__pos,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54          = ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left_pos
% 0.29/0.54  thf(fact_3415_mult__less__cancel__left__pos,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54          = ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left_pos
% 0.29/0.54  thf(fact_3416_mult__less__cancel__left__neg,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54          = ( ord_less_real @ B @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left_neg
% 0.29/0.54  thf(fact_3417_mult__less__cancel__left__neg,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54          = ( ord_less_rat @ B @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left_neg
% 0.29/0.54  thf(fact_3418_mult__less__cancel__left__neg,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.54       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54          = ( ord_less_int @ B @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left_neg
% 0.29/0.54  thf(fact_3419_zero__less__mult__pos2,axiom,
% 0.29/0.54      ! [B: real,A: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_pos2
% 0.29/0.54  thf(fact_3420_zero__less__mult__pos2,axiom,
% 0.29/0.54      ! [B: rat,A: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_pos2
% 0.29/0.54  thf(fact_3421_zero__less__mult__pos2,axiom,
% 0.29/0.54      ! [B: nat,A: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.54         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_pos2
% 0.29/0.54  thf(fact_3422_zero__less__mult__pos2,axiom,
% 0.29/0.54      ! [B: int,A: int] :
% 0.29/0.54        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 0.29/0.54       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.54         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_pos2
% 0.29/0.54  thf(fact_3423_zero__less__mult__pos,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_pos
% 0.29/0.54  thf(fact_3424_zero__less__mult__pos,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_pos
% 0.29/0.54  thf(fact_3425_zero__less__mult__pos,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.54         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_pos
% 0.29/0.54  thf(fact_3426_zero__less__mult__pos,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.29/0.54       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.54         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_pos
% 0.29/0.54  thf(fact_3427_zero__less__mult__iff,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.54        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54            & ( ord_less_real @ zero_zero_real @ B ) )
% 0.29/0.54          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.54            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_iff
% 0.29/0.54  thf(fact_3428_zero__less__mult__iff,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.29/0.54          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.54            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_iff
% 0.29/0.54  thf(fact_3429_zero__less__mult__iff,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.29/0.54        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.54            & ( ord_less_int @ zero_zero_int @ B ) )
% 0.29/0.54          | ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.54            & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_less_mult_iff
% 0.29/0.54  thf(fact_3430_mult__pos__neg2,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.54         => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_neg2
% 0.29/0.54  thf(fact_3431_mult__pos__neg2,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_neg2
% 0.29/0.54  thf(fact_3432_mult__pos__neg2,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.54       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.29/0.54         => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_neg2
% 0.29/0.54  thf(fact_3433_mult__pos__neg2,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.54       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.29/0.54         => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_neg2
% 0.29/0.54  thf(fact_3434_mult__pos__pos,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.54         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_pos
% 0.29/0.54  thf(fact_3435_mult__pos__pos,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.54         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_pos
% 0.29/0.54  thf(fact_3436_mult__pos__pos,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.54         => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_pos
% 0.29/0.54  thf(fact_3437_mult__pos__pos,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.54       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.54         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_pos
% 0.29/0.54  thf(fact_3438_mult__pos__neg,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.54         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_neg
% 0.29/0.54  thf(fact_3439_mult__pos__neg,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_neg
% 0.29/0.54  thf(fact_3440_mult__pos__neg,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.54       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.29/0.54         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_neg
% 0.29/0.54  thf(fact_3441_mult__pos__neg,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.54       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.29/0.54         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_pos_neg
% 0.29/0.54  thf(fact_3442_mult__neg__pos,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.54         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_neg_pos
% 0.29/0.54  thf(fact_3443_mult__neg__pos,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.54         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_neg_pos
% 0.29/0.54  thf(fact_3444_mult__neg__pos,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.54         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_neg_pos
% 0.29/0.54  thf(fact_3445_mult__neg__pos,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.54       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.54         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_neg_pos
% 0.29/0.54  thf(fact_3446_mult__less__0__iff,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.29/0.54        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54            & ( ord_less_real @ B @ zero_zero_real ) )
% 0.29/0.54          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.54            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_0_iff
% 0.29/0.54  thf(fact_3447_mult__less__0__iff,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.29/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.29/0.54          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.54            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_0_iff
% 0.29/0.54  thf(fact_3448_mult__less__0__iff,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.29/0.54        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.54            & ( ord_less_int @ B @ zero_zero_int ) )
% 0.29/0.54          | ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.54            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_0_iff
% 0.29/0.54  thf(fact_3449_not__square__less__zero,axiom,
% 0.29/0.54      ! [A: real] :
% 0.29/0.54        ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 0.29/0.54  
% 0.29/0.54  % not_square_less_zero
% 0.29/0.54  thf(fact_3450_not__square__less__zero,axiom,
% 0.29/0.54      ! [A: rat] :
% 0.29/0.54        ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 0.29/0.54  
% 0.29/0.54  % not_square_less_zero
% 0.29/0.54  thf(fact_3451_not__square__less__zero,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 0.29/0.54  
% 0.29/0.54  % not_square_less_zero
% 0.29/0.54  thf(fact_3452_mult__neg__neg,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.54         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_neg_neg
% 0.29/0.54  thf(fact_3453_mult__neg__neg,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.54       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_neg_neg
% 0.29/0.54  thf(fact_3454_mult__neg__neg,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.54       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.29/0.54         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_neg_neg
% 0.29/0.54  thf(fact_3455_less__1__mult,axiom,
% 0.29/0.54      ! [M: real,N: real] :
% 0.29/0.54        ( ( ord_less_real @ one_one_real @ M )
% 0.29/0.54       => ( ( ord_less_real @ one_one_real @ N )
% 0.29/0.54         => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_1_mult
% 0.29/0.54  thf(fact_3456_less__1__mult,axiom,
% 0.29/0.54      ! [M: rat,N: rat] :
% 0.29/0.54        ( ( ord_less_rat @ one_one_rat @ M )
% 0.29/0.54       => ( ( ord_less_rat @ one_one_rat @ N )
% 0.29/0.54         => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_1_mult
% 0.29/0.54  thf(fact_3457_less__1__mult,axiom,
% 0.29/0.54      ! [M: nat,N: nat] :
% 0.29/0.54        ( ( ord_less_nat @ one_one_nat @ M )
% 0.29/0.54       => ( ( ord_less_nat @ one_one_nat @ N )
% 0.29/0.54         => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_1_mult
% 0.29/0.54  thf(fact_3458_less__1__mult,axiom,
% 0.29/0.54      ! [M: int,N: int] :
% 0.29/0.54        ( ( ord_less_int @ one_one_int @ M )
% 0.29/0.54       => ( ( ord_less_int @ one_one_int @ N )
% 0.29/0.54         => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_1_mult
% 0.29/0.54  thf(fact_3459_nonzero__eq__divide__eq,axiom,
% 0.29/0.54      ! [C: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( C != zero_zero_complex )
% 0.29/0.54       => ( ( A
% 0.29/0.54            = ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.54          = ( ( times_times_complex @ A @ C )
% 0.29/0.54            = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_eq_divide_eq
% 0.29/0.54  thf(fact_3460_nonzero__eq__divide__eq,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( C != zero_zero_real )
% 0.29/0.54       => ( ( A
% 0.29/0.54            = ( divide_divide_real @ B @ C ) )
% 0.29/0.54          = ( ( times_times_real @ A @ C )
% 0.29/0.54            = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_eq_divide_eq
% 0.29/0.54  thf(fact_3461_nonzero__eq__divide__eq,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( C != zero_zero_rat )
% 0.29/0.54       => ( ( A
% 0.29/0.54            = ( divide_divide_rat @ B @ C ) )
% 0.29/0.54          = ( ( times_times_rat @ A @ C )
% 0.29/0.54            = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_eq_divide_eq
% 0.29/0.54  thf(fact_3462_nonzero__divide__eq__eq,axiom,
% 0.29/0.54      ! [C: complex,B: complex,A: complex] :
% 0.29/0.54        ( ( C != zero_zero_complex )
% 0.29/0.54       => ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.29/0.54            = A )
% 0.29/0.54          = ( B
% 0.29/0.54            = ( times_times_complex @ A @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_divide_eq_eq
% 0.29/0.54  thf(fact_3463_nonzero__divide__eq__eq,axiom,
% 0.29/0.54      ! [C: real,B: real,A: real] :
% 0.29/0.54        ( ( C != zero_zero_real )
% 0.29/0.54       => ( ( ( divide_divide_real @ B @ C )
% 0.29/0.54            = A )
% 0.29/0.54          = ( B
% 0.29/0.54            = ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_divide_eq_eq
% 0.29/0.54  thf(fact_3464_nonzero__divide__eq__eq,axiom,
% 0.29/0.54      ! [C: rat,B: rat,A: rat] :
% 0.29/0.54        ( ( C != zero_zero_rat )
% 0.29/0.54       => ( ( ( divide_divide_rat @ B @ C )
% 0.29/0.54            = A )
% 0.29/0.54          = ( B
% 0.29/0.54            = ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_divide_eq_eq
% 0.29/0.54  thf(fact_3465_eq__divide__imp,axiom,
% 0.29/0.54      ! [C: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( C != zero_zero_complex )
% 0.29/0.54       => ( ( ( times_times_complex @ A @ C )
% 0.29/0.54            = B )
% 0.29/0.54         => ( A
% 0.29/0.54            = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_imp
% 0.29/0.54  thf(fact_3466_eq__divide__imp,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( C != zero_zero_real )
% 0.29/0.54       => ( ( ( times_times_real @ A @ C )
% 0.29/0.54            = B )
% 0.29/0.54         => ( A
% 0.29/0.54            = ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_imp
% 0.29/0.54  thf(fact_3467_eq__divide__imp,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( C != zero_zero_rat )
% 0.29/0.54       => ( ( ( times_times_rat @ A @ C )
% 0.29/0.54            = B )
% 0.29/0.54         => ( A
% 0.29/0.54            = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_imp
% 0.29/0.54  thf(fact_3468_divide__eq__imp,axiom,
% 0.29/0.54      ! [C: complex,B: complex,A: complex] :
% 0.29/0.54        ( ( C != zero_zero_complex )
% 0.29/0.54       => ( ( B
% 0.29/0.54            = ( times_times_complex @ A @ C ) )
% 0.29/0.54         => ( ( divide1717551699836669952omplex @ B @ C )
% 0.29/0.54            = A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_imp
% 0.29/0.54  thf(fact_3469_divide__eq__imp,axiom,
% 0.29/0.54      ! [C: real,B: real,A: real] :
% 0.29/0.54        ( ( C != zero_zero_real )
% 0.29/0.54       => ( ( B
% 0.29/0.54            = ( times_times_real @ A @ C ) )
% 0.29/0.54         => ( ( divide_divide_real @ B @ C )
% 0.29/0.54            = A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_imp
% 0.29/0.54  thf(fact_3470_divide__eq__imp,axiom,
% 0.29/0.54      ! [C: rat,B: rat,A: rat] :
% 0.29/0.54        ( ( C != zero_zero_rat )
% 0.29/0.54       => ( ( B
% 0.29/0.54            = ( times_times_rat @ A @ C ) )
% 0.29/0.54         => ( ( divide_divide_rat @ B @ C )
% 0.29/0.54            = A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_imp
% 0.29/0.54  thf(fact_3471_eq__divide__eq,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( A
% 0.29/0.54          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.54        = ( ( ( C != zero_zero_complex )
% 0.29/0.54           => ( ( times_times_complex @ A @ C )
% 0.29/0.54              = B ) )
% 0.29/0.54          & ( ( C = zero_zero_complex )
% 0.29/0.54           => ( A = zero_zero_complex ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_eq
% 0.29/0.54  thf(fact_3472_eq__divide__eq,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( A
% 0.29/0.54          = ( divide_divide_real @ B @ C ) )
% 0.29/0.54        = ( ( ( C != zero_zero_real )
% 0.29/0.54           => ( ( times_times_real @ A @ C )
% 0.29/0.54              = B ) )
% 0.29/0.54          & ( ( C = zero_zero_real )
% 0.29/0.54           => ( A = zero_zero_real ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_eq
% 0.29/0.54  thf(fact_3473_eq__divide__eq,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( A
% 0.29/0.54          = ( divide_divide_rat @ B @ C ) )
% 0.29/0.54        = ( ( ( C != zero_zero_rat )
% 0.29/0.54           => ( ( times_times_rat @ A @ C )
% 0.29/0.54              = B ) )
% 0.29/0.54          & ( ( C = zero_zero_rat )
% 0.29/0.54           => ( A = zero_zero_rat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_eq
% 0.29/0.54  thf(fact_3474_divide__eq__eq,axiom,
% 0.29/0.54      ! [B: complex,C: complex,A: complex] :
% 0.29/0.54        ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.29/0.54          = A )
% 0.29/0.54        = ( ( ( C != zero_zero_complex )
% 0.29/0.54           => ( B
% 0.29/0.54              = ( times_times_complex @ A @ C ) ) )
% 0.29/0.54          & ( ( C = zero_zero_complex )
% 0.29/0.54           => ( A = zero_zero_complex ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_eq
% 0.29/0.54  thf(fact_3475_divide__eq__eq,axiom,
% 0.29/0.54      ! [B: real,C: real,A: real] :
% 0.29/0.54        ( ( ( divide_divide_real @ B @ C )
% 0.29/0.54          = A )
% 0.29/0.54        = ( ( ( C != zero_zero_real )
% 0.29/0.54           => ( B
% 0.29/0.54              = ( times_times_real @ A @ C ) ) )
% 0.29/0.54          & ( ( C = zero_zero_real )
% 0.29/0.54           => ( A = zero_zero_real ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_eq
% 0.29/0.54  thf(fact_3476_divide__eq__eq,axiom,
% 0.29/0.54      ! [B: rat,C: rat,A: rat] :
% 0.29/0.54        ( ( ( divide_divide_rat @ B @ C )
% 0.29/0.54          = A )
% 0.29/0.54        = ( ( ( C != zero_zero_rat )
% 0.29/0.54           => ( B
% 0.29/0.54              = ( times_times_rat @ A @ C ) ) )
% 0.29/0.54          & ( ( C = zero_zero_rat )
% 0.29/0.54           => ( A = zero_zero_rat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_eq
% 0.29/0.54  thf(fact_3477_frac__eq__eq,axiom,
% 0.29/0.54      ! [Y: complex,Z: complex,X: complex,W: complex] :
% 0.29/0.54        ( ( Y != zero_zero_complex )
% 0.29/0.54       => ( ( Z != zero_zero_complex )
% 0.29/0.54         => ( ( ( divide1717551699836669952omplex @ X @ Y )
% 0.29/0.54              = ( divide1717551699836669952omplex @ W @ Z ) )
% 0.29/0.54            = ( ( times_times_complex @ X @ Z )
% 0.29/0.54              = ( times_times_complex @ W @ Y ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % frac_eq_eq
% 0.29/0.54  thf(fact_3478_frac__eq__eq,axiom,
% 0.29/0.54      ! [Y: real,Z: real,X: real,W: real] :
% 0.29/0.54        ( ( Y != zero_zero_real )
% 0.29/0.54       => ( ( Z != zero_zero_real )
% 0.29/0.54         => ( ( ( divide_divide_real @ X @ Y )
% 0.29/0.54              = ( divide_divide_real @ W @ Z ) )
% 0.29/0.54            = ( ( times_times_real @ X @ Z )
% 0.29/0.54              = ( times_times_real @ W @ Y ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % frac_eq_eq
% 0.29/0.54  thf(fact_3479_frac__eq__eq,axiom,
% 0.29/0.54      ! [Y: rat,Z: rat,X: rat,W: rat] :
% 0.29/0.54        ( ( Y != zero_zero_rat )
% 0.29/0.54       => ( ( Z != zero_zero_rat )
% 0.29/0.54         => ( ( ( divide_divide_rat @ X @ Y )
% 0.29/0.54              = ( divide_divide_rat @ W @ Z ) )
% 0.29/0.54            = ( ( times_times_rat @ X @ Z )
% 0.29/0.54              = ( times_times_rat @ W @ Y ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % frac_eq_eq
% 0.29/0.54  thf(fact_3480_mult__numeral__1,axiom,
% 0.29/0.54      ! [A: complex] :
% 0.29/0.54        ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_numeral_1
% 0.29/0.54  thf(fact_3481_mult__numeral__1,axiom,
% 0.29/0.54      ! [A: real] :
% 0.29/0.54        ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_numeral_1
% 0.29/0.54  thf(fact_3482_mult__numeral__1,axiom,
% 0.29/0.54      ! [A: rat] :
% 0.29/0.54        ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_numeral_1
% 0.29/0.54  thf(fact_3483_mult__numeral__1,axiom,
% 0.29/0.54      ! [A: nat] :
% 0.29/0.54        ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_numeral_1
% 0.29/0.54  thf(fact_3484_mult__numeral__1,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_numeral_1
% 0.29/0.54  thf(fact_3485_mult__numeral__1__right,axiom,
% 0.29/0.54      ! [A: complex] :
% 0.29/0.54        ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_numeral_1_right
% 0.29/0.54  thf(fact_3486_mult__numeral__1__right,axiom,
% 0.29/0.54      ! [A: real] :
% 0.29/0.54        ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_numeral_1_right
% 0.29/0.54  thf(fact_3487_mult__numeral__1__right,axiom,
% 0.29/0.54      ! [A: rat] :
% 0.29/0.54        ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_numeral_1_right
% 0.29/0.54  thf(fact_3488_mult__numeral__1__right,axiom,
% 0.29/0.54      ! [A: nat] :
% 0.29/0.54        ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_numeral_1_right
% 0.29/0.54  thf(fact_3489_mult__numeral__1__right,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_numeral_1_right
% 0.29/0.54  thf(fact_3490_left__right__inverse__power,axiom,
% 0.29/0.54      ! [X: complex,Y: complex,N: nat] :
% 0.29/0.54        ( ( ( times_times_complex @ X @ Y )
% 0.29/0.54          = one_one_complex )
% 0.29/0.54       => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
% 0.29/0.54          = one_one_complex ) ) ).
% 0.29/0.54  
% 0.29/0.54  % left_right_inverse_power
% 0.29/0.54  thf(fact_3491_left__right__inverse__power,axiom,
% 0.29/0.54      ! [X: real,Y: real,N: nat] :
% 0.29/0.54        ( ( ( times_times_real @ X @ Y )
% 0.29/0.54          = one_one_real )
% 0.29/0.54       => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
% 0.29/0.54          = one_one_real ) ) ).
% 0.29/0.54  
% 0.29/0.54  % left_right_inverse_power
% 0.29/0.54  thf(fact_3492_left__right__inverse__power,axiom,
% 0.29/0.54      ! [X: rat,Y: rat,N: nat] :
% 0.29/0.54        ( ( ( times_times_rat @ X @ Y )
% 0.29/0.54          = one_one_rat )
% 0.29/0.54       => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
% 0.29/0.54          = one_one_rat ) ) ).
% 0.29/0.54  
% 0.29/0.54  % left_right_inverse_power
% 0.29/0.54  thf(fact_3493_left__right__inverse__power,axiom,
% 0.29/0.54      ! [X: nat,Y: nat,N: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ X @ Y )
% 0.29/0.54          = one_one_nat )
% 0.29/0.54       => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
% 0.29/0.54          = one_one_nat ) ) ).
% 0.29/0.54  
% 0.29/0.54  % left_right_inverse_power
% 0.29/0.54  thf(fact_3494_left__right__inverse__power,axiom,
% 0.29/0.54      ! [X: int,Y: int,N: nat] :
% 0.29/0.54        ( ( ( times_times_int @ X @ Y )
% 0.29/0.54          = one_one_int )
% 0.29/0.54       => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
% 0.29/0.54          = one_one_int ) ) ).
% 0.29/0.54  
% 0.29/0.54  % left_right_inverse_power
% 0.29/0.54  thf(fact_3495_Suc__diff__Suc,axiom,
% 0.29/0.54      ! [N: nat,M: nat] :
% 0.29/0.54        ( ( ord_less_nat @ N @ M )
% 0.29/0.54       => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 0.29/0.54          = ( minus_minus_nat @ M @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Suc_diff_Suc
% 0.29/0.54  thf(fact_3496_diff__less__Suc,axiom,
% 0.29/0.54      ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.29/0.54  
% 0.29/0.54  % diff_less_Suc
% 0.29/0.54  thf(fact_3497_diff__less,axiom,
% 0.29/0.54      ! [N: nat,M: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.54         => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % diff_less
% 0.29/0.54  thf(fact_3498_Suc__diff__le,axiom,
% 0.29/0.54      ! [N: nat,M: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.54       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.29/0.54          = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Suc_diff_le
% 0.29/0.54  thf(fact_3499_power__Suc,axiom,
% 0.29/0.54      ! [A: complex,N: nat] :
% 0.29/0.54        ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.29/0.54        = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc
% 0.29/0.54  thf(fact_3500_power__Suc,axiom,
% 0.29/0.54      ! [A: real,N: nat] :
% 0.29/0.54        ( ( power_power_real @ A @ ( suc @ N ) )
% 0.29/0.54        = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc
% 0.29/0.54  thf(fact_3501_power__Suc,axiom,
% 0.29/0.54      ! [A: rat,N: nat] :
% 0.29/0.54        ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.29/0.54        = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc
% 0.29/0.54  thf(fact_3502_power__Suc,axiom,
% 0.29/0.54      ! [A: nat,N: nat] :
% 0.29/0.54        ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.29/0.54        = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc
% 0.29/0.54  thf(fact_3503_power__Suc,axiom,
% 0.29/0.54      ! [A: int,N: nat] :
% 0.29/0.54        ( ( power_power_int @ A @ ( suc @ N ) )
% 0.29/0.54        = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc
% 0.29/0.54  thf(fact_3504_power__Suc2,axiom,
% 0.29/0.54      ! [A: complex,N: nat] :
% 0.29/0.54        ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.29/0.54        = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc2
% 0.29/0.54  thf(fact_3505_power__Suc2,axiom,
% 0.29/0.54      ! [A: real,N: nat] :
% 0.29/0.54        ( ( power_power_real @ A @ ( suc @ N ) )
% 0.29/0.54        = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc2
% 0.29/0.54  thf(fact_3506_power__Suc2,axiom,
% 0.29/0.54      ! [A: rat,N: nat] :
% 0.29/0.54        ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.29/0.54        = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc2
% 0.29/0.54  thf(fact_3507_power__Suc2,axiom,
% 0.29/0.54      ! [A: nat,N: nat] :
% 0.29/0.54        ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.29/0.54        = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc2
% 0.29/0.54  thf(fact_3508_power__Suc2,axiom,
% 0.29/0.54      ! [A: int,N: nat] :
% 0.29/0.54        ( ( power_power_int @ A @ ( suc @ N ) )
% 0.29/0.54        = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc2
% 0.29/0.54  thf(fact_3509_less__diff__iff,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.54       => ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.54         => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.29/0.54            = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_diff_iff
% 0.29/0.54  thf(fact_3510_diff__less__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( ord_less_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ C @ A )
% 0.29/0.54         => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % diff_less_mono
% 0.29/0.54  thf(fact_3511_diff__add__0,axiom,
% 0.29/0.54      ! [N: nat,M: nat] :
% 0.29/0.54        ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.29/0.54        = zero_zero_nat ) ).
% 0.29/0.54  
% 0.29/0.54  % diff_add_0
% 0.29/0.54  thf(fact_3512_div__mult2__eq_H,axiom,
% 0.29/0.54      ! [A: nat,M: nat,N: nat] :
% 0.29/0.54        ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.29/0.54        = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult2_eq'
% 0.29/0.54  thf(fact_3513_div__mult2__eq_H,axiom,
% 0.29/0.54      ! [A: int,M: nat,N: nat] :
% 0.29/0.54        ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.29/0.54        = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult2_eq'
% 0.29/0.54  thf(fact_3514_add__diff__inverse__nat,axiom,
% 0.29/0.54      ! [M: nat,N: nat] :
% 0.29/0.54        ( ~ ( ord_less_nat @ M @ N )
% 0.29/0.54       => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.54          = M ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_diff_inverse_nat
% 0.29/0.54  thf(fact_3515_less__diff__conv,axiom,
% 0.29/0.54      ! [I: nat,J: nat,K: nat] :
% 0.29/0.54        ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.29/0.54        = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_diff_conv
% 0.29/0.54  thf(fact_3516_Nat_Ole__imp__diff__is__add,axiom,
% 0.29/0.54      ! [I: nat,J: nat,K: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.54       => ( ( ( minus_minus_nat @ J @ I )
% 0.29/0.54            = K )
% 0.29/0.54          = ( J
% 0.29/0.54            = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Nat.le_imp_diff_is_add
% 0.29/0.54  thf(fact_3517_Nat_Odiff__add__assoc2,axiom,
% 0.29/0.54      ! [K: nat,J: nat,I: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.54       => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.29/0.54          = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Nat.diff_add_assoc2
% 0.29/0.54  thf(fact_3518_Nat_Odiff__add__assoc,axiom,
% 0.29/0.54      ! [K: nat,J: nat,I: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.54       => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.29/0.54          = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Nat.diff_add_assoc
% 0.29/0.54  thf(fact_3519_Nat_Ole__diff__conv2,axiom,
% 0.29/0.54      ! [K: nat,J: nat,I: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.54       => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.29/0.54          = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Nat.le_diff_conv2
% 0.29/0.54  thf(fact_3520_le__diff__conv,axiom,
% 0.29/0.54      ! [J: nat,K: nat,I: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.29/0.54        = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % le_diff_conv
% 0.29/0.54  thf(fact_3521_power__add,axiom,
% 0.29/0.54      ! [A: complex,M: nat,N: nat] :
% 0.29/0.54        ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 0.29/0.54        = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_add
% 0.29/0.54  thf(fact_3522_power__add,axiom,
% 0.29/0.54      ! [A: real,M: nat,N: nat] :
% 0.29/0.54        ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 0.29/0.54        = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_add
% 0.29/0.54  thf(fact_3523_power__add,axiom,
% 0.29/0.54      ! [A: rat,M: nat,N: nat] :
% 0.29/0.54        ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.29/0.54        = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_add
% 0.29/0.54  thf(fact_3524_power__add,axiom,
% 0.29/0.54      ! [A: nat,M: nat,N: nat] :
% 0.29/0.54        ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.29/0.54        = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_add
% 0.29/0.54  thf(fact_3525_power__add,axiom,
% 0.29/0.54      ! [A: int,M: nat,N: nat] :
% 0.29/0.54        ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 0.29/0.54        = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_add
% 0.29/0.54  thf(fact_3526_Suc__mult__less__cancel1,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N: nat] :
% 0.29/0.54        ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.29/0.54        = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Suc_mult_less_cancel1
% 0.29/0.54  thf(fact_3527_mult__less__mono1,axiom,
% 0.29/0.54      ! [I: nat,J: nat,K: nat] :
% 0.29/0.54        ( ( ord_less_nat @ I @ J )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.54         => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_mono1
% 0.29/0.54  thf(fact_3528_mult__less__mono2,axiom,
% 0.29/0.54      ! [I: nat,J: nat,K: nat] :
% 0.29/0.54        ( ( ord_less_nat @ I @ J )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.54         => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_mono2
% 0.29/0.54  thf(fact_3529_nat__mult__eq__cancel1,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.54       => ( ( ( times_times_nat @ K @ M )
% 0.29/0.54            = ( times_times_nat @ K @ N ) )
% 0.29/0.54          = ( M = N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_mult_eq_cancel1
% 0.29/0.54  thf(fact_3530_nat__mult__less__cancel1,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.54       => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.54          = ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_mult_less_cancel1
% 0.29/0.54  thf(fact_3531_Suc__mult__le__cancel1,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.29/0.54        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Suc_mult_le_cancel1
% 0.29/0.54  thf(fact_3532_diff__Suc__eq__diff__pred,axiom,
% 0.29/0.54      ! [M: nat,N: nat] :
% 0.29/0.54        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.29/0.54        = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 0.29/0.54  
% 0.29/0.54  % diff_Suc_eq_diff_pred
% 0.29/0.54  thf(fact_3533_mult__Suc,axiom,
% 0.29/0.54      ! [M: nat,N: nat] :
% 0.29/0.54        ( ( times_times_nat @ ( suc @ M ) @ N )
% 0.29/0.54        = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_Suc
% 0.29/0.54  thf(fact_3534_mult__eq__self__implies__10,axiom,
% 0.29/0.54      ! [M: nat,N: nat] :
% 0.29/0.54        ( ( M
% 0.29/0.54          = ( times_times_nat @ M @ N ) )
% 0.29/0.54       => ( ( N = one_one_nat )
% 0.29/0.54          | ( M = zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_eq_self_implies_10
% 0.29/0.54  thf(fact_3535_less__mult__imp__div__less,axiom,
% 0.29/0.54      ! [M: nat,I: nat,N: nat] :
% 0.29/0.54        ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 0.29/0.54       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_mult_imp_div_less
% 0.29/0.54  thf(fact_3536_times__div__less__eq__dividend,axiom,
% 0.29/0.54      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 0.29/0.54  
% 0.29/0.54  % times_div_less_eq_dividend
% 0.29/0.54  thf(fact_3537_div__times__less__eq__dividend,axiom,
% 0.29/0.54      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 0.29/0.54  
% 0.29/0.54  % div_times_less_eq_dividend
% 0.29/0.54  thf(fact_3538_floor__divide__lower,axiom,
% 0.29/0.54      ! [Q3: real,P5: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ Q3 )
% 0.29/0.54       => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ Q3 ) @ P5 ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_divide_lower
% 0.29/0.54  thf(fact_3539_floor__divide__lower,axiom,
% 0.29/0.54      ! [Q3: rat,P5: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 0.29/0.54       => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q3 ) ) ) @ Q3 ) @ P5 ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_divide_lower
% 0.29/0.54  thf(fact_3540_power__odd__eq,axiom,
% 0.29/0.54      ! [A: complex,N: nat] :
% 0.29/0.54        ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.54        = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_odd_eq
% 0.29/0.54  thf(fact_3541_power__odd__eq,axiom,
% 0.29/0.54      ! [A: real,N: nat] :
% 0.29/0.54        ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.54        = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_odd_eq
% 0.29/0.54  thf(fact_3542_power__odd__eq,axiom,
% 0.29/0.54      ! [A: rat,N: nat] :
% 0.29/0.54        ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.54        = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_odd_eq
% 0.29/0.54  thf(fact_3543_power__odd__eq,axiom,
% 0.29/0.54      ! [A: nat,N: nat] :
% 0.29/0.54        ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.54        = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_odd_eq
% 0.29/0.54  thf(fact_3544_power__odd__eq,axiom,
% 0.29/0.54      ! [A: int,N: nat] :
% 0.29/0.54        ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.54        = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_odd_eq
% 0.29/0.54  thf(fact_3545_floor__divide__upper,axiom,
% 0.29/0.54      ! [Q3: real,P5: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ Q3 )
% 0.29/0.54       => ( ord_less_real @ P5 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_divide_upper
% 0.29/0.54  thf(fact_3546_floor__divide__upper,axiom,
% 0.29/0.54      ! [Q3: rat,P5: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 0.29/0.54       => ( ord_less_rat @ P5 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_divide_upper
% 0.29/0.54  thf(fact_3547_le__floor__iff,axiom,
% 0.29/0.54      ! [Z: int,X: real] :
% 0.29/0.54        ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
% 0.29/0.54        = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% 0.29/0.54  
% 0.29/0.54  % le_floor_iff
% 0.29/0.54  thf(fact_3548_le__floor__iff,axiom,
% 0.29/0.54      ! [Z: int,X: rat] :
% 0.29/0.54        ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
% 0.29/0.54        = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% 0.29/0.54  
% 0.29/0.54  % le_floor_iff
% 0.29/0.54  thf(fact_3549_floor__less__iff,axiom,
% 0.29/0.54      ! [X: real,Z: int] :
% 0.29/0.54        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z )
% 0.29/0.54        = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_less_iff
% 0.29/0.54  thf(fact_3550_floor__less__iff,axiom,
% 0.29/0.54      ! [X: rat,Z: int] :
% 0.29/0.54        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
% 0.29/0.54        = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_less_iff
% 0.29/0.54  thf(fact_3551_powr__mono2_H,axiom,
% 0.29/0.54      ! [A: real,X: real,Y: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.54         => ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.54           => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % powr_mono2'
% 0.29/0.54  thf(fact_3552_powr__less__mono2,axiom,
% 0.29/0.54      ! [A: real,X: real,Y: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.54         => ( ( ord_less_real @ X @ Y )
% 0.29/0.54           => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % powr_less_mono2
% 0.29/0.54  thf(fact_3553_le__floor__add,axiom,
% 0.29/0.54      ! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % le_floor_add
% 0.29/0.54  thf(fact_3554_le__floor__add,axiom,
% 0.29/0.54      ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % le_floor_add
% 0.29/0.54  thf(fact_3555_powr__inj,axiom,
% 0.29/0.54      ! [A: real,X: real,Y: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( A != one_one_real )
% 0.29/0.54         => ( ( ( powr_real @ A @ X )
% 0.29/0.54              = ( powr_real @ A @ Y ) )
% 0.29/0.54            = ( X = Y ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % powr_inj
% 0.29/0.54  thf(fact_3556_gr__one__powr,axiom,
% 0.29/0.54      ! [X: real,Y: real] :
% 0.29/0.54        ( ( ord_less_real @ one_one_real @ X )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.54         => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % gr_one_powr
% 0.29/0.54  thf(fact_3557_floor__add__int,axiom,
% 0.29/0.54      ! [X: real,Z: int] :
% 0.29/0.54        ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z )
% 0.29/0.54        = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_add_int
% 0.29/0.54  thf(fact_3558_floor__add__int,axiom,
% 0.29/0.54      ! [X: rat,Z: int] :
% 0.29/0.54        ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
% 0.29/0.54        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_add_int
% 0.29/0.54  thf(fact_3559_int__add__floor,axiom,
% 0.29/0.54      ! [Z: int,X: real] :
% 0.29/0.54        ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X ) )
% 0.29/0.54        = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % int_add_floor
% 0.29/0.54  thf(fact_3560_int__add__floor,axiom,
% 0.29/0.54      ! [Z: int,X: rat] :
% 0.29/0.54        ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
% 0.29/0.54        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % int_add_floor
% 0.29/0.54  thf(fact_3561_ge__one__powr__ge__zero,axiom,
% 0.29/0.54      ! [X: real,A: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ one_one_real @ X )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54         => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ge_one_powr_ge_zero
% 0.29/0.54  thf(fact_3562_powr__mono__both,axiom,
% 0.29/0.54      ! [A: real,B: real,X: real,Y: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ A @ B )
% 0.29/0.54         => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.29/0.54           => ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.54             => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % powr_mono_both
% 0.29/0.54  thf(fact_3563_powr__le1,axiom,
% 0.29/0.54      ! [A: real,X: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.54         => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.54           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % powr_le1
% 0.29/0.54  thf(fact_3564_powr__divide,axiom,
% 0.29/0.54      ! [X: real,Y: real,A: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.54         => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
% 0.29/0.54            = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % powr_divide
% 0.29/0.54  thf(fact_3565_power2__diff,axiom,
% 0.29/0.54      ! [X: complex,Y: complex] :
% 0.29/0.54        ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.54        = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power2_diff
% 0.29/0.54  thf(fact_3566_power2__diff,axiom,
% 0.29/0.54      ! [X: real,Y: real] :
% 0.29/0.54        ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.54        = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power2_diff
% 0.29/0.54  thf(fact_3567_power2__diff,axiom,
% 0.29/0.54      ! [X: rat,Y: rat] :
% 0.29/0.54        ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.54        = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power2_diff
% 0.29/0.54  thf(fact_3568_power2__diff,axiom,
% 0.29/0.54      ! [X: int,Y: int] :
% 0.29/0.54        ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.54        = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power2_diff
% 0.29/0.54  thf(fact_3569_floor__divide__of__int__eq,axiom,
% 0.29/0.54      ! [K: int,L: int] :
% 0.29/0.54        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
% 0.29/0.54        = ( divide_divide_int @ K @ L ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_divide_of_int_eq
% 0.29/0.54  thf(fact_3570_floor__divide__of__int__eq,axiom,
% 0.29/0.54      ! [K: int,L: int] :
% 0.29/0.54        ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
% 0.29/0.54        = ( divide_divide_int @ K @ L ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_divide_of_int_eq
% 0.29/0.54  thf(fact_3571_floor__power,axiom,
% 0.29/0.54      ! [X: real,N: nat] :
% 0.29/0.54        ( ( X
% 0.29/0.54          = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
% 0.29/0.54       => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N ) )
% 0.29/0.54          = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_power
% 0.29/0.54  thf(fact_3572_floor__power,axiom,
% 0.29/0.54      ! [X: rat,N: nat] :
% 0.29/0.54        ( ( X
% 0.29/0.54          = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
% 0.29/0.54       => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N ) )
% 0.29/0.54          = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % floor_power
% 0.29/0.54  thf(fact_3573_ceiling__divide__lower,axiom,
% 0.29/0.54      ! [Q3: real,P5: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ Q3 )
% 0.29/0.54       => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P5 ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ceiling_divide_lower
% 0.29/0.54  thf(fact_3574_ceiling__divide__lower,axiom,
% 0.29/0.54      ! [Q3: rat,P5: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 0.29/0.54       => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P5 ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ceiling_divide_lower
% 0.29/0.54  thf(fact_3575_log__base__powr,axiom,
% 0.29/0.54      ! [A: real,B: real,X: real] :
% 0.29/0.54        ( ( A != zero_zero_real )
% 0.29/0.54       => ( ( log @ ( powr_real @ A @ B ) @ X )
% 0.29/0.54          = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % log_base_powr
% 0.29/0.54  thf(fact_3576_round__mono,axiom,
% 0.29/0.54      ! [X: rat,Y: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ X @ Y )
% 0.29/0.54       => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % round_mono
% 0.29/0.54  thf(fact_3577_mult__less__le__imp__less,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_real @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54           => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_le_imp_less
% 0.29/0.54  thf(fact_3578_mult__less__le__imp__less,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_rat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54           => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_le_imp_less
% 0.29/0.54  thf(fact_3579_mult__less__le__imp__less,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.54        ( ( ord_less_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54           => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.54             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_le_imp_less
% 0.29/0.54  thf(fact_3580_mult__less__le__imp__less,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.54        ( ( ord_less_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_int @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54           => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_le_imp_less
% 0.29/0.54  thf(fact_3581_mult__le__less__imp__less,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_real @ C @ D )
% 0.29/0.54         => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_less_imp_less
% 0.29/0.54  thf(fact_3582_mult__le__less__imp__less,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_rat @ C @ D )
% 0.29/0.54         => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_less_imp_less
% 0.29/0.54  thf(fact_3583_mult__le__less__imp__less,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_nat @ C @ D )
% 0.29/0.54         => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.54           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_less_imp_less
% 0.29/0.54  thf(fact_3584_mult__le__less__imp__less,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_int @ C @ D )
% 0.29/0.54         => ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.54           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_less_imp_less
% 0.29/0.54  thf(fact_3585_mult__right__le__imp__le,axiom,
% 0.29/0.54      ! [A: real,C: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_le_imp_le
% 0.29/0.54  thf(fact_3586_mult__right__le__imp__le,axiom,
% 0.29/0.54      ! [A: rat,C: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_le_imp_le
% 0.29/0.54  thf(fact_3587_mult__right__le__imp__le,axiom,
% 0.29/0.54      ! [A: nat,C: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_le_imp_le
% 0.29/0.54  thf(fact_3588_mult__right__le__imp__le,axiom,
% 0.29/0.54      ! [A: int,C: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.54       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_le_imp_le
% 0.29/0.54  thf(fact_3589_mult__left__le__imp__le,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le_imp_le
% 0.29/0.54  thf(fact_3590_mult__left__le__imp__le,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le_imp_le
% 0.29/0.54  thf(fact_3591_mult__left__le__imp__le,axiom,
% 0.29/0.54      ! [C: nat,A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le_imp_le
% 0.29/0.54  thf(fact_3592_mult__left__le__imp__le,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le_imp_le
% 0.29/0.54  thf(fact_3593_mult__le__cancel__left__pos,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_left_pos
% 0.29/0.54  thf(fact_3594_mult__le__cancel__left__pos,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_left_pos
% 0.29/0.54  thf(fact_3595_mult__le__cancel__left__pos,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_left_pos
% 0.29/0.54  thf(fact_3596_mult__le__cancel__left__neg,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_left_neg
% 0.29/0.54  thf(fact_3597_mult__le__cancel__left__neg,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_left_neg
% 0.29/0.54  thf(fact_3598_mult__le__cancel__left__neg,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.54       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_left_neg
% 0.29/0.54  thf(fact_3599_mult__less__cancel__right,axiom,
% 0.29/0.54      ! [A: real,C: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54           => ( ord_less_real @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.54           => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_right
% 0.29/0.54  thf(fact_3600_mult__less__cancel__right,axiom,
% 0.29/0.54      ! [A: rat,C: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54           => ( ord_less_rat @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.54           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_right
% 0.29/0.54  thf(fact_3601_mult__less__cancel__right,axiom,
% 0.29/0.54      ! [A: int,C: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54           => ( ord_less_int @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.54           => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_right
% 0.29/0.54  thf(fact_3602_mult__strict__mono_H,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_real @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_mono'
% 0.29/0.54  thf(fact_3603_mult__strict__mono_H,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_rat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_mono'
% 0.29/0.54  thf(fact_3604_mult__strict__mono_H,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.54        ( ( ord_less_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_nat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_mono'
% 0.29/0.54  thf(fact_3605_mult__strict__mono_H,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.54        ( ( ord_less_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_int @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_mono'
% 0.29/0.54  thf(fact_3606_mult__right__less__imp__less,axiom,
% 0.29/0.54      ! [A: real,C: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_less_imp_less
% 0.29/0.54  thf(fact_3607_mult__right__less__imp__less,axiom,
% 0.29/0.54      ! [A: rat,C: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_less_imp_less
% 0.29/0.54  thf(fact_3608_mult__right__less__imp__less,axiom,
% 0.29/0.54      ! [A: nat,C: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_less_imp_less
% 0.29/0.54  thf(fact_3609_mult__right__less__imp__less,axiom,
% 0.29/0.54      ! [A: int,C: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_less_imp_less
% 0.29/0.54  thf(fact_3610_mult__less__cancel__left,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54           => ( ord_less_real @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.54           => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left
% 0.29/0.54  thf(fact_3611_mult__less__cancel__left,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54           => ( ord_less_rat @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.54           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left
% 0.29/0.54  thf(fact_3612_mult__less__cancel__left,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54           => ( ord_less_int @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.54           => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel_left
% 0.29/0.54  thf(fact_3613_mult__strict__mono,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_real @ C @ D )
% 0.29/0.54         => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.54           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_mono
% 0.29/0.54  thf(fact_3614_mult__strict__mono,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_rat @ C @ D )
% 0.29/0.54         => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.54           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_mono
% 0.29/0.54  thf(fact_3615_mult__strict__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.54        ( ( ord_less_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_nat @ C @ D )
% 0.29/0.54         => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.54           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_mono
% 0.29/0.54  thf(fact_3616_mult__strict__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.54        ( ( ord_less_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_int @ C @ D )
% 0.29/0.54         => ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.54           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_strict_mono
% 0.29/0.54  thf(fact_3617_mult__left__less__imp__less,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_less_imp_less
% 0.29/0.54  thf(fact_3618_mult__left__less__imp__less,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_less_imp_less
% 0.29/0.54  thf(fact_3619_mult__left__less__imp__less,axiom,
% 0.29/0.54      ! [C: nat,A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_less_imp_less
% 0.29/0.54  thf(fact_3620_mult__left__less__imp__less,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_less_imp_less
% 0.29/0.54  thf(fact_3621_mult__le__cancel__right,axiom,
% 0.29/0.54      ! [A: real,C: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54           => ( ord_less_eq_real @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_right
% 0.29/0.54  thf(fact_3622_mult__le__cancel__right,axiom,
% 0.29/0.54      ! [A: rat,C: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54           => ( ord_less_eq_rat @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_right
% 0.29/0.54  thf(fact_3623_mult__le__cancel__right,axiom,
% 0.29/0.54      ! [A: int,C: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54           => ( ord_less_eq_int @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.54           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_right
% 0.29/0.54  thf(fact_3624_mult__le__cancel__left,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54           => ( ord_less_eq_real @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_left
% 0.29/0.54  thf(fact_3625_mult__le__cancel__left,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54           => ( ord_less_eq_rat @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_left
% 0.29/0.54  thf(fact_3626_mult__le__cancel__left,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.54           => ( ord_less_eq_int @ A @ B ) )
% 0.29/0.54          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.54           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel_left
% 0.29/0.54  thf(fact_3627_mult__left__le__one__le,axiom,
% 0.29/0.54      ! [X: real,Y: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.54         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.54           => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le_one_le
% 0.29/0.54  thf(fact_3628_mult__left__le__one__le,axiom,
% 0.29/0.54      ! [X: rat,Y: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.29/0.54         => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 0.29/0.54           => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le_one_le
% 0.29/0.54  thf(fact_3629_mult__left__le__one__le,axiom,
% 0.29/0.54      ! [X: int,Y: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.54         => ( ( ord_less_eq_int @ Y @ one_one_int )
% 0.29/0.54           => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le_one_le
% 0.29/0.54  thf(fact_3630_mult__right__le__one__le,axiom,
% 0.29/0.54      ! [X: real,Y: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.54         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.54           => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_le_one_le
% 0.29/0.54  thf(fact_3631_mult__right__le__one__le,axiom,
% 0.29/0.54      ! [X: rat,Y: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.29/0.54         => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 0.29/0.54           => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_le_one_le
% 0.29/0.54  thf(fact_3632_mult__right__le__one__le,axiom,
% 0.29/0.54      ! [X: int,Y: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.54         => ( ( ord_less_eq_int @ Y @ one_one_int )
% 0.29/0.54           => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_le_one_le
% 0.29/0.54  thf(fact_3633_mult__le__one,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ one_one_real )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.54         => ( ( ord_less_eq_real @ B @ one_one_real )
% 0.29/0.54           => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_one
% 0.29/0.54  thf(fact_3634_mult__le__one,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.54         => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 0.29/0.54           => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_one
% 0.29/0.54  thf(fact_3635_mult__le__one,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.54         => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.29/0.54           => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_one
% 0.29/0.54  thf(fact_3636_mult__le__one,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ one_one_int )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.54         => ( ( ord_less_eq_int @ B @ one_one_int )
% 0.29/0.54           => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_one
% 0.29/0.54  thf(fact_3637_mult__left__le,axiom,
% 0.29/0.54      ! [C: real,A: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ C @ one_one_real )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le
% 0.29/0.54  thf(fact_3638_mult__left__le,axiom,
% 0.29/0.54      ! [C: rat,A: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le
% 0.29/0.54  thf(fact_3639_mult__left__le,axiom,
% 0.29/0.54      ! [C: nat,A: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le
% 0.29/0.54  thf(fact_3640_mult__left__le,axiom,
% 0.29/0.54      ! [C: int,A: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ C @ one_one_int )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_le
% 0.29/0.54  thf(fact_3641_sum__squares__le__zero__iff,axiom,
% 0.29/0.54      ! [X: real,Y: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
% 0.29/0.54        = ( ( X = zero_zero_real )
% 0.29/0.54          & ( Y = zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_le_zero_iff
% 0.29/0.54  thf(fact_3642_sum__squares__le__zero__iff,axiom,
% 0.29/0.54      ! [X: rat,Y: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
% 0.29/0.54        = ( ( X = zero_zero_rat )
% 0.29/0.54          & ( Y = zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_le_zero_iff
% 0.29/0.54  thf(fact_3643_sum__squares__le__zero__iff,axiom,
% 0.29/0.54      ! [X: int,Y: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
% 0.29/0.54        = ( ( X = zero_zero_int )
% 0.29/0.54          & ( Y = zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_le_zero_iff
% 0.29/0.54  thf(fact_3644_sum__squares__ge__zero,axiom,
% 0.29/0.54      ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_ge_zero
% 0.29/0.54  thf(fact_3645_sum__squares__ge__zero,axiom,
% 0.29/0.54      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_ge_zero
% 0.29/0.54  thf(fact_3646_sum__squares__ge__zero,axiom,
% 0.29/0.54      ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_ge_zero
% 0.29/0.54  thf(fact_3647_sum__squares__gt__zero__iff,axiom,
% 0.29/0.54      ! [X: real,Y: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
% 0.29/0.54        = ( ( X != zero_zero_real )
% 0.29/0.54          | ( Y != zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_gt_zero_iff
% 0.29/0.54  thf(fact_3648_sum__squares__gt__zero__iff,axiom,
% 0.29/0.54      ! [X: rat,Y: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
% 0.29/0.54        = ( ( X != zero_zero_rat )
% 0.29/0.54          | ( Y != zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_gt_zero_iff
% 0.29/0.54  thf(fact_3649_sum__squares__gt__zero__iff,axiom,
% 0.29/0.54      ! [X: int,Y: int] :
% 0.29/0.54        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
% 0.29/0.54        = ( ( X != zero_zero_int )
% 0.29/0.54          | ( Y != zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_gt_zero_iff
% 0.29/0.54  thf(fact_3650_not__sum__squares__lt__zero,axiom,
% 0.29/0.54      ! [X: real,Y: real] :
% 0.29/0.54        ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% 0.29/0.54  
% 0.29/0.54  % not_sum_squares_lt_zero
% 0.29/0.54  thf(fact_3651_not__sum__squares__lt__zero,axiom,
% 0.29/0.54      ! [X: rat,Y: rat] :
% 0.29/0.54        ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% 0.29/0.54  
% 0.29/0.54  % not_sum_squares_lt_zero
% 0.29/0.54  thf(fact_3652_not__sum__squares__lt__zero,axiom,
% 0.29/0.54      ! [X: int,Y: int] :
% 0.29/0.54        ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% 0.29/0.54  
% 0.29/0.54  % not_sum_squares_lt_zero
% 0.29/0.54  thf(fact_3653_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.29/0.54      ! [C: nat,A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.29/0.54          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.29/0.54  thf(fact_3654_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.29/0.54          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.29/0.54  thf(fact_3655_divide__strict__left__mono__neg,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.54           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_strict_left_mono_neg
% 0.29/0.54  thf(fact_3656_divide__strict__left__mono__neg,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.54           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_strict_left_mono_neg
% 0.29/0.54  thf(fact_3657_divide__strict__left__mono,axiom,
% 0.29/0.54      ! [B: real,A: real,C: real] :
% 0.29/0.54        ( ( ord_less_real @ B @ A )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.54           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_strict_left_mono
% 0.29/0.54  thf(fact_3658_divide__strict__left__mono,axiom,
% 0.29/0.54      ! [B: rat,A: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_rat @ B @ A )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.54           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_strict_left_mono
% 0.29/0.54  thf(fact_3659_mult__imp__less__div__pos,axiom,
% 0.29/0.54      ! [Y: real,Z: real,X: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.54       => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
% 0.29/0.54         => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_imp_less_div_pos
% 0.29/0.54  thf(fact_3660_mult__imp__less__div__pos,axiom,
% 0.29/0.54      ! [Y: rat,Z: rat,X: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.29/0.54       => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
% 0.29/0.54         => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_imp_less_div_pos
% 0.29/0.54  thf(fact_3661_mult__imp__div__pos__less,axiom,
% 0.29/0.54      ! [Y: real,X: real,Z: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.54       => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
% 0.29/0.54         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_imp_div_pos_less
% 0.29/0.54  thf(fact_3662_mult__imp__div__pos__less,axiom,
% 0.29/0.54      ! [Y: rat,X: rat,Z: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.29/0.54       => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
% 0.29/0.54         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_imp_div_pos_less
% 0.29/0.54  thf(fact_3663_pos__less__divide__eq,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.54          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % pos_less_divide_eq
% 0.29/0.54  thf(fact_3664_pos__less__divide__eq,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.54          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % pos_less_divide_eq
% 0.29/0.54  thf(fact_3665_pos__divide__less__eq,axiom,
% 0.29/0.54      ! [C: real,B: real,A: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.54          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % pos_divide_less_eq
% 0.29/0.54  thf(fact_3666_pos__divide__less__eq,axiom,
% 0.29/0.54      ! [C: rat,B: rat,A: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.54          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % pos_divide_less_eq
% 0.29/0.54  thf(fact_3667_neg__less__divide__eq,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.54          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % neg_less_divide_eq
% 0.29/0.54  thf(fact_3668_neg__less__divide__eq,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.54          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % neg_less_divide_eq
% 0.29/0.54  thf(fact_3669_neg__divide__less__eq,axiom,
% 0.29/0.54      ! [C: real,B: real,A: real] :
% 0.29/0.54        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.54          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % neg_divide_less_eq
% 0.29/0.54  thf(fact_3670_neg__divide__less__eq,axiom,
% 0.29/0.54      ! [C: rat,B: rat,A: rat] :
% 0.29/0.54        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.54          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % neg_divide_less_eq
% 0.29/0.54  thf(fact_3671_less__divide__eq,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54           => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.29/0.54          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54               => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.29/0.54              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_divide_eq
% 0.29/0.54  thf(fact_3672_less__divide__eq,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.29/0.54          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54               => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.54              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_divide_eq
% 0.29/0.54  thf(fact_3673_divide__less__eq,axiom,
% 0.29/0.54      ! [B: real,C: real,A: real] :
% 0.29/0.54        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54           => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.29/0.54          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.54           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54               => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.29/0.54              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.54               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_less_eq
% 0.29/0.54  thf(fact_3674_divide__less__eq,axiom,
% 0.29/0.54      ! [B: rat,C: rat,A: rat] :
% 0.29/0.54        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54           => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.54          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.54           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.29/0.54              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.54               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_less_eq
% 0.29/0.54  thf(fact_3675_eq__divide__eq__numeral_I1_J,axiom,
% 0.29/0.54      ! [W: num,B: complex,C: complex] :
% 0.29/0.54        ( ( ( numera6690914467698888265omplex @ W )
% 0.29/0.54          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.54        = ( ( ( C != zero_zero_complex )
% 0.29/0.54           => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 0.29/0.54              = B ) )
% 0.29/0.54          & ( ( C = zero_zero_complex )
% 0.29/0.54           => ( ( numera6690914467698888265omplex @ W )
% 0.29/0.54              = zero_zero_complex ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_eq_numeral(1)
% 0.29/0.54  thf(fact_3676_eq__divide__eq__numeral_I1_J,axiom,
% 0.29/0.54      ! [W: num,B: real,C: real] :
% 0.29/0.54        ( ( ( numeral_numeral_real @ W )
% 0.29/0.54          = ( divide_divide_real @ B @ C ) )
% 0.29/0.54        = ( ( ( C != zero_zero_real )
% 0.29/0.54           => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 0.29/0.54              = B ) )
% 0.29/0.54          & ( ( C = zero_zero_real )
% 0.29/0.54           => ( ( numeral_numeral_real @ W )
% 0.29/0.54              = zero_zero_real ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_eq_numeral(1)
% 0.29/0.54  thf(fact_3677_eq__divide__eq__numeral_I1_J,axiom,
% 0.29/0.54      ! [W: num,B: rat,C: rat] :
% 0.29/0.54        ( ( ( numeral_numeral_rat @ W )
% 0.29/0.54          = ( divide_divide_rat @ B @ C ) )
% 0.29/0.54        = ( ( ( C != zero_zero_rat )
% 0.29/0.54           => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 0.29/0.54              = B ) )
% 0.29/0.54          & ( ( C = zero_zero_rat )
% 0.29/0.54           => ( ( numeral_numeral_rat @ W )
% 0.29/0.54              = zero_zero_rat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_eq_numeral(1)
% 0.29/0.54  thf(fact_3678_divide__eq__eq__numeral_I1_J,axiom,
% 0.29/0.54      ! [B: complex,C: complex,W: num] :
% 0.29/0.54        ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.29/0.54          = ( numera6690914467698888265omplex @ W ) )
% 0.29/0.54        = ( ( ( C != zero_zero_complex )
% 0.29/0.54           => ( B
% 0.29/0.54              = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 0.29/0.54          & ( ( C = zero_zero_complex )
% 0.29/0.54           => ( ( numera6690914467698888265omplex @ W )
% 0.29/0.54              = zero_zero_complex ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_eq_numeral(1)
% 0.29/0.54  thf(fact_3679_divide__eq__eq__numeral_I1_J,axiom,
% 0.29/0.54      ! [B: real,C: real,W: num] :
% 0.29/0.54        ( ( ( divide_divide_real @ B @ C )
% 0.29/0.54          = ( numeral_numeral_real @ W ) )
% 0.29/0.54        = ( ( ( C != zero_zero_real )
% 0.29/0.54           => ( B
% 0.29/0.54              = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.29/0.54          & ( ( C = zero_zero_real )
% 0.29/0.54           => ( ( numeral_numeral_real @ W )
% 0.29/0.54              = zero_zero_real ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_eq_numeral(1)
% 0.29/0.54  thf(fact_3680_divide__eq__eq__numeral_I1_J,axiom,
% 0.29/0.54      ! [B: rat,C: rat,W: num] :
% 0.29/0.54        ( ( ( divide_divide_rat @ B @ C )
% 0.29/0.54          = ( numeral_numeral_rat @ W ) )
% 0.29/0.54        = ( ( ( C != zero_zero_rat )
% 0.29/0.54           => ( B
% 0.29/0.54              = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.29/0.54          & ( ( C = zero_zero_rat )
% 0.29/0.54           => ( ( numeral_numeral_rat @ W )
% 0.29/0.54              = zero_zero_rat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_eq_numeral(1)
% 0.29/0.54  thf(fact_3681_add__divide__eq__if__simps_I2_J,axiom,
% 0.29/0.54      ! [Z: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( ( Z = zero_zero_complex )
% 0.29/0.54         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.29/0.54            = B ) )
% 0.29/0.54        & ( ( Z != zero_zero_complex )
% 0.29/0.54         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.29/0.54            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_divide_eq_if_simps(2)
% 0.29/0.54  thf(fact_3682_add__divide__eq__if__simps_I2_J,axiom,
% 0.29/0.54      ! [Z: real,A: real,B: real] :
% 0.29/0.54        ( ( ( Z = zero_zero_real )
% 0.29/0.54         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.29/0.54            = B ) )
% 0.29/0.54        & ( ( Z != zero_zero_real )
% 0.29/0.54         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.29/0.54            = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_divide_eq_if_simps(2)
% 0.29/0.54  thf(fact_3683_add__divide__eq__if__simps_I2_J,axiom,
% 0.29/0.54      ! [Z: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ( Z = zero_zero_rat )
% 0.29/0.54         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.29/0.54            = B ) )
% 0.29/0.54        & ( ( Z != zero_zero_rat )
% 0.29/0.54         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.29/0.54            = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_divide_eq_if_simps(2)
% 0.29/0.54  thf(fact_3684_add__divide__eq__if__simps_I1_J,axiom,
% 0.29/0.54      ! [Z: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( ( Z = zero_zero_complex )
% 0.29/0.54         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.29/0.54            = A ) )
% 0.29/0.54        & ( ( Z != zero_zero_complex )
% 0.29/0.54         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.29/0.54            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_divide_eq_if_simps(1)
% 0.29/0.54  thf(fact_3685_add__divide__eq__if__simps_I1_J,axiom,
% 0.29/0.54      ! [Z: real,A: real,B: real] :
% 0.29/0.54        ( ( ( Z = zero_zero_real )
% 0.29/0.54         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.29/0.54            = A ) )
% 0.29/0.54        & ( ( Z != zero_zero_real )
% 0.29/0.54         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.29/0.54            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_divide_eq_if_simps(1)
% 0.29/0.54  thf(fact_3686_add__divide__eq__if__simps_I1_J,axiom,
% 0.29/0.54      ! [Z: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ( Z = zero_zero_rat )
% 0.29/0.54         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.29/0.54            = A ) )
% 0.29/0.54        & ( ( Z != zero_zero_rat )
% 0.29/0.54         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.29/0.54            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_divide_eq_if_simps(1)
% 0.29/0.54  thf(fact_3687_add__frac__eq,axiom,
% 0.29/0.54      ! [Y: complex,Z: complex,X: complex,W: complex] :
% 0.29/0.54        ( ( Y != zero_zero_complex )
% 0.29/0.54       => ( ( Z != zero_zero_complex )
% 0.29/0.54         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.29/0.54            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_frac_eq
% 0.29/0.54  thf(fact_3688_add__frac__eq,axiom,
% 0.29/0.54      ! [Y: real,Z: real,X: real,W: real] :
% 0.29/0.54        ( ( Y != zero_zero_real )
% 0.29/0.54       => ( ( Z != zero_zero_real )
% 0.29/0.54         => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 0.29/0.54            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_frac_eq
% 0.29/0.54  thf(fact_3689_add__frac__eq,axiom,
% 0.29/0.54      ! [Y: rat,Z: rat,X: rat,W: rat] :
% 0.29/0.54        ( ( Y != zero_zero_rat )
% 0.29/0.54       => ( ( Z != zero_zero_rat )
% 0.29/0.54         => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 0.29/0.54            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_frac_eq
% 0.29/0.54  thf(fact_3690_add__frac__num,axiom,
% 0.29/0.54      ! [Y: complex,X: complex,Z: complex] :
% 0.29/0.54        ( ( Y != zero_zero_complex )
% 0.29/0.54       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
% 0.29/0.54          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_frac_num
% 0.29/0.54  thf(fact_3691_add__frac__num,axiom,
% 0.29/0.54      ! [Y: real,X: real,Z: real] :
% 0.29/0.54        ( ( Y != zero_zero_real )
% 0.29/0.54       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
% 0.29/0.54          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_frac_num
% 0.29/0.54  thf(fact_3692_add__frac__num,axiom,
% 0.29/0.54      ! [Y: rat,X: rat,Z: rat] :
% 0.29/0.54        ( ( Y != zero_zero_rat )
% 0.29/0.54       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
% 0.29/0.54          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_frac_num
% 0.29/0.54  thf(fact_3693_add__num__frac,axiom,
% 0.29/0.54      ! [Y: complex,Z: complex,X: complex] :
% 0.29/0.54        ( ( Y != zero_zero_complex )
% 0.29/0.54       => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.29/0.54          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_num_frac
% 0.29/0.54  thf(fact_3694_add__num__frac,axiom,
% 0.29/0.54      ! [Y: real,Z: real,X: real] :
% 0.29/0.54        ( ( Y != zero_zero_real )
% 0.29/0.54       => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
% 0.29/0.54          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_num_frac
% 0.29/0.54  thf(fact_3695_add__num__frac,axiom,
% 0.29/0.54      ! [Y: rat,Z: rat,X: rat] :
% 0.29/0.54        ( ( Y != zero_zero_rat )
% 0.29/0.54       => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
% 0.29/0.54          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_num_frac
% 0.29/0.54  thf(fact_3696_add__divide__eq__iff,axiom,
% 0.29/0.54      ! [Z: complex,X: complex,Y: complex] :
% 0.29/0.54        ( ( Z != zero_zero_complex )
% 0.29/0.54       => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
% 0.29/0.54          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_divide_eq_iff
% 0.29/0.54  thf(fact_3697_add__divide__eq__iff,axiom,
% 0.29/0.54      ! [Z: real,X: real,Y: real] :
% 0.29/0.54        ( ( Z != zero_zero_real )
% 0.29/0.54       => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
% 0.29/0.54          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_divide_eq_iff
% 0.29/0.54  thf(fact_3698_add__divide__eq__iff,axiom,
% 0.29/0.54      ! [Z: rat,X: rat,Y: rat] :
% 0.29/0.54        ( ( Z != zero_zero_rat )
% 0.29/0.54       => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
% 0.29/0.54          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_divide_eq_iff
% 0.29/0.54  thf(fact_3699_divide__add__eq__iff,axiom,
% 0.29/0.54      ! [Z: complex,X: complex,Y: complex] :
% 0.29/0.54        ( ( Z != zero_zero_complex )
% 0.29/0.54       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
% 0.29/0.54          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_add_eq_iff
% 0.29/0.54  thf(fact_3700_divide__add__eq__iff,axiom,
% 0.29/0.54      ! [Z: real,X: real,Y: real] :
% 0.29/0.54        ( ( Z != zero_zero_real )
% 0.29/0.54       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
% 0.29/0.54          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_add_eq_iff
% 0.29/0.54  thf(fact_3701_divide__add__eq__iff,axiom,
% 0.29/0.54      ! [Z: rat,X: rat,Y: rat] :
% 0.29/0.54        ( ( Z != zero_zero_rat )
% 0.29/0.54       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
% 0.29/0.54          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_add_eq_iff
% 0.29/0.54  thf(fact_3702_ex__less__of__nat__mult,axiom,
% 0.29/0.54      ! [X: real,Y: real] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.54       => ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ex_less_of_nat_mult
% 0.29/0.54  thf(fact_3703_ex__less__of__nat__mult,axiom,
% 0.29/0.54      ! [X: rat,Y: rat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.29/0.54       => ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ex_less_of_nat_mult
% 0.29/0.54  thf(fact_3704_power__gt1__lemma,axiom,
% 0.29/0.54      ! [A: real,N: nat] :
% 0.29/0.54        ( ( ord_less_real @ one_one_real @ A )
% 0.29/0.54       => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_gt1_lemma
% 0.29/0.54  thf(fact_3705_power__gt1__lemma,axiom,
% 0.29/0.54      ! [A: rat,N: nat] :
% 0.29/0.54        ( ( ord_less_rat @ one_one_rat @ A )
% 0.29/0.54       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_gt1_lemma
% 0.29/0.54  thf(fact_3706_power__gt1__lemma,axiom,
% 0.29/0.54      ! [A: nat,N: nat] :
% 0.29/0.54        ( ( ord_less_nat @ one_one_nat @ A )
% 0.29/0.54       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_gt1_lemma
% 0.29/0.54  thf(fact_3707_power__gt1__lemma,axiom,
% 0.29/0.54      ! [A: int,N: nat] :
% 0.29/0.54        ( ( ord_less_int @ one_one_int @ A )
% 0.29/0.54       => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_gt1_lemma
% 0.29/0.54  thf(fact_3708_power__less__power__Suc,axiom,
% 0.29/0.54      ! [A: real,N: nat] :
% 0.29/0.54        ( ( ord_less_real @ one_one_real @ A )
% 0.29/0.54       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_less_power_Suc
% 0.29/0.54  thf(fact_3709_power__less__power__Suc,axiom,
% 0.29/0.54      ! [A: rat,N: nat] :
% 0.29/0.54        ( ( ord_less_rat @ one_one_rat @ A )
% 0.29/0.54       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_less_power_Suc
% 0.29/0.54  thf(fact_3710_power__less__power__Suc,axiom,
% 0.29/0.54      ! [A: nat,N: nat] :
% 0.29/0.54        ( ( ord_less_nat @ one_one_nat @ A )
% 0.29/0.54       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_less_power_Suc
% 0.29/0.54  thf(fact_3711_power__less__power__Suc,axiom,
% 0.29/0.54      ! [A: int,N: nat] :
% 0.29/0.54        ( ( ord_less_int @ one_one_int @ A )
% 0.29/0.54       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_less_power_Suc
% 0.29/0.54  thf(fact_3712_diff__Suc__less,axiom,
% 0.29/0.54      ! [N: nat,I: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.54       => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.29/0.54  
% 0.29/0.54  % diff_Suc_less
% 0.29/0.54  thf(fact_3713_round__def,axiom,
% 0.29/0.54      ( archim8280529875227126926d_real
% 0.29/0.54      = ( ^ [X4: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % round_def
% 0.29/0.54  thf(fact_3714_round__def,axiom,
% 0.29/0.54      ( archim7778729529865785530nd_rat
% 0.29/0.54      = ( ^ [X4: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % round_def
% 0.29/0.54  thf(fact_3715_nat__diff__split,axiom,
% 0.29/0.54      ! [P: nat > $o,A: nat,B: nat] :
% 0.29/0.54        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.29/0.54        = ( ( ( ord_less_nat @ A @ B )
% 0.29/0.54           => ( P @ zero_zero_nat ) )
% 0.29/0.54          & ! [D3: nat] :
% 0.29/0.54              ( ( A
% 0.29/0.55                = ( plus_plus_nat @ B @ D3 ) )
% 0.29/0.55             => ( P @ D3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nat_diff_split
% 0.29/0.55  thf(fact_3716_nat__diff__split__asm,axiom,
% 0.29/0.55      ! [P: nat > $o,A: nat,B: nat] :
% 0.29/0.55        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.29/0.55        = ( ~ ( ( ( ord_less_nat @ A @ B )
% 0.29/0.55                & ~ ( P @ zero_zero_nat ) )
% 0.29/0.55              | ? [D3: nat] :
% 0.29/0.55                  ( ( A
% 0.29/0.55                    = ( plus_plus_nat @ B @ D3 ) )
% 0.29/0.55                  & ~ ( P @ D3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nat_diff_split_asm
% 0.29/0.55  thf(fact_3717_less__diff__conv2,axiom,
% 0.29/0.55      ! [K: nat,J: nat,I: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.55       => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.29/0.55          = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_diff_conv2
% 0.29/0.55  thf(fact_3718_n__less__n__mult__m,axiom,
% 0.29/0.55      ! [N: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.55         => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % n_less_n_mult_m
% 0.29/0.55  thf(fact_3719_n__less__m__mult__n,axiom,
% 0.29/0.55      ! [N: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.55         => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % n_less_m_mult_n
% 0.29/0.55  thf(fact_3720_one__less__mult,axiom,
% 0.29/0.55      ! [N: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.55       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.55         => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % one_less_mult
% 0.29/0.55  thf(fact_3721_nat__mult__le__cancel1,axiom,
% 0.29/0.55      ! [K: nat,M: nat,N: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.55       => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.55          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nat_mult_le_cancel1
% 0.29/0.55  thf(fact_3722_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 0.29/0.55      ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 0.29/0.55        ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 0.29/0.55  
% 0.29/0.55  % VEBT_internal.naive_member.simps(2)
% 0.29/0.55  thf(fact_3723_ceiling__ge__round,axiom,
% 0.29/0.55      ! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_ge_round
% 0.29/0.55  thf(fact_3724_ceiling__ge__round,axiom,
% 0.29/0.55      ! [X: rat] : ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_ge_round
% 0.29/0.55  thf(fact_3725_div__less__iff__less__mult,axiom,
% 0.29/0.55      ! [Q3: nat,M: nat,N: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 0.29/0.55       => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
% 0.29/0.55          = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_less_iff_less_mult
% 0.29/0.55  thf(fact_3726_nat__mult__div__cancel1,axiom,
% 0.29/0.55      ! [K: nat,M: nat,N: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.55       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.55          = ( divide_divide_nat @ M @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nat_mult_div_cancel1
% 0.29/0.55  thf(fact_3727_one__add__floor,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 0.29/0.55        = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % one_add_floor
% 0.29/0.55  thf(fact_3728_one__add__floor,axiom,
% 0.29/0.55      ! [X: rat] :
% 0.29/0.55        ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 0.29/0.55        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % one_add_floor
% 0.29/0.55  thf(fact_3729_floor__log__eq__powr__iff,axiom,
% 0.29/0.55      ! [X: real,B: real,K: int] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.55         => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
% 0.29/0.55              = K )
% 0.29/0.55            = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 0.29/0.55              & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_log_eq_powr_iff
% 0.29/0.55  thf(fact_3730_floor__divide__of__nat__eq,axiom,
% 0.29/0.55      ! [M: nat,N: nat] :
% 0.29/0.55        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.29/0.55        = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_divide_of_nat_eq
% 0.29/0.55  thf(fact_3731_floor__divide__of__nat__eq,axiom,
% 0.29/0.55      ! [M: nat,N: nat] :
% 0.29/0.55        ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
% 0.29/0.55        = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_divide_of_nat_eq
% 0.29/0.55  thf(fact_3732_powr__realpow,axiom,
% 0.29/0.55      ! [X: real,N: nat] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 0.29/0.55          = ( power_power_real @ X @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % powr_realpow
% 0.29/0.55  thf(fact_3733_powr__less__iff,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
% 0.29/0.55            = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % powr_less_iff
% 0.29/0.55  thf(fact_3734_less__powr__iff,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
% 0.29/0.55            = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_powr_iff
% 0.29/0.55  thf(fact_3735_log__less__iff,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
% 0.29/0.55            = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % log_less_iff
% 0.29/0.55  thf(fact_3736_less__log__iff,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
% 0.29/0.55            = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_log_iff
% 0.29/0.55  thf(fact_3737_ceiling__altdef,axiom,
% 0.29/0.55      ( archim7802044766580827645g_real
% 0.29/0.55      = ( ^ [X4: real] :
% 0.29/0.55            ( if_int
% 0.29/0.55            @ ( X4
% 0.29/0.55              = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) )
% 0.29/0.55            @ ( archim6058952711729229775r_real @ X4 )
% 0.29/0.55            @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_altdef
% 0.29/0.55  thf(fact_3738_ceiling__altdef,axiom,
% 0.29/0.55      ( archim2889992004027027881ng_rat
% 0.29/0.55      = ( ^ [X4: rat] :
% 0.29/0.55            ( if_int
% 0.29/0.55            @ ( X4
% 0.29/0.55              = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) )
% 0.29/0.55            @ ( archim3151403230148437115or_rat @ X4 )
% 0.29/0.55            @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_altdef
% 0.29/0.55  thf(fact_3739_of__int__round__ge,axiom,
% 0.29/0.55      ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_round_ge
% 0.29/0.55  thf(fact_3740_of__int__round__ge,axiom,
% 0.29/0.55      ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_round_ge
% 0.29/0.55  thf(fact_3741_of__int__round__gt,axiom,
% 0.29/0.55      ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_round_gt
% 0.29/0.55  thf(fact_3742_of__int__round__gt,axiom,
% 0.29/0.55      ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_round_gt
% 0.29/0.55  thf(fact_3743_real__of__int__floor__add__one__gt,axiom,
% 0.29/0.55      ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % real_of_int_floor_add_one_gt
% 0.29/0.55  thf(fact_3744_floor__eq,axiom,
% 0.29/0.55      ! [N: int,X: real] :
% 0.29/0.55        ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.29/0.55       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.29/0.55         => ( ( archim6058952711729229775r_real @ X )
% 0.29/0.55            = N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_eq
% 0.29/0.55  thf(fact_3745_real__of__int__floor__add__one__ge,axiom,
% 0.29/0.55      ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % real_of_int_floor_add_one_ge
% 0.29/0.55  thf(fact_3746_power2__commute,axiom,
% 0.29/0.55      ! [X: complex,Y: complex] :
% 0.29/0.55        ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_commute
% 0.29/0.55  thf(fact_3747_power2__commute,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_commute
% 0.29/0.55  thf(fact_3748_power2__commute,axiom,
% 0.29/0.55      ! [X: rat,Y: rat] :
% 0.29/0.55        ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_commute
% 0.29/0.55  thf(fact_3749_power2__commute,axiom,
% 0.29/0.55      ! [X: int,Y: int] :
% 0.29/0.55        ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_commute
% 0.29/0.55  thf(fact_3750_field__le__mult__one__interval,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ! [Z3: real] :
% 0.29/0.55            ( ( ord_less_real @ zero_zero_real @ Z3 )
% 0.29/0.55           => ( ( ord_less_real @ Z3 @ one_one_real )
% 0.29/0.55             => ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ Y ) ) )
% 0.29/0.55       => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.29/0.55  
% 0.29/0.55  % field_le_mult_one_interval
% 0.29/0.55  thf(fact_3751_field__le__mult__one__interval,axiom,
% 0.29/0.55      ! [X: rat,Y: rat] :
% 0.29/0.55        ( ! [Z3: rat] :
% 0.29/0.55            ( ( ord_less_rat @ zero_zero_rat @ Z3 )
% 0.29/0.55           => ( ( ord_less_rat @ Z3 @ one_one_rat )
% 0.29/0.55             => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y ) ) )
% 0.29/0.55       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.29/0.55  
% 0.29/0.55  % field_le_mult_one_interval
% 0.29/0.55  thf(fact_3752_mult__less__cancel__right2,axiom,
% 0.29/0.55      ! [A: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ A @ one_one_real ) )
% 0.29/0.55          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right2
% 0.29/0.55  thf(fact_3753_mult__less__cancel__right2,axiom,
% 0.29/0.55      ! [A: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ A @ one_one_rat ) )
% 0.29/0.55          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right2
% 0.29/0.55  thf(fact_3754_mult__less__cancel__right2,axiom,
% 0.29/0.55      ! [A: int,C: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_int @ A @ one_one_int ) )
% 0.29/0.55          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right2
% 0.29/0.55  thf(fact_3755_mult__less__cancel__right1,axiom,
% 0.29/0.55      ! [C: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ one_one_real @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right1
% 0.29/0.55  thf(fact_3756_mult__less__cancel__right1,axiom,
% 0.29/0.55      ! [C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ one_one_rat @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right1
% 0.29/0.55  thf(fact_3757_mult__less__cancel__right1,axiom,
% 0.29/0.55      ! [C: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_int @ one_one_int @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right1
% 0.29/0.55  thf(fact_3758_mult__less__cancel__left2,axiom,
% 0.29/0.55      ! [C: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ A @ one_one_real ) )
% 0.29/0.55          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left2
% 0.29/0.55  thf(fact_3759_mult__less__cancel__left2,axiom,
% 0.29/0.55      ! [C: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ A @ one_one_rat ) )
% 0.29/0.55          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left2
% 0.29/0.55  thf(fact_3760_mult__less__cancel__left2,axiom,
% 0.29/0.55      ! [C: int,A: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_int @ A @ one_one_int ) )
% 0.29/0.55          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left2
% 0.29/0.55  thf(fact_3761_mult__less__cancel__left1,axiom,
% 0.29/0.55      ! [C: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ one_one_real @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left1
% 0.29/0.55  thf(fact_3762_mult__less__cancel__left1,axiom,
% 0.29/0.55      ! [C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ one_one_rat @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left1
% 0.29/0.55  thf(fact_3763_mult__less__cancel__left1,axiom,
% 0.29/0.55      ! [C: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_int @ one_one_int @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left1
% 0.29/0.55  thf(fact_3764_mult__le__cancel__right2,axiom,
% 0.29/0.55      ! [A: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.29/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right2
% 0.29/0.55  thf(fact_3765_mult__le__cancel__right2,axiom,
% 0.29/0.55      ! [A: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.29/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right2
% 0.29/0.55  thf(fact_3766_mult__le__cancel__right2,axiom,
% 0.29/0.55      ! [A: int,C: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.29/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right2
% 0.29/0.55  thf(fact_3767_mult__le__cancel__right1,axiom,
% 0.29/0.55      ! [C: real,B: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.29/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right1
% 0.29/0.55  thf(fact_3768_mult__le__cancel__right1,axiom,
% 0.29/0.55      ! [C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.29/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right1
% 0.29/0.55  thf(fact_3769_mult__le__cancel__right1,axiom,
% 0.29/0.55      ! [C: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.29/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right1
% 0.29/0.55  thf(fact_3770_mult__le__cancel__left2,axiom,
% 0.29/0.55      ! [C: real,A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.29/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left2
% 0.29/0.55  thf(fact_3771_mult__le__cancel__left2,axiom,
% 0.29/0.55      ! [C: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.29/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left2
% 0.29/0.55  thf(fact_3772_mult__le__cancel__left2,axiom,
% 0.29/0.55      ! [C: int,A: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.29/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left2
% 0.29/0.55  thf(fact_3773_mult__le__cancel__left1,axiom,
% 0.29/0.55      ! [C: real,B: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.29/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left1
% 0.29/0.55  thf(fact_3774_mult__le__cancel__left1,axiom,
% 0.29/0.55      ! [C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.29/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left1
% 0.29/0.55  thf(fact_3775_mult__le__cancel__left1,axiom,
% 0.29/0.55      ! [C: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.29/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left1
% 0.29/0.55  thf(fact_3776_divide__left__mono__neg,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.55       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.55           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_left_mono_neg
% 0.29/0.55  thf(fact_3777_divide__left__mono__neg,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.55       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.55           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_left_mono_neg
% 0.29/0.55  thf(fact_3778_mult__imp__le__div__pos,axiom,
% 0.29/0.55      ! [Y: real,Z: real,X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
% 0.29/0.55         => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_le_div_pos
% 0.29/0.55  thf(fact_3779_mult__imp__le__div__pos,axiom,
% 0.29/0.55      ! [Y: rat,Z: rat,X: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
% 0.29/0.55         => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_le_div_pos
% 0.29/0.55  thf(fact_3780_mult__imp__div__pos__le,axiom,
% 0.29/0.55      ! [Y: real,X: real,Z: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.55       => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
% 0.29/0.55         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_div_pos_le
% 0.29/0.55  thf(fact_3781_mult__imp__div__pos__le,axiom,
% 0.29/0.55      ! [Y: rat,X: rat,Z: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.29/0.55       => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
% 0.29/0.55         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_div_pos_le
% 0.29/0.55  thf(fact_3782_pos__le__divide__eq,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_le_divide_eq
% 0.29/0.55  thf(fact_3783_pos__le__divide__eq,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_le_divide_eq
% 0.29/0.55  thf(fact_3784_pos__divide__le__eq,axiom,
% 0.29/0.55      ! [C: real,B: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_divide_le_eq
% 0.29/0.55  thf(fact_3785_pos__divide__le__eq,axiom,
% 0.29/0.55      ! [C: rat,B: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_divide_le_eq
% 0.29/0.55  thf(fact_3786_neg__le__divide__eq,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_le_divide_eq
% 0.29/0.55  thf(fact_3787_neg__le__divide__eq,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_le_divide_eq
% 0.29/0.55  thf(fact_3788_neg__divide__le__eq,axiom,
% 0.29/0.55      ! [C: real,B: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_divide_le_eq
% 0.29/0.55  thf(fact_3789_neg__divide__le__eq,axiom,
% 0.29/0.55      ! [C: rat,B: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_divide_le_eq
% 0.29/0.55  thf(fact_3790_divide__left__mono,axiom,
% 0.29/0.55      ! [B: real,A: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ B @ A )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.55           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_left_mono
% 0.29/0.55  thf(fact_3791_divide__left__mono,axiom,
% 0.29/0.55      ! [B: rat,A: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.55           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_left_mono
% 0.29/0.55  thf(fact_3792_le__divide__eq,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_divide_eq
% 0.29/0.55  thf(fact_3793_le__divide__eq,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_divide_eq
% 0.29/0.55  thf(fact_3794_divide__le__eq,axiom,
% 0.29/0.55      ! [B: real,C: real,A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_le_eq
% 0.29/0.55  thf(fact_3795_divide__le__eq,axiom,
% 0.29/0.55      ! [B: rat,C: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_le_eq
% 0.29/0.55  thf(fact_3796_convex__bound__le,axiom,
% 0.29/0.55      ! [X: real,A: real,Y: real,U: real,V: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ X @ A )
% 0.29/0.55       => ( ( ord_less_eq_real @ Y @ A )
% 0.29/0.55         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.29/0.55           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.29/0.55             => ( ( ( plus_plus_real @ U @ V )
% 0.29/0.55                  = one_one_real )
% 0.29/0.55               => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_le
% 0.29/0.55  thf(fact_3797_convex__bound__le,axiom,
% 0.29/0.55      ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ X @ A )
% 0.29/0.55       => ( ( ord_less_eq_rat @ Y @ A )
% 0.29/0.55         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.29/0.55           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.29/0.55             => ( ( ( plus_plus_rat @ U @ V )
% 0.29/0.55                  = one_one_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_le
% 0.29/0.55  thf(fact_3798_convex__bound__le,axiom,
% 0.29/0.55      ! [X: int,A: int,Y: int,U: int,V: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ X @ A )
% 0.29/0.55       => ( ( ord_less_eq_int @ Y @ A )
% 0.29/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.29/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.29/0.55             => ( ( ( plus_plus_int @ U @ V )
% 0.29/0.55                  = one_one_int )
% 0.29/0.55               => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_le
% 0.29/0.55  thf(fact_3799_less__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W: num,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3800_less__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W: num,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3801_divide__less__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: real,C: real,W: num] :
% 0.29/0.55        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_less_eq_numeral(1)
% 0.29/0.55  thf(fact_3802_divide__less__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: rat,C: rat,W: num] :
% 0.29/0.55        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_less_eq_numeral(1)
% 0.29/0.55  thf(fact_3803_power__Suc__less,axiom,
% 0.29/0.55      ! [A: real,N: nat] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55       => ( ( ord_less_real @ A @ one_one_real )
% 0.29/0.55         => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc_less
% 0.29/0.55  thf(fact_3804_power__Suc__less,axiom,
% 0.29/0.55      ! [A: rat,N: nat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.55       => ( ( ord_less_rat @ A @ one_one_rat )
% 0.29/0.55         => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc_less
% 0.29/0.55  thf(fact_3805_power__Suc__less,axiom,
% 0.29/0.55      ! [A: nat,N: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.55       => ( ( ord_less_nat @ A @ one_one_nat )
% 0.29/0.55         => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc_less
% 0.29/0.55  thf(fact_3806_power__Suc__less,axiom,
% 0.29/0.55      ! [A: int,N: nat] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.55       => ( ( ord_less_int @ A @ one_one_int )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc_less
% 0.29/0.55  thf(fact_3807_mult__2,axiom,
% 0.29/0.55      ! [Z: complex] :
% 0.29/0.55        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 0.29/0.55        = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3808_mult__2,axiom,
% 0.29/0.55      ! [Z: real] :
% 0.29/0.55        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 0.29/0.55        = ( plus_plus_real @ Z @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3809_mult__2,axiom,
% 0.29/0.55      ! [Z: rat] :
% 0.29/0.55        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 0.29/0.55        = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3810_mult__2,axiom,
% 0.29/0.55      ! [Z: nat] :
% 0.29/0.55        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 0.29/0.55        = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3811_mult__2,axiom,
% 0.29/0.55      ! [Z: int] :
% 0.29/0.55        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 0.29/0.55        = ( plus_plus_int @ Z @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3812_mult__2__right,axiom,
% 0.29/0.55      ! [Z: complex] :
% 0.29/0.55        ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3813_mult__2__right,axiom,
% 0.29/0.55      ! [Z: real] :
% 0.29/0.55        ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_real @ Z @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3814_mult__2__right,axiom,
% 0.29/0.55      ! [Z: rat] :
% 0.29/0.55        ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3815_mult__2__right,axiom,
% 0.29/0.55      ! [Z: nat] :
% 0.29/0.55        ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3816_mult__2__right,axiom,
% 0.29/0.55      ! [Z: int] :
% 0.29/0.55        ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_int @ Z @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3817_left__add__twice,axiom,
% 0.29/0.55      ! [A: complex,B: complex] :
% 0.29/0.55        ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 0.29/0.55        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3818_left__add__twice,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.29/0.55        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3819_left__add__twice,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.55        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3820_left__add__twice,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.55        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3821_left__add__twice,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.29/0.55        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3822_power__diff,axiom,
% 0.29/0.55      ! [A: complex,N: nat,M: nat] :
% 0.29/0.55        ( ( A != zero_zero_complex )
% 0.29/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.55         => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.55            = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_diff
% 0.29/0.55  thf(fact_3823_power__diff,axiom,
% 0.29/0.55      ! [A: real,N: nat,M: nat] :
% 0.29/0.55        ( ( A != zero_zero_real )
% 0.29/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.55         => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.55            = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_diff
% 0.29/0.55  thf(fact_3824_power__diff,axiom,
% 0.29/0.55      ! [A: rat,N: nat,M: nat] :
% 0.29/0.55        ( ( A != zero_zero_rat )
% 0.29/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.55         => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.55            = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_diff
% 0.29/0.55  thf(fact_3825_power__diff,axiom,
% 0.29/0.55      ! [A: nat,N: nat,M: nat] :
% 0.29/0.55        ( ( A != zero_zero_nat )
% 0.29/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.55         => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.55            = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_diff
% 0.29/0.55  thf(fact_3826_power__diff,axiom,
% 0.29/0.55      ! [A: int,N: nat,M: nat] :
% 0.29/0.55        ( ( A != zero_zero_int )
% 0.29/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.55         => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.55            = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_diff
% 0.29/0.55  thf(fact_3827_power4__eq__xxxx,axiom,
% 0.29/0.55      ! [X: complex] :
% 0.29/0.55        ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.55        = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power4_eq_xxxx
% 0.29/0.55  thf(fact_3828_power4__eq__xxxx,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.55        = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power4_eq_xxxx
% 0.29/0.55  thf(fact_3829_power4__eq__xxxx,axiom,
% 0.29/0.55      ! [X: rat] :
% 0.29/0.55        ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.55        = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power4_eq_xxxx
% 0.29/0.55  thf(fact_3830_power4__eq__xxxx,axiom,
% 0.29/0.55      ! [X: nat] :
% 0.29/0.55        ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.55        = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power4_eq_xxxx
% 0.29/0.55  thf(fact_3831_power4__eq__xxxx,axiom,
% 0.29/0.55      ! [X: int] :
% 0.29/0.55        ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.55        = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power4_eq_xxxx
% 0.29/0.55  thf(fact_3832_power2__eq__square,axiom,
% 0.29/0.55      ! [A: complex] :
% 0.29/0.55        ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( times_times_complex @ A @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_eq_square
% 0.29/0.55  thf(fact_3833_power2__eq__square,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( times_times_real @ A @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_eq_square
% 0.29/0.55  thf(fact_3834_power2__eq__square,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( times_times_rat @ A @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_eq_square
% 0.29/0.55  thf(fact_3835_power2__eq__square,axiom,
% 0.29/0.55      ! [A: nat] :
% 0.29/0.55        ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( times_times_nat @ A @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_eq_square
% 0.29/0.55  thf(fact_3836_power2__eq__square,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( times_times_int @ A @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_eq_square
% 0.29/0.55  thf(fact_3837_of__int__ceiling__diff__one__le,axiom,
% 0.29/0.55      ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_ceiling_diff_one_le
% 0.29/0.55  thf(fact_3838_of__int__ceiling__diff__one__le,axiom,
% 0.29/0.55      ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_ceiling_diff_one_le
% 0.29/0.55  thf(fact_3839_power__even__eq,axiom,
% 0.29/0.55      ! [A: nat,N: nat] :
% 0.29/0.55        ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.55        = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_even_eq
% 0.29/0.55  thf(fact_3840_power__even__eq,axiom,
% 0.29/0.55      ! [A: real,N: nat] :
% 0.29/0.55        ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.55        = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_even_eq
% 0.29/0.55  thf(fact_3841_power__even__eq,axiom,
% 0.29/0.55      ! [A: int,N: nat] :
% 0.29/0.55        ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.55        = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_even_eq
% 0.29/0.55  thf(fact_3842_power__even__eq,axiom,
% 0.29/0.55      ! [A: complex,N: nat] :
% 0.29/0.55        ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.55        = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_even_eq
% 0.29/0.55  thf(fact_3843_Suc__double__not__eq__double,axiom,
% 0.29/0.55      ! [M: nat,N: nat] :
% 0.29/0.55        ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.55       != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Suc_double_not_eq_double
% 0.29/0.55  thf(fact_3844_double__not__eq__Suc__double,axiom,
% 0.29/0.55      ! [M: nat,N: nat] :
% 0.29/0.55        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.29/0.55       != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % double_not_eq_Suc_double
% 0.29/0.55  thf(fact_3845_Suc__pred_H,axiom,
% 0.29/0.55      ! [N: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55       => ( N
% 0.29/0.55          = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Suc_pred'
% 0.29/0.55  thf(fact_3846_Suc__diff__eq__diff__pred,axiom,
% 0.29/0.55      ! [N: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.29/0.55          = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Suc_diff_eq_diff_pred
% 0.29/0.55  thf(fact_3847_div__if,axiom,
% 0.29/0.55      ( divide_divide_nat
% 0.29/0.55      = ( ^ [M6: nat,N3: nat] :
% 0.29/0.55            ( if_nat
% 0.29/0.55            @ ( ( ord_less_nat @ M6 @ N3 )
% 0.29/0.55              | ( N3 = zero_zero_nat ) )
% 0.29/0.55            @ zero_zero_nat
% 0.29/0.55            @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_if
% 0.29/0.55  thf(fact_3848_div__geq,axiom,
% 0.29/0.55      ! [N: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55       => ( ~ ( ord_less_nat @ M @ N )
% 0.29/0.55         => ( ( divide_divide_nat @ M @ N )
% 0.29/0.55            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_geq
% 0.29/0.55  thf(fact_3849_add__eq__if,axiom,
% 0.29/0.55      ( plus_plus_nat
% 0.29/0.55      = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_eq_if
% 0.29/0.55  thf(fact_3850_div__nat__eqI,axiom,
% 0.29/0.55      ! [N: nat,Q3: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
% 0.29/0.55       => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
% 0.29/0.55         => ( ( divide_divide_nat @ M @ N )
% 0.29/0.55            = Q3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_nat_eqI
% 0.29/0.55  thf(fact_3851_less__eq__div__iff__mult__less__eq,axiom,
% 0.29/0.55      ! [Q3: nat,M: nat,N: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 0.29/0.55       => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
% 0.29/0.55          = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_eq_div_iff_mult_less_eq
% 0.29/0.55  thf(fact_3852_split__div,axiom,
% 0.29/0.55      ! [P: nat > $o,M: nat,N: nat] :
% 0.29/0.55        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.29/0.55        = ( ( ( N = zero_zero_nat )
% 0.29/0.55           => ( P @ zero_zero_nat ) )
% 0.29/0.55          & ( ( N != zero_zero_nat )
% 0.29/0.55           => ! [I3: nat,J2: nat] :
% 0.29/0.55                ( ( ord_less_nat @ J2 @ N )
% 0.29/0.55               => ( ( M
% 0.29/0.55                    = ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J2 ) )
% 0.29/0.55                 => ( P @ I3 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % split_div
% 0.29/0.55  thf(fact_3853_dividend__less__div__times,axiom,
% 0.29/0.55      ! [N: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % dividend_less_div_times
% 0.29/0.55  thf(fact_3854_dividend__less__times__div,axiom,
% 0.29/0.55      ! [N: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % dividend_less_times_div
% 0.29/0.55  thf(fact_3855_floor__split,axiom,
% 0.29/0.55      ! [P: int > $o,T: real] :
% 0.29/0.55        ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 0.29/0.55        = ( ! [I3: int] :
% 0.29/0.55              ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I3 ) @ T )
% 0.29/0.55                & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) ) )
% 0.29/0.55             => ( P @ I3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_split
% 0.29/0.55  thf(fact_3856_floor__split,axiom,
% 0.29/0.55      ! [P: int > $o,T: rat] :
% 0.29/0.55        ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 0.29/0.55        = ( ! [I3: int] :
% 0.29/0.55              ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I3 ) @ T )
% 0.29/0.55                & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) ) )
% 0.29/0.55             => ( P @ I3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_split
% 0.29/0.55  thf(fact_3857_floor__eq__iff,axiom,
% 0.29/0.55      ! [X: real,A: int] :
% 0.29/0.55        ( ( ( archim6058952711729229775r_real @ X )
% 0.29/0.55          = A )
% 0.29/0.55        = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
% 0.29/0.55          & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_eq_iff
% 0.29/0.55  thf(fact_3858_floor__eq__iff,axiom,
% 0.29/0.55      ! [X: rat,A: int] :
% 0.29/0.55        ( ( ( archim3151403230148437115or_rat @ X )
% 0.29/0.55          = A )
% 0.29/0.55        = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
% 0.29/0.55          & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_eq_iff
% 0.29/0.55  thf(fact_3859_floor__unique,axiom,
% 0.29/0.55      ! [Z: int,X: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
% 0.29/0.55       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
% 0.29/0.55         => ( ( archim6058952711729229775r_real @ X )
% 0.29/0.55            = Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_unique
% 0.29/0.55  thf(fact_3860_floor__unique,axiom,
% 0.29/0.55      ! [Z: int,X: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
% 0.29/0.55       => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
% 0.29/0.55         => ( ( archim3151403230148437115or_rat @ X )
% 0.29/0.55            = Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_unique
% 0.29/0.55  thf(fact_3861_less__floor__iff,axiom,
% 0.29/0.55      ! [Z: int,X: real] :
% 0.29/0.55        ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
% 0.29/0.55        = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_floor_iff
% 0.29/0.55  thf(fact_3862_less__floor__iff,axiom,
% 0.29/0.55      ! [Z: int,X: rat] :
% 0.29/0.55        ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
% 0.29/0.55        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_floor_iff
% 0.29/0.55  thf(fact_3863_floor__le__iff,axiom,
% 0.29/0.55      ! [X: real,Z: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
% 0.29/0.55        = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_le_iff
% 0.29/0.55  thf(fact_3864_floor__le__iff,axiom,
% 0.29/0.55      ! [X: rat,Z: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
% 0.29/0.55        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_le_iff
% 0.29/0.55  thf(fact_3865_floor__correct,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
% 0.29/0.55        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_correct
% 0.29/0.55  thf(fact_3866_floor__correct,axiom,
% 0.29/0.55      ! [X: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
% 0.29/0.55        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_correct
% 0.29/0.55  thf(fact_3867_le__log__iff,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
% 0.29/0.55            = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_log_iff
% 0.29/0.55  thf(fact_3868_log__le__iff,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
% 0.29/0.55            = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % log_le_iff
% 0.29/0.55  thf(fact_3869_le__powr__iff,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
% 0.29/0.55            = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_powr_iff
% 0.29/0.55  thf(fact_3870_powr__le__iff,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
% 0.29/0.55            = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % powr_le_iff
% 0.29/0.55  thf(fact_3871_floor__eq2,axiom,
% 0.29/0.55      ! [N: int,X: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.29/0.55       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.29/0.55         => ( ( archim6058952711729229775r_real @ X )
% 0.29/0.55            = N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_eq2
% 0.29/0.55  thf(fact_3872_floor__divide__real__eq__div,axiom,
% 0.29/0.55      ! [B: int,A: real] :
% 0.29/0.55        ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.55       => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 0.29/0.55          = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_divide_real_eq_div
% 0.29/0.55  thf(fact_3873_convex__bound__lt,axiom,
% 0.29/0.55      ! [X: real,A: real,Y: real,U: real,V: real] :
% 0.29/0.55        ( ( ord_less_real @ X @ A )
% 0.29/0.55       => ( ( ord_less_real @ Y @ A )
% 0.29/0.55         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.29/0.55           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.29/0.55             => ( ( ( plus_plus_real @ U @ V )
% 0.29/0.55                  = one_one_real )
% 0.29/0.55               => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_lt
% 0.29/0.55  thf(fact_3874_convex__bound__lt,axiom,
% 0.29/0.55      ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 0.29/0.55        ( ( ord_less_rat @ X @ A )
% 0.29/0.55       => ( ( ord_less_rat @ Y @ A )
% 0.29/0.55         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.29/0.55           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.29/0.55             => ( ( ( plus_plus_rat @ U @ V )
% 0.29/0.55                  = one_one_rat )
% 0.29/0.55               => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_lt
% 0.29/0.55  thf(fact_3875_convex__bound__lt,axiom,
% 0.29/0.55      ! [X: int,A: int,Y: int,U: int,V: int] :
% 0.29/0.55        ( ( ord_less_int @ X @ A )
% 0.29/0.55       => ( ( ord_less_int @ Y @ A )
% 0.29/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.29/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.29/0.55             => ( ( ( plus_plus_int @ U @ V )
% 0.29/0.55                  = one_one_int )
% 0.29/0.55               => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_lt
% 0.29/0.55  thf(fact_3876_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.29/0.55      ! [N: nat,M: nat] :
% 0.29/0.55        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.55         != zero_zero_nat )
% 0.29/0.55       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.29/0.55         != zero_zero_nat ) ) ).
% 0.29/0.55  
% 0.29/0.55  % exp_not_zero_imp_exp_diff_not_zero
% 0.29/0.55  thf(fact_3877_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.29/0.55      ! [N: nat,M: nat] :
% 0.29/0.55        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.29/0.55         != zero_zero_int )
% 0.29/0.55       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.29/0.55         != zero_zero_int ) ) ).
% 0.29/0.55  
% 0.29/0.55  % exp_not_zero_imp_exp_diff_not_zero
% 0.29/0.55  thf(fact_3878_le__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W: num,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3879_le__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W: num,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3880_divide__le__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: real,C: real,W: num] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_le_eq_numeral(1)
% 0.29/0.55  thf(fact_3881_divide__le__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: rat,C: rat,W: num] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_le_eq_numeral(1)
% 0.29/0.55  thf(fact_3882_round__unique,axiom,
% 0.29/0.55      ! [X: real,Y: int] :
% 0.29/0.55        ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.29/0.55         => ( ( archim8280529875227126926d_real @ X )
% 0.29/0.55            = Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % round_unique
% 0.29/0.55  thf(fact_3883_round__unique,axiom,
% 0.29/0.55      ! [X: rat,Y: int] :
% 0.29/0.55        ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.55         => ( ( archim7778729529865785530nd_rat @ X )
% 0.29/0.55            = Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % round_unique
% 0.29/0.55  thf(fact_3884_power__diff__power__eq,axiom,
% 0.29/0.55      ! [A: nat,N: nat,M: nat] :
% 0.29/0.55        ( ( A != zero_zero_nat )
% 0.29/0.55       => ( ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.55           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.29/0.55              = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.29/0.55          & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.29/0.55           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.29/0.55              = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_diff_power_eq
% 0.29/0.55  thf(fact_3885_power__diff__power__eq,axiom,
% 0.29/0.55      ! [A: int,N: nat,M: nat] :
% 0.29/0.55        ( ( A != zero_zero_int )
% 0.29/0.55       => ( ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.55           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.29/0.55              = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.29/0.55          & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.29/0.55           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.29/0.55              = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_diff_power_eq
% 0.29/0.55  thf(fact_3886_ceiling__correct,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
% 0.29/0.55        & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_correct
% 0.29/0.55  thf(fact_3887_ceiling__correct,axiom,
% 0.29/0.55      ! [X: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
% 0.29/0.55        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_correct
% 0.29/0.55  thf(fact_3888_ceiling__unique,axiom,
% 0.29/0.55      ! [Z: int,X: real] :
% 0.29/0.55        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
% 0.29/0.55       => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
% 0.29/0.55         => ( ( archim7802044766580827645g_real @ X )
% 0.29/0.55            = Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_unique
% 0.29/0.55  thf(fact_3889_ceiling__unique,axiom,
% 0.29/0.55      ! [Z: int,X: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
% 0.29/0.55       => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
% 0.29/0.55         => ( ( archim2889992004027027881ng_rat @ X )
% 0.29/0.55            = Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_unique
% 0.29/0.55  thf(fact_3890_ceiling__eq__iff,axiom,
% 0.29/0.55      ! [X: real,A: int] :
% 0.29/0.55        ( ( ( archim7802044766580827645g_real @ X )
% 0.29/0.55          = A )
% 0.29/0.55        = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
% 0.29/0.55          & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_eq_iff
% 0.29/0.55  thf(fact_3891_ceiling__eq__iff,axiom,
% 0.29/0.55      ! [X: rat,A: int] :
% 0.29/0.55        ( ( ( archim2889992004027027881ng_rat @ X )
% 0.29/0.55          = A )
% 0.29/0.55        = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
% 0.29/0.55          & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_eq_iff
% 0.29/0.55  thf(fact_3892_ceiling__split,axiom,
% 0.29/0.55      ! [P: int > $o,T: real] :
% 0.29/0.55        ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 0.29/0.55        = ( ! [I3: int] :
% 0.29/0.55              ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) @ T )
% 0.29/0.55                & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I3 ) ) )
% 0.29/0.55             => ( P @ I3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_split
% 0.29/0.55  thf(fact_3893_ceiling__split,axiom,
% 0.29/0.55      ! [P: int > $o,T: rat] :
% 0.29/0.55        ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 0.29/0.55        = ( ! [I3: int] :
% 0.29/0.55              ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T )
% 0.29/0.55                & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I3 ) ) )
% 0.29/0.55             => ( P @ I3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_split
% 0.29/0.55  thf(fact_3894_ceiling__less__iff,axiom,
% 0.29/0.55      ! [X: real,Z: int] :
% 0.29/0.55        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
% 0.29/0.55        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_less_iff
% 0.29/0.55  thf(fact_3895_ceiling__less__iff,axiom,
% 0.29/0.55      ! [X: rat,Z: int] :
% 0.29/0.55        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
% 0.29/0.55        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_less_iff
% 0.29/0.55  thf(fact_3896_diff__le__diff__pow,axiom,
% 0.29/0.55      ! [K: nat,M: nat,N: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.29/0.55       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % diff_le_diff_pow
% 0.29/0.55  thf(fact_3897_le__ceiling__iff,axiom,
% 0.29/0.55      ! [Z: int,X: real] :
% 0.29/0.55        ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
% 0.29/0.55        = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_ceiling_iff
% 0.29/0.55  thf(fact_3898_le__ceiling__iff,axiom,
% 0.29/0.55      ! [Z: int,X: rat] :
% 0.29/0.55        ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
% 0.29/0.55        = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_ceiling_iff
% 0.29/0.55  thf(fact_3899_le__div__geq,axiom,
% 0.29/0.55      ! [N: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.55         => ( ( divide_divide_nat @ M @ N )
% 0.29/0.55            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_div_geq
% 0.29/0.55  thf(fact_3900_split__div_H,axiom,
% 0.29/0.55      ! [P: nat > $o,M: nat,N: nat] :
% 0.29/0.55        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.29/0.55        = ( ( ( N = zero_zero_nat )
% 0.29/0.55            & ( P @ zero_zero_nat ) )
% 0.29/0.55          | ? [Q4: nat] :
% 0.29/0.55              ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
% 0.29/0.55              & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
% 0.29/0.55              & ( P @ Q4 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % split_div'
% 0.29/0.55  thf(fact_3901_power2__sum,axiom,
% 0.29/0.55      ! [X: complex,Y: complex] :
% 0.29/0.55        ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_sum
% 0.29/0.55  thf(fact_3902_power2__sum,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_sum
% 0.29/0.55  thf(fact_3903_power2__sum,axiom,
% 0.29/0.55      ! [X: rat,Y: rat] :
% 0.29/0.55        ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_sum
% 0.29/0.55  thf(fact_3904_power2__sum,axiom,
% 0.29/0.55      ! [X: nat,Y: nat] :
% 0.29/0.55        ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_sum
% 0.29/0.55  thf(fact_3905_power2__sum,axiom,
% 0.29/0.55      ! [X: int,Y: int] :
% 0.29/0.55        ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power2_sum
% 0.29/0.55  thf(fact_3906_zero__le__even__power_H,axiom,
% 0.29/0.55      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_le_even_power'
% 0.29/0.55  thf(fact_3907_zero__le__even__power_H,axiom,
% 0.29/0.55      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_le_even_power'
% 0.29/0.55  thf(fact_3908_zero__le__even__power_H,axiom,
% 0.29/0.55      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_le_even_power'
% 0.29/0.55  thf(fact_3909_nat__bit__induct,axiom,
% 0.29/0.55      ! [P: nat > $o,N: nat] :
% 0.29/0.55        ( ( P @ zero_zero_nat )
% 0.29/0.55       => ( ! [N2: nat] :
% 0.29/0.55              ( ( P @ N2 )
% 0.29/0.55             => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.55               => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.29/0.55         => ( ! [N2: nat] :
% 0.29/0.55                ( ( P @ N2 )
% 0.29/0.55               => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.29/0.55           => ( P @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nat_bit_induct
% 0.29/0.55  thf(fact_3910_ceiling__divide__upper,axiom,
% 0.29/0.55      ! [Q3: real,P5: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ Q3 )
% 0.29/0.55       => ( ord_less_eq_real @ P5 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_divide_upper
% 0.29/0.55  thf(fact_3911_ceiling__divide__upper,axiom,
% 0.29/0.55      ! [Q3: rat,P5: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 0.29/0.55       => ( ord_less_eq_rat @ P5 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_divide_upper
% 0.29/0.55  thf(fact_3912_sum__squares__bound,axiom,
% 0.29/0.55      ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_bound
% 0.29/0.55  thf(fact_3913_sum__squares__bound,axiom,
% 0.29/0.55      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_bound
% 0.29/0.55  thf(fact_3914_odd__0__le__power__imp__0__le,axiom,
% 0.29/0.55      ! [A: real,N: nat] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.29/0.55       => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % odd_0_le_power_imp_0_le
% 0.29/0.55  thf(fact_3915_odd__0__le__power__imp__0__le,axiom,
% 0.29/0.55      ! [A: rat,N: nat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.29/0.55       => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % odd_0_le_power_imp_0_le
% 0.29/0.55  thf(fact_3916_odd__0__le__power__imp__0__le,axiom,
% 0.29/0.55      ! [A: int,N: nat] :
% 0.29/0.55        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.29/0.55       => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % odd_0_le_power_imp_0_le
% 0.29/0.55  thf(fact_3917_odd__power__less__zero,axiom,
% 0.29/0.55      ! [A: real,N: nat] :
% 0.29/0.55        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.55       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % odd_power_less_zero
% 0.29/0.55  thf(fact_3918_odd__power__less__zero,axiom,
% 0.29/0.55      ! [A: rat,N: nat] :
% 0.29/0.55        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.55       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 0.29/0.55  
% 0.29/0.55  % odd_power_less_zero
% 0.29/0.55  thf(fact_3919_odd__power__less__zero,axiom,
% 0.29/0.55      ! [A: int,N: nat] :
% 0.29/0.55        ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.55       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 0.29/0.55  
% 0.29/0.55  % odd_power_less_zero
% 0.29/0.55  thf(fact_3920_int__power__div__base,axiom,
% 0.29/0.55      ! [M: nat,K: int] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.55         => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.29/0.55            = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_power_div_base
% 0.29/0.55  thf(fact_3921_arith__geo__mean,axiom,
% 0.29/0.55      ! [U: real,X: real,Y: real] :
% 0.29/0.55        ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55          = ( times_times_real @ X @ Y ) )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.55           => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % arith_geo_mean
% 0.29/0.55  thf(fact_3922_arith__geo__mean,axiom,
% 0.29/0.55      ! [U: rat,X: rat,Y: rat] :
% 0.29/0.55        ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55          = ( times_times_rat @ X @ Y ) )
% 0.29/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.29/0.55         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.29/0.55           => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % arith_geo_mean
% 0.29/0.55  thf(fact_3923_floor__log2__div2,axiom,
% 0.29/0.55      ! [N: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.55       => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.29/0.55          = ( 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 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_log2_div2
% 0.29/0.55  thf(fact_3924_floor__log__nat__eq__if,axiom,
% 0.29/0.55      ! [B: nat,N: nat,K: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.29/0.55       => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.29/0.55         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.29/0.55           => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.29/0.55              = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % floor_log_nat_eq_if
% 0.29/0.55  thf(fact_3925_low__inv,axiom,
% 0.29/0.55      ! [X: nat,N: nat,Y: nat] :
% 0.29/0.55        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.55       => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 0.29/0.55          = X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % low_inv
% 0.29/0.55  thf(fact_3926_set__bit__0,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 0.29/0.55        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % set_bit_0
% 0.29/0.55  thf(fact_3927_set__bit__0,axiom,
% 0.29/0.55      ! [A: nat] :
% 0.29/0.55        ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 0.29/0.55        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % set_bit_0
% 0.29/0.55  thf(fact_3928_artanh__def,axiom,
% 0.29/0.55      ( artanh_real
% 0.29/0.55      = ( ^ [X4: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % artanh_def
% 0.29/0.55  thf(fact_3929_unset__bit__0,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 0.29/0.55        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % unset_bit_0
% 0.29/0.55  thf(fact_3930_unset__bit__0,axiom,
% 0.29/0.55      ! [A: nat] :
% 0.29/0.55        ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 0.29/0.55        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % unset_bit_0
% 0.29/0.55  thf(fact_3931_lemma__termdiff3,axiom,
% 0.29/0.55      ! [H: real,Z: real,K4: real,N: nat] :
% 0.29/0.55        ( ( H != zero_zero_real )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K4 )
% 0.29/0.55         => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H ) ) @ K4 )
% 0.29/0.55           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H ) @ ( 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 @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % lemma_termdiff3
% 0.29/0.55  thf(fact_3932_lemma__termdiff3,axiom,
% 0.29/0.55      ! [H: complex,Z: complex,K4: real,N: nat] :
% 0.29/0.55        ( ( H != zero_zero_complex )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K4 )
% 0.29/0.55         => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H ) ) @ K4 )
% 0.29/0.55           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H ) @ ( 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 @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % lemma_termdiff3
% 0.29/0.55  thf(fact_3933_both__member__options__def,axiom,
% 0.29/0.55      ( vEBT_V8194947554948674370ptions
% 0.29/0.55      = ( ^ [T2: vEBT_VEBT,X4: nat] :
% 0.29/0.55            ( ( vEBT_V5719532721284313246member @ T2 @ X4 )
% 0.29/0.55            | ( vEBT_VEBT_membermima @ T2 @ X4 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % both_member_options_def
% 0.29/0.55  thf(fact_3934_mult__le__cancel__iff2,axiom,
% 0.29/0.55      ! [Z: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ Z )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
% 0.29/0.55          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_iff2
% 0.29/0.55  thf(fact_3935_mult__le__cancel__iff2,axiom,
% 0.29/0.55      ! [Z: rat,X: rat,Y: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
% 0.29/0.55          = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_iff2
% 0.29/0.55  thf(fact_3936_mult__le__cancel__iff2,axiom,
% 0.29/0.55      ! [Z: int,X: int,Y: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.29/0.55       => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
% 0.29/0.55          = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_iff2
% 0.29/0.55  thf(fact_3937_mult__le__cancel__iff1,axiom,
% 0.29/0.55      ! [Z: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ Z )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
% 0.29/0.55          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_iff1
% 0.29/0.55  thf(fact_3938_mult__le__cancel__iff1,axiom,
% 0.29/0.55      ! [Z: rat,X: rat,Y: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 0.29/0.55          = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_iff1
% 0.29/0.55  thf(fact_3939_mult__le__cancel__iff1,axiom,
% 0.29/0.55      ! [Z: int,X: int,Y: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.29/0.55       => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 0.29/0.55          = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_iff1
% 0.29/0.55  thf(fact_3940_buildup__nothing__in__min__max,axiom,
% 0.29/0.55      ! [N: nat,X: nat] :
% 0.29/0.55        ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 0.29/0.55  
% 0.29/0.55  % buildup_nothing_in_min_max
% 0.29/0.55  thf(fact_3941_Diff__insert0,axiom,
% 0.29/0.55      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.29/0.55        ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 0.29/0.55       => ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B2 ) )
% 0.29/0.55          = ( minus_5127226145743854075T_VEBT @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_insert0
% 0.29/0.55  thf(fact_3942_Diff__insert0,axiom,
% 0.29/0.55      ! [X: $o,A2: set_o,B2: set_o] :
% 0.29/0.55        ( ~ ( member_o @ X @ A2 )
% 0.29/0.55       => ( ( minus_minus_set_o @ A2 @ ( insert_o @ X @ B2 ) )
% 0.29/0.55          = ( minus_minus_set_o @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_insert0
% 0.29/0.55  thf(fact_3943_Diff__insert0,axiom,
% 0.29/0.55      ! [X: complex,A2: set_complex,B2: set_complex] :
% 0.29/0.55        ( ~ ( member_complex @ X @ A2 )
% 0.29/0.55       => ( ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ B2 ) )
% 0.29/0.55          = ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_insert0
% 0.29/0.55  thf(fact_3944_Diff__insert0,axiom,
% 0.29/0.55      ! [X: real,A2: set_real,B2: set_real] :
% 0.29/0.55        ( ~ ( member_real @ X @ A2 )
% 0.29/0.55       => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 0.29/0.55          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_insert0
% 0.29/0.55  thf(fact_3945_Diff__insert0,axiom,
% 0.29/0.55      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.29/0.55        ( ~ ( member_set_nat @ X @ A2 )
% 0.29/0.55       => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 0.29/0.55          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_insert0
% 0.29/0.55  thf(fact_3946_Diff__insert0,axiom,
% 0.29/0.55      ! [X: int,A2: set_int,B2: set_int] :
% 0.29/0.55        ( ~ ( member_int @ X @ A2 )
% 0.29/0.55       => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 0.29/0.55          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_insert0
% 0.29/0.55  thf(fact_3947_Diff__insert0,axiom,
% 0.29/0.55      ! [X: nat,A2: set_nat,B2: set_nat] :
% 0.29/0.55        ( ~ ( member_nat @ X @ A2 )
% 0.29/0.55       => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 0.29/0.55          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_insert0
% 0.29/0.55  thf(fact_3948_insert__Diff1,axiom,
% 0.29/0.55      ! [X: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.29/0.55        ( ( member_VEBT_VEBT @ X @ B2 )
% 0.29/0.55       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B2 )
% 0.29/0.55          = ( minus_5127226145743854075T_VEBT @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff1
% 0.29/0.55  thf(fact_3949_insert__Diff1,axiom,
% 0.29/0.55      ! [X: $o,B2: set_o,A2: set_o] :
% 0.29/0.55        ( ( member_o @ X @ B2 )
% 0.29/0.55       => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
% 0.29/0.55          = ( minus_minus_set_o @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff1
% 0.29/0.55  thf(fact_3950_insert__Diff1,axiom,
% 0.29/0.55      ! [X: complex,B2: set_complex,A2: set_complex] :
% 0.29/0.55        ( ( member_complex @ X @ B2 )
% 0.29/0.55       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
% 0.29/0.55          = ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff1
% 0.29/0.55  thf(fact_3951_insert__Diff1,axiom,
% 0.29/0.55      ! [X: real,B2: set_real,A2: set_real] :
% 0.29/0.55        ( ( member_real @ X @ B2 )
% 0.29/0.55       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 0.29/0.55          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff1
% 0.29/0.55  thf(fact_3952_insert__Diff1,axiom,
% 0.29/0.55      ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 0.29/0.55        ( ( member_set_nat @ X @ B2 )
% 0.29/0.55       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 0.29/0.55          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff1
% 0.29/0.55  thf(fact_3953_insert__Diff1,axiom,
% 0.29/0.55      ! [X: int,B2: set_int,A2: set_int] :
% 0.29/0.55        ( ( member_int @ X @ B2 )
% 0.29/0.55       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 0.29/0.55          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff1
% 0.29/0.55  thf(fact_3954_insert__Diff1,axiom,
% 0.29/0.55      ! [X: nat,B2: set_nat,A2: set_nat] :
% 0.29/0.55        ( ( member_nat @ X @ B2 )
% 0.29/0.55       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 0.29/0.55          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff1
% 0.29/0.55  thf(fact_3955_idiff__0,axiom,
% 0.29/0.55      ! [N: extended_enat] :
% 0.29/0.55        ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.29/0.55        = zero_z5237406670263579293d_enat ) ).
% 0.29/0.55  
% 0.29/0.55  % idiff_0
% 0.29/0.55  thf(fact_3956_idiff__0__right,axiom,
% 0.29/0.55      ! [N: extended_enat] :
% 0.29/0.55        ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.29/0.55        = N ) ).
% 0.29/0.55  
% 0.29/0.55  % idiff_0_right
% 0.29/0.55  thf(fact_3957_ln__less__cancel__iff,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.55         => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 0.29/0.55            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_less_cancel_iff
% 0.29/0.55  thf(fact_3958_ln__inj__iff,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.55         => ( ( ( ln_ln_real @ X )
% 0.29/0.55              = ( ln_ln_real @ Y ) )
% 0.29/0.55            = ( X = Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_inj_iff
% 0.29/0.55  thf(fact_3959_unset__bit__nonnegative__int__iff,axiom,
% 0.29/0.55      ! [N: nat,K: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 0.29/0.55        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.29/0.55  
% 0.29/0.55  % unset_bit_nonnegative_int_iff
% 0.29/0.55  thf(fact_3960_set__bit__nonnegative__int__iff,axiom,
% 0.29/0.55      ! [N: nat,K: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 0.29/0.55        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.29/0.55  
% 0.29/0.55  % set_bit_nonnegative_int_iff
% 0.29/0.55  thf(fact_3961_unset__bit__negative__int__iff,axiom,
% 0.29/0.55      ! [N: nat,K: int] :
% 0.29/0.55        ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 0.29/0.55        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.29/0.55  
% 0.29/0.55  % unset_bit_negative_int_iff
% 0.29/0.55  thf(fact_3962_set__bit__negative__int__iff,axiom,
% 0.29/0.55      ! [N: nat,K: int] :
% 0.29/0.55        ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 0.29/0.55        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.29/0.55  
% 0.29/0.55  % set_bit_negative_int_iff
% 0.29/0.55  thf(fact_3963_real__divide__square__eq,axiom,
% 0.29/0.55      ! [R2: real,A: real] :
% 0.29/0.55        ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
% 0.29/0.55        = ( divide_divide_real @ A @ R2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % real_divide_square_eq
% 0.29/0.55  thf(fact_3964_semiring__norm_I13_J,axiom,
% 0.29/0.55      ! [M: num,N: num] :
% 0.29/0.55        ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.29/0.55        = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % semiring_norm(13)
% 0.29/0.55  thf(fact_3965_semiring__norm_I11_J,axiom,
% 0.29/0.55      ! [M: num] :
% 0.29/0.55        ( ( times_times_num @ M @ one )
% 0.29/0.55        = M ) ).
% 0.29/0.55  
% 0.29/0.55  % semiring_norm(11)
% 0.29/0.55  thf(fact_3966_semiring__norm_I12_J,axiom,
% 0.29/0.55      ! [N: num] :
% 0.29/0.55        ( ( times_times_num @ one @ N )
% 0.29/0.55        = N ) ).
% 0.29/0.55  
% 0.29/0.55  % semiring_norm(12)
% 0.29/0.55  thf(fact_3967_ln__le__cancel__iff,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.55         => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 0.29/0.55            = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_le_cancel_iff
% 0.29/0.55  thf(fact_3968_ln__less__zero__iff,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.29/0.55          = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_less_zero_iff
% 0.29/0.55  thf(fact_3969_ln__gt__zero__iff,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.29/0.55          = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_gt_zero_iff
% 0.29/0.55  thf(fact_3970_ln__eq__zero__iff,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ( ln_ln_real @ X )
% 0.29/0.55            = zero_zero_real )
% 0.29/0.55          = ( X = one_one_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_eq_zero_iff
% 0.29/0.55  thf(fact_3971_not__real__square__gt__zero,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 0.29/0.55        = ( X = zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % not_real_square_gt_zero
% 0.29/0.55  thf(fact_3972_ln__one,axiom,
% 0.29/0.55      ( ( ln_ln_real @ one_one_real )
% 0.29/0.55      = zero_zero_real ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_one
% 0.29/0.55  thf(fact_3973_num__double,axiom,
% 0.29/0.55      ! [N: num] :
% 0.29/0.55        ( ( times_times_num @ ( bit0 @ one ) @ N )
% 0.29/0.55        = ( bit0 @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % num_double
% 0.29/0.55  thf(fact_3974_power__mult__numeral,axiom,
% 0.29/0.55      ! [A: nat,M: num,N: num] :
% 0.29/0.55        ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.55        = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_mult_numeral
% 0.29/0.55  thf(fact_3975_power__mult__numeral,axiom,
% 0.29/0.55      ! [A: real,M: num,N: num] :
% 0.29/0.55        ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.55        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_mult_numeral
% 0.29/0.55  thf(fact_3976_power__mult__numeral,axiom,
% 0.29/0.55      ! [A: int,M: num,N: num] :
% 0.29/0.55        ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.55        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_mult_numeral
% 0.29/0.55  thf(fact_3977_power__mult__numeral,axiom,
% 0.29/0.55      ! [A: complex,M: num,N: num] :
% 0.29/0.55        ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.55        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_mult_numeral
% 0.29/0.55  thf(fact_3978_ln__le__zero__iff,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.29/0.55          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_le_zero_iff
% 0.29/0.55  thf(fact_3979_ln__ge__zero__iff,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.29/0.55          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_ge_zero_iff
% 0.29/0.55  thf(fact_3980_zle__diff1__eq,axiom,
% 0.29/0.55      ! [W: int,Z: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 0.29/0.55        = ( ord_less_int @ W @ Z ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zle_diff1_eq
% 0.29/0.55  thf(fact_3981_int__distrib_I4_J,axiom,
% 0.29/0.55      ! [W: int,Z1: int,Z22: int] :
% 0.29/0.55        ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 0.29/0.55        = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_distrib(4)
% 0.29/0.55  thf(fact_3982_int__distrib_I3_J,axiom,
% 0.29/0.55      ! [Z1: int,Z22: int,W: int] :
% 0.29/0.55        ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 0.29/0.55        = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_distrib(3)
% 0.29/0.55  thf(fact_3983_ln__powr,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( X != zero_zero_real )
% 0.29/0.55       => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
% 0.29/0.55          = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_powr
% 0.29/0.55  thf(fact_3984_double__diff,axiom,
% 0.29/0.55      ! [A2: set_nat,B2: set_nat,C6: set_nat] :
% 0.29/0.55        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.55       => ( ( ord_less_eq_set_nat @ B2 @ C6 )
% 0.29/0.55         => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C6 @ A2 ) )
% 0.29/0.55            = A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % double_diff
% 0.29/0.55  thf(fact_3985_double__diff,axiom,
% 0.29/0.55      ! [A2: set_int,B2: set_int,C6: set_int] :
% 0.29/0.55        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.55       => ( ( ord_less_eq_set_int @ B2 @ C6 )
% 0.29/0.55         => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C6 @ A2 ) )
% 0.29/0.55            = A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % double_diff
% 0.29/0.55  thf(fact_3986_Diff__subset,axiom,
% 0.29/0.55      ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_subset
% 0.29/0.55  thf(fact_3987_Diff__subset,axiom,
% 0.29/0.55      ! [A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ A2 ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_subset
% 0.29/0.55  thf(fact_3988_Diff__mono,axiom,
% 0.29/0.55      ! [A2: set_nat,C6: set_nat,D2: set_nat,B2: set_nat] :
% 0.29/0.55        ( ( ord_less_eq_set_nat @ A2 @ C6 )
% 0.29/0.55       => ( ( ord_less_eq_set_nat @ D2 @ B2 )
% 0.29/0.55         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C6 @ D2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_mono
% 0.29/0.55  thf(fact_3989_Diff__mono,axiom,
% 0.29/0.55      ! [A2: set_int,C6: set_int,D2: set_int,B2: set_int] :
% 0.29/0.55        ( ( ord_less_eq_set_int @ A2 @ C6 )
% 0.29/0.55       => ( ( ord_less_eq_set_int @ D2 @ B2 )
% 0.29/0.55         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( minus_minus_set_int @ C6 @ D2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_mono
% 0.29/0.55  thf(fact_3990_insert__Diff__if,axiom,
% 0.29/0.55      ! [X: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.29/0.55        ( ( ( member_VEBT_VEBT @ X @ B2 )
% 0.29/0.55         => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( minus_5127226145743854075T_VEBT @ A2 @ B2 ) ) )
% 0.29/0.55        & ( ~ ( member_VEBT_VEBT @ X @ B2 )
% 0.29/0.55         => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( insert_VEBT_VEBT @ X @ ( minus_5127226145743854075T_VEBT @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff_if
% 0.29/0.55  thf(fact_3991_insert__Diff__if,axiom,
% 0.29/0.55      ! [X: $o,B2: set_o,A2: set_o] :
% 0.29/0.55        ( ( ( member_o @ X @ B2 )
% 0.29/0.55         => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( minus_minus_set_o @ A2 @ B2 ) ) )
% 0.29/0.55        & ( ~ ( member_o @ X @ B2 )
% 0.29/0.55         => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( insert_o @ X @ ( minus_minus_set_o @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff_if
% 0.29/0.55  thf(fact_3992_insert__Diff__if,axiom,
% 0.29/0.55      ! [X: complex,B2: set_complex,A2: set_complex] :
% 0.29/0.55        ( ( ( member_complex @ X @ B2 )
% 0.29/0.55         => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( minus_811609699411566653omplex @ A2 @ B2 ) ) )
% 0.29/0.55        & ( ~ ( member_complex @ X @ B2 )
% 0.29/0.55         => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( insert_complex @ X @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff_if
% 0.29/0.55  thf(fact_3993_insert__Diff__if,axiom,
% 0.29/0.55      ! [X: real,B2: set_real,A2: set_real] :
% 0.29/0.55        ( ( ( member_real @ X @ B2 )
% 0.29/0.55         => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( minus_minus_set_real @ A2 @ B2 ) ) )
% 0.29/0.55        & ( ~ ( member_real @ X @ B2 )
% 0.29/0.55         => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff_if
% 0.29/0.55  thf(fact_3994_insert__Diff__if,axiom,
% 0.29/0.55      ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 0.29/0.55        ( ( ( member_set_nat @ X @ B2 )
% 0.29/0.55         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
% 0.29/0.55        & ( ~ ( member_set_nat @ X @ B2 )
% 0.29/0.55         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff_if
% 0.29/0.55  thf(fact_3995_insert__Diff__if,axiom,
% 0.29/0.55      ! [X: int,B2: set_int,A2: set_int] :
% 0.29/0.55        ( ( ( member_int @ X @ B2 )
% 0.29/0.55         => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( minus_minus_set_int @ A2 @ B2 ) ) )
% 0.29/0.55        & ( ~ ( member_int @ X @ B2 )
% 0.29/0.55         => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff_if
% 0.29/0.55  thf(fact_3996_insert__Diff__if,axiom,
% 0.29/0.55      ! [X: nat,B2: set_nat,A2: set_nat] :
% 0.29/0.55        ( ( ( member_nat @ X @ B2 )
% 0.29/0.55         => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( minus_minus_set_nat @ A2 @ B2 ) ) )
% 0.29/0.55        & ( ~ ( member_nat @ X @ B2 )
% 0.29/0.55         => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 0.29/0.55            = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % insert_Diff_if
% 0.29/0.55  thf(fact_3997_minus__int__code_I1_J,axiom,
% 0.29/0.55      ! [K: int] :
% 0.29/0.55        ( ( minus_minus_int @ K @ zero_zero_int )
% 0.29/0.55        = K ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_int_code(1)
% 0.29/0.55  thf(fact_3998_times__int__code_I2_J,axiom,
% 0.29/0.55      ! [L: int] :
% 0.29/0.55        ( ( times_times_int @ zero_zero_int @ L )
% 0.29/0.55        = zero_zero_int ) ).
% 0.29/0.55  
% 0.29/0.55  % times_int_code(2)
% 0.29/0.55  thf(fact_3999_times__int__code_I1_J,axiom,
% 0.29/0.55      ! [K: int] :
% 0.29/0.55        ( ( times_times_int @ K @ zero_zero_int )
% 0.29/0.55        = zero_zero_int ) ).
% 0.29/0.55  
% 0.29/0.55  % times_int_code(1)
% 0.29/0.55  thf(fact_4000_int__diff__cases,axiom,
% 0.29/0.55      ! [Z: int] :
% 0.29/0.55        ~ ! [M5: nat,N2: nat] :
% 0.29/0.55            ( Z
% 0.29/0.55           != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_diff_cases
% 0.29/0.55  thf(fact_4001_psubset__imp__ex__mem,axiom,
% 0.29/0.55      ! [A2: set_complex,B2: set_complex] :
% 0.29/0.55        ( ( ord_less_set_complex @ A2 @ B2 )
% 0.29/0.55       => ? [B6: complex] : ( member_complex @ B6 @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % psubset_imp_ex_mem
% 0.29/0.55  thf(fact_4002_psubset__imp__ex__mem,axiom,
% 0.29/0.55      ! [A2: set_real,B2: set_real] :
% 0.29/0.55        ( ( ord_less_set_real @ A2 @ B2 )
% 0.29/0.55       => ? [B6: real] : ( member_real @ B6 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % psubset_imp_ex_mem
% 0.29/0.55  thf(fact_4003_psubset__imp__ex__mem,axiom,
% 0.29/0.55      ! [A2: set_set_nat,B2: set_set_nat] :
% 0.29/0.55        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 0.29/0.55       => ? [B6: set_nat] : ( member_set_nat @ B6 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % psubset_imp_ex_mem
% 0.29/0.55  thf(fact_4004_psubset__imp__ex__mem,axiom,
% 0.29/0.55      ! [A2: set_int,B2: set_int] :
% 0.29/0.55        ( ( ord_less_set_int @ A2 @ B2 )
% 0.29/0.55       => ? [B6: int] : ( member_int @ B6 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % psubset_imp_ex_mem
% 0.29/0.55  thf(fact_4005_psubset__imp__ex__mem,axiom,
% 0.29/0.55      ! [A2: set_nat,B2: set_nat] :
% 0.29/0.55        ( ( ord_less_set_nat @ A2 @ B2 )
% 0.29/0.55       => ? [B6: nat] : ( member_nat @ B6 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % psubset_imp_ex_mem
% 0.29/0.55  thf(fact_4006_int__distrib_I1_J,axiom,
% 0.29/0.55      ! [Z1: int,Z22: int,W: int] :
% 0.29/0.55        ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 0.29/0.55        = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_distrib(1)
% 0.29/0.55  thf(fact_4007_int__distrib_I2_J,axiom,
% 0.29/0.55      ! [W: int,Z1: int,Z22: int] :
% 0.29/0.55        ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 0.29/0.55        = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_distrib(2)
% 0.29/0.55  thf(fact_4008_powr__powr,axiom,
% 0.29/0.55      ! [X: real,A: real,B: real] :
% 0.29/0.55        ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 0.29/0.55        = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % powr_powr
% 0.29/0.55  thf(fact_4009_imult__is__0,axiom,
% 0.29/0.55      ! [M: extended_enat,N: extended_enat] :
% 0.29/0.55        ( ( ( times_7803423173614009249d_enat @ M @ N )
% 0.29/0.55          = zero_z5237406670263579293d_enat )
% 0.29/0.55        = ( ( M = zero_z5237406670263579293d_enat )
% 0.29/0.55          | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % imult_is_0
% 0.29/0.55  thf(fact_4010_ln__eq__minus__one,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ( ln_ln_real @ X )
% 0.29/0.55            = ( minus_minus_real @ X @ one_one_real ) )
% 0.29/0.55         => ( X = one_one_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_eq_minus_one
% 0.29/0.55  thf(fact_4011_ln__div,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.55         => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
% 0.29/0.55            = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_div
% 0.29/0.55  thf(fact_4012_ln__mult,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.55         => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
% 0.29/0.55            = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_mult
% 0.29/0.55  thf(fact_4013_unset__bit__less__eq,axiom,
% 0.29/0.55      ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 0.29/0.55  
% 0.29/0.55  % unset_bit_less_eq
% 0.29/0.55  thf(fact_4014_set__bit__greater__eq,axiom,
% 0.29/0.55      ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 0.29/0.55  
% 0.29/0.55  % set_bit_greater_eq
% 0.29/0.55  thf(fact_4015_ln__le__minus__one,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_le_minus_one
% 0.29/0.55  thf(fact_4016_ln__diff__le,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.55         => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_diff_le
% 0.29/0.55  thf(fact_4017_ln__less__self,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_less_self
% 0.29/0.55  thf(fact_4018_ln__realpow,axiom,
% 0.29/0.55      ! [X: real,N: nat] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
% 0.29/0.55          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_realpow
% 0.29/0.55  thf(fact_4019_log__def,axiom,
% 0.29/0.55      ( log
% 0.29/0.55      = ( ^ [A3: real,X4: real] : ( divide_divide_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ A3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % log_def
% 0.29/0.55  thf(fact_4020_image__diff__subset,axiom,
% 0.29/0.55      ! [F: nat > set_nat,A2: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % image_diff_subset
% 0.29/0.55  thf(fact_4021_image__diff__subset,axiom,
% 0.29/0.55      ! [F: int > nat,A2: set_int,B2: set_int] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_int_nat @ F @ A2 ) @ ( image_int_nat @ F @ B2 ) ) @ ( image_int_nat @ F @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % image_diff_subset
% 0.29/0.55  thf(fact_4022_image__diff__subset,axiom,
% 0.29/0.55      ! [F: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % image_diff_subset
% 0.29/0.55  thf(fact_4023_image__diff__subset,axiom,
% 0.29/0.55      ! [F: int > int,A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_int_int @ F @ A2 ) @ ( image_int_int @ F @ B2 ) ) @ ( image_int_int @ F @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % image_diff_subset
% 0.29/0.55  thf(fact_4024_image__diff__subset,axiom,
% 0.29/0.55      ! [F: nat > int,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B2 ) ) @ ( image_nat_int @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % image_diff_subset
% 0.29/0.55  thf(fact_4025_div__mult2__numeral__eq,axiom,
% 0.29/0.55      ! [A: nat,K: num,L: num] :
% 0.29/0.55        ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
% 0.29/0.55        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_mult2_numeral_eq
% 0.29/0.55  thf(fact_4026_div__mult2__numeral__eq,axiom,
% 0.29/0.55      ! [A: int,K: num,L: num] :
% 0.29/0.55        ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
% 0.29/0.55        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_mult2_numeral_eq
% 0.29/0.55  thf(fact_4027_subset__Diff__insert,axiom,
% 0.29/0.55      ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,X: vEBT_VEBT,C6: set_VEBT_VEBT] :
% 0.29/0.55        ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( minus_5127226145743854075T_VEBT @ B2 @ ( insert_VEBT_VEBT @ X @ C6 ) ) )
% 0.29/0.55        = ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( minus_5127226145743854075T_VEBT @ B2 @ C6 ) )
% 0.29/0.55          & ~ ( member_VEBT_VEBT @ X @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % subset_Diff_insert
% 0.29/0.55  thf(fact_4028_subset__Diff__insert,axiom,
% 0.29/0.55      ! [A2: set_o,B2: set_o,X: $o,C6: set_o] :
% 0.29/0.55        ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ ( insert_o @ X @ C6 ) ) )
% 0.29/0.55        = ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ C6 ) )
% 0.29/0.55          & ~ ( member_o @ X @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % subset_Diff_insert
% 0.29/0.55  thf(fact_4029_subset__Diff__insert,axiom,
% 0.29/0.55      ! [A2: set_complex,B2: set_complex,X: complex,C6: set_complex] :
% 0.29/0.55        ( ( ord_le211207098394363844omplex @ A2 @ ( minus_811609699411566653omplex @ B2 @ ( insert_complex @ X @ C6 ) ) )
% 0.29/0.55        = ( ( ord_le211207098394363844omplex @ A2 @ ( minus_811609699411566653omplex @ B2 @ C6 ) )
% 0.29/0.55          & ~ ( member_complex @ X @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % subset_Diff_insert
% 0.29/0.55  thf(fact_4030_subset__Diff__insert,axiom,
% 0.29/0.55      ! [A2: set_real,B2: set_real,X: real,C6: set_real] :
% 0.29/0.55        ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X @ C6 ) ) )
% 0.29/0.55        = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C6 ) )
% 0.29/0.55          & ~ ( member_real @ X @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % subset_Diff_insert
% 0.29/0.55  thf(fact_4031_subset__Diff__insert,axiom,
% 0.29/0.55      ! [A2: set_set_nat,B2: set_set_nat,X: set_nat,C6: set_set_nat] :
% 0.29/0.55        ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X @ C6 ) ) )
% 0.29/0.55        = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C6 ) )
% 0.29/0.55          & ~ ( member_set_nat @ X @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % subset_Diff_insert
% 0.29/0.55  thf(fact_4032_subset__Diff__insert,axiom,
% 0.29/0.55      ! [A2: set_nat,B2: set_nat,X: nat,C6: set_nat] :
% 0.29/0.55        ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C6 ) ) )
% 0.29/0.55        = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C6 ) )
% 0.29/0.55          & ~ ( member_nat @ X @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % subset_Diff_insert
% 0.29/0.55  thf(fact_4033_subset__Diff__insert,axiom,
% 0.29/0.55      ! [A2: set_int,B2: set_int,X: int,C6: set_int] :
% 0.29/0.55        ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X @ C6 ) ) )
% 0.29/0.55        = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C6 ) )
% 0.29/0.55          & ~ ( member_int @ X @ A2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % subset_Diff_insert
% 0.29/0.55  thf(fact_4034_int__le__induct,axiom,
% 0.29/0.55      ! [I: int,K: int,P: int > $o] :
% 0.29/0.55        ( ( ord_less_eq_int @ I @ K )
% 0.29/0.55       => ( ( P @ K )
% 0.29/0.55         => ( ! [I2: int] :
% 0.29/0.55                ( ( ord_less_eq_int @ I2 @ K )
% 0.29/0.55               => ( ( P @ I2 )
% 0.29/0.55                 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.29/0.55           => ( P @ I ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_le_induct
% 0.29/0.55  thf(fact_4035_int__less__induct,axiom,
% 0.29/0.55      ! [I: int,K: int,P: int > $o] :
% 0.29/0.55        ( ( ord_less_int @ I @ K )
% 0.29/0.55       => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.29/0.55         => ( ! [I2: int] :
% 0.29/0.55                ( ( ord_less_int @ I2 @ K )
% 0.29/0.55               => ( ( P @ I2 )
% 0.29/0.55                 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.29/0.55           => ( P @ I ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_less_induct
% 0.29/0.55  thf(fact_4036_zmult__zless__mono2,axiom,
% 0.29/0.55      ! [I: int,J: int,K: int] :
% 0.29/0.55        ( ( ord_less_int @ I @ J )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zmult_zless_mono2
% 0.29/0.55  thf(fact_4037_int__ops_I7_J,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 0.29/0.55        = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_ops(7)
% 0.29/0.55  thf(fact_4038_ln__powr__bound2,axiom,
% 0.29/0.55      ! [X: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55         => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_powr_bound2
% 0.29/0.55  thf(fact_4039_log__eq__div__ln__mult__log,axiom,
% 0.29/0.55      ! [A: real,B: real,X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55       => ( ( A != one_one_real )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.55           => ( ( B != one_one_real )
% 0.29/0.55             => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55               => ( ( log @ A @ X )
% 0.29/0.55                  = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % log_eq_div_ln_mult_log
% 0.29/0.55  thf(fact_4040_enat__0__less__mult__iff,axiom,
% 0.29/0.55      ! [M: extended_enat,N: extended_enat] :
% 0.29/0.55        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 0.29/0.55        = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.29/0.55          & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % enat_0_less_mult_iff
% 0.29/0.55  thf(fact_4041_add__diff__assoc__enat,axiom,
% 0.29/0.55      ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
% 0.29/0.55        ( ( ord_le2932123472753598470d_enat @ Z @ Y )
% 0.29/0.55       => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
% 0.29/0.55          = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_diff_assoc_enat
% 0.29/0.55  thf(fact_4042_ln__bound,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_bound
% 0.29/0.55  thf(fact_4043_ln__gt__zero__imp__gt__one,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_gt_zero_imp_gt_one
% 0.29/0.55  thf(fact_4044_ln__less__zero,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.55         => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_less_zero
% 0.29/0.55  thf(fact_4045_ln__gt__zero,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ X )
% 0.29/0.55       => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_gt_zero
% 0.29/0.55  thf(fact_4046_ln__ge__zero,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ one_one_real @ X )
% 0.29/0.55       => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_ge_zero
% 0.29/0.55  thf(fact_4047_int__induct,axiom,
% 0.29/0.55      ! [P: int > $o,K: int,I: int] :
% 0.29/0.55        ( ( P @ K )
% 0.29/0.55       => ( ! [I2: int] :
% 0.29/0.55              ( ( ord_less_eq_int @ K @ I2 )
% 0.29/0.55             => ( ( P @ I2 )
% 0.29/0.55               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.29/0.55         => ( ! [I2: int] :
% 0.29/0.55                ( ( ord_less_eq_int @ I2 @ K )
% 0.29/0.55               => ( ( P @ I2 )
% 0.29/0.55                 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.29/0.55           => ( P @ I ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_induct
% 0.29/0.55  thf(fact_4048_reals__Archimedean3,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ! [Y5: real] :
% 0.29/0.55          ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % reals_Archimedean3
% 0.29/0.55  thf(fact_4049_pos__zmult__eq__1__iff,axiom,
% 0.29/0.55      ! [M: int,N: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ M )
% 0.29/0.55       => ( ( ( times_times_int @ M @ N )
% 0.29/0.55            = one_one_int )
% 0.29/0.55          = ( ( M = one_one_int )
% 0.29/0.55            & ( N = one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_zmult_eq_1_iff
% 0.29/0.55  thf(fact_4050_powr__mult,axiom,
% 0.29/0.55      ! [X: real,Y: real,A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.55         => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
% 0.29/0.55            = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % powr_mult
% 0.29/0.55  thf(fact_4051_zdiv__zmult2__eq,axiom,
% 0.29/0.55      ! [C: int,A: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.29/0.55          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zdiv_zmult2_eq
% 0.29/0.55  thf(fact_4052_log__powr,axiom,
% 0.29/0.55      ! [X: real,B: real,Y: real] :
% 0.29/0.55        ( ( X != zero_zero_real )
% 0.29/0.55       => ( ( log @ B @ ( powr_real @ X @ Y ) )
% 0.29/0.55          = ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % log_powr
% 0.29/0.55  thf(fact_4053_ln__ge__zero__imp__ge__one,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55         => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_ge_zero_imp_ge_one
% 0.29/0.55  thf(fact_4054_ln__one__plus__pos__lower__bound,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.55         => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_one_plus_pos_lower_bound
% 0.29/0.55  thf(fact_4055_ln__add__one__self__le__self,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_add_one_self_le_self
% 0.29/0.55  thf(fact_4056_int__ops_I6_J,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.29/0.55         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.29/0.55            = zero_zero_int ) )
% 0.29/0.55        & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.29/0.55         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.29/0.55            = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_ops(6)
% 0.29/0.55  thf(fact_4057_zmult__zless__mono2__lemma,axiom,
% 0.29/0.55      ! [I: int,J: int,K: nat] :
% 0.29/0.55        ( ( ord_less_int @ I @ J )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zmult_zless_mono2_lemma
% 0.29/0.55  thf(fact_4058_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 0.29/0.55      ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 0.29/0.55        ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 0.29/0.55  
% 0.29/0.55  % VEBT_internal.membermima.simps(2)
% 0.29/0.55  thf(fact_4059_unique__quotient__lemma__neg,axiom,
% 0.29/0.55      ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.29/0.55       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.29/0.55         => ( ( ord_less_int @ B @ R2 )
% 0.29/0.55           => ( ( ord_less_int @ B @ R4 )
% 0.29/0.55             => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % unique_quotient_lemma_neg
% 0.29/0.55  thf(fact_4060_unique__quotient__lemma,axiom,
% 0.29/0.55      ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.29/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.29/0.55         => ( ( ord_less_int @ R4 @ B )
% 0.29/0.55           => ( ( ord_less_int @ R2 @ B )
% 0.29/0.55             => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % unique_quotient_lemma
% 0.29/0.55  thf(fact_4061_zdiv__mono2__neg__lemma,axiom,
% 0.29/0.55      ! [B: int,Q3: int,R2: int,B7: int,Q5: int,R4: int] :
% 0.29/0.55        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
% 0.29/0.55          = ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 0.29/0.55       => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) @ zero_zero_int )
% 0.29/0.55         => ( ( ord_less_int @ R2 @ B )
% 0.29/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.29/0.55             => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.29/0.55               => ( ( ord_less_eq_int @ B7 @ B )
% 0.29/0.55                 => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zdiv_mono2_neg_lemma
% 0.29/0.55  thf(fact_4062_zdiv__mono2__lemma,axiom,
% 0.29/0.55      ! [B: int,Q3: int,R2: int,B7: int,Q5: int,R4: int] :
% 0.29/0.55        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
% 0.29/0.55          = ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 0.29/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 0.29/0.55         => ( ( ord_less_int @ R4 @ B7 )
% 0.29/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.29/0.55             => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.29/0.55               => ( ( ord_less_eq_int @ B7 @ B )
% 0.29/0.55                 => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zdiv_mono2_lemma
% 0.29/0.55  thf(fact_4063_q__pos__lemma,axiom,
% 0.29/0.55      ! [B7: int,Q5: int,R4: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 0.29/0.55       => ( ( ord_less_int @ R4 @ B7 )
% 0.29/0.55         => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.29/0.55           => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % q_pos_lemma
% 0.29/0.55  thf(fact_4064_ceiling__diff__floor__le__1,axiom,
% 0.29/0.55      ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_diff_floor_le_1
% 0.29/0.55  thf(fact_4065_ceiling__diff__floor__le__1,axiom,
% 0.29/0.55      ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).
% 0.29/0.55  
% 0.29/0.55  % ceiling_diff_floor_le_1
% 0.29/0.55  thf(fact_4066_real__of__int__floor__gt__diff__one,axiom,
% 0.29/0.55      ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % real_of_int_floor_gt_diff_one
% 0.29/0.55  thf(fact_4067_real__of__int__floor__ge__diff__one,axiom,
% 0.29/0.55      ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % real_of_int_floor_ge_diff_one
% 0.29/0.55  thf(fact_4068_real__of__nat__div2,axiom,
% 0.29/0.55      ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % real_of_nat_div2
% 0.29/0.55  thf(fact_4069_log__divide,axiom,
% 0.29/0.55      ! [A: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55       => ( ( A != one_one_real )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55           => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.55             => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
% 0.29/0.55                = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % log_divide
% 0.29/0.55  thf(fact_4070_four__x__squared,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.55        = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % four_x_squared
% 0.29/0.55  thf(fact_4071_real__of__nat__div3,axiom,
% 0.29/0.55      ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% 0.29/0.55  
% 0.29/0.55  % real_of_nat_div3
% 0.29/0.55  thf(fact_4072_real__archimedian__rdiv__eq__0,axiom,
% 0.29/0.55      ! [X: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ! [M5: nat] :
% 0.29/0.55                ( ( ord_less_nat @ zero_zero_nat @ M5 )
% 0.29/0.55               => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
% 0.29/0.55           => ( X = zero_zero_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % real_archimedian_rdiv_eq_0
% 0.29/0.55  thf(fact_4073_real__of__int__div2,axiom,
% 0.29/0.55      ! [N: int,X: 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 @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % real_of_int_div2
% 0.29/0.55  thf(fact_4074_real__of__int__div3,axiom,
% 0.29/0.55      ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).
% 0.29/0.55  
% 0.29/0.55  % real_of_int_div3
% 0.29/0.55  thf(fact_4075_powr__mult__base,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
% 0.29/0.55          = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % powr_mult_base
% 0.29/0.55  thf(fact_4076_log__mult,axiom,
% 0.29/0.55      ! [A: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55       => ( ( A != one_one_real )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55           => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.55             => ( ( log @ A @ ( times_times_real @ X @ Y ) )
% 0.29/0.55                = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % log_mult
% 0.29/0.55  thf(fact_4077_split__zdiv,axiom,
% 0.29/0.55      ! [P: int > $o,N: int,K: int] :
% 0.29/0.55        ( ( P @ ( divide_divide_int @ N @ K ) )
% 0.29/0.55        = ( ( ( K = zero_zero_int )
% 0.29/0.55           => ( P @ zero_zero_int ) )
% 0.29/0.55          & ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.55           => ! [I3: int,J2: int] :
% 0.29/0.55                ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
% 0.29/0.55                  & ( ord_less_int @ J2 @ K )
% 0.29/0.55                  & ( N
% 0.29/0.55                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 0.29/0.55               => ( P @ I3 ) ) )
% 0.29/0.55          & ( ( ord_less_int @ K @ zero_zero_int )
% 0.29/0.55           => ! [I3: int,J2: int] :
% 0.29/0.55                ( ( ( ord_less_int @ K @ J2 )
% 0.29/0.55                  & ( ord_less_eq_int @ J2 @ zero_zero_int )
% 0.29/0.55                  & ( N
% 0.29/0.55                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 0.29/0.55               => ( P @ I3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % split_zdiv
% 0.29/0.55  thf(fact_4078_int__div__neg__eq,axiom,
% 0.29/0.55      ! [A: int,B: int,Q3: int,R2: int] :
% 0.29/0.55        ( ( A
% 0.29/0.55          = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.29/0.55       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.29/0.55         => ( ( ord_less_int @ B @ R2 )
% 0.29/0.55           => ( ( divide_divide_int @ A @ B )
% 0.29/0.55              = Q3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_div_neg_eq
% 0.29/0.55  thf(fact_4079_int__div__pos__eq,axiom,
% 0.29/0.55      ! [A: int,B: int,Q3: int,R2: int] :
% 0.29/0.55        ( ( A
% 0.29/0.55          = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.29/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.29/0.55         => ( ( ord_less_int @ R2 @ B )
% 0.29/0.55           => ( ( divide_divide_int @ A @ B )
% 0.29/0.55              = Q3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % int_div_pos_eq
% 0.29/0.55  thf(fact_4080_log__nat__power,axiom,
% 0.29/0.55      ! [X: real,B: real,N: nat] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( log @ B @ ( power_power_real @ X @ N ) )
% 0.29/0.55          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % log_nat_power
% 0.29/0.55  thf(fact_4081_ln__powr__bound,axiom,
% 0.29/0.55      ! [X: real,A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ one_one_real @ X )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55         => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_powr_bound
% 0.29/0.55  thf(fact_4082_minus__log__eq__powr,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.55       => ( ( B != one_one_real )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55           => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
% 0.29/0.55              = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_log_eq_powr
% 0.29/0.55  thf(fact_4083_div__pos__geq,axiom,
% 0.29/0.55      ! [L: int,K: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ L )
% 0.29/0.55       => ( ( ord_less_eq_int @ L @ K )
% 0.29/0.55         => ( ( divide_divide_int @ K @ L )
% 0.29/0.55            = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_pos_geq
% 0.29/0.55  thf(fact_4084_add__log__eq__powr,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.55       => ( ( B != one_one_real )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55           => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
% 0.29/0.55              = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_log_eq_powr
% 0.29/0.55  thf(fact_4085_log__add__eq__powr,axiom,
% 0.29/0.55      ! [B: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.55       => ( ( B != one_one_real )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55           => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
% 0.29/0.55              = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % log_add_eq_powr
% 0.29/0.55  thf(fact_4086_L2__set__mult__ineq__lemma,axiom,
% 0.29/0.55      ! [A: real,C: real,B: 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 @ B @ 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 @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % L2_set_mult_ineq_lemma
% 0.29/0.55  thf(fact_4087_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 0.29/0.55      ! [X: nat,N: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.55           => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % VEBT_internal.exp_split_high_low(2)
% 0.29/0.55  thf(fact_4088_pos__zdiv__mult__2,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.55       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.55          = ( divide_divide_int @ B @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_zdiv_mult_2
% 0.29/0.55  thf(fact_4089_neg__zdiv__mult__2,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.55       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.55          = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_zdiv_mult_2
% 0.29/0.55  thf(fact_4090_mult__less__iff1,axiom,
% 0.29/0.55      ! [Z: real,X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ Z )
% 0.29/0.55       => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
% 0.29/0.55          = ( ord_less_real @ X @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_iff1
% 0.29/0.55  thf(fact_4091_mult__less__iff1,axiom,
% 0.29/0.55      ! [Z: rat,X: rat,Y: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.29/0.55       => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 0.29/0.55          = ( ord_less_rat @ X @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_iff1
% 0.29/0.55  thf(fact_4092_mult__less__iff1,axiom,
% 0.29/0.55      ! [Z: int,X: int,Y: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.29/0.55       => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 0.29/0.55          = ( ord_less_int @ X @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_iff1
% 0.29/0.55  thf(fact_4093_real__average__minus__second,axiom,
% 0.29/0.55      ! [B: real,A: real] :
% 0.29/0.55        ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.29/0.55        = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % real_average_minus_second
% 0.29/0.55  thf(fact_4094_real__average__minus__first,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.29/0.55        = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % real_average_minus_first
% 0.29/0.55  thf(fact_4095_norm__divide__numeral,axiom,
% 0.29/0.55      ! [A: real,W: num] :
% 0.29/0.55        ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.55        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_divide_numeral
% 0.29/0.55  thf(fact_4096_norm__divide__numeral,axiom,
% 0.29/0.55      ! [A: complex,W: num] :
% 0.29/0.55        ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.55        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_divide_numeral
% 0.29/0.55  thf(fact_4097_norm__mult__numeral1,axiom,
% 0.29/0.55      ! [W: num,A: real] :
% 0.29/0.55        ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.29/0.55        = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_mult_numeral1
% 0.29/0.55  thf(fact_4098_norm__mult__numeral1,axiom,
% 0.29/0.55      ! [W: num,A: complex] :
% 0.29/0.55        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.29/0.55        = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_mult_numeral1
% 0.29/0.55  thf(fact_4099_norm__mult__numeral2,axiom,
% 0.29/0.55      ! [A: real,W: num] :
% 0.29/0.55        ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.55        = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_mult_numeral2
% 0.29/0.55  thf(fact_4100_norm__mult__numeral2,axiom,
% 0.29/0.55      ! [A: complex,W: num] :
% 0.29/0.55        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.55        = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_mult_numeral2
% 0.29/0.55  thf(fact_4101_norm__le__zero__iff,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 0.29/0.55        = ( X = zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_le_zero_iff
% 0.29/0.55  thf(fact_4102_norm__le__zero__iff,axiom,
% 0.29/0.55      ! [X: complex] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 0.29/0.55        = ( X = zero_zero_complex ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_le_zero_iff
% 0.29/0.55  thf(fact_4103_zero__less__norm__iff,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 0.29/0.55        = ( X != zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_norm_iff
% 0.29/0.55  thf(fact_4104_zero__less__norm__iff,axiom,
% 0.29/0.55      ! [X: complex] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 0.29/0.55        = ( X != zero_zero_complex ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_norm_iff
% 0.29/0.55  thf(fact_4105_norm__of__nat,axiom,
% 0.29/0.55      ! [N: nat] :
% 0.29/0.55        ( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.29/0.55        = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_of_nat
% 0.29/0.55  thf(fact_4106_norm__of__nat,axiom,
% 0.29/0.55      ! [N: nat] :
% 0.29/0.55        ( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N ) )
% 0.29/0.55        = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_of_nat
% 0.29/0.55  thf(fact_4107_DiffI,axiom,
% 0.29/0.55      ! [C: complex,A2: set_complex,B2: set_complex] :
% 0.29/0.55        ( ( member_complex @ C @ A2 )
% 0.29/0.55       => ( ~ ( member_complex @ C @ B2 )
% 0.29/0.55         => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffI
% 0.29/0.55  thf(fact_4108_DiffI,axiom,
% 0.29/0.55      ! [C: real,A2: set_real,B2: set_real] :
% 0.29/0.55        ( ( member_real @ C @ A2 )
% 0.29/0.55       => ( ~ ( member_real @ C @ B2 )
% 0.29/0.55         => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffI
% 0.29/0.55  thf(fact_4109_DiffI,axiom,
% 0.29/0.55      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.29/0.55        ( ( member_set_nat @ C @ A2 )
% 0.29/0.55       => ( ~ ( member_set_nat @ C @ B2 )
% 0.29/0.55         => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffI
% 0.29/0.55  thf(fact_4110_DiffI,axiom,
% 0.29/0.55      ! [C: int,A2: set_int,B2: set_int] :
% 0.29/0.55        ( ( member_int @ C @ A2 )
% 0.29/0.55       => ( ~ ( member_int @ C @ B2 )
% 0.29/0.55         => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffI
% 0.29/0.55  thf(fact_4111_DiffI,axiom,
% 0.29/0.55      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.29/0.55        ( ( member_nat @ C @ A2 )
% 0.29/0.55       => ( ~ ( member_nat @ C @ B2 )
% 0.29/0.55         => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffI
% 0.29/0.55  thf(fact_4112_Diff__iff,axiom,
% 0.29/0.55      ! [C: complex,A2: set_complex,B2: set_complex] :
% 0.29/0.55        ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.29/0.55        = ( ( member_complex @ C @ A2 )
% 0.29/0.55          & ~ ( member_complex @ C @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_iff
% 0.29/0.55  thf(fact_4113_Diff__iff,axiom,
% 0.29/0.55      ! [C: real,A2: set_real,B2: set_real] :
% 0.29/0.55        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.29/0.55        = ( ( member_real @ C @ A2 )
% 0.29/0.55          & ~ ( member_real @ C @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_iff
% 0.29/0.55  thf(fact_4114_Diff__iff,axiom,
% 0.29/0.55      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.29/0.55        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.29/0.55        = ( ( member_set_nat @ C @ A2 )
% 0.29/0.55          & ~ ( member_set_nat @ C @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_iff
% 0.29/0.55  thf(fact_4115_Diff__iff,axiom,
% 0.29/0.55      ! [C: int,A2: set_int,B2: set_int] :
% 0.29/0.55        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.29/0.55        = ( ( member_int @ C @ A2 )
% 0.29/0.55          & ~ ( member_int @ C @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_iff
% 0.29/0.55  thf(fact_4116_Diff__iff,axiom,
% 0.29/0.55      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.29/0.55        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.29/0.55        = ( ( member_nat @ C @ A2 )
% 0.29/0.55          & ~ ( member_nat @ C @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_iff
% 0.29/0.55  thf(fact_4117_Diff__idemp,axiom,
% 0.29/0.55      ! [A2: set_nat,B2: set_nat] :
% 0.29/0.55        ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
% 0.29/0.55        = ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Diff_idemp
% 0.29/0.55  thf(fact_4118_norm__eq__zero,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ( real_V7735802525324610683m_real @ X )
% 0.29/0.55          = zero_zero_real )
% 0.29/0.55        = ( X = zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_eq_zero
% 0.29/0.55  thf(fact_4119_norm__eq__zero,axiom,
% 0.29/0.55      ! [X: complex] :
% 0.29/0.55        ( ( ( real_V1022390504157884413omplex @ X )
% 0.29/0.55          = zero_zero_real )
% 0.29/0.55        = ( X = zero_zero_complex ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_eq_zero
% 0.29/0.55  thf(fact_4120_norm__zero,axiom,
% 0.29/0.55      ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 0.29/0.55      = zero_zero_real ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_zero
% 0.29/0.55  thf(fact_4121_norm__zero,axiom,
% 0.29/0.55      ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 0.29/0.55      = zero_zero_real ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_zero
% 0.29/0.55  thf(fact_4122_norm__one,axiom,
% 0.29/0.55      ( ( real_V7735802525324610683m_real @ one_one_real )
% 0.29/0.55      = one_one_real ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_one
% 0.29/0.55  thf(fact_4123_norm__one,axiom,
% 0.29/0.55      ( ( real_V1022390504157884413omplex @ one_one_complex )
% 0.29/0.55      = one_one_real ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_one
% 0.29/0.55  thf(fact_4124_norm__numeral,axiom,
% 0.29/0.55      ! [W: num] :
% 0.29/0.55        ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 0.29/0.55        = ( numeral_numeral_real @ W ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_numeral
% 0.29/0.55  thf(fact_4125_norm__numeral,axiom,
% 0.29/0.55      ! [W: num] :
% 0.29/0.55        ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.29/0.55        = ( numeral_numeral_real @ W ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_numeral
% 0.29/0.55  thf(fact_4126_DiffE,axiom,
% 0.29/0.55      ! [C: complex,A2: set_complex,B2: set_complex] :
% 0.29/0.55        ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.29/0.55       => ~ ( ( member_complex @ C @ A2 )
% 0.29/0.55           => ( member_complex @ C @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffE
% 0.29/0.55  thf(fact_4127_DiffE,axiom,
% 0.29/0.55      ! [C: real,A2: set_real,B2: set_real] :
% 0.29/0.55        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.29/0.55       => ~ ( ( member_real @ C @ A2 )
% 0.29/0.55           => ( member_real @ C @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffE
% 0.29/0.55  thf(fact_4128_DiffE,axiom,
% 0.29/0.55      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.29/0.55        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.29/0.55       => ~ ( ( member_set_nat @ C @ A2 )
% 0.29/0.55           => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffE
% 0.29/0.55  thf(fact_4129_DiffE,axiom,
% 0.29/0.55      ! [C: int,A2: set_int,B2: set_int] :
% 0.29/0.55        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.29/0.55       => ~ ( ( member_int @ C @ A2 )
% 0.29/0.55           => ( member_int @ C @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffE
% 0.29/0.55  thf(fact_4130_DiffE,axiom,
% 0.29/0.55      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.29/0.55        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.29/0.55       => ~ ( ( member_nat @ C @ A2 )
% 0.29/0.55           => ( member_nat @ C @ B2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffE
% 0.29/0.55  thf(fact_4131_DiffD1,axiom,
% 0.29/0.55      ! [C: complex,A2: set_complex,B2: set_complex] :
% 0.29/0.55        ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.29/0.55       => ( member_complex @ C @ A2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffD1
% 0.29/0.55  thf(fact_4132_DiffD1,axiom,
% 0.29/0.55      ! [C: real,A2: set_real,B2: set_real] :
% 0.29/0.55        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.29/0.55       => ( member_real @ C @ A2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffD1
% 0.29/0.55  thf(fact_4133_DiffD1,axiom,
% 0.29/0.55      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.29/0.55        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.29/0.55       => ( member_set_nat @ C @ A2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffD1
% 0.29/0.55  thf(fact_4134_DiffD1,axiom,
% 0.29/0.55      ! [C: int,A2: set_int,B2: set_int] :
% 0.29/0.55        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.29/0.55       => ( member_int @ C @ A2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffD1
% 0.29/0.55  thf(fact_4135_DiffD1,axiom,
% 0.29/0.55      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.29/0.55        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.29/0.55       => ( member_nat @ C @ A2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffD1
% 0.29/0.55  thf(fact_4136_DiffD2,axiom,
% 0.29/0.55      ! [C: complex,A2: set_complex,B2: set_complex] :
% 0.29/0.55        ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.29/0.55       => ~ ( member_complex @ C @ B2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffD2
% 0.29/0.55  thf(fact_4137_DiffD2,axiom,
% 0.29/0.55      ! [C: real,A2: set_real,B2: set_real] :
% 0.29/0.55        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.29/0.55       => ~ ( member_real @ C @ B2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffD2
% 0.29/0.55  thf(fact_4138_DiffD2,axiom,
% 0.29/0.55      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.29/0.55        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.29/0.55       => ~ ( member_set_nat @ C @ B2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffD2
% 0.29/0.55  thf(fact_4139_DiffD2,axiom,
% 0.29/0.55      ! [C: int,A2: set_int,B2: set_int] :
% 0.29/0.55        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.29/0.55       => ~ ( member_int @ C @ B2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffD2
% 0.29/0.55  thf(fact_4140_DiffD2,axiom,
% 0.29/0.55      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.29/0.55        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.29/0.55       => ~ ( member_nat @ C @ B2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % DiffD2
% 0.29/0.55  thf(fact_4141_norm__diff__ineq,axiom,
% 0.29/0.55      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_diff_ineq
% 0.29/0.55  thf(fact_4142_norm__diff__ineq,axiom,
% 0.29/0.55      ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_diff_ineq
% 0.29/0.55  thf(fact_4143_norm__triangle__ineq2,axiom,
% 0.29/0.55      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_ineq2
% 0.29/0.55  thf(fact_4144_norm__triangle__ineq2,axiom,
% 0.29/0.55      ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_ineq2
% 0.29/0.55  thf(fact_4145_norm__not__less__zero,axiom,
% 0.29/0.55      ! [X: complex] :
% 0.29/0.55        ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_not_less_zero
% 0.29/0.55  thf(fact_4146_norm__ge__zero,axiom,
% 0.29/0.55      ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_ge_zero
% 0.29/0.55  thf(fact_4147_norm__divide,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.55        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_divide
% 0.29/0.55  thf(fact_4148_norm__divide,axiom,
% 0.29/0.55      ! [A: complex,B: complex] :
% 0.29/0.55        ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.55        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_divide
% 0.29/0.55  thf(fact_4149_norm__power,axiom,
% 0.29/0.55      ! [X: real,N: nat] :
% 0.29/0.55        ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
% 0.29/0.55        = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_power
% 0.29/0.55  thf(fact_4150_norm__power,axiom,
% 0.29/0.55      ! [X: complex,N: nat] :
% 0.29/0.55        ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
% 0.29/0.55        = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_power
% 0.29/0.55  thf(fact_4151_nonzero__norm__divide,axiom,
% 0.29/0.55      ! [B: real,A: real] :
% 0.29/0.55        ( ( B != zero_zero_real )
% 0.29/0.55       => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.55          = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nonzero_norm_divide
% 0.29/0.55  thf(fact_4152_nonzero__norm__divide,axiom,
% 0.29/0.55      ! [B: complex,A: complex] :
% 0.29/0.55        ( ( B != zero_zero_complex )
% 0.29/0.55       => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.55          = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nonzero_norm_divide
% 0.29/0.55  thf(fact_4153_power__eq__imp__eq__norm,axiom,
% 0.29/0.55      ! [W: real,N: nat,Z: real] :
% 0.29/0.55        ( ( ( power_power_real @ W @ N )
% 0.29/0.55          = ( power_power_real @ Z @ N ) )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55         => ( ( real_V7735802525324610683m_real @ W )
% 0.29/0.55            = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_eq_imp_eq_norm
% 0.29/0.55  thf(fact_4154_power__eq__imp__eq__norm,axiom,
% 0.29/0.55      ! [W: complex,N: nat,Z: complex] :
% 0.29/0.55        ( ( ( power_power_complex @ W @ N )
% 0.29/0.55          = ( power_power_complex @ Z @ N ) )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.55         => ( ( real_V1022390504157884413omplex @ W )
% 0.29/0.55            = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_eq_imp_eq_norm
% 0.29/0.55  thf(fact_4155_norm__mult__less,axiom,
% 0.29/0.55      ! [X: real,R2: real,Y: real,S: real] :
% 0.29/0.55        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
% 0.29/0.55       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 0.29/0.55         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_mult_less
% 0.29/0.55  thf(fact_4156_norm__mult__less,axiom,
% 0.29/0.55      ! [X: complex,R2: real,Y: complex,S: real] :
% 0.29/0.55        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
% 0.29/0.55       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 0.29/0.55         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_mult_less
% 0.29/0.55  thf(fact_4157_norm__mult__ineq,axiom,
% 0.29/0.55      ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_mult_ineq
% 0.29/0.55  thf(fact_4158_norm__mult__ineq,axiom,
% 0.29/0.55      ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_mult_ineq
% 0.29/0.55  thf(fact_4159_norm__triangle__lt,axiom,
% 0.29/0.55      ! [X: real,Y: real,E2: real] :
% 0.29/0.55        ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
% 0.29/0.55       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_lt
% 0.29/0.55  thf(fact_4160_norm__triangle__lt,axiom,
% 0.29/0.55      ! [X: complex,Y: complex,E2: real] :
% 0.29/0.55        ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
% 0.29/0.55       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_lt
% 0.29/0.55  thf(fact_4161_norm__add__less,axiom,
% 0.29/0.55      ! [X: real,R2: real,Y: real,S: real] :
% 0.29/0.55        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
% 0.29/0.55       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 0.29/0.55         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_add_less
% 0.29/0.55  thf(fact_4162_norm__add__less,axiom,
% 0.29/0.55      ! [X: complex,R2: real,Y: complex,S: real] :
% 0.29/0.55        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
% 0.29/0.55       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 0.29/0.55         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_add_less
% 0.29/0.55  thf(fact_4163_norm__add__leD,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 0.29/0.55       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_add_leD
% 0.29/0.55  thf(fact_4164_norm__add__leD,axiom,
% 0.29/0.55      ! [A: complex,B: complex,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 0.29/0.55       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_add_leD
% 0.29/0.55  thf(fact_4165_norm__triangle__le,axiom,
% 0.29/0.55      ! [X: real,Y: real,E2: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
% 0.29/0.55       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_le
% 0.29/0.55  thf(fact_4166_norm__triangle__le,axiom,
% 0.29/0.55      ! [X: complex,Y: complex,E2: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
% 0.29/0.55       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_le
% 0.29/0.55  thf(fact_4167_norm__triangle__ineq,axiom,
% 0.29/0.55      ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_ineq
% 0.29/0.55  thf(fact_4168_norm__triangle__ineq,axiom,
% 0.29/0.55      ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_ineq
% 0.29/0.55  thf(fact_4169_norm__triangle__mono,axiom,
% 0.29/0.55      ! [A: real,R2: real,B: real,S: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
% 0.29/0.55         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_mono
% 0.29/0.55  thf(fact_4170_norm__triangle__mono,axiom,
% 0.29/0.55      ! [A: complex,R2: real,B: complex,S: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
% 0.29/0.55         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_mono
% 0.29/0.55  thf(fact_4171_norm__power__ineq,axiom,
% 0.29/0.55      ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_power_ineq
% 0.29/0.55  thf(fact_4172_norm__power__ineq,axiom,
% 0.29/0.55      ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_power_ineq
% 0.29/0.55  thf(fact_4173_norm__diff__triangle__less,axiom,
% 0.29/0.55      ! [X: real,Y: real,E1: real,Z: real,E22: real] :
% 0.29/0.55        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 0.29/0.55       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
% 0.29/0.55         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_diff_triangle_less
% 0.29/0.55  thf(fact_4174_norm__diff__triangle__less,axiom,
% 0.29/0.55      ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
% 0.29/0.55        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 0.29/0.55       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
% 0.29/0.55         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_diff_triangle_less
% 0.29/0.55  thf(fact_4175_norm__triangle__le__diff,axiom,
% 0.29/0.55      ! [X: real,Y: real,E2: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
% 0.29/0.55       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_le_diff
% 0.29/0.55  thf(fact_4176_norm__triangle__le__diff,axiom,
% 0.29/0.55      ! [X: complex,Y: complex,E2: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
% 0.29/0.55       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_le_diff
% 0.29/0.55  thf(fact_4177_norm__diff__triangle__le,axiom,
% 0.29/0.55      ! [X: real,Y: real,E1: real,Z: real,E22: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
% 0.29/0.55         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_diff_triangle_le
% 0.29/0.55  thf(fact_4178_norm__diff__triangle__le,axiom,
% 0.29/0.55      ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
% 0.29/0.55         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_diff_triangle_le
% 0.29/0.55  thf(fact_4179_norm__triangle__ineq4,axiom,
% 0.29/0.55      ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_ineq4
% 0.29/0.55  thf(fact_4180_norm__triangle__ineq4,axiom,
% 0.29/0.55      ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_ineq4
% 0.29/0.55  thf(fact_4181_norm__triangle__sub,axiom,
% 0.29/0.55      ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_sub
% 0.29/0.55  thf(fact_4182_norm__triangle__sub,axiom,
% 0.29/0.55      ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_triangle_sub
% 0.29/0.55  thf(fact_4183_power__eq__1__iff,axiom,
% 0.29/0.55      ! [W: real,N: nat] :
% 0.29/0.55        ( ( ( power_power_real @ W @ N )
% 0.29/0.55          = one_one_real )
% 0.29/0.55       => ( ( ( real_V7735802525324610683m_real @ W )
% 0.29/0.55            = one_one_real )
% 0.29/0.55          | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_eq_1_iff
% 0.29/0.55  thf(fact_4184_power__eq__1__iff,axiom,
% 0.29/0.55      ! [W: complex,N: nat] :
% 0.29/0.55        ( ( ( power_power_complex @ W @ N )
% 0.29/0.55          = one_one_complex )
% 0.29/0.55       => ( ( ( real_V1022390504157884413omplex @ W )
% 0.29/0.55            = one_one_real )
% 0.29/0.55          | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_eq_1_iff
% 0.29/0.55  thf(fact_4185_norm__diff__triangle__ineq,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_diff_triangle_ineq
% 0.29/0.55  thf(fact_4186_norm__diff__triangle__ineq,axiom,
% 0.29/0.55      ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_diff_triangle_ineq
% 0.29/0.55  thf(fact_4187_square__norm__one,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55          = one_one_real )
% 0.29/0.55       => ( ( real_V7735802525324610683m_real @ X )
% 0.29/0.55          = one_one_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % square_norm_one
% 0.29/0.55  thf(fact_4188_square__norm__one,axiom,
% 0.29/0.55      ! [X: complex] :
% 0.29/0.55        ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55          = one_one_complex )
% 0.29/0.55       => ( ( real_V1022390504157884413omplex @ X )
% 0.29/0.55          = one_one_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % square_norm_one
% 0.29/0.55  thf(fact_4189_norm__power__diff,axiom,
% 0.29/0.55      ! [Z: real,W: real,M: nat] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 0.29/0.55         => ( 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 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_power_diff
% 0.29/0.55  thf(fact_4190_norm__power__diff,axiom,
% 0.29/0.55      ! [Z: complex,W: complex,M: nat] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 0.29/0.55         => ( 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 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % norm_power_diff
% 0.29/0.55  thf(fact_4191_both__member__options__ding,axiom,
% 0.29/0.55      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
% 0.29/0.55        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 0.29/0.55       => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.29/0.55         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.55           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % both_member_options_ding
% 0.29/0.55  thf(fact_4192_linear__plus__1__le__power,axiom,
% 0.29/0.55      ! [X: real,N: nat] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % linear_plus_1_le_power
% 0.29/0.55  thf(fact_4193_zdiff__int__split,axiom,
% 0.29/0.55      ! [P: int > $o,X: nat,Y: nat] :
% 0.29/0.55        ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
% 0.29/0.55        = ( ( ( ord_less_eq_nat @ Y @ X )
% 0.29/0.55           => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
% 0.29/0.55          & ( ( ord_less_nat @ X @ Y )
% 0.29/0.55           => ( P @ zero_zero_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zdiff_int_split
% 0.29/0.55  thf(fact_4194_ln__2__less__1,axiom,
% 0.29/0.55      ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.29/0.55  
% 0.29/0.55  % ln_2_less_1
% 0.29/0.55  thf(fact_4195_decr__mult__lemma,axiom,
% 0.29/0.55      ! [D: int,P: int > $o,K: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.55       => ( ! [X3: int] :
% 0.29/0.55              ( ( P @ X3 )
% 0.29/0.55             => ( P @ ( minus_minus_int @ X3 @ D ) ) )
% 0.29/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.55           => ! [X2: int] :
% 0.29/0.55                ( ( P @ X2 )
% 0.29/0.55               => ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % decr_mult_lemma
% 0.29/0.55  thf(fact_4196_incr__mult__lemma,axiom,
% 0.29/0.55      ! [D: int,P: int > $o,K: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.55       => ( ! [X3: int] :
% 0.29/0.55              ( ( P @ X3 )
% 0.29/0.55             => ( P @ ( plus_plus_int @ X3 @ D ) ) )
% 0.29/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.55           => ! [X2: int] :
% 0.29/0.55                ( ( P @ X2 )
% 0.29/0.55               => ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % incr_mult_lemma
% 0.29/0.55  thf(fact_4197_inthall,axiom,
% 0.29/0.55      ! [Xs: list_complex,P: complex > $o,N: nat] :
% 0.29/0.55        ( ! [X3: complex] :
% 0.29/0.55            ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
% 0.29/0.55           => ( P @ X3 ) )
% 0.29/0.55       => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.29/0.55         => ( P @ ( nth_complex @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % inthall
% 0.29/0.55  thf(fact_4198_inthall,axiom,
% 0.29/0.55      ! [Xs: list_real,P: real > $o,N: nat] :
% 0.29/0.55        ( ! [X3: real] :
% 0.29/0.55            ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.55           => ( P @ X3 ) )
% 0.29/0.55       => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.29/0.55         => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % inthall
% 0.29/0.55  thf(fact_4199_inthall,axiom,
% 0.29/0.55      ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
% 0.29/0.55        ( ! [X3: set_nat] :
% 0.29/0.55            ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 0.29/0.55           => ( P @ X3 ) )
% 0.29/0.55       => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.29/0.55         => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % inthall
% 0.29/0.55  thf(fact_4200_inthall,axiom,
% 0.29/0.55      ! [Xs: list_nat,P: nat > $o,N: nat] :
% 0.29/0.55        ( ! [X3: nat] :
% 0.29/0.55            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.29/0.55           => ( P @ X3 ) )
% 0.29/0.55       => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.29/0.55         => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % inthall
% 0.29/0.55  thf(fact_4201_inthall,axiom,
% 0.29/0.55      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 0.29/0.55        ( ! [X3: vEBT_VEBT] :
% 0.29/0.55            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.55           => ( P @ X3 ) )
% 0.29/0.55       => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.55         => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % inthall
% 0.29/0.55  thf(fact_4202_inthall,axiom,
% 0.29/0.55      ! [Xs: list_o,P: $o > $o,N: nat] :
% 0.29/0.55        ( ! [X3: $o] :
% 0.29/0.55            ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 0.29/0.55           => ( P @ X3 ) )
% 0.29/0.55       => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.29/0.55         => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % inthall
% 0.29/0.55  thf(fact_4203_inthall,axiom,
% 0.29/0.55      ! [Xs: list_int,P: int > $o,N: nat] :
% 0.29/0.55        ( ! [X3: int] :
% 0.29/0.55            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.29/0.55           => ( P @ X3 ) )
% 0.29/0.55       => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.29/0.55         => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % inthall
% 0.29/0.55  thf(fact_4204_nth__equalityI,axiom,
% 0.29/0.55      ! [Xs: list_nat,Ys2: list_nat] :
% 0.29/0.55        ( ( ( size_size_list_nat @ Xs )
% 0.29/0.55          = ( size_size_list_nat @ Ys2 ) )
% 0.29/0.55       => ( ! [I2: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.55             => ( ( nth_nat @ Xs @ I2 )
% 0.29/0.55                = ( nth_nat @ Ys2 @ I2 ) ) )
% 0.29/0.55         => ( Xs = Ys2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_equalityI
% 0.29/0.55  thf(fact_4205_nth__equalityI,axiom,
% 0.29/0.55      ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.29/0.55        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.55          = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 0.29/0.55       => ( ! [I2: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.55             => ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 0.29/0.55                = ( nth_VEBT_VEBT @ Ys2 @ I2 ) ) )
% 0.29/0.55         => ( Xs = Ys2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_equalityI
% 0.29/0.55  thf(fact_4206_nth__equalityI,axiom,
% 0.29/0.55      ! [Xs: list_o,Ys2: list_o] :
% 0.29/0.55        ( ( ( size_size_list_o @ Xs )
% 0.29/0.55          = ( size_size_list_o @ Ys2 ) )
% 0.29/0.55       => ( ! [I2: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.29/0.55             => ( ( nth_o @ Xs @ I2 )
% 0.29/0.55                = ( nth_o @ Ys2 @ I2 ) ) )
% 0.29/0.55         => ( Xs = Ys2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_equalityI
% 0.29/0.55  thf(fact_4207_nth__equalityI,axiom,
% 0.29/0.55      ! [Xs: list_int,Ys2: list_int] :
% 0.29/0.55        ( ( ( size_size_list_int @ Xs )
% 0.29/0.55          = ( size_size_list_int @ Ys2 ) )
% 0.29/0.55       => ( ! [I2: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.29/0.55             => ( ( nth_int @ Xs @ I2 )
% 0.29/0.55                = ( nth_int @ Ys2 @ I2 ) ) )
% 0.29/0.55         => ( Xs = Ys2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_equalityI
% 0.29/0.55  thf(fact_4208_Skolem__list__nth,axiom,
% 0.29/0.55      ! [K: nat,P: nat > nat > $o] :
% 0.29/0.55        ( ( ! [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ K )
% 0.29/0.55             => ? [X6: nat] : ( P @ I3 @ X6 ) ) )
% 0.29/0.55        = ( ? [Xs3: list_nat] :
% 0.29/0.55              ( ( ( size_size_list_nat @ Xs3 )
% 0.29/0.55                = K )
% 0.29/0.55              & ! [I3: nat] :
% 0.29/0.55                  ( ( ord_less_nat @ I3 @ K )
% 0.29/0.55                 => ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Skolem_list_nth
% 0.29/0.55  thf(fact_4209_Skolem__list__nth,axiom,
% 0.29/0.55      ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 0.29/0.55        ( ( ! [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ K )
% 0.29/0.55             => ? [X6: vEBT_VEBT] : ( P @ I3 @ X6 ) ) )
% 0.29/0.55        = ( ? [Xs3: list_VEBT_VEBT] :
% 0.29/0.55              ( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 0.29/0.55                = K )
% 0.29/0.55              & ! [I3: nat] :
% 0.29/0.55                  ( ( ord_less_nat @ I3 @ K )
% 0.29/0.55                 => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Skolem_list_nth
% 0.29/0.55  thf(fact_4210_Skolem__list__nth,axiom,
% 0.29/0.55      ! [K: nat,P: nat > $o > $o] :
% 0.29/0.55        ( ( ! [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ K )
% 0.29/0.55             => ? [X6: $o] : ( P @ I3 @ X6 ) ) )
% 0.29/0.55        = ( ? [Xs3: list_o] :
% 0.29/0.55              ( ( ( size_size_list_o @ Xs3 )
% 0.29/0.55                = K )
% 0.29/0.55              & ! [I3: nat] :
% 0.29/0.55                  ( ( ord_less_nat @ I3 @ K )
% 0.29/0.55                 => ( P @ I3 @ ( nth_o @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Skolem_list_nth
% 0.29/0.55  thf(fact_4211_Skolem__list__nth,axiom,
% 0.29/0.55      ! [K: nat,P: nat > int > $o] :
% 0.29/0.55        ( ( ! [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ K )
% 0.29/0.55             => ? [X6: int] : ( P @ I3 @ X6 ) ) )
% 0.29/0.55        = ( ? [Xs3: list_int] :
% 0.29/0.55              ( ( ( size_size_list_int @ Xs3 )
% 0.29/0.55                = K )
% 0.29/0.55              & ! [I3: nat] :
% 0.29/0.55                  ( ( ord_less_nat @ I3 @ K )
% 0.29/0.55                 => ( P @ I3 @ ( nth_int @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Skolem_list_nth
% 0.29/0.55  thf(fact_4212_list__eq__iff__nth__eq,axiom,
% 0.29/0.55      ( ( ^ [Y3: list_nat,Z2: list_nat] : ( Y3 = Z2 ) )
% 0.29/0.55      = ( ^ [Xs3: list_nat,Ys3: list_nat] :
% 0.29/0.55            ( ( ( size_size_list_nat @ Xs3 )
% 0.29/0.55              = ( size_size_list_nat @ Ys3 ) )
% 0.29/0.55            & ! [I3: nat] :
% 0.29/0.55                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
% 0.29/0.55               => ( ( nth_nat @ Xs3 @ I3 )
% 0.29/0.55                  = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % list_eq_iff_nth_eq
% 0.29/0.55  thf(fact_4213_list__eq__iff__nth__eq,axiom,
% 0.29/0.55      ( ( ^ [Y3: list_VEBT_VEBT,Z2: list_VEBT_VEBT] : ( Y3 = Z2 ) )
% 0.29/0.55      = ( ^ [Xs3: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.29/0.55            ( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 0.29/0.55              = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 0.29/0.55            & ! [I3: nat] :
% 0.29/0.55                ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 0.29/0.55               => ( ( nth_VEBT_VEBT @ Xs3 @ I3 )
% 0.29/0.55                  = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % list_eq_iff_nth_eq
% 0.29/0.55  thf(fact_4214_list__eq__iff__nth__eq,axiom,
% 0.29/0.55      ( ( ^ [Y3: list_o,Z2: list_o] : ( Y3 = Z2 ) )
% 0.29/0.55      = ( ^ [Xs3: list_o,Ys3: list_o] :
% 0.29/0.55            ( ( ( size_size_list_o @ Xs3 )
% 0.29/0.55              = ( size_size_list_o @ Ys3 ) )
% 0.29/0.55            & ! [I3: nat] :
% 0.29/0.55                ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs3 ) )
% 0.29/0.55               => ( ( nth_o @ Xs3 @ I3 )
% 0.29/0.55                  = ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % list_eq_iff_nth_eq
% 0.29/0.55  thf(fact_4215_list__eq__iff__nth__eq,axiom,
% 0.29/0.55      ( ( ^ [Y3: list_int,Z2: list_int] : ( Y3 = Z2 ) )
% 0.29/0.55      = ( ^ [Xs3: list_int,Ys3: list_int] :
% 0.29/0.55            ( ( ( size_size_list_int @ Xs3 )
% 0.29/0.55              = ( size_size_list_int @ Ys3 ) )
% 0.29/0.55            & ! [I3: nat] :
% 0.29/0.55                ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs3 ) )
% 0.29/0.55               => ( ( nth_int @ Xs3 @ I3 )
% 0.29/0.55                  = ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % list_eq_iff_nth_eq
% 0.29/0.55  thf(fact_4216_all__set__conv__all__nth,axiom,
% 0.29/0.55      ! [Xs: list_nat,P: nat > $o] :
% 0.29/0.55        ( ( ! [X4: nat] :
% 0.29/0.55              ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.29/0.55             => ( P @ X4 ) ) )
% 0.29/0.55        = ( ! [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.55             => ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_set_conv_all_nth
% 0.29/0.55  thf(fact_4217_all__set__conv__all__nth,axiom,
% 0.29/0.55      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.29/0.55        ( ( ! [X4: vEBT_VEBT] :
% 0.29/0.55              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.55             => ( P @ X4 ) ) )
% 0.29/0.55        = ( ! [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.55             => ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_set_conv_all_nth
% 0.29/0.55  thf(fact_4218_all__set__conv__all__nth,axiom,
% 0.29/0.55      ! [Xs: list_o,P: $o > $o] :
% 0.29/0.55        ( ( ! [X4: $o] :
% 0.29/0.55              ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.29/0.55             => ( P @ X4 ) ) )
% 0.29/0.55        = ( ! [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 0.29/0.55             => ( P @ ( nth_o @ Xs @ I3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_set_conv_all_nth
% 0.29/0.55  thf(fact_4219_all__set__conv__all__nth,axiom,
% 0.29/0.55      ! [Xs: list_int,P: int > $o] :
% 0.29/0.55        ( ( ! [X4: int] :
% 0.29/0.55              ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.29/0.55             => ( P @ X4 ) ) )
% 0.29/0.55        = ( ! [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 0.29/0.55             => ( P @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_set_conv_all_nth
% 0.29/0.55  thf(fact_4220_all__nth__imp__all__set,axiom,
% 0.29/0.55      ! [Xs: list_complex,P: complex > $o,X: complex] :
% 0.29/0.55        ( ! [I2: nat] :
% 0.29/0.55            ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.29/0.55           => ( P @ ( nth_complex @ Xs @ I2 ) ) )
% 0.29/0.55       => ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 0.29/0.55         => ( P @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_nth_imp_all_set
% 0.29/0.55  thf(fact_4221_all__nth__imp__all__set,axiom,
% 0.29/0.55      ! [Xs: list_real,P: real > $o,X: real] :
% 0.29/0.55        ( ! [I2: nat] :
% 0.29/0.55            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
% 0.29/0.55           => ( P @ ( nth_real @ Xs @ I2 ) ) )
% 0.29/0.55       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.29/0.55         => ( P @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_nth_imp_all_set
% 0.29/0.55  thf(fact_4222_all__nth__imp__all__set,axiom,
% 0.29/0.55      ! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
% 0.29/0.55        ( ! [I2: nat] :
% 0.29/0.55            ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.29/0.55           => ( P @ ( nth_set_nat @ Xs @ I2 ) ) )
% 0.29/0.55       => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.29/0.55         => ( P @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_nth_imp_all_set
% 0.29/0.55  thf(fact_4223_all__nth__imp__all__set,axiom,
% 0.29/0.55      ! [Xs: list_nat,P: nat > $o,X: nat] :
% 0.29/0.55        ( ! [I2: nat] :
% 0.29/0.55            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.55           => ( P @ ( nth_nat @ Xs @ I2 ) ) )
% 0.29/0.55       => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.29/0.55         => ( P @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_nth_imp_all_set
% 0.29/0.55  thf(fact_4224_all__nth__imp__all__set,axiom,
% 0.29/0.55      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 0.29/0.55        ( ! [I2: nat] :
% 0.29/0.55            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.55           => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
% 0.29/0.55       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.55         => ( P @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_nth_imp_all_set
% 0.29/0.55  thf(fact_4225_all__nth__imp__all__set,axiom,
% 0.29/0.55      ! [Xs: list_o,P: $o > $o,X: $o] :
% 0.29/0.55        ( ! [I2: nat] :
% 0.29/0.55            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.29/0.55           => ( P @ ( nth_o @ Xs @ I2 ) ) )
% 0.29/0.55       => ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.29/0.55         => ( P @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_nth_imp_all_set
% 0.29/0.55  thf(fact_4226_all__nth__imp__all__set,axiom,
% 0.29/0.55      ! [Xs: list_int,P: int > $o,X: int] :
% 0.29/0.55        ( ! [I2: nat] :
% 0.29/0.55            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.29/0.55           => ( P @ ( nth_int @ Xs @ I2 ) ) )
% 0.29/0.55       => ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.29/0.55         => ( P @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % all_nth_imp_all_set
% 0.29/0.55  thf(fact_4227_in__set__conv__nth,axiom,
% 0.29/0.55      ! [X: complex,Xs: list_complex] :
% 0.29/0.55        ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 0.29/0.55        = ( ? [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.29/0.55              & ( ( nth_complex @ Xs @ I3 )
% 0.29/0.55                = X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % in_set_conv_nth
% 0.29/0.55  thf(fact_4228_in__set__conv__nth,axiom,
% 0.29/0.55      ! [X: real,Xs: list_real] :
% 0.29/0.55        ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.29/0.55        = ( ? [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
% 0.29/0.55              & ( ( nth_real @ Xs @ I3 )
% 0.29/0.55                = X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % in_set_conv_nth
% 0.29/0.55  thf(fact_4229_in__set__conv__nth,axiom,
% 0.29/0.55      ! [X: set_nat,Xs: list_set_nat] :
% 0.29/0.55        ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.29/0.55        = ( ? [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.29/0.55              & ( ( nth_set_nat @ Xs @ I3 )
% 0.29/0.55                = X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % in_set_conv_nth
% 0.29/0.55  thf(fact_4230_in__set__conv__nth,axiom,
% 0.29/0.55      ! [X: nat,Xs: list_nat] :
% 0.29/0.55        ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.29/0.55        = ( ? [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.55              & ( ( nth_nat @ Xs @ I3 )
% 0.29/0.55                = X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % in_set_conv_nth
% 0.29/0.55  thf(fact_4231_in__set__conv__nth,axiom,
% 0.29/0.55      ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.29/0.55        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.55        = ( ? [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.55              & ( ( nth_VEBT_VEBT @ Xs @ I3 )
% 0.29/0.55                = X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % in_set_conv_nth
% 0.29/0.55  thf(fact_4232_in__set__conv__nth,axiom,
% 0.29/0.55      ! [X: $o,Xs: list_o] :
% 0.29/0.55        ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.29/0.55        = ( ? [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 0.29/0.55              & ( ( nth_o @ Xs @ I3 )
% 0.29/0.55                = X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % in_set_conv_nth
% 0.29/0.55  thf(fact_4233_in__set__conv__nth,axiom,
% 0.29/0.55      ! [X: int,Xs: list_int] :
% 0.29/0.55        ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.29/0.55        = ( ? [I3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 0.29/0.55              & ( ( nth_int @ Xs @ I3 )
% 0.29/0.55                = X ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % in_set_conv_nth
% 0.29/0.55  thf(fact_4234_list__ball__nth,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_nat,P: nat > $o] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.29/0.55       => ( ! [X3: nat] :
% 0.29/0.55              ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.29/0.55             => ( P @ X3 ) )
% 0.29/0.55         => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % list_ball_nth
% 0.29/0.55  thf(fact_4235_list__ball__nth,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.55       => ( ! [X3: vEBT_VEBT] :
% 0.29/0.55              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.55             => ( P @ X3 ) )
% 0.29/0.55         => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % list_ball_nth
% 0.29/0.55  thf(fact_4236_list__ball__nth,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_o,P: $o > $o] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.29/0.55       => ( ! [X3: $o] :
% 0.29/0.55              ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 0.29/0.55             => ( P @ X3 ) )
% 0.29/0.55         => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % list_ball_nth
% 0.29/0.55  thf(fact_4237_list__ball__nth,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_int,P: int > $o] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.29/0.55       => ( ! [X3: int] :
% 0.29/0.55              ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.29/0.55             => ( P @ X3 ) )
% 0.29/0.55         => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % list_ball_nth
% 0.29/0.55  thf(fact_4238_nth__mem,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_complex] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.29/0.55       => ( member_complex @ ( nth_complex @ Xs @ N ) @ ( set_complex2 @ Xs ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_mem
% 0.29/0.55  thf(fact_4239_nth__mem,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_real] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.29/0.55       => ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_mem
% 0.29/0.55  thf(fact_4240_nth__mem,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_set_nat] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.29/0.55       => ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_mem
% 0.29/0.55  thf(fact_4241_nth__mem,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_nat] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.29/0.55       => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_mem
% 0.29/0.55  thf(fact_4242_nth__mem,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_VEBT_VEBT] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.55       => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_mem
% 0.29/0.55  thf(fact_4243_nth__mem,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_o] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.29/0.55       => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_mem
% 0.29/0.55  thf(fact_4244_nth__mem,axiom,
% 0.29/0.55      ! [N: nat,Xs: list_int] :
% 0.29/0.55        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.29/0.55       => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nth_mem
% 0.29/0.55  thf(fact_4245_minf_I7_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.55       => ~ ( ord_less_real @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(7)
% 0.29/0.55  thf(fact_4246_minf_I7_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.55       => ~ ( ord_less_rat @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(7)
% 0.29/0.55  thf(fact_4247_minf_I7_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.55       => ~ ( ord_less_num @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(7)
% 0.29/0.55  thf(fact_4248_minf_I7_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.55       => ~ ( ord_less_nat @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(7)
% 0.29/0.55  thf(fact_4249_minf_I7_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.55       => ~ ( ord_less_int @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(7)
% 0.29/0.55  thf(fact_4250_minf_I5_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.55       => ( ord_less_real @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(5)
% 0.29/0.55  thf(fact_4251_minf_I5_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.55       => ( ord_less_rat @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(5)
% 0.29/0.55  thf(fact_4252_minf_I5_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.55       => ( ord_less_num @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(5)
% 0.29/0.55  thf(fact_4253_minf_I5_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.55       => ( ord_less_nat @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(5)
% 0.29/0.55  thf(fact_4254_minf_I5_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.55       => ( ord_less_int @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(5)
% 0.29/0.55  thf(fact_4255_minf_I4_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(4)
% 0.29/0.55  thf(fact_4256_minf_I4_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(4)
% 0.29/0.55  thf(fact_4257_minf_I4_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(4)
% 0.29/0.55  thf(fact_4258_minf_I4_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(4)
% 0.29/0.55  thf(fact_4259_minf_I4_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(4)
% 0.29/0.55  thf(fact_4260_minf_I3_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(3)
% 0.29/0.55  thf(fact_4261_minf_I3_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(3)
% 0.29/0.55  thf(fact_4262_minf_I3_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(3)
% 0.29/0.55  thf(fact_4263_minf_I3_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(3)
% 0.29/0.55  thf(fact_4264_minf_I3_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(3)
% 0.29/0.55  thf(fact_4265_minf_I2_J,axiom,
% 0.29/0.55      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.29/0.55        ( ? [Z5: real] :
% 0.29/0.55          ! [X3: real] :
% 0.29/0.55            ( ( ord_less_real @ X3 @ Z5 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: real] :
% 0.29/0.55            ! [X3: real] :
% 0.29/0.55              ( ( ord_less_real @ X3 @ Z5 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: real] :
% 0.29/0.55            ! [X2: real] :
% 0.29/0.55              ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  | ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(2)
% 0.29/0.55  thf(fact_4266_minf_I2_J,axiom,
% 0.29/0.55      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.29/0.55        ( ? [Z5: rat] :
% 0.29/0.55          ! [X3: rat] :
% 0.29/0.55            ( ( ord_less_rat @ X3 @ Z5 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: rat] :
% 0.29/0.55            ! [X3: rat] :
% 0.29/0.55              ( ( ord_less_rat @ X3 @ Z5 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: rat] :
% 0.29/0.55            ! [X2: rat] :
% 0.29/0.55              ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  | ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(2)
% 0.29/0.55  thf(fact_4267_minf_I2_J,axiom,
% 0.29/0.55      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.29/0.55        ( ? [Z5: num] :
% 0.29/0.55          ! [X3: num] :
% 0.29/0.55            ( ( ord_less_num @ X3 @ Z5 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: num] :
% 0.29/0.55            ! [X3: num] :
% 0.29/0.55              ( ( ord_less_num @ X3 @ Z5 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: num] :
% 0.29/0.55            ! [X2: num] :
% 0.29/0.55              ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  | ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(2)
% 0.29/0.55  thf(fact_4268_minf_I2_J,axiom,
% 0.29/0.55      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.29/0.55        ( ? [Z5: nat] :
% 0.29/0.55          ! [X3: nat] :
% 0.29/0.55            ( ( ord_less_nat @ X3 @ Z5 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: nat] :
% 0.29/0.55            ! [X3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ X3 @ Z5 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: nat] :
% 0.29/0.55            ! [X2: nat] :
% 0.29/0.55              ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  | ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(2)
% 0.29/0.55  thf(fact_4269_minf_I2_J,axiom,
% 0.29/0.55      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.29/0.55        ( ? [Z5: int] :
% 0.29/0.55          ! [X3: int] :
% 0.29/0.55            ( ( ord_less_int @ X3 @ Z5 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: int] :
% 0.29/0.55            ! [X3: int] :
% 0.29/0.55              ( ( ord_less_int @ X3 @ Z5 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: int] :
% 0.29/0.55            ! [X2: int] :
% 0.29/0.55              ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  | ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(2)
% 0.29/0.55  thf(fact_4270_minf_I1_J,axiom,
% 0.29/0.55      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.29/0.55        ( ? [Z5: real] :
% 0.29/0.55          ! [X3: real] :
% 0.29/0.55            ( ( ord_less_real @ X3 @ Z5 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: real] :
% 0.29/0.55            ! [X3: real] :
% 0.29/0.55              ( ( ord_less_real @ X3 @ Z5 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: real] :
% 0.29/0.55            ! [X2: real] :
% 0.29/0.55              ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  & ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(1)
% 0.29/0.55  thf(fact_4271_minf_I1_J,axiom,
% 0.29/0.55      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.29/0.55        ( ? [Z5: rat] :
% 0.29/0.55          ! [X3: rat] :
% 0.29/0.55            ( ( ord_less_rat @ X3 @ Z5 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: rat] :
% 0.29/0.55            ! [X3: rat] :
% 0.29/0.55              ( ( ord_less_rat @ X3 @ Z5 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: rat] :
% 0.29/0.55            ! [X2: rat] :
% 0.29/0.55              ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  & ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(1)
% 0.29/0.55  thf(fact_4272_minf_I1_J,axiom,
% 0.29/0.55      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.29/0.55        ( ? [Z5: num] :
% 0.29/0.55          ! [X3: num] :
% 0.29/0.55            ( ( ord_less_num @ X3 @ Z5 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: num] :
% 0.29/0.55            ! [X3: num] :
% 0.29/0.55              ( ( ord_less_num @ X3 @ Z5 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: num] :
% 0.29/0.55            ! [X2: num] :
% 0.29/0.55              ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  & ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(1)
% 0.29/0.55  thf(fact_4273_minf_I1_J,axiom,
% 0.29/0.55      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.29/0.55        ( ? [Z5: nat] :
% 0.29/0.55          ! [X3: nat] :
% 0.29/0.55            ( ( ord_less_nat @ X3 @ Z5 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: nat] :
% 0.29/0.55            ! [X3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ X3 @ Z5 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: nat] :
% 0.29/0.55            ! [X2: nat] :
% 0.29/0.55              ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  & ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(1)
% 0.29/0.55  thf(fact_4274_minf_I1_J,axiom,
% 0.29/0.55      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.29/0.55        ( ? [Z5: int] :
% 0.29/0.55          ! [X3: int] :
% 0.29/0.55            ( ( ord_less_int @ X3 @ Z5 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: int] :
% 0.29/0.55            ! [X3: int] :
% 0.29/0.55              ( ( ord_less_int @ X3 @ Z5 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: int] :
% 0.29/0.55            ! [X2: int] :
% 0.29/0.55              ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  & ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(1)
% 0.29/0.55  thf(fact_4275_pinf_I7_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.55       => ( ord_less_real @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(7)
% 0.29/0.55  thf(fact_4276_pinf_I7_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.55       => ( ord_less_rat @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(7)
% 0.29/0.55  thf(fact_4277_pinf_I7_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.55       => ( ord_less_num @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(7)
% 0.29/0.55  thf(fact_4278_pinf_I7_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.55       => ( ord_less_nat @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(7)
% 0.29/0.55  thf(fact_4279_pinf_I7_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.55       => ( ord_less_int @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(7)
% 0.29/0.55  thf(fact_4280_pinf_I5_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.55       => ~ ( ord_less_real @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(5)
% 0.29/0.55  thf(fact_4281_pinf_I5_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.55       => ~ ( ord_less_rat @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(5)
% 0.29/0.55  thf(fact_4282_pinf_I5_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.55       => ~ ( ord_less_num @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(5)
% 0.29/0.55  thf(fact_4283_pinf_I5_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.55       => ~ ( ord_less_nat @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(5)
% 0.29/0.55  thf(fact_4284_pinf_I5_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.55       => ~ ( ord_less_int @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(5)
% 0.29/0.55  thf(fact_4285_pinf_I4_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(4)
% 0.29/0.55  thf(fact_4286_pinf_I4_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(4)
% 0.29/0.55  thf(fact_4287_pinf_I4_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(4)
% 0.29/0.55  thf(fact_4288_pinf_I4_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(4)
% 0.29/0.55  thf(fact_4289_pinf_I4_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(4)
% 0.29/0.55  thf(fact_4290_pinf_I3_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(3)
% 0.29/0.55  thf(fact_4291_pinf_I3_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(3)
% 0.29/0.55  thf(fact_4292_pinf_I3_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(3)
% 0.29/0.55  thf(fact_4293_pinf_I3_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(3)
% 0.29/0.55  thf(fact_4294_pinf_I3_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.55       => ( X2 != T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(3)
% 0.29/0.55  thf(fact_4295_pinf_I2_J,axiom,
% 0.29/0.55      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.29/0.55        ( ? [Z5: real] :
% 0.29/0.55          ! [X3: real] :
% 0.29/0.55            ( ( ord_less_real @ Z5 @ X3 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: real] :
% 0.29/0.55            ! [X3: real] :
% 0.29/0.55              ( ( ord_less_real @ Z5 @ X3 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: real] :
% 0.29/0.55            ! [X2: real] :
% 0.29/0.55              ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  | ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(2)
% 0.29/0.55  thf(fact_4296_pinf_I2_J,axiom,
% 0.29/0.55      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.29/0.55        ( ? [Z5: rat] :
% 0.29/0.55          ! [X3: rat] :
% 0.29/0.55            ( ( ord_less_rat @ Z5 @ X3 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: rat] :
% 0.29/0.55            ! [X3: rat] :
% 0.29/0.55              ( ( ord_less_rat @ Z5 @ X3 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: rat] :
% 0.29/0.55            ! [X2: rat] :
% 0.29/0.55              ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  | ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(2)
% 0.29/0.55  thf(fact_4297_pinf_I2_J,axiom,
% 0.29/0.55      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.29/0.55        ( ? [Z5: num] :
% 0.29/0.55          ! [X3: num] :
% 0.29/0.55            ( ( ord_less_num @ Z5 @ X3 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: num] :
% 0.29/0.55            ! [X3: num] :
% 0.29/0.55              ( ( ord_less_num @ Z5 @ X3 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: num] :
% 0.29/0.55            ! [X2: num] :
% 0.29/0.55              ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  | ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(2)
% 0.29/0.55  thf(fact_4298_pinf_I2_J,axiom,
% 0.29/0.55      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.29/0.55        ( ? [Z5: nat] :
% 0.29/0.55          ! [X3: nat] :
% 0.29/0.55            ( ( ord_less_nat @ Z5 @ X3 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: nat] :
% 0.29/0.55            ! [X3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ Z5 @ X3 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: nat] :
% 0.29/0.55            ! [X2: nat] :
% 0.29/0.55              ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  | ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(2)
% 0.29/0.55  thf(fact_4299_pinf_I2_J,axiom,
% 0.29/0.55      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.29/0.55        ( ? [Z5: int] :
% 0.29/0.55          ! [X3: int] :
% 0.29/0.55            ( ( ord_less_int @ Z5 @ X3 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: int] :
% 0.29/0.55            ! [X3: int] :
% 0.29/0.55              ( ( ord_less_int @ Z5 @ X3 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: int] :
% 0.29/0.55            ! [X2: int] :
% 0.29/0.55              ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  | ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(2)
% 0.29/0.55  thf(fact_4300_pinf_I1_J,axiom,
% 0.29/0.55      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.29/0.55        ( ? [Z5: real] :
% 0.29/0.55          ! [X3: real] :
% 0.29/0.55            ( ( ord_less_real @ Z5 @ X3 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: real] :
% 0.29/0.55            ! [X3: real] :
% 0.29/0.55              ( ( ord_less_real @ Z5 @ X3 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: real] :
% 0.29/0.55            ! [X2: real] :
% 0.29/0.55              ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  & ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(1)
% 0.29/0.55  thf(fact_4301_pinf_I1_J,axiom,
% 0.29/0.55      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.29/0.55        ( ? [Z5: rat] :
% 0.29/0.55          ! [X3: rat] :
% 0.29/0.55            ( ( ord_less_rat @ Z5 @ X3 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: rat] :
% 0.29/0.55            ! [X3: rat] :
% 0.29/0.55              ( ( ord_less_rat @ Z5 @ X3 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: rat] :
% 0.29/0.55            ! [X2: rat] :
% 0.29/0.55              ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  & ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(1)
% 0.29/0.55  thf(fact_4302_pinf_I1_J,axiom,
% 0.29/0.55      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.29/0.55        ( ? [Z5: num] :
% 0.29/0.55          ! [X3: num] :
% 0.29/0.55            ( ( ord_less_num @ Z5 @ X3 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: num] :
% 0.29/0.55            ! [X3: num] :
% 0.29/0.55              ( ( ord_less_num @ Z5 @ X3 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: num] :
% 0.29/0.55            ! [X2: num] :
% 0.29/0.55              ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  & ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(1)
% 0.29/0.55  thf(fact_4303_pinf_I1_J,axiom,
% 0.29/0.55      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.29/0.55        ( ? [Z5: nat] :
% 0.29/0.55          ! [X3: nat] :
% 0.29/0.55            ( ( ord_less_nat @ Z5 @ X3 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: nat] :
% 0.29/0.55            ! [X3: nat] :
% 0.29/0.55              ( ( ord_less_nat @ Z5 @ X3 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: nat] :
% 0.29/0.55            ! [X2: nat] :
% 0.29/0.55              ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  & ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(1)
% 0.29/0.55  thf(fact_4304_pinf_I1_J,axiom,
% 0.29/0.55      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.29/0.55        ( ? [Z5: int] :
% 0.29/0.55          ! [X3: int] :
% 0.29/0.55            ( ( ord_less_int @ Z5 @ X3 )
% 0.29/0.55           => ( ( P @ X3 )
% 0.29/0.55              = ( P6 @ X3 ) ) )
% 0.29/0.55       => ( ? [Z5: int] :
% 0.29/0.55            ! [X3: int] :
% 0.29/0.55              ( ( ord_less_int @ Z5 @ X3 )
% 0.29/0.55             => ( ( Q @ X3 )
% 0.29/0.55                = ( Q6 @ X3 ) ) )
% 0.29/0.55         => ? [Z3: int] :
% 0.29/0.55            ! [X2: int] :
% 0.29/0.55              ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.55             => ( ( ( P @ X2 )
% 0.29/0.55                  & ( Q @ X2 ) )
% 0.29/0.55                = ( ( P6 @ X2 )
% 0.29/0.55                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(1)
% 0.29/0.55  thf(fact_4305_minf_I8_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.55       => ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(8)
% 0.29/0.55  thf(fact_4306_minf_I8_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.55       => ~ ( ord_less_eq_rat @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(8)
% 0.29/0.55  thf(fact_4307_minf_I8_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.55       => ~ ( ord_less_eq_num @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(8)
% 0.29/0.55  thf(fact_4308_minf_I8_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.55       => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(8)
% 0.29/0.55  thf(fact_4309_minf_I8_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.55       => ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(8)
% 0.29/0.55  thf(fact_4310_minf_I6_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.55       => ( ord_less_eq_real @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(6)
% 0.29/0.55  thf(fact_4311_minf_I6_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.55       => ( ord_less_eq_rat @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(6)
% 0.29/0.55  thf(fact_4312_minf_I6_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.55       => ( ord_less_eq_num @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(6)
% 0.29/0.55  thf(fact_4313_minf_I6_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.55       => ( ord_less_eq_nat @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(6)
% 0.29/0.55  thf(fact_4314_minf_I6_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.55       => ( ord_less_eq_int @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minf(6)
% 0.29/0.55  thf(fact_4315_pinf_I8_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.55       => ( ord_less_eq_real @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(8)
% 0.29/0.55  thf(fact_4316_pinf_I8_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.55       => ( ord_less_eq_rat @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(8)
% 0.29/0.55  thf(fact_4317_pinf_I8_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.55       => ( ord_less_eq_num @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(8)
% 0.29/0.55  thf(fact_4318_pinf_I8_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.55       => ( ord_less_eq_nat @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(8)
% 0.29/0.55  thf(fact_4319_pinf_I8_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.55       => ( ord_less_eq_int @ T @ X2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(8)
% 0.29/0.55  thf(fact_4320_pinf_I6_J,axiom,
% 0.29/0.55      ! [T: real] :
% 0.29/0.55      ? [Z3: real] :
% 0.29/0.55      ! [X2: real] :
% 0.29/0.55        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.55       => ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(6)
% 0.29/0.55  thf(fact_4321_pinf_I6_J,axiom,
% 0.29/0.55      ! [T: rat] :
% 0.29/0.55      ? [Z3: rat] :
% 0.29/0.55      ! [X2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.55       => ~ ( ord_less_eq_rat @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(6)
% 0.29/0.55  thf(fact_4322_pinf_I6_J,axiom,
% 0.29/0.55      ! [T: num] :
% 0.29/0.55      ? [Z3: num] :
% 0.29/0.55      ! [X2: num] :
% 0.29/0.55        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.55       => ~ ( ord_less_eq_num @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(6)
% 0.29/0.55  thf(fact_4323_pinf_I6_J,axiom,
% 0.29/0.55      ! [T: nat] :
% 0.29/0.55      ? [Z3: nat] :
% 0.29/0.55      ! [X2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.55       => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(6)
% 0.29/0.55  thf(fact_4324_pinf_I6_J,axiom,
% 0.29/0.55      ! [T: int] :
% 0.29/0.55      ? [Z3: int] :
% 0.29/0.55      ! [X2: int] :
% 0.29/0.55        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.55       => ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pinf(6)
% 0.29/0.55  thf(fact_4325_imp__le__cong,axiom,
% 0.29/0.55      ! [X: int,X7: int,P: $o,P6: $o] :
% 0.29/0.55        ( ( X = X7 )
% 0.29/0.55       => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.29/0.55           => ( P = P6 ) )
% 0.29/0.55         => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.55             => P )
% 0.29/0.55            = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.29/0.55             => P6 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % imp_le_cong
% 0.29/0.55  thf(fact_4326_conj__le__cong,axiom,
% 0.29/0.55      ! [X: int,X7: int,P: $o,P6: $o] :
% 0.29/0.55        ( ( X = X7 )
% 0.29/0.55       => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.29/0.55           => ( P = P6 ) )
% 0.29/0.55         => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.55              & P )
% 0.29/0.55            = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.29/0.55              & P6 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % conj_le_cong
% 0.29/0.55  thf(fact_4327_Bolzano,axiom,
% 0.29/0.55      ! [A: real,B: real,P: real > real > $o] :
% 0.29/0.55        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.55       => ( ! [A5: real,B6: real,C4: real] :
% 0.29/0.55              ( ( P @ A5 @ B6 )
% 0.29/0.55             => ( ( P @ B6 @ C4 )
% 0.29/0.55               => ( ( ord_less_eq_real @ A5 @ B6 )
% 0.29/0.55                 => ( ( ord_less_eq_real @ B6 @ C4 )
% 0.29/0.55                   => ( P @ A5 @ C4 ) ) ) ) )
% 0.29/0.55         => ( ! [X3: real] :
% 0.29/0.55                ( ( ord_less_eq_real @ A @ X3 )
% 0.29/0.55               => ( ( ord_less_eq_real @ X3 @ B )
% 0.29/0.55                 => ? [D4: real] :
% 0.29/0.55                      ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.29/0.55                      & ! [A5: real,B6: real] :
% 0.29/0.55                          ( ( ( ord_less_eq_real @ A5 @ X3 )
% 0.29/0.55                            & ( ord_less_eq_real @ X3 @ B6 )
% 0.29/0.55                            & ( ord_less_real @ ( minus_minus_real @ B6 @ A5 ) @ D4 ) )
% 0.29/0.55                         => ( P @ A5 @ B6 ) ) ) ) )
% 0.29/0.55           => ( P @ A @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Bolzano
% 0.29/0.55  thf(fact_4328_minusinfinity,axiom,
% 0.29/0.55      ! [D: int,P1: int > $o,P: int > $o] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.55       => ( ! [X3: int,K3: int] :
% 0.29/0.55              ( ( P1 @ X3 )
% 0.29/0.55              = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.29/0.55         => ( ? [Z5: int] :
% 0.29/0.55              ! [X3: int] :
% 0.29/0.55                ( ( ord_less_int @ X3 @ Z5 )
% 0.29/0.55               => ( ( P @ X3 )
% 0.29/0.55                  = ( P1 @ X3 ) ) )
% 0.29/0.55           => ( ? [X_1: int] : ( P1 @ X_1 )
% 0.29/0.55             => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minusinfinity
% 0.29/0.55  thf(fact_4329_plusinfinity,axiom,
% 0.29/0.55      ! [D: int,P6: int > $o,P: int > $o] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.55       => ( ! [X3: int,K3: int] :
% 0.29/0.55              ( ( P6 @ X3 )
% 0.29/0.55              = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.29/0.55         => ( ? [Z5: int] :
% 0.29/0.55              ! [X3: int] :
% 0.29/0.55                ( ( ord_less_int @ Z5 @ X3 )
% 0.29/0.55               => ( ( P @ X3 )
% 0.29/0.55                  = ( P6 @ X3 ) ) )
% 0.29/0.55           => ( ? [X_1: int] : ( P6 @ X_1 )
% 0.29/0.55             => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % plusinfinity
% 0.29/0.55  thf(fact_4330_in__children__def,axiom,
% 0.29/0.55      ( vEBT_V5917875025757280293ildren
% 0.29/0.55      = ( ^ [N3: nat,TreeList2: list_VEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ N3 ) ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % in_children_def
% 0.29/0.55  thf(fact_4331_low__def,axiom,
% 0.29/0.55      ( vEBT_VEBT_low
% 0.29/0.55      = ( ^ [X4: nat,N3: nat] : ( modulo_modulo_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % low_def
% 0.29/0.55  thf(fact_4332_tanh__ln__real,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 0.29/0.55          = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % tanh_ln_real
% 0.29/0.55  thf(fact_4333_ln__one__minus__pos__lower__bound,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.55       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.55         => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % ln_one_minus_pos_lower_bound
% 0.29/0.55  thf(fact_4334_arcosh__def,axiom,
% 0.29/0.55      ( arcosh_real
% 0.29/0.55      = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % arcosh_def
% 0.29/0.55  thf(fact_4335_abs__ln__one__plus__x__minus__x__bound,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.55       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_ln_one_plus_x_minus_x_bound
% 0.29/0.55  thf(fact_4336_add_Oinverse__inverse,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.29/0.55        = A ) ).
% 0.29/0.55  
% 0.29/0.55  % add.inverse_inverse
% 0.29/0.55  thf(fact_4337_add_Oinverse__inverse,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.29/0.55        = A ) ).
% 0.29/0.55  
% 0.29/0.55  % add.inverse_inverse
% 0.29/0.55  thf(fact_4338_add_Oinverse__inverse,axiom,
% 0.29/0.55      ! [A: complex] :
% 0.29/0.55        ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.29/0.55        = A ) ).
% 0.29/0.55  
% 0.29/0.55  % add.inverse_inverse
% 0.29/0.55  thf(fact_4339_add_Oinverse__inverse,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 0.29/0.55        = A ) ).
% 0.29/0.55  
% 0.29/0.55  % add.inverse_inverse
% 0.29/0.55  thf(fact_4340_add_Oinverse__inverse,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.55        = A ) ).
% 0.29/0.55  
% 0.29/0.55  % add.inverse_inverse
% 0.29/0.55  thf(fact_4341_neg__equal__iff__equal,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( ( uminus_uminus_real @ A )
% 0.29/0.55          = ( uminus_uminus_real @ B ) )
% 0.29/0.55        = ( A = B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_iff_equal
% 0.29/0.55  thf(fact_4342_neg__equal__iff__equal,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ( uminus_uminus_int @ A )
% 0.29/0.55          = ( uminus_uminus_int @ B ) )
% 0.29/0.55        = ( A = B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_iff_equal
% 0.29/0.55  thf(fact_4343_neg__equal__iff__equal,axiom,
% 0.29/0.55      ! [A: complex,B: complex] :
% 0.29/0.55        ( ( ( uminus1482373934393186551omplex @ A )
% 0.29/0.55          = ( uminus1482373934393186551omplex @ B ) )
% 0.29/0.55        = ( A = B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_iff_equal
% 0.29/0.55  thf(fact_4344_neg__equal__iff__equal,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( ( uminus_uminus_rat @ A )
% 0.29/0.55          = ( uminus_uminus_rat @ B ) )
% 0.29/0.55        = ( A = B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_iff_equal
% 0.29/0.55  thf(fact_4345_neg__equal__iff__equal,axiom,
% 0.29/0.55      ! [A: code_integer,B: code_integer] :
% 0.29/0.55        ( ( ( uminus1351360451143612070nteger @ A )
% 0.29/0.55          = ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.55        = ( A = B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_iff_equal
% 0.29/0.55  thf(fact_4346_verit__minus__simplify_I4_J,axiom,
% 0.29/0.55      ! [B: real] :
% 0.29/0.55        ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % verit_minus_simplify(4)
% 0.29/0.55  thf(fact_4347_verit__minus__simplify_I4_J,axiom,
% 0.29/0.55      ! [B: int] :
% 0.29/0.55        ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % verit_minus_simplify(4)
% 0.29/0.55  thf(fact_4348_verit__minus__simplify_I4_J,axiom,
% 0.29/0.55      ! [B: complex] :
% 0.29/0.55        ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % verit_minus_simplify(4)
% 0.29/0.55  thf(fact_4349_verit__minus__simplify_I4_J,axiom,
% 0.29/0.55      ! [B: rat] :
% 0.29/0.55        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % verit_minus_simplify(4)
% 0.29/0.55  thf(fact_4350_verit__minus__simplify_I4_J,axiom,
% 0.29/0.55      ! [B: code_integer] :
% 0.29/0.55        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % verit_minus_simplify(4)
% 0.29/0.55  thf(fact_4351_abs__idempotent,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.29/0.55        = ( abs_abs_real @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_idempotent
% 0.29/0.55  thf(fact_4352_abs__idempotent,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.29/0.55        = ( abs_abs_int @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_idempotent
% 0.29/0.55  thf(fact_4353_abs__idempotent,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.29/0.55        = ( abs_abs_rat @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_idempotent
% 0.29/0.55  thf(fact_4354_abs__idempotent,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.29/0.55        = ( abs_abs_Code_integer @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_idempotent
% 0.29/0.55  thf(fact_4355_arsinh__minus__real,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( arsinh_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.55        = ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % arsinh_minus_real
% 0.29/0.55  thf(fact_4356_VEBT_Oinject_I2_J,axiom,
% 0.29/0.55      ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
% 0.29/0.55        ( ( ( vEBT_Leaf @ X21 @ X222 )
% 0.29/0.55          = ( vEBT_Leaf @ Y21 @ Y22 ) )
% 0.29/0.55        = ( ( X21 = Y21 )
% 0.29/0.55          & ( X222 = Y22 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % VEBT.inject(2)
% 0.29/0.55  thf(fact_4357_tanh__real__eq__iff,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( ( tanh_real @ X )
% 0.29/0.55          = ( tanh_real @ Y ) )
% 0.29/0.55        = ( X = Y ) ) ).
% 0.29/0.55  
% 0.29/0.55  % tanh_real_eq_iff
% 0.29/0.55  thf(fact_4358_neg__le__iff__le,axiom,
% 0.29/0.55      ! [B: real,A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.29/0.55        = ( ord_less_eq_real @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_le_iff_le
% 0.29/0.55  thf(fact_4359_neg__le__iff__le,axiom,
% 0.29/0.55      ! [B: code_integer,A: code_integer] :
% 0.29/0.55        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.55        = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_le_iff_le
% 0.29/0.55  thf(fact_4360_neg__le__iff__le,axiom,
% 0.29/0.55      ! [B: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.29/0.55        = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_le_iff_le
% 0.29/0.55  thf(fact_4361_neg__le__iff__le,axiom,
% 0.29/0.55      ! [B: int,A: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.29/0.55        = ( ord_less_eq_int @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_le_iff_le
% 0.29/0.55  thf(fact_4362_add_Oinverse__neutral,axiom,
% 0.29/0.55      ( ( uminus_uminus_real @ zero_zero_real )
% 0.29/0.55      = zero_zero_real ) ).
% 0.29/0.55  
% 0.29/0.55  % add.inverse_neutral
% 0.29/0.55  thf(fact_4363_add_Oinverse__neutral,axiom,
% 0.29/0.55      ( ( uminus_uminus_int @ zero_zero_int )
% 0.29/0.55      = zero_zero_int ) ).
% 0.29/0.55  
% 0.29/0.55  % add.inverse_neutral
% 0.29/0.55  thf(fact_4364_add_Oinverse__neutral,axiom,
% 0.29/0.55      ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 0.29/0.55      = zero_zero_complex ) ).
% 0.29/0.55  
% 0.29/0.55  % add.inverse_neutral
% 0.29/0.55  thf(fact_4365_add_Oinverse__neutral,axiom,
% 0.29/0.55      ( ( uminus_uminus_rat @ zero_zero_rat )
% 0.29/0.55      = zero_zero_rat ) ).
% 0.29/0.55  
% 0.29/0.55  % add.inverse_neutral
% 0.29/0.55  thf(fact_4366_add_Oinverse__neutral,axiom,
% 0.29/0.55      ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.29/0.55      = zero_z3403309356797280102nteger ) ).
% 0.29/0.55  
% 0.29/0.55  % add.inverse_neutral
% 0.29/0.55  thf(fact_4367_neg__0__equal__iff__equal,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( zero_zero_real
% 0.29/0.55          = ( uminus_uminus_real @ A ) )
% 0.29/0.55        = ( zero_zero_real = A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_0_equal_iff_equal
% 0.29/0.55  thf(fact_4368_neg__0__equal__iff__equal,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( zero_zero_int
% 0.29/0.55          = ( uminus_uminus_int @ A ) )
% 0.29/0.55        = ( zero_zero_int = A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_0_equal_iff_equal
% 0.29/0.55  thf(fact_4369_neg__0__equal__iff__equal,axiom,
% 0.29/0.55      ! [A: complex] :
% 0.29/0.55        ( ( zero_zero_complex
% 0.29/0.55          = ( uminus1482373934393186551omplex @ A ) )
% 0.29/0.55        = ( zero_zero_complex = A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_0_equal_iff_equal
% 0.29/0.55  thf(fact_4370_neg__0__equal__iff__equal,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( zero_zero_rat
% 0.29/0.55          = ( uminus_uminus_rat @ A ) )
% 0.29/0.55        = ( zero_zero_rat = A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_0_equal_iff_equal
% 0.29/0.55  thf(fact_4371_neg__0__equal__iff__equal,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( zero_z3403309356797280102nteger
% 0.29/0.55          = ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.55        = ( zero_z3403309356797280102nteger = A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_0_equal_iff_equal
% 0.29/0.55  thf(fact_4372_neg__equal__0__iff__equal,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( ( uminus_uminus_real @ A )
% 0.29/0.55          = zero_zero_real )
% 0.29/0.55        = ( A = zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_0_iff_equal
% 0.29/0.55  thf(fact_4373_neg__equal__0__iff__equal,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( ( uminus_uminus_int @ A )
% 0.29/0.55          = zero_zero_int )
% 0.29/0.55        = ( A = zero_zero_int ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_0_iff_equal
% 0.29/0.55  thf(fact_4374_neg__equal__0__iff__equal,axiom,
% 0.29/0.55      ! [A: complex] :
% 0.29/0.55        ( ( ( uminus1482373934393186551omplex @ A )
% 0.29/0.55          = zero_zero_complex )
% 0.29/0.55        = ( A = zero_zero_complex ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_0_iff_equal
% 0.29/0.55  thf(fact_4375_neg__equal__0__iff__equal,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( ( uminus_uminus_rat @ A )
% 0.29/0.55          = zero_zero_rat )
% 0.29/0.55        = ( A = zero_zero_rat ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_0_iff_equal
% 0.29/0.55  thf(fact_4376_neg__equal__0__iff__equal,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( ( uminus1351360451143612070nteger @ A )
% 0.29/0.55          = zero_z3403309356797280102nteger )
% 0.29/0.55        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_0_iff_equal
% 0.29/0.55  thf(fact_4377_equal__neg__zero,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( A
% 0.29/0.55          = ( uminus_uminus_real @ A ) )
% 0.29/0.55        = ( A = zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % equal_neg_zero
% 0.29/0.55  thf(fact_4378_equal__neg__zero,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( A
% 0.29/0.55          = ( uminus_uminus_int @ A ) )
% 0.29/0.55        = ( A = zero_zero_int ) ) ).
% 0.29/0.55  
% 0.29/0.55  % equal_neg_zero
% 0.29/0.55  thf(fact_4379_equal__neg__zero,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( A
% 0.29/0.55          = ( uminus_uminus_rat @ A ) )
% 0.29/0.55        = ( A = zero_zero_rat ) ) ).
% 0.29/0.55  
% 0.29/0.55  % equal_neg_zero
% 0.29/0.55  thf(fact_4380_equal__neg__zero,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( A
% 0.29/0.55          = ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.55        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.55  
% 0.29/0.55  % equal_neg_zero
% 0.29/0.55  thf(fact_4381_neg__equal__zero,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( ( uminus_uminus_real @ A )
% 0.29/0.55          = A )
% 0.29/0.55        = ( A = zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_zero
% 0.29/0.55  thf(fact_4382_neg__equal__zero,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( ( uminus_uminus_int @ A )
% 0.29/0.55          = A )
% 0.29/0.55        = ( A = zero_zero_int ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_zero
% 0.29/0.55  thf(fact_4383_neg__equal__zero,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( ( uminus_uminus_rat @ A )
% 0.29/0.55          = A )
% 0.29/0.55        = ( A = zero_zero_rat ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_zero
% 0.29/0.55  thf(fact_4384_neg__equal__zero,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( ( uminus1351360451143612070nteger @ A )
% 0.29/0.55          = A )
% 0.29/0.55        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_equal_zero
% 0.29/0.55  thf(fact_4385_neg__less__iff__less,axiom,
% 0.29/0.55      ! [B: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.29/0.55        = ( ord_less_real @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_less_iff_less
% 0.29/0.55  thf(fact_4386_neg__less__iff__less,axiom,
% 0.29/0.55      ! [B: int,A: int] :
% 0.29/0.55        ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.29/0.55        = ( ord_less_int @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_less_iff_less
% 0.29/0.55  thf(fact_4387_neg__less__iff__less,axiom,
% 0.29/0.55      ! [B: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.29/0.55        = ( ord_less_rat @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_less_iff_less
% 0.29/0.55  thf(fact_4388_neg__less__iff__less,axiom,
% 0.29/0.55      ! [B: code_integer,A: code_integer] :
% 0.29/0.55        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.55        = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_less_iff_less
% 0.29/0.55  thf(fact_4389_neg__numeral__eq__iff,axiom,
% 0.29/0.55      ! [M: num,N: num] :
% 0.29/0.55        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.29/0.55          = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.55        = ( M = N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_numeral_eq_iff
% 0.29/0.55  thf(fact_4390_neg__numeral__eq__iff,axiom,
% 0.29/0.55      ! [M: num,N: num] :
% 0.29/0.55        ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.29/0.55          = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.55        = ( M = N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_numeral_eq_iff
% 0.29/0.55  thf(fact_4391_neg__numeral__eq__iff,axiom,
% 0.29/0.55      ! [M: num,N: num] :
% 0.29/0.55        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.29/0.55          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.29/0.55        = ( M = N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_numeral_eq_iff
% 0.29/0.55  thf(fact_4392_neg__numeral__eq__iff,axiom,
% 0.29/0.55      ! [M: num,N: num] :
% 0.29/0.55        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.29/0.55          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.55        = ( M = N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_numeral_eq_iff
% 0.29/0.55  thf(fact_4393_neg__numeral__eq__iff,axiom,
% 0.29/0.55      ! [M: num,N: num] :
% 0.29/0.55        ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.29/0.55          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.55        = ( M = N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_numeral_eq_iff
% 0.29/0.55  thf(fact_4394_add__minus__cancel,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % add_minus_cancel
% 0.29/0.55  thf(fact_4395_add__minus__cancel,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % add_minus_cancel
% 0.29/0.55  thf(fact_4396_add__minus__cancel,axiom,
% 0.29/0.55      ! [A: complex,B: complex] :
% 0.29/0.55        ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % add_minus_cancel
% 0.29/0.55  thf(fact_4397_add__minus__cancel,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % add_minus_cancel
% 0.29/0.55  thf(fact_4398_add__minus__cancel,axiom,
% 0.29/0.55      ! [A: code_integer,B: code_integer] :
% 0.29/0.55        ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % add_minus_cancel
% 0.29/0.55  thf(fact_4399_minus__add__cancel,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_add_cancel
% 0.29/0.55  thf(fact_4400_minus__add__cancel,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_add_cancel
% 0.29/0.55  thf(fact_4401_minus__add__cancel,axiom,
% 0.29/0.55      ! [A: complex,B: complex] :
% 0.29/0.55        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_add_cancel
% 0.29/0.55  thf(fact_4402_minus__add__cancel,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_add_cancel
% 0.29/0.55  thf(fact_4403_minus__add__cancel,axiom,
% 0.29/0.55      ! [A: code_integer,B: code_integer] :
% 0.29/0.55        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.29/0.55        = B ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_add_cancel
% 0.29/0.55  thf(fact_4404_minus__add__distrib,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.29/0.55        = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_add_distrib
% 0.29/0.55  thf(fact_4405_minus__add__distrib,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.29/0.55        = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_add_distrib
% 0.29/0.55  thf(fact_4406_minus__add__distrib,axiom,
% 0.29/0.55      ! [A: complex,B: complex] :
% 0.29/0.55        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.29/0.55        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_add_distrib
% 0.29/0.55  thf(fact_4407_minus__add__distrib,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.55        = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_add_distrib
% 0.29/0.55  thf(fact_4408_minus__add__distrib,axiom,
% 0.29/0.55      ! [A: code_integer,B: code_integer] :
% 0.29/0.55        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.29/0.55        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_add_distrib
% 0.29/0.55  thf(fact_4409_minus__diff__eq,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 0.29/0.55        = ( minus_minus_real @ B @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_diff_eq
% 0.29/0.55  thf(fact_4410_minus__diff__eq,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.29/0.55        = ( minus_minus_int @ B @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_diff_eq
% 0.29/0.55  thf(fact_4411_minus__diff__eq,axiom,
% 0.29/0.55      ! [A: complex,B: complex] :
% 0.29/0.55        ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 0.29/0.55        = ( minus_minus_complex @ B @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_diff_eq
% 0.29/0.55  thf(fact_4412_minus__diff__eq,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 0.29/0.55        = ( minus_minus_rat @ B @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_diff_eq
% 0.29/0.55  thf(fact_4413_minus__diff__eq,axiom,
% 0.29/0.55      ! [A: code_integer,B: code_integer] :
% 0.29/0.55        ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.29/0.55        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % minus_diff_eq
% 0.29/0.55  thf(fact_4414_div__minus__minus,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.29/0.55        = ( divide_divide_int @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_minus_minus
% 0.29/0.55  thf(fact_4415_div__minus__minus,axiom,
% 0.29/0.55      ! [A: code_integer,B: code_integer] :
% 0.29/0.55        ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.55        = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_minus_minus
% 0.29/0.55  thf(fact_4416_abs__0__eq,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( zero_z3403309356797280102nteger
% 0.29/0.55          = ( abs_abs_Code_integer @ A ) )
% 0.29/0.55        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_0_eq
% 0.29/0.55  thf(fact_4417_abs__0__eq,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( zero_zero_real
% 0.29/0.55          = ( abs_abs_real @ A ) )
% 0.29/0.55        = ( A = zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_0_eq
% 0.29/0.55  thf(fact_4418_abs__0__eq,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( zero_zero_rat
% 0.29/0.55          = ( abs_abs_rat @ A ) )
% 0.29/0.55        = ( A = zero_zero_rat ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_0_eq
% 0.29/0.55  thf(fact_4419_abs__0__eq,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( zero_zero_int
% 0.29/0.55          = ( abs_abs_int @ A ) )
% 0.29/0.55        = ( A = zero_zero_int ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_0_eq
% 0.29/0.55  thf(fact_4420_abs__eq__0,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( ( abs_abs_Code_integer @ A )
% 0.29/0.55          = zero_z3403309356797280102nteger )
% 0.29/0.55        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_eq_0
% 0.29/0.55  thf(fact_4421_abs__eq__0,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( ( abs_abs_real @ A )
% 0.29/0.55          = zero_zero_real )
% 0.29/0.55        = ( A = zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_eq_0
% 0.29/0.55  thf(fact_4422_abs__eq__0,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( ( abs_abs_rat @ A )
% 0.29/0.55          = zero_zero_rat )
% 0.29/0.55        = ( A = zero_zero_rat ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_eq_0
% 0.29/0.55  thf(fact_4423_abs__eq__0,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( ( abs_abs_int @ A )
% 0.29/0.55          = zero_zero_int )
% 0.29/0.55        = ( A = zero_zero_int ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_eq_0
% 0.29/0.55  thf(fact_4424_abs__zero,axiom,
% 0.29/0.55      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.29/0.55      = zero_z3403309356797280102nteger ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_zero
% 0.29/0.55  thf(fact_4425_abs__zero,axiom,
% 0.29/0.55      ( ( abs_abs_real @ zero_zero_real )
% 0.29/0.55      = zero_zero_real ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_zero
% 0.29/0.55  thf(fact_4426_abs__zero,axiom,
% 0.29/0.55      ( ( abs_abs_rat @ zero_zero_rat )
% 0.29/0.55      = zero_zero_rat ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_zero
% 0.29/0.55  thf(fact_4427_abs__zero,axiom,
% 0.29/0.55      ( ( abs_abs_int @ zero_zero_int )
% 0.29/0.55      = zero_zero_int ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_zero
% 0.29/0.55  thf(fact_4428_abs__0,axiom,
% 0.29/0.55      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.29/0.55      = zero_z3403309356797280102nteger ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_0
% 0.29/0.55  thf(fact_4429_abs__0,axiom,
% 0.29/0.55      ( ( abs_abs_complex @ zero_zero_complex )
% 0.29/0.55      = zero_zero_complex ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_0
% 0.29/0.55  thf(fact_4430_abs__0,axiom,
% 0.29/0.55      ( ( abs_abs_real @ zero_zero_real )
% 0.29/0.55      = zero_zero_real ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_0
% 0.29/0.55  thf(fact_4431_abs__0,axiom,
% 0.29/0.55      ( ( abs_abs_rat @ zero_zero_rat )
% 0.29/0.55      = zero_zero_rat ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_0
% 0.29/0.55  thf(fact_4432_abs__0,axiom,
% 0.29/0.55      ( ( abs_abs_int @ zero_zero_int )
% 0.29/0.55      = zero_zero_int ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_0
% 0.29/0.55  thf(fact_4433_bits__mod__0,axiom,
% 0.29/0.55      ! [A: nat] :
% 0.29/0.55        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.29/0.55        = zero_zero_nat ) ).
% 0.29/0.55  
% 0.29/0.55  % bits_mod_0
% 0.29/0.55  thf(fact_4434_bits__mod__0,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.29/0.55        = zero_zero_int ) ).
% 0.29/0.55  
% 0.29/0.55  % bits_mod_0
% 0.29/0.55  thf(fact_4435_bits__mod__0,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.55        = zero_z3403309356797280102nteger ) ).
% 0.29/0.55  
% 0.29/0.55  % bits_mod_0
% 0.29/0.55  thf(fact_4436_mod__self,axiom,
% 0.29/0.55      ! [A: nat] :
% 0.29/0.55        ( ( modulo_modulo_nat @ A @ A )
% 0.29/0.55        = zero_zero_nat ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_self
% 0.29/0.55  thf(fact_4437_mod__self,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( modulo_modulo_int @ A @ A )
% 0.29/0.55        = zero_zero_int ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_self
% 0.29/0.55  thf(fact_4438_mod__self,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( modulo364778990260209775nteger @ A @ A )
% 0.29/0.55        = zero_z3403309356797280102nteger ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_self
% 0.29/0.55  thf(fact_4439_mod__by__0,axiom,
% 0.29/0.55      ! [A: nat] :
% 0.29/0.55        ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 0.29/0.55        = A ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_by_0
% 0.29/0.55  thf(fact_4440_mod__by__0,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( modulo_modulo_int @ A @ zero_zero_int )
% 0.29/0.55        = A ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_by_0
% 0.29/0.55  thf(fact_4441_mod__by__0,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 0.29/0.55        = A ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_by_0
% 0.29/0.55  thf(fact_4442_mod__0,axiom,
% 0.29/0.55      ! [A: nat] :
% 0.29/0.55        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.29/0.55        = zero_zero_nat ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_0
% 0.29/0.55  thf(fact_4443_mod__0,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.29/0.55        = zero_zero_int ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_0
% 0.29/0.55  thf(fact_4444_mod__0,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.55        = zero_z3403309356797280102nteger ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_0
% 0.29/0.55  thf(fact_4445_abs__numeral,axiom,
% 0.29/0.55      ! [N: num] :
% 0.29/0.55        ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 0.29/0.55        = ( numera6620942414471956472nteger @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_numeral
% 0.29/0.55  thf(fact_4446_abs__numeral,axiom,
% 0.29/0.55      ! [N: num] :
% 0.29/0.55        ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 0.29/0.55        = ( numeral_numeral_real @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_numeral
% 0.29/0.55  thf(fact_4447_abs__numeral,axiom,
% 0.29/0.55      ! [N: num] :
% 0.29/0.55        ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 0.29/0.55        = ( numeral_numeral_rat @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_numeral
% 0.29/0.55  thf(fact_4448_abs__numeral,axiom,
% 0.29/0.55      ! [N: num] :
% 0.29/0.55        ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 0.29/0.55        = ( numeral_numeral_int @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_numeral
% 0.29/0.55  thf(fact_4449_abs__1,axiom,
% 0.29/0.55      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.29/0.55      = one_one_Code_integer ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_1
% 0.29/0.55  thf(fact_4450_abs__1,axiom,
% 0.29/0.55      ( ( abs_abs_complex @ one_one_complex )
% 0.29/0.55      = one_one_complex ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_1
% 0.29/0.55  thf(fact_4451_abs__1,axiom,
% 0.29/0.55      ( ( abs_abs_real @ one_one_real )
% 0.29/0.55      = one_one_real ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_1
% 0.29/0.55  thf(fact_4452_abs__1,axiom,
% 0.29/0.55      ( ( abs_abs_rat @ one_one_rat )
% 0.29/0.55      = one_one_rat ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_1
% 0.29/0.55  thf(fact_4453_abs__1,axiom,
% 0.29/0.55      ( ( abs_abs_int @ one_one_int )
% 0.29/0.55      = one_one_int ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_1
% 0.29/0.55  thf(fact_4454_abs__add__abs,axiom,
% 0.29/0.55      ! [A: code_integer,B: code_integer] :
% 0.29/0.55        ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 0.29/0.55        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_add_abs
% 0.29/0.55  thf(fact_4455_abs__add__abs,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.29/0.55        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_add_abs
% 0.29/0.55  thf(fact_4456_abs__add__abs,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 0.29/0.55        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_add_abs
% 0.29/0.55  thf(fact_4457_abs__add__abs,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 0.29/0.55        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_add_abs
% 0.29/0.55  thf(fact_4458_mod__add__self1,axiom,
% 0.29/0.55      ! [B: nat,A: nat] :
% 0.29/0.55        ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.29/0.55        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_add_self1
% 0.29/0.55  thf(fact_4459_mod__add__self1,axiom,
% 0.29/0.55      ! [B: int,A: int] :
% 0.29/0.55        ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.29/0.55        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_add_self1
% 0.29/0.55  thf(fact_4460_mod__add__self1,axiom,
% 0.29/0.55      ! [B: code_integer,A: code_integer] :
% 0.29/0.55        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 0.29/0.55        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_add_self1
% 0.29/0.55  thf(fact_4461_mod__add__self2,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.29/0.55        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_add_self2
% 0.29/0.55  thf(fact_4462_mod__add__self2,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.29/0.55        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_add_self2
% 0.29/0.55  thf(fact_4463_mod__add__self2,axiom,
% 0.29/0.55      ! [A: code_integer,B: code_integer] :
% 0.29/0.55        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 0.29/0.55        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_add_self2
% 0.29/0.55  thf(fact_4464_abs__divide,axiom,
% 0.29/0.55      ! [A: complex,B: complex] :
% 0.29/0.55        ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.55        = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_divide
% 0.29/0.55  thf(fact_4465_abs__divide,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.55        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_divide
% 0.29/0.55  thf(fact_4466_abs__divide,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.29/0.55        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_divide
% 0.29/0.55  thf(fact_4467_abs__minus__cancel,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.29/0.55        = ( abs_abs_real @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_minus_cancel
% 0.29/0.55  thf(fact_4468_abs__minus__cancel,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.29/0.55        = ( abs_abs_int @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_minus_cancel
% 0.29/0.55  thf(fact_4469_abs__minus__cancel,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.29/0.55        = ( abs_abs_rat @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_minus_cancel
% 0.29/0.55  thf(fact_4470_abs__minus__cancel,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.55        = ( abs_abs_Code_integer @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_minus_cancel
% 0.29/0.55  thf(fact_4471_of__int__minus,axiom,
% 0.29/0.55      ! [Z: int] :
% 0.29/0.55        ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
% 0.29/0.55        = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_minus
% 0.29/0.55  thf(fact_4472_of__int__minus,axiom,
% 0.29/0.55      ! [Z: int] :
% 0.29/0.55        ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
% 0.29/0.55        = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_minus
% 0.29/0.55  thf(fact_4473_of__int__minus,axiom,
% 0.29/0.55      ! [Z: int] :
% 0.29/0.55        ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z ) )
% 0.29/0.55        = ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_minus
% 0.29/0.55  thf(fact_4474_of__int__minus,axiom,
% 0.29/0.55      ! [Z: int] :
% 0.29/0.55        ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z ) )
% 0.29/0.55        = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_minus
% 0.29/0.55  thf(fact_4475_of__int__minus,axiom,
% 0.29/0.55      ! [Z: int] :
% 0.29/0.55        ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z ) )
% 0.29/0.55        = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_minus
% 0.29/0.55  thf(fact_4476_abs__of__nat,axiom,
% 0.29/0.55      ! [N: nat] :
% 0.29/0.55        ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.29/0.55        = ( semiri4939895301339042750nteger @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_of_nat
% 0.29/0.55  thf(fact_4477_abs__of__nat,axiom,
% 0.29/0.55      ! [N: nat] :
% 0.29/0.55        ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.29/0.55        = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_of_nat
% 0.29/0.55  thf(fact_4478_abs__of__nat,axiom,
% 0.29/0.55      ! [N: nat] :
% 0.29/0.55        ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.29/0.55        = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_of_nat
% 0.29/0.55  thf(fact_4479_abs__of__nat,axiom,
% 0.29/0.55      ! [N: nat] :
% 0.29/0.55        ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.29/0.55        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.29/0.55  
% 0.29/0.55  % abs_of_nat
% 0.29/0.55  thf(fact_4480_of__int__abs,axiom,
% 0.29/0.55      ! [X: int] :
% 0.29/0.55        ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
% 0.29/0.55        = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_abs
% 0.29/0.55  thf(fact_4481_of__int__abs,axiom,
% 0.29/0.55      ! [X: int] :
% 0.29/0.55        ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
% 0.29/0.55        = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_abs
% 0.29/0.55  thf(fact_4482_of__int__abs,axiom,
% 0.29/0.55      ! [X: int] :
% 0.29/0.55        ( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
% 0.29/0.55        = ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_abs
% 0.29/0.55  thf(fact_4483_of__int__abs,axiom,
% 0.29/0.55      ! [X: int] :
% 0.29/0.55        ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
% 0.29/0.55        = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % of_int_abs
% 0.29/0.55  thf(fact_4484_mod__less,axiom,
% 0.29/0.55      ! [M: nat,N: nat] :
% 0.29/0.55        ( ( ord_less_nat @ M @ N )
% 0.29/0.55       => ( ( modulo_modulo_nat @ M @ N )
% 0.29/0.55          = M ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mod_less
% 0.29/0.55  thf(fact_4485_tanh__0,axiom,
% 0.29/0.55      ( ( tanh_complex @ zero_zero_complex )
% 0.29/0.55      = zero_zero_complex ) ).
% 0.29/0.55  
% 0.29/0.55  % tanh_0
% 0.29/0.55  thf(fact_4486_tanh__0,axiom,
% 0.29/0.55      ( ( tanh_real @ zero_zero_real )
% 0.29/0.55      = zero_zero_real ) ).
% 0.29/0.55  
% 0.29/0.55  % tanh_0
% 0.29/0.55  thf(fact_4487_tanh__minus,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( tanh_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.55        = ( uminus_uminus_real @ ( tanh_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % tanh_minus
% 0.29/0.55  thf(fact_4488_tanh__minus,axiom,
% 0.29/0.55      ! [X: complex] :
% 0.29/0.55        ( ( tanh_complex @ ( uminus1482373934393186551omplex @ X ) )
% 0.29/0.55        = ( uminus1482373934393186551omplex @ ( tanh_complex @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % tanh_minus
% 0.29/0.55  thf(fact_4489_tanh__real__zero__iff,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( ( tanh_real @ X )
% 0.29/0.55          = zero_zero_real )
% 0.29/0.55        = ( X = zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % tanh_real_zero_iff
% 0.29/0.55  thf(fact_4490_tanh__real__less__iff,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
% 0.29/0.55        = ( ord_less_real @ X @ Y ) ) ).
% 0.29/0.55  
% 0.29/0.55  % tanh_real_less_iff
% 0.29/0.55  thf(fact_4491_tanh__real__le__iff,axiom,
% 0.29/0.55      ! [X: real,Y: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
% 0.29/0.55        = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.29/0.55  
% 0.29/0.55  % tanh_real_le_iff
% 0.29/0.55  thf(fact_4492_tanh__real__abs,axiom,
% 0.29/0.55      ! [X: real] :
% 0.29/0.55        ( ( tanh_real @ ( abs_abs_real @ X ) )
% 0.29/0.55        = ( abs_abs_real @ ( tanh_real @ X ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % tanh_real_abs
% 0.29/0.55  thf(fact_4493_neg__less__eq__nonneg,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.29/0.55        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_less_eq_nonneg
% 0.29/0.55  thf(fact_4494_neg__less__eq__nonneg,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.29/0.55        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_less_eq_nonneg
% 0.29/0.55  thf(fact_4495_neg__less__eq__nonneg,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.29/0.55        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_less_eq_nonneg
% 0.29/0.55  thf(fact_4496_neg__less__eq__nonneg,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.29/0.55        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_less_eq_nonneg
% 0.29/0.55  thf(fact_4497_less__eq__neg__nonpos,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.29/0.55        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_eq_neg_nonpos
% 0.29/0.55  thf(fact_4498_less__eq__neg__nonpos,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.55        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_eq_neg_nonpos
% 0.29/0.55  thf(fact_4499_less__eq__neg__nonpos,axiom,
% 0.29/0.55      ! [A: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.29/0.55        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_eq_neg_nonpos
% 0.29/0.55  thf(fact_4500_less__eq__neg__nonpos,axiom,
% 0.29/0.55      ! [A: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.29/0.55        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_eq_neg_nonpos
% 0.29/0.55  thf(fact_4501_neg__le__0__iff__le,axiom,
% 0.29/0.55      ! [A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.29/0.55        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_le_0_iff_le
% 0.29/0.55  thf(fact_4502_neg__le__0__iff__le,axiom,
% 0.29/0.55      ! [A: code_integer] :
% 0.29/0.55        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.29/0.55        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_le_0_iff_le
% 0.29/0.55  thf(fact_4503_neg__le__0__iff__le,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.29/0.56        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_le_0_iff_le
% 0.29/0.56  thf(fact_4504_neg__le__0__iff__le,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.29/0.56        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_le_0_iff_le
% 0.29/0.56  thf(fact_4505_neg__0__le__iff__le,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.29/0.56        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_0_le_iff_le
% 0.29/0.56  thf(fact_4506_neg__0__le__iff__le,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.56        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_0_le_iff_le
% 0.29/0.56  thf(fact_4507_neg__0__le__iff__le,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.29/0.56        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_0_le_iff_le
% 0.29/0.56  thf(fact_4508_neg__0__le__iff__le,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.29/0.56        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_0_le_iff_le
% 0.29/0.56  thf(fact_4509_less__neg__neg,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.29/0.56        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_neg_neg
% 0.29/0.56  thf(fact_4510_less__neg__neg,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.29/0.56        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_neg_neg
% 0.29/0.56  thf(fact_4511_less__neg__neg,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.29/0.56        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_neg_neg
% 0.29/0.56  thf(fact_4512_less__neg__neg,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.56        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_neg_neg
% 0.29/0.56  thf(fact_4513_neg__less__pos,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.29/0.56        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_less_pos
% 0.29/0.56  thf(fact_4514_neg__less__pos,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.29/0.56        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_less_pos
% 0.29/0.56  thf(fact_4515_neg__less__pos,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.29/0.56        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_less_pos
% 0.29/0.56  thf(fact_4516_neg__less__pos,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.29/0.56        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_less_pos
% 0.29/0.56  thf(fact_4517_neg__0__less__iff__less,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.29/0.56        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_0_less_iff_less
% 0.29/0.56  thf(fact_4518_neg__0__less__iff__less,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.29/0.56        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_0_less_iff_less
% 0.29/0.56  thf(fact_4519_neg__0__less__iff__less,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.29/0.56        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_0_less_iff_less
% 0.29/0.56  thf(fact_4520_neg__0__less__iff__less,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.56        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_0_less_iff_less
% 0.29/0.56  thf(fact_4521_neg__less__0__iff__less,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.29/0.56        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_less_0_iff_less
% 0.29/0.56  thf(fact_4522_neg__less__0__iff__less,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.29/0.56        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_less_0_iff_less
% 0.29/0.56  thf(fact_4523_neg__less__0__iff__less,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.29/0.56        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_less_0_iff_less
% 0.29/0.56  thf(fact_4524_neg__less__0__iff__less,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.29/0.56        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_less_0_iff_less
% 0.29/0.56  thf(fact_4525_add_Oright__inverse,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.29/0.56        = zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % add.right_inverse
% 0.29/0.56  thf(fact_4526_add_Oright__inverse,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % add.right_inverse
% 0.29/0.56  thf(fact_4527_add_Oright__inverse,axiom,
% 0.29/0.56      ! [A: complex] :
% 0.29/0.56        ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 0.29/0.56        = zero_zero_complex ) ).
% 0.29/0.56  
% 0.29/0.56  % add.right_inverse
% 0.29/0.56  thf(fact_4528_add_Oright__inverse,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.29/0.56        = zero_zero_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % add.right_inverse
% 0.29/0.56  thf(fact_4529_add_Oright__inverse,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % add.right_inverse
% 0.29/0.56  thf(fact_4530_ab__left__minus,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.29/0.56        = zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_left_minus
% 0.29/0.56  thf(fact_4531_ab__left__minus,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_left_minus
% 0.29/0.56  thf(fact_4532_ab__left__minus,axiom,
% 0.29/0.56      ! [A: complex] :
% 0.29/0.56        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.29/0.56        = zero_zero_complex ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_left_minus
% 0.29/0.56  thf(fact_4533_ab__left__minus,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.29/0.56        = zero_zero_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_left_minus
% 0.29/0.56  thf(fact_4534_ab__left__minus,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_left_minus
% 0.29/0.56  thf(fact_4535_verit__minus__simplify_I3_J,axiom,
% 0.29/0.56      ! [B: real] :
% 0.29/0.56        ( ( minus_minus_real @ zero_zero_real @ B )
% 0.29/0.56        = ( uminus_uminus_real @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_minus_simplify(3)
% 0.29/0.56  thf(fact_4536_verit__minus__simplify_I3_J,axiom,
% 0.29/0.56      ! [B: int] :
% 0.29/0.56        ( ( minus_minus_int @ zero_zero_int @ B )
% 0.29/0.56        = ( uminus_uminus_int @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_minus_simplify(3)
% 0.29/0.56  thf(fact_4537_verit__minus__simplify_I3_J,axiom,
% 0.29/0.56      ! [B: complex] :
% 0.29/0.56        ( ( minus_minus_complex @ zero_zero_complex @ B )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_minus_simplify(3)
% 0.29/0.56  thf(fact_4538_verit__minus__simplify_I3_J,axiom,
% 0.29/0.56      ! [B: rat] :
% 0.29/0.56        ( ( minus_minus_rat @ zero_zero_rat @ B )
% 0.29/0.56        = ( uminus_uminus_rat @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_minus_simplify(3)
% 0.29/0.56  thf(fact_4539_verit__minus__simplify_I3_J,axiom,
% 0.29/0.56      ! [B: code_integer] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_minus_simplify(3)
% 0.29/0.56  thf(fact_4540_diff__0,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( minus_minus_real @ zero_zero_real @ A )
% 0.29/0.56        = ( uminus_uminus_real @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_0
% 0.29/0.56  thf(fact_4541_diff__0,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( minus_minus_int @ zero_zero_int @ A )
% 0.29/0.56        = ( uminus_uminus_int @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_0
% 0.29/0.56  thf(fact_4542_diff__0,axiom,
% 0.29/0.56      ! [A: complex] :
% 0.29/0.56        ( ( minus_minus_complex @ zero_zero_complex @ A )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_0
% 0.29/0.56  thf(fact_4543_diff__0,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( minus_minus_rat @ zero_zero_rat @ A )
% 0.29/0.56        = ( uminus_uminus_rat @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_0
% 0.29/0.56  thf(fact_4544_diff__0,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_0
% 0.29/0.56  thf(fact_4545_add__neg__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.56        = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_simps(3)
% 0.29/0.56  thf(fact_4546_add__neg__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_simps(3)
% 0.29/0.56  thf(fact_4547_add__neg__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_simps(3)
% 0.29/0.56  thf(fact_4548_add__neg__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.56        = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_simps(3)
% 0.29/0.56  thf(fact_4549_add__neg__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_simps(3)
% 0.29/0.56  thf(fact_4550_mult__minus1,axiom,
% 0.29/0.56      ! [Z: real] :
% 0.29/0.56        ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 0.29/0.56        = ( uminus_uminus_real @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_minus1
% 0.29/0.56  thf(fact_4551_mult__minus1,axiom,
% 0.29/0.56      ! [Z: int] :
% 0.29/0.56        ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 0.29/0.56        = ( uminus_uminus_int @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_minus1
% 0.29/0.56  thf(fact_4552_mult__minus1,axiom,
% 0.29/0.56      ! [Z: complex] :
% 0.29/0.56        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_minus1
% 0.29/0.56  thf(fact_4553_mult__minus1,axiom,
% 0.29/0.56      ! [Z: rat] :
% 0.29/0.56        ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 0.29/0.56        = ( uminus_uminus_rat @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_minus1
% 0.29/0.56  thf(fact_4554_mult__minus1,axiom,
% 0.29/0.56      ! [Z: code_integer] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_minus1
% 0.29/0.56  thf(fact_4555_mult__minus1__right,axiom,
% 0.29/0.56      ! [Z: real] :
% 0.29/0.56        ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.56        = ( uminus_uminus_real @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_minus1_right
% 0.29/0.56  thf(fact_4556_mult__minus1__right,axiom,
% 0.29/0.56      ! [Z: int] :
% 0.29/0.56        ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56        = ( uminus_uminus_int @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_minus1_right
% 0.29/0.56  thf(fact_4557_mult__minus1__right,axiom,
% 0.29/0.56      ! [Z: complex] :
% 0.29/0.56        ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_minus1_right
% 0.29/0.56  thf(fact_4558_mult__minus1__right,axiom,
% 0.29/0.56      ! [Z: rat] :
% 0.29/0.56        ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.56        = ( uminus_uminus_rat @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_minus1_right
% 0.29/0.56  thf(fact_4559_mult__minus1__right,axiom,
% 0.29/0.56      ! [Z: code_integer] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_minus1_right
% 0.29/0.56  thf(fact_4560_abs__le__zero__iff,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.29/0.56        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_zero_iff
% 0.29/0.56  thf(fact_4561_abs__le__zero__iff,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.29/0.56        = ( A = zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_zero_iff
% 0.29/0.56  thf(fact_4562_abs__le__zero__iff,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 0.29/0.56        = ( A = zero_zero_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_zero_iff
% 0.29/0.56  thf(fact_4563_abs__le__zero__iff,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 0.29/0.56        = ( A = zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_zero_iff
% 0.29/0.56  thf(fact_4564_abs__le__self__iff,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 0.29/0.56        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_self_iff
% 0.29/0.56  thf(fact_4565_abs__le__self__iff,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.29/0.56        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_self_iff
% 0.29/0.56  thf(fact_4566_abs__le__self__iff,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 0.29/0.56        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_self_iff
% 0.29/0.56  thf(fact_4567_abs__le__self__iff,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 0.29/0.56        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_self_iff
% 0.29/0.56  thf(fact_4568_abs__of__nonneg,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.56       => ( ( abs_abs_Code_integer @ A )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_nonneg
% 0.29/0.56  thf(fact_4569_abs__of__nonneg,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.56       => ( ( abs_abs_real @ A )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_nonneg
% 0.29/0.56  thf(fact_4570_abs__of__nonneg,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.56       => ( ( abs_abs_rat @ A )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_nonneg
% 0.29/0.56  thf(fact_4571_abs__of__nonneg,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.56       => ( ( abs_abs_int @ A )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_nonneg
% 0.29/0.56  thf(fact_4572_uminus__add__conv__diff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 0.29/0.56        = ( minus_minus_real @ B @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % uminus_add_conv_diff
% 0.29/0.56  thf(fact_4573_uminus__add__conv__diff,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.29/0.56        = ( minus_minus_int @ B @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % uminus_add_conv_diff
% 0.29/0.56  thf(fact_4574_uminus__add__conv__diff,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.29/0.56        = ( minus_minus_complex @ B @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % uminus_add_conv_diff
% 0.29/0.56  thf(fact_4575_uminus__add__conv__diff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.29/0.56        = ( minus_minus_rat @ B @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % uminus_add_conv_diff
% 0.29/0.56  thf(fact_4576_uminus__add__conv__diff,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.29/0.56        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % uminus_add_conv_diff
% 0.29/0.56  thf(fact_4577_diff__minus__eq__add,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 0.29/0.56        = ( plus_plus_real @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_minus_eq_add
% 0.29/0.56  thf(fact_4578_diff__minus__eq__add,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.29/0.56        = ( plus_plus_int @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_minus_eq_add
% 0.29/0.56  thf(fact_4579_diff__minus__eq__add,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.29/0.56        = ( plus_plus_complex @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_minus_eq_add
% 0.29/0.56  thf(fact_4580_diff__minus__eq__add,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.29/0.56        = ( plus_plus_rat @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_minus_eq_add
% 0.29/0.56  thf(fact_4581_diff__minus__eq__add,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.56        = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_minus_eq_add
% 0.29/0.56  thf(fact_4582_zero__less__abs__iff,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 0.29/0.56        = ( A != zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_less_abs_iff
% 0.29/0.56  thf(fact_4583_zero__less__abs__iff,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.29/0.56        = ( A != zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_less_abs_iff
% 0.29/0.56  thf(fact_4584_zero__less__abs__iff,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 0.29/0.56        = ( A != zero_zero_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_less_abs_iff
% 0.29/0.56  thf(fact_4585_zero__less__abs__iff,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 0.29/0.56        = ( A != zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_less_abs_iff
% 0.29/0.56  thf(fact_4586_divide__minus1,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.56        = ( uminus_uminus_real @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_minus1
% 0.29/0.56  thf(fact_4587_divide__minus1,axiom,
% 0.29/0.56      ! [X: complex] :
% 0.29/0.56        ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_minus1
% 0.29/0.56  thf(fact_4588_divide__minus1,axiom,
% 0.29/0.56      ! [X: rat] :
% 0.29/0.56        ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.56        = ( uminus_uminus_rat @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_minus1
% 0.29/0.56  thf(fact_4589_div__minus1__right,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56        = ( uminus_uminus_int @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % div_minus1_right
% 0.29/0.56  thf(fact_4590_div__minus1__right,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % div_minus1_right
% 0.29/0.56  thf(fact_4591_mod__mult__self1__is__0,axiom,
% 0.29/0.56      ! [B: nat,A: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self1_is_0
% 0.29/0.56  thf(fact_4592_mod__mult__self1__is__0,axiom,
% 0.29/0.56      ! [B: int,A: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self1_is_0
% 0.29/0.56  thf(fact_4593_mod__mult__self1__is__0,axiom,
% 0.29/0.56      ! [B: code_integer,A: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self1_is_0
% 0.29/0.56  thf(fact_4594_mod__mult__self2__is__0,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self2_is_0
% 0.29/0.56  thf(fact_4595_mod__mult__self2__is__0,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self2_is_0
% 0.29/0.56  thf(fact_4596_mod__mult__self2__is__0,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self2_is_0
% 0.29/0.56  thf(fact_4597_bits__mod__by__1,axiom,
% 0.29/0.56      ! [A: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_mod_by_1
% 0.29/0.56  thf(fact_4598_bits__mod__by__1,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ A @ one_one_int )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_mod_by_1
% 0.29/0.56  thf(fact_4599_bits__mod__by__1,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_mod_by_1
% 0.29/0.56  thf(fact_4600_mod__by__1,axiom,
% 0.29/0.56      ! [A: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_by_1
% 0.29/0.56  thf(fact_4601_mod__by__1,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ A @ one_one_int )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_by_1
% 0.29/0.56  thf(fact_4602_mod__by__1,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_by_1
% 0.29/0.56  thf(fact_4603_mod__div__trivial,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_div_trivial
% 0.29/0.56  thf(fact_4604_mod__div__trivial,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_div_trivial
% 0.29/0.56  thf(fact_4605_mod__div__trivial,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_div_trivial
% 0.29/0.56  thf(fact_4606_bits__mod__div__trivial,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_mod_div_trivial
% 0.29/0.56  thf(fact_4607_bits__mod__div__trivial,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_mod_div_trivial
% 0.29/0.56  thf(fact_4608_bits__mod__div__trivial,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_mod_div_trivial
% 0.29/0.56  thf(fact_4609_mod__mult__self4,axiom,
% 0.29/0.56      ! [B: nat,C: nat,A: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.29/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self4
% 0.29/0.56  thf(fact_4610_mod__mult__self4,axiom,
% 0.29/0.56      ! [B: int,C: int,A: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.29/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self4
% 0.29/0.56  thf(fact_4611_mod__mult__self4,axiom,
% 0.29/0.56      ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 0.29/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self4
% 0.29/0.56  thf(fact_4612_mod__mult__self3,axiom,
% 0.29/0.56      ! [C: nat,B: nat,A: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.29/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self3
% 0.29/0.56  thf(fact_4613_mod__mult__self3,axiom,
% 0.29/0.56      ! [C: int,B: int,A: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.29/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self3
% 0.29/0.56  thf(fact_4614_mod__mult__self3,axiom,
% 0.29/0.56      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 0.29/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self3
% 0.29/0.56  thf(fact_4615_mod__mult__self2,axiom,
% 0.29/0.56      ! [A: nat,B: nat,C: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.29/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self2
% 0.29/0.56  thf(fact_4616_mod__mult__self2,axiom,
% 0.29/0.56      ! [A: int,B: int,C: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.29/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self2
% 0.29/0.56  thf(fact_4617_mod__mult__self2,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 0.29/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self2
% 0.29/0.56  thf(fact_4618_mod__mult__self1,axiom,
% 0.29/0.56      ! [A: nat,C: nat,B: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.29/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self1
% 0.29/0.56  thf(fact_4619_mod__mult__self1,axiom,
% 0.29/0.56      ! [A: int,C: int,B: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.29/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self1
% 0.29/0.56  thf(fact_4620_mod__mult__self1,axiom,
% 0.29/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 0.29/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_self1
% 0.29/0.56  thf(fact_4621_abs__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_real @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_neg_numeral
% 0.29/0.56  thf(fact_4622_abs__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_int @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_neg_numeral
% 0.29/0.56  thf(fact_4623_abs__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_rat @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_neg_numeral
% 0.29/0.56  thf(fact_4624_abs__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.56        = ( numera6620942414471956472nteger @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_neg_numeral
% 0.29/0.56  thf(fact_4625_abs__neg__one,axiom,
% 0.29/0.56      ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.56      = one_one_real ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_neg_one
% 0.29/0.56  thf(fact_4626_abs__neg__one,axiom,
% 0.29/0.56      ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56      = one_one_int ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_neg_one
% 0.29/0.56  thf(fact_4627_abs__neg__one,axiom,
% 0.29/0.56      ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.56      = one_one_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_neg_one
% 0.29/0.56  thf(fact_4628_abs__neg__one,axiom,
% 0.29/0.56      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56      = one_one_Code_integer ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_neg_one
% 0.29/0.56  thf(fact_4629_abs__power__minus,axiom,
% 0.29/0.56      ! [A: real,N: nat] :
% 0.29/0.56        ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.29/0.56        = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_power_minus
% 0.29/0.56  thf(fact_4630_abs__power__minus,axiom,
% 0.29/0.56      ! [A: int,N: nat] :
% 0.29/0.56        ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.29/0.56        = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_power_minus
% 0.29/0.56  thf(fact_4631_abs__power__minus,axiom,
% 0.29/0.56      ! [A: rat,N: nat] :
% 0.29/0.56        ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.29/0.56        = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_power_minus
% 0.29/0.56  thf(fact_4632_abs__power__minus,axiom,
% 0.29/0.56      ! [A: code_integer,N: nat] :
% 0.29/0.56        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.29/0.56        = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_power_minus
% 0.29/0.56  thf(fact_4633_of__real__eq__0__iff,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ( real_V1803761363581548252l_real @ X )
% 0.29/0.56          = zero_zero_real )
% 0.29/0.56        = ( X = zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_eq_0_iff
% 0.29/0.56  thf(fact_4634_of__real__eq__0__iff,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ( real_V4546457046886955230omplex @ X )
% 0.29/0.56          = zero_zero_complex )
% 0.29/0.56        = ( X = zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_eq_0_iff
% 0.29/0.56  thf(fact_4635_of__real__0,axiom,
% 0.29/0.56      ( ( real_V1803761363581548252l_real @ zero_zero_real )
% 0.29/0.56      = zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_0
% 0.29/0.56  thf(fact_4636_of__real__0,axiom,
% 0.29/0.56      ( ( real_V4546457046886955230omplex @ zero_zero_real )
% 0.29/0.56      = zero_zero_complex ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_0
% 0.29/0.56  thf(fact_4637_of__real__1,axiom,
% 0.29/0.56      ( ( real_V1803761363581548252l_real @ one_one_real )
% 0.29/0.56      = one_one_real ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_1
% 0.29/0.56  thf(fact_4638_of__real__1,axiom,
% 0.29/0.56      ( ( real_V4546457046886955230omplex @ one_one_real )
% 0.29/0.56      = one_one_complex ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_1
% 0.29/0.56  thf(fact_4639_of__real__eq__1__iff,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ( real_V1803761363581548252l_real @ X )
% 0.29/0.56          = one_one_real )
% 0.29/0.56        = ( X = one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_eq_1_iff
% 0.29/0.56  thf(fact_4640_of__real__eq__1__iff,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ( real_V4546457046886955230omplex @ X )
% 0.29/0.56          = one_one_complex )
% 0.29/0.56        = ( X = one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_eq_1_iff
% 0.29/0.56  thf(fact_4641_mod__by__Suc__0,axiom,
% 0.29/0.56      ! [M: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_by_Suc_0
% 0.29/0.56  thf(fact_4642_of__real__numeral,axiom,
% 0.29/0.56      ! [W: num] :
% 0.29/0.56        ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
% 0.29/0.56        = ( numeral_numeral_real @ W ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_numeral
% 0.29/0.56  thf(fact_4643_of__real__numeral,axiom,
% 0.29/0.56      ! [W: num] :
% 0.29/0.56        ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
% 0.29/0.56        = ( numera6690914467698888265omplex @ W ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_numeral
% 0.29/0.56  thf(fact_4644_of__real__divide,axiom,
% 0.29/0.56      ! [X: real,Y: real] :
% 0.29/0.56        ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
% 0.29/0.56        = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_divide
% 0.29/0.56  thf(fact_4645_of__real__divide,axiom,
% 0.29/0.56      ! [X: real,Y: real] :
% 0.29/0.56        ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
% 0.29/0.56        = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_divide
% 0.29/0.56  thf(fact_4646_real__add__minus__iff,axiom,
% 0.29/0.56      ! [X: real,A: real] :
% 0.29/0.56        ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
% 0.29/0.56          = zero_zero_real )
% 0.29/0.56        = ( X = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % real_add_minus_iff
% 0.29/0.56  thf(fact_4647_of__real__add,axiom,
% 0.29/0.56      ! [X: real,Y: real] :
% 0.29/0.56        ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.56        = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_add
% 0.29/0.56  thf(fact_4648_of__real__add,axiom,
% 0.29/0.56      ! [X: real,Y: real] :
% 0.29/0.56        ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.56        = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_add
% 0.29/0.56  thf(fact_4649_of__real__power,axiom,
% 0.29/0.56      ! [X: real,N: nat] :
% 0.29/0.56        ( ( real_V1803761363581548252l_real @ ( power_power_real @ X @ N ) )
% 0.29/0.56        = ( power_power_real @ ( real_V1803761363581548252l_real @ X ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_power
% 0.29/0.56  thf(fact_4650_of__real__power,axiom,
% 0.29/0.56      ! [X: real,N: nat] :
% 0.29/0.56        ( ( real_V4546457046886955230omplex @ ( power_power_real @ X @ N ) )
% 0.29/0.56        = ( power_power_complex @ ( real_V4546457046886955230omplex @ X ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_power
% 0.29/0.56  thf(fact_4651_floor__uminus__of__int,axiom,
% 0.29/0.56      ! [Z: int] :
% 0.29/0.56        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % floor_uminus_of_int
% 0.29/0.56  thf(fact_4652_floor__uminus__of__int,axiom,
% 0.29/0.56      ! [Z: int] :
% 0.29/0.56        ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % floor_uminus_of_int
% 0.29/0.56  thf(fact_4653_of__real__of__nat__eq,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( real_V4546457046886955230omplex @ ( semiri5074537144036343181t_real @ N ) )
% 0.29/0.56        = ( semiri8010041392384452111omplex @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_of_nat_eq
% 0.29/0.56  thf(fact_4654_of__real__of__nat__eq,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( real_V1803761363581548252l_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.29/0.56        = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_of_nat_eq
% 0.29/0.56  thf(fact_4655_of__real__of__int__eq,axiom,
% 0.29/0.56      ! [Z: int] :
% 0.29/0.56        ( ( real_V1803761363581548252l_real @ ( ring_1_of_int_real @ Z ) )
% 0.29/0.56        = ( ring_1_of_int_real @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_of_int_eq
% 0.29/0.56  thf(fact_4656_of__real__of__int__eq,axiom,
% 0.29/0.56      ! [Z: int] :
% 0.29/0.56        ( ( real_V4546457046886955230omplex @ ( ring_1_of_int_real @ Z ) )
% 0.29/0.56        = ( ring_17405671764205052669omplex @ Z ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_of_int_eq
% 0.29/0.56  thf(fact_4657_tanh__real__neg__iff,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
% 0.29/0.56        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % tanh_real_neg_iff
% 0.29/0.56  thf(fact_4658_tanh__real__pos__iff,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
% 0.29/0.56        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % tanh_real_pos_iff
% 0.29/0.56  thf(fact_4659_tanh__real__nonneg__iff,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
% 0.29/0.56        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % tanh_real_nonneg_iff
% 0.29/0.56  thf(fact_4660_tanh__real__nonpos__iff,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
% 0.29/0.56        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % tanh_real_nonpos_iff
% 0.29/0.56  thf(fact_4661_dbl__simps_I1_J,axiom,
% 0.29/0.56      ! [K: num] :
% 0.29/0.56        ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.29/0.56        = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dbl_simps(1)
% 0.29/0.56  thf(fact_4662_dbl__simps_I1_J,axiom,
% 0.29/0.56      ! [K: num] :
% 0.29/0.56        ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dbl_simps(1)
% 0.29/0.56  thf(fact_4663_dbl__simps_I1_J,axiom,
% 0.29/0.56      ! [K: num] :
% 0.29/0.56        ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dbl_simps(1)
% 0.29/0.56  thf(fact_4664_dbl__simps_I1_J,axiom,
% 0.29/0.56      ! [K: num] :
% 0.29/0.56        ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.29/0.56        = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dbl_simps(1)
% 0.29/0.56  thf(fact_4665_dbl__simps_I1_J,axiom,
% 0.29/0.56      ! [K: num] :
% 0.29/0.56        ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dbl_simps(1)
% 0.29/0.56  thf(fact_4666_add__neg__numeral__special_I8_J,axiom,
% 0.29/0.56      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.29/0.56      = zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(8)
% 0.29/0.56  thf(fact_4667_add__neg__numeral__special_I8_J,axiom,
% 0.29/0.56      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.29/0.56      = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(8)
% 0.29/0.56  thf(fact_4668_add__neg__numeral__special_I8_J,axiom,
% 0.29/0.56      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.29/0.56      = zero_zero_complex ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(8)
% 0.29/0.56  thf(fact_4669_add__neg__numeral__special_I8_J,axiom,
% 0.29/0.56      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.29/0.56      = zero_zero_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(8)
% 0.29/0.56  thf(fact_4670_add__neg__numeral__special_I8_J,axiom,
% 0.29/0.56      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.29/0.56      = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(8)
% 0.29/0.56  thf(fact_4671_add__neg__numeral__special_I7_J,axiom,
% 0.29/0.56      ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.56      = zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(7)
% 0.29/0.56  thf(fact_4672_add__neg__numeral__special_I7_J,axiom,
% 0.29/0.56      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56      = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(7)
% 0.29/0.56  thf(fact_4673_add__neg__numeral__special_I7_J,axiom,
% 0.29/0.56      ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.56      = zero_zero_complex ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(7)
% 0.29/0.56  thf(fact_4674_add__neg__numeral__special_I7_J,axiom,
% 0.29/0.56      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.56      = zero_zero_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(7)
% 0.29/0.56  thf(fact_4675_add__neg__numeral__special_I7_J,axiom,
% 0.29/0.56      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56      = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(7)
% 0.29/0.56  thf(fact_4676_diff__numeral__special_I12_J,axiom,
% 0.29/0.56      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.56      = zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(12)
% 0.29/0.56  thf(fact_4677_diff__numeral__special_I12_J,axiom,
% 0.29/0.56      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56      = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(12)
% 0.29/0.56  thf(fact_4678_diff__numeral__special_I12_J,axiom,
% 0.29/0.56      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.56      = zero_zero_complex ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(12)
% 0.29/0.56  thf(fact_4679_diff__numeral__special_I12_J,axiom,
% 0.29/0.56      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.56      = zero_zero_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(12)
% 0.29/0.56  thf(fact_4680_diff__numeral__special_I12_J,axiom,
% 0.29/0.56      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56      = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(12)
% 0.29/0.56  thf(fact_4681_neg__one__eq__numeral__iff,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( ( uminus_uminus_real @ one_one_real )
% 0.29/0.56          = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.56        = ( N = one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_eq_numeral_iff
% 0.29/0.56  thf(fact_4682_neg__one__eq__numeral__iff,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( ( uminus_uminus_int @ one_one_int )
% 0.29/0.56          = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.56        = ( N = one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_eq_numeral_iff
% 0.29/0.56  thf(fact_4683_neg__one__eq__numeral__iff,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 0.29/0.56          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.29/0.56        = ( N = one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_eq_numeral_iff
% 0.29/0.56  thf(fact_4684_neg__one__eq__numeral__iff,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( ( uminus_uminus_rat @ one_one_rat )
% 0.29/0.56          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.56        = ( N = one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_eq_numeral_iff
% 0.29/0.56  thf(fact_4685_neg__one__eq__numeral__iff,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 0.29/0.56          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.56        = ( N = one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_eq_numeral_iff
% 0.29/0.56  thf(fact_4686_numeral__eq__neg__one__iff,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 0.29/0.56          = ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.56        = ( N = one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_eq_neg_one_iff
% 0.29/0.56  thf(fact_4687_numeral__eq__neg__one__iff,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 0.29/0.56          = ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56        = ( N = one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_eq_neg_one_iff
% 0.29/0.56  thf(fact_4688_numeral__eq__neg__one__iff,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 0.29/0.56          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.56        = ( N = one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_eq_neg_one_iff
% 0.29/0.56  thf(fact_4689_numeral__eq__neg__one__iff,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 0.29/0.56          = ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.56        = ( N = one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_eq_neg_one_iff
% 0.29/0.56  thf(fact_4690_numeral__eq__neg__one__iff,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 0.29/0.56          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56        = ( N = one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_eq_neg_one_iff
% 0.29/0.56  thf(fact_4691_zero__le__divide__abs__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 0.29/0.56        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.56          | ( B = zero_zero_real ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_le_divide_abs_iff
% 0.29/0.56  thf(fact_4692_zero__le__divide__abs__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 0.29/0.56        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.56          | ( B = zero_zero_rat ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_le_divide_abs_iff
% 0.29/0.56  thf(fact_4693_divide__le__0__abs__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 0.29/0.56        = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.56          | ( B = zero_zero_real ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_le_0_abs_iff
% 0.29/0.56  thf(fact_4694_divide__le__0__abs__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 0.29/0.56        = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.56          | ( B = zero_zero_rat ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_le_0_abs_iff
% 0.29/0.56  thf(fact_4695_abs__of__nonpos,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.56       => ( ( abs_abs_real @ A )
% 0.29/0.56          = ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_nonpos
% 0.29/0.56  thf(fact_4696_abs__of__nonpos,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 0.29/0.56       => ( ( abs_abs_Code_integer @ A )
% 0.29/0.56          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_nonpos
% 0.29/0.56  thf(fact_4697_abs__of__nonpos,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.56       => ( ( abs_abs_rat @ A )
% 0.29/0.56          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_nonpos
% 0.29/0.56  thf(fact_4698_abs__of__nonpos,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.56       => ( ( abs_abs_int @ A )
% 0.29/0.56          = ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_nonpos
% 0.29/0.56  thf(fact_4699_minus__one__mult__self,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 0.29/0.56        = one_one_real ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_one_mult_self
% 0.29/0.56  thf(fact_4700_minus__one__mult__self,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 0.29/0.56        = one_one_int ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_one_mult_self
% 0.29/0.56  thf(fact_4701_minus__one__mult__self,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 0.29/0.56        = one_one_complex ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_one_mult_self
% 0.29/0.56  thf(fact_4702_minus__one__mult__self,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 0.29/0.56        = one_one_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_one_mult_self
% 0.29/0.56  thf(fact_4703_minus__one__mult__self,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 0.29/0.56        = one_one_Code_integer ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_one_mult_self
% 0.29/0.56  thf(fact_4704_left__minus__one__mult__self,axiom,
% 0.29/0.56      ! [N: nat,A: real] :
% 0.29/0.56        ( ( 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 ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % left_minus_one_mult_self
% 0.29/0.56  thf(fact_4705_left__minus__one__mult__self,axiom,
% 0.29/0.56      ! [N: nat,A: int] :
% 0.29/0.56        ( ( 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 ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % left_minus_one_mult_self
% 0.29/0.56  thf(fact_4706_left__minus__one__mult__self,axiom,
% 0.29/0.56      ! [N: nat,A: complex] :
% 0.29/0.56        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % left_minus_one_mult_self
% 0.29/0.56  thf(fact_4707_left__minus__one__mult__self,axiom,
% 0.29/0.56      ! [N: nat,A: rat] :
% 0.29/0.56        ( ( 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 ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % left_minus_one_mult_self
% 0.29/0.56  thf(fact_4708_left__minus__one__mult__self,axiom,
% 0.29/0.56      ! [N: nat,A: code_integer] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % left_minus_one_mult_self
% 0.29/0.56  thf(fact_4709_mod__minus1__right,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_minus1_right
% 0.29/0.56  thf(fact_4710_mod__minus1__right,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_minus1_right
% 0.29/0.56  thf(fact_4711_norm__neg__numeral,axiom,
% 0.29/0.56      ! [W: num] :
% 0.29/0.56        ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.56        = ( numeral_numeral_real @ W ) ) ).
% 0.29/0.56  
% 0.29/0.56  % norm_neg_numeral
% 0.29/0.56  thf(fact_4712_norm__neg__numeral,axiom,
% 0.29/0.56      ! [W: num] :
% 0.29/0.56        ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.56        = ( numeral_numeral_real @ W ) ) ).
% 0.29/0.56  
% 0.29/0.56  % norm_neg_numeral
% 0.29/0.56  thf(fact_4713_floor__neg__numeral,axiom,
% 0.29/0.56      ! [V: num] :
% 0.29/0.56        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % floor_neg_numeral
% 0.29/0.56  thf(fact_4714_floor__neg__numeral,axiom,
% 0.29/0.56      ! [V: num] :
% 0.29/0.56        ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % floor_neg_numeral
% 0.29/0.56  thf(fact_4715_semiring__norm_I168_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: real] :
% 0.29/0.56        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 0.29/0.56        = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(168)
% 0.29/0.56  thf(fact_4716_semiring__norm_I168_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: int] :
% 0.29/0.56        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 0.29/0.56        = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(168)
% 0.29/0.56  thf(fact_4717_semiring__norm_I168_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: complex] :
% 0.29/0.56        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 0.29/0.56        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(168)
% 0.29/0.56  thf(fact_4718_semiring__norm_I168_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: rat] :
% 0.29/0.56        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 0.29/0.56        = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(168)
% 0.29/0.56  thf(fact_4719_semiring__norm_I168_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 0.29/0.56        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(168)
% 0.29/0.56  thf(fact_4720_ceiling__neg__numeral,axiom,
% 0.29/0.56      ! [V: num] :
% 0.29/0.56        ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_neg_numeral
% 0.29/0.56  thf(fact_4721_ceiling__neg__numeral,axiom,
% 0.29/0.56      ! [V: num] :
% 0.29/0.56        ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_neg_numeral
% 0.29/0.56  thf(fact_4722_diff__numeral__simps_I2_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_simps(2)
% 0.29/0.56  thf(fact_4723_diff__numeral__simps_I2_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_simps(2)
% 0.29/0.56  thf(fact_4724_diff__numeral__simps_I2_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.29/0.56        = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_simps(2)
% 0.29/0.56  thf(fact_4725_diff__numeral__simps_I2_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_simps(2)
% 0.29/0.56  thf(fact_4726_diff__numeral__simps_I2_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.56        = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_simps(2)
% 0.29/0.56  thf(fact_4727_diff__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.29/0.56        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_simps(3)
% 0.29/0.56  thf(fact_4728_diff__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_simps(3)
% 0.29/0.56  thf(fact_4729_diff__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_simps(3)
% 0.29/0.56  thf(fact_4730_diff__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.29/0.56        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_simps(3)
% 0.29/0.56  thf(fact_4731_diff__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_simps(3)
% 0.29/0.56  thf(fact_4732_Suc__mod__mult__self1,axiom,
% 0.29/0.56      ! [M: nat,K: nat,N: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 0.29/0.56        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % Suc_mod_mult_self1
% 0.29/0.56  thf(fact_4733_Suc__mod__mult__self2,axiom,
% 0.29/0.56      ! [M: nat,N: nat,K: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 0.29/0.56        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % Suc_mod_mult_self2
% 0.29/0.56  thf(fact_4734_Suc__mod__mult__self3,axiom,
% 0.29/0.56      ! [K: nat,N: nat,M: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 0.29/0.56        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % Suc_mod_mult_self3
% 0.29/0.56  thf(fact_4735_Suc__mod__mult__self4,axiom,
% 0.29/0.56      ! [N: nat,K: nat,M: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 0.29/0.56        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % Suc_mod_mult_self4
% 0.29/0.56  thf(fact_4736_semiring__norm_I172_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: real] :
% 0.29/0.56        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 0.29/0.56        = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(172)
% 0.29/0.56  thf(fact_4737_semiring__norm_I172_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: int] :
% 0.29/0.56        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 0.29/0.56        = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(172)
% 0.29/0.56  thf(fact_4738_semiring__norm_I172_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: complex] :
% 0.29/0.56        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 0.29/0.56        = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(172)
% 0.29/0.56  thf(fact_4739_semiring__norm_I172_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: rat] :
% 0.29/0.56        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 0.29/0.56        = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(172)
% 0.29/0.56  thf(fact_4740_semiring__norm_I172_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: code_integer] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 0.29/0.56        = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(172)
% 0.29/0.56  thf(fact_4741_semiring__norm_I171_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: real] :
% 0.29/0.56        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 0.29/0.56        = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(171)
% 0.29/0.56  thf(fact_4742_semiring__norm_I171_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: int] :
% 0.29/0.56        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 0.29/0.56        = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(171)
% 0.29/0.56  thf(fact_4743_semiring__norm_I171_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: complex] :
% 0.29/0.56        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 0.29/0.56        = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(171)
% 0.29/0.56  thf(fact_4744_semiring__norm_I171_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: rat] :
% 0.29/0.56        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 0.29/0.56        = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(171)
% 0.29/0.56  thf(fact_4745_semiring__norm_I171_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: code_integer] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 0.29/0.56        = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(171)
% 0.29/0.56  thf(fact_4746_semiring__norm_I170_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: real] :
% 0.29/0.56        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
% 0.29/0.56        = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(170)
% 0.29/0.56  thf(fact_4747_semiring__norm_I170_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: int] :
% 0.29/0.56        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
% 0.29/0.56        = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(170)
% 0.29/0.56  thf(fact_4748_semiring__norm_I170_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: complex] :
% 0.29/0.56        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
% 0.29/0.56        = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(170)
% 0.29/0.56  thf(fact_4749_semiring__norm_I170_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: rat] :
% 0.29/0.56        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
% 0.29/0.56        = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(170)
% 0.29/0.56  thf(fact_4750_semiring__norm_I170_J,axiom,
% 0.29/0.56      ! [V: num,W: num,Y: code_integer] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
% 0.29/0.56        = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % semiring_norm(170)
% 0.29/0.56  thf(fact_4751_mult__neg__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.56        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(3)
% 0.29/0.56  thf(fact_4752_mult__neg__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(3)
% 0.29/0.56  thf(fact_4753_mult__neg__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(3)
% 0.29/0.56  thf(fact_4754_mult__neg__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.56        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(3)
% 0.29/0.56  thf(fact_4755_mult__neg__numeral__simps_I3_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(3)
% 0.29/0.56  thf(fact_4756_mult__neg__numeral__simps_I2_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.29/0.56        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(2)
% 0.29/0.56  thf(fact_4757_mult__neg__numeral__simps_I2_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(2)
% 0.29/0.56  thf(fact_4758_mult__neg__numeral__simps_I2_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(2)
% 0.29/0.56  thf(fact_4759_mult__neg__numeral__simps_I2_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.29/0.56        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(2)
% 0.29/0.56  thf(fact_4760_mult__neg__numeral__simps_I2_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(2)
% 0.29/0.56  thf(fact_4761_mult__neg__numeral__simps_I1_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(1)
% 0.29/0.56  thf(fact_4762_mult__neg__numeral__simps_I1_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(1)
% 0.29/0.56  thf(fact_4763_mult__neg__numeral__simps_I1_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.29/0.56        = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(1)
% 0.29/0.56  thf(fact_4764_mult__neg__numeral__simps_I1_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(1)
% 0.29/0.56  thf(fact_4765_mult__neg__numeral__simps_I1_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.56        = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_neg_numeral_simps(1)
% 0.29/0.56  thf(fact_4766_artanh__minus__real,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.56       => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.56          = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % artanh_minus_real
% 0.29/0.56  thf(fact_4767_neg__numeral__le__iff,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.56        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_iff
% 0.29/0.56  thf(fact_4768_neg__numeral__le__iff,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.56        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_iff
% 0.29/0.56  thf(fact_4769_neg__numeral__le__iff,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.56        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_iff
% 0.29/0.56  thf(fact_4770_neg__numeral__le__iff,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.56        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_iff
% 0.29/0.56  thf(fact_4771_neg__numeral__less__iff,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.56        = ( ord_less_num @ N @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_iff
% 0.29/0.56  thf(fact_4772_neg__numeral__less__iff,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.56        = ( ord_less_num @ N @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_iff
% 0.29/0.56  thf(fact_4773_neg__numeral__less__iff,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.56        = ( ord_less_num @ N @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_iff
% 0.29/0.56  thf(fact_4774_neg__numeral__less__iff,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.56        = ( ord_less_num @ N @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_iff
% 0.29/0.56  thf(fact_4775_round__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % round_neg_numeral
% 0.29/0.56  thf(fact_4776_round__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % round_neg_numeral
% 0.29/0.56  thf(fact_4777_norm__of__int,axiom,
% 0.29/0.56      ! [Z: int] :
% 0.29/0.56        ( ( real_V7735802525324610683m_real @ ( ring_1_of_int_real @ Z ) )
% 0.29/0.56        = ( abs_abs_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % norm_of_int
% 0.29/0.56  thf(fact_4778_norm__of__int,axiom,
% 0.29/0.56      ! [Z: int] :
% 0.29/0.56        ( ( real_V1022390504157884413omplex @ ( ring_17405671764205052669omplex @ Z ) )
% 0.29/0.56        = ( abs_abs_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % norm_of_int
% 0.29/0.56  thf(fact_4779_not__neg__one__le__neg__numeral__iff,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.29/0.56        = ( M != one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_neg_one_le_neg_numeral_iff
% 0.29/0.56  thf(fact_4780_not__neg__one__le__neg__numeral__iff,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.29/0.56        = ( M != one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_neg_one_le_neg_numeral_iff
% 0.29/0.56  thf(fact_4781_not__neg__one__le__neg__numeral__iff,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.29/0.56        = ( M != one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_neg_one_le_neg_numeral_iff
% 0.29/0.56  thf(fact_4782_not__neg__one__le__neg__numeral__iff,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.29/0.56        = ( M != one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_neg_one_le_neg_numeral_iff
% 0.29/0.56  thf(fact_4783_le__divide__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [A: real,B: real,W: num] :
% 0.29/0.56        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.29/0.56        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_divide_eq_numeral1(2)
% 0.29/0.56  thf(fact_4784_le__divide__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [A: rat,B: rat,W: num] :
% 0.29/0.56        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.29/0.56        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_divide_eq_numeral1(2)
% 0.29/0.56  thf(fact_4785_divide__le__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [B: real,W: num,A: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.29/0.56        = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_le_eq_numeral1(2)
% 0.29/0.56  thf(fact_4786_divide__le__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [B: rat,W: num,A: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.29/0.56        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_le_eq_numeral1(2)
% 0.29/0.56  thf(fact_4787_eq__divide__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [A: real,B: real,W: num] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.29/0.56        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.29/0.56             != zero_zero_real )
% 0.29/0.56           => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.56              = B ) )
% 0.29/0.56          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.29/0.56              = zero_zero_real )
% 0.29/0.56           => ( A = zero_zero_real ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_divide_eq_numeral1(2)
% 0.29/0.56  thf(fact_4788_eq__divide__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [A: complex,B: complex,W: num] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.29/0.56        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.29/0.56             != zero_zero_complex )
% 0.29/0.56           => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.56              = B ) )
% 0.29/0.56          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.29/0.56              = zero_zero_complex )
% 0.29/0.56           => ( A = zero_zero_complex ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_divide_eq_numeral1(2)
% 0.29/0.56  thf(fact_4789_eq__divide__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [A: rat,B: rat,W: num] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.29/0.56        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.29/0.56             != zero_zero_rat )
% 0.29/0.56           => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.29/0.56              = B ) )
% 0.29/0.56          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.29/0.56              = zero_zero_rat )
% 0.29/0.56           => ( A = zero_zero_rat ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_divide_eq_numeral1(2)
% 0.29/0.56  thf(fact_4790_divide__eq__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [B: real,W: num,A: real] :
% 0.29/0.56        ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.56          = A )
% 0.29/0.56        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.29/0.56             != zero_zero_real )
% 0.29/0.56           => ( B
% 0.29/0.56              = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 0.29/0.56          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.29/0.56              = zero_zero_real )
% 0.29/0.56           => ( A = zero_zero_real ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_eq_eq_numeral1(2)
% 0.29/0.56  thf(fact_4791_divide__eq__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [B: complex,W: num,A: complex] :
% 0.29/0.56        ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.56          = A )
% 0.29/0.56        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.29/0.56             != zero_zero_complex )
% 0.29/0.56           => ( B
% 0.29/0.56              = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 0.29/0.56          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.29/0.56              = zero_zero_complex )
% 0.29/0.56           => ( A = zero_zero_complex ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_eq_eq_numeral1(2)
% 0.29/0.56  thf(fact_4792_divide__eq__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [B: rat,W: num,A: rat] :
% 0.29/0.56        ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.29/0.56          = A )
% 0.29/0.56        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.29/0.56             != zero_zero_rat )
% 0.29/0.56           => ( B
% 0.29/0.56              = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 0.29/0.56          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.29/0.56              = zero_zero_rat )
% 0.29/0.56           => ( A = zero_zero_rat ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_eq_eq_numeral1(2)
% 0.29/0.56  thf(fact_4793_neg__numeral__less__neg__one__iff,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.56        = ( M != one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_neg_one_iff
% 0.29/0.56  thf(fact_4794_neg__numeral__less__neg__one__iff,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56        = ( M != one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_neg_one_iff
% 0.29/0.56  thf(fact_4795_neg__numeral__less__neg__one__iff,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.56        = ( M != one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_neg_one_iff
% 0.29/0.56  thf(fact_4796_neg__numeral__less__neg__one__iff,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56        = ( M != one ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_neg_one_iff
% 0.29/0.56  thf(fact_4797_less__divide__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [A: real,B: real,W: num] :
% 0.29/0.56        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.29/0.56        = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_divide_eq_numeral1(2)
% 0.29/0.56  thf(fact_4798_less__divide__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [A: rat,B: rat,W: num] :
% 0.29/0.56        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.29/0.56        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_divide_eq_numeral1(2)
% 0.29/0.56  thf(fact_4799_divide__less__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [B: real,W: num,A: real] :
% 0.29/0.56        ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.29/0.56        = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_less_eq_numeral1(2)
% 0.29/0.56  thf(fact_4800_divide__less__eq__numeral1_I2_J,axiom,
% 0.29/0.56      ! [B: rat,W: num,A: rat] :
% 0.29/0.56        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.29/0.56        = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % divide_less_eq_numeral1(2)
% 0.29/0.56  thf(fact_4801_bits__one__mod__two__eq__one,axiom,
% 0.29/0.56      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56      = one_one_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_one_mod_two_eq_one
% 0.29/0.56  thf(fact_4802_bits__one__mod__two__eq__one,axiom,
% 0.29/0.56      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.56      = one_one_int ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_one_mod_two_eq_one
% 0.29/0.56  thf(fact_4803_bits__one__mod__two__eq__one,axiom,
% 0.29/0.56      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.56      = one_one_Code_integer ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_one_mod_two_eq_one
% 0.29/0.56  thf(fact_4804_one__mod__two__eq__one,axiom,
% 0.29/0.56      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56      = one_one_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % one_mod_two_eq_one
% 0.29/0.56  thf(fact_4805_one__mod__two__eq__one,axiom,
% 0.29/0.56      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.56      = one_one_int ) ).
% 0.29/0.56  
% 0.29/0.56  % one_mod_two_eq_one
% 0.29/0.56  thf(fact_4806_one__mod__two__eq__one,axiom,
% 0.29/0.56      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.56      = one_one_Code_integer ) ).
% 0.29/0.56  
% 0.29/0.56  % one_mod_two_eq_one
% 0.29/0.56  thf(fact_4807_zero__less__power__abs__iff,axiom,
% 0.29/0.56      ! [A: code_integer,N: nat] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 0.29/0.56        = ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.56          | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_less_power_abs_iff
% 0.29/0.56  thf(fact_4808_zero__less__power__abs__iff,axiom,
% 0.29/0.56      ! [A: real,N: nat] :
% 0.29/0.56        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.29/0.56        = ( ( A != zero_zero_real )
% 0.29/0.56          | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_less_power_abs_iff
% 0.29/0.56  thf(fact_4809_zero__less__power__abs__iff,axiom,
% 0.29/0.56      ! [A: rat,N: nat] :
% 0.29/0.56        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 0.29/0.56        = ( ( A != zero_zero_rat )
% 0.29/0.56          | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_less_power_abs_iff
% 0.29/0.56  thf(fact_4810_zero__less__power__abs__iff,axiom,
% 0.29/0.56      ! [A: int,N: nat] :
% 0.29/0.56        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 0.29/0.56        = ( ( A != zero_zero_int )
% 0.29/0.56          | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_less_power_abs_iff
% 0.29/0.56  thf(fact_4811_power2__minus,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power2_minus
% 0.29/0.56  thf(fact_4812_power2__minus,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power2_minus
% 0.29/0.56  thf(fact_4813_power2__minus,axiom,
% 0.29/0.56      ! [A: complex] :
% 0.29/0.56        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power2_minus
% 0.29/0.56  thf(fact_4814_power2__minus,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power2_minus
% 0.29/0.56  thf(fact_4815_power2__minus,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power2_minus
% 0.29/0.56  thf(fact_4816_abs__power2,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.56        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_power2
% 0.29/0.56  thf(fact_4817_abs__power2,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.56        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_power2
% 0.29/0.56  thf(fact_4818_abs__power2,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.56        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_power2
% 0.29/0.56  thf(fact_4819_abs__power2,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.56        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_power2
% 0.29/0.56  thf(fact_4820_power2__abs,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power2_abs
% 0.29/0.56  thf(fact_4821_power2__abs,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power2_abs
% 0.29/0.56  thf(fact_4822_power2__abs,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power2_abs
% 0.29/0.56  thf(fact_4823_power2__abs,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power2_abs
% 0.29/0.56  thf(fact_4824_mod2__Suc__Suc,axiom,
% 0.29/0.56      ! [M: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod2_Suc_Suc
% 0.29/0.56  thf(fact_4825_of__real__neg__numeral,axiom,
% 0.29/0.56      ! [W: num] :
% 0.29/0.56        ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.56        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_neg_numeral
% 0.29/0.56  thf(fact_4826_of__real__neg__numeral,axiom,
% 0.29/0.56      ! [W: num] :
% 0.29/0.56        ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_real_neg_numeral
% 0.29/0.56  thf(fact_4827_Suc__times__numeral__mod__eq,axiom,
% 0.29/0.56      ! [K: num,N: nat] :
% 0.29/0.56        ( ( ( numeral_numeral_nat @ K )
% 0.29/0.56         != one_one_nat )
% 0.29/0.56       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 0.29/0.56          = one_one_nat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % Suc_times_numeral_mod_eq
% 0.29/0.56  thf(fact_4828_add__neg__numeral__special_I9_J,axiom,
% 0.29/0.56      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.56      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(9)
% 0.29/0.56  thf(fact_4829_add__neg__numeral__special_I9_J,axiom,
% 0.29/0.56      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(9)
% 0.29/0.56  thf(fact_4830_add__neg__numeral__special_I9_J,axiom,
% 0.29/0.56      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.56      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(9)
% 0.29/0.56  thf(fact_4831_add__neg__numeral__special_I9_J,axiom,
% 0.29/0.56      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.56      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(9)
% 0.29/0.56  thf(fact_4832_add__neg__numeral__special_I9_J,axiom,
% 0.29/0.56      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_neg_numeral_special(9)
% 0.29/0.56  thf(fact_4833_diff__numeral__special_I11_J,axiom,
% 0.29/0.56      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.56      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(11)
% 0.29/0.56  thf(fact_4834_diff__numeral__special_I11_J,axiom,
% 0.29/0.56      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(11)
% 0.29/0.56  thf(fact_4835_diff__numeral__special_I11_J,axiom,
% 0.29/0.56      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.56      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(11)
% 0.29/0.56  thf(fact_4836_diff__numeral__special_I11_J,axiom,
% 0.29/0.56      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.56      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(11)
% 0.29/0.56  thf(fact_4837_diff__numeral__special_I11_J,axiom,
% 0.29/0.56      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56      = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(11)
% 0.29/0.56  thf(fact_4838_diff__numeral__special_I10_J,axiom,
% 0.29/0.56      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.29/0.56      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(10)
% 0.29/0.56  thf(fact_4839_diff__numeral__special_I10_J,axiom,
% 0.29/0.56      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.29/0.56      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(10)
% 0.29/0.56  thf(fact_4840_diff__numeral__special_I10_J,axiom,
% 0.29/0.56      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.29/0.56      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(10)
% 0.29/0.56  thf(fact_4841_diff__numeral__special_I10_J,axiom,
% 0.29/0.56      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.29/0.56      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(10)
% 0.29/0.56  thf(fact_4842_diff__numeral__special_I10_J,axiom,
% 0.29/0.56      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.29/0.56      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(10)
% 0.29/0.56  thf(fact_4843_minus__1__div__2__eq,axiom,
% 0.29/0.56      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.56      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_1_div_2_eq
% 0.29/0.56  thf(fact_4844_minus__1__div__2__eq,axiom,
% 0.29/0.56      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.56      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_1_div_2_eq
% 0.29/0.56  thf(fact_4845_not__mod__2__eq__0__eq__1,axiom,
% 0.29/0.56      ! [A: nat] :
% 0.29/0.56        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56         != zero_zero_nat )
% 0.29/0.56        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56          = one_one_nat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_mod_2_eq_0_eq_1
% 0.29/0.56  thf(fact_4846_not__mod__2__eq__0__eq__1,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.56         != zero_zero_int )
% 0.29/0.56        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.56          = one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_mod_2_eq_0_eq_1
% 0.29/0.56  thf(fact_4847_not__mod__2__eq__0__eq__1,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.56         != zero_z3403309356797280102nteger )
% 0.29/0.56        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.56          = one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_mod_2_eq_0_eq_1
% 0.29/0.56  thf(fact_4848_not__mod__2__eq__1__eq__0,axiom,
% 0.29/0.56      ! [A: nat] :
% 0.29/0.56        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56         != one_one_nat )
% 0.29/0.56        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56          = zero_zero_nat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_mod_2_eq_1_eq_0
% 0.29/0.56  thf(fact_4849_not__mod__2__eq__1__eq__0,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.56         != one_one_int )
% 0.29/0.56        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.56          = zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_mod_2_eq_1_eq_0
% 0.29/0.56  thf(fact_4850_not__mod__2__eq__1__eq__0,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.56         != one_one_Code_integer )
% 0.29/0.56        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.56          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_mod_2_eq_1_eq_0
% 0.29/0.56  thf(fact_4851_bits__minus__1__mod__2__eq,axiom,
% 0.29/0.56      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.56      = one_one_int ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_minus_1_mod_2_eq
% 0.29/0.56  thf(fact_4852_bits__minus__1__mod__2__eq,axiom,
% 0.29/0.56      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.56      = one_one_Code_integer ) ).
% 0.29/0.56  
% 0.29/0.56  % bits_minus_1_mod_2_eq
% 0.29/0.56  thf(fact_4853_minus__1__mod__2__eq,axiom,
% 0.29/0.56      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.56      = one_one_int ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_1_mod_2_eq
% 0.29/0.56  thf(fact_4854_minus__1__mod__2__eq,axiom,
% 0.29/0.56      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.56      = one_one_Code_integer ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_1_mod_2_eq
% 0.29/0.56  thf(fact_4855_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.29/0.56      ! [A: real,N: nat] :
% 0.29/0.56        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.56        = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % Power.ring_1_class.power_minus_even
% 0.29/0.56  thf(fact_4856_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.29/0.56      ! [A: int,N: nat] :
% 0.29/0.56        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.56        = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % Power.ring_1_class.power_minus_even
% 0.29/0.56  thf(fact_4857_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.29/0.56      ! [A: complex,N: nat] :
% 0.29/0.56        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.56        = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % Power.ring_1_class.power_minus_even
% 0.29/0.56  thf(fact_4858_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.29/0.56      ! [A: rat,N: nat] :
% 0.29/0.56        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.56        = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % Power.ring_1_class.power_minus_even
% 0.29/0.56  thf(fact_4859_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.29/0.56      ! [A: code_integer,N: nat] :
% 0.29/0.56        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.56        = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % Power.ring_1_class.power_minus_even
% 0.29/0.56  thf(fact_4860_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56         != ( suc @ zero_zero_nat ) )
% 0.29/0.56        = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56          = zero_zero_nat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_mod2_eq_Suc_0_eq_0
% 0.29/0.56  thf(fact_4861_diff__numeral__special_I4_J,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 0.29/0.56        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(4)
% 0.29/0.56  thf(fact_4862_diff__numeral__special_I4_J,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(4)
% 0.29/0.56  thf(fact_4863_diff__numeral__special_I4_J,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(4)
% 0.29/0.56  thf(fact_4864_diff__numeral__special_I4_J,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 0.29/0.56        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(4)
% 0.29/0.56  thf(fact_4865_diff__numeral__special_I4_J,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(4)
% 0.29/0.56  thf(fact_4866_diff__numeral__special_I3_J,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(3)
% 0.29/0.56  thf(fact_4867_diff__numeral__special_I3_J,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(3)
% 0.29/0.56  thf(fact_4868_diff__numeral__special_I3_J,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.29/0.56        = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(3)
% 0.29/0.56  thf(fact_4869_diff__numeral__special_I3_J,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.56        = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(3)
% 0.29/0.56  thf(fact_4870_diff__numeral__special_I3_J,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.56        = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_numeral_special(3)
% 0.29/0.56  thf(fact_4871_add__self__mod__2,axiom,
% 0.29/0.56      ! [M: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % add_self_mod_2
% 0.29/0.56  thf(fact_4872_neg__numeral__le__floor,axiom,
% 0.29/0.56      ! [V: num,X: real] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 0.29/0.56        = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_floor
% 0.29/0.56  thf(fact_4873_neg__numeral__le__floor,axiom,
% 0.29/0.56      ! [V: num,X: rat] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.29/0.56        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_floor
% 0.29/0.56  thf(fact_4874_floor__less__neg__numeral,axiom,
% 0.29/0.56      ! [X: real,V: num] :
% 0.29/0.56        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.29/0.56        = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % floor_less_neg_numeral
% 0.29/0.56  thf(fact_4875_floor__less__neg__numeral,axiom,
% 0.29/0.56      ! [X: rat,V: num] :
% 0.29/0.56        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.29/0.56        = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % floor_less_neg_numeral
% 0.29/0.56  thf(fact_4876_ceiling__le__neg__numeral,axiom,
% 0.29/0.56      ! [X: real,V: num] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.29/0.56        = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_le_neg_numeral
% 0.29/0.56  thf(fact_4877_ceiling__le__neg__numeral,axiom,
% 0.29/0.56      ! [X: rat,V: num] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.29/0.56        = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_le_neg_numeral
% 0.29/0.56  thf(fact_4878_ceiling__less__zero,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 0.29/0.56        = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_less_zero
% 0.29/0.56  thf(fact_4879_ceiling__less__zero,axiom,
% 0.29/0.56      ! [X: rat] :
% 0.29/0.56        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 0.29/0.56        = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_less_zero
% 0.29/0.56  thf(fact_4880_zero__le__ceiling,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 0.29/0.56        = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_le_ceiling
% 0.29/0.56  thf(fact_4881_zero__le__ceiling,axiom,
% 0.29/0.56      ! [X: rat] :
% 0.29/0.56        ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.29/0.56        = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_le_ceiling
% 0.29/0.56  thf(fact_4882_neg__numeral__less__ceiling,axiom,
% 0.29/0.56      ! [V: num,X: real] :
% 0.29/0.56        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 0.29/0.56        = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_ceiling
% 0.29/0.56  thf(fact_4883_neg__numeral__less__ceiling,axiom,
% 0.29/0.56      ! [V: num,X: rat] :
% 0.29/0.56        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.29/0.56        = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_ceiling
% 0.29/0.56  thf(fact_4884_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [Y: int,X: num,N: nat] :
% 0.29/0.56        ( ( ( ring_1_of_int_real @ Y )
% 0.29/0.56          = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.29/0.56        = ( Y
% 0.29/0.56          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_eq_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4885_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [Y: int,X: num,N: nat] :
% 0.29/0.56        ( ( ( ring_1_of_int_int @ Y )
% 0.29/0.56          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.29/0.56        = ( Y
% 0.29/0.56          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_eq_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4886_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [Y: int,X: num,N: nat] :
% 0.29/0.56        ( ( ( ring_17405671764205052669omplex @ Y )
% 0.29/0.56          = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
% 0.29/0.56        = ( Y
% 0.29/0.56          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_eq_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4887_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [Y: int,X: num,N: nat] :
% 0.29/0.56        ( ( ( ring_1_of_int_rat @ Y )
% 0.29/0.56          = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.29/0.56        = ( Y
% 0.29/0.56          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_eq_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4888_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [Y: int,X: num,N: nat] :
% 0.29/0.56        ( ( ( ring_18347121197199848620nteger @ Y )
% 0.29/0.56          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.29/0.56        = ( Y
% 0.29/0.56          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_eq_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4889_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,Y: int] :
% 0.29/0.56        ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
% 0.29/0.56          = ( ring_1_of_int_real @ Y ) )
% 0.29/0.56        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.29/0.56          = Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_eq_of_int_cancel_iff
% 0.29/0.56  thf(fact_4890_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,Y: int] :
% 0.29/0.56        ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.29/0.56          = ( ring_1_of_int_int @ Y ) )
% 0.29/0.56        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.29/0.56          = Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_eq_of_int_cancel_iff
% 0.29/0.56  thf(fact_4891_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,Y: int] :
% 0.29/0.56        ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
% 0.29/0.56          = ( ring_17405671764205052669omplex @ Y ) )
% 0.29/0.56        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.29/0.56          = Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_eq_of_int_cancel_iff
% 0.29/0.56  thf(fact_4892_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,Y: int] :
% 0.29/0.56        ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
% 0.29/0.56          = ( ring_1_of_int_rat @ Y ) )
% 0.29/0.56        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.29/0.56          = Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_eq_of_int_cancel_iff
% 0.29/0.56  thf(fact_4893_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,Y: int] :
% 0.29/0.56        ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
% 0.29/0.56          = ( ring_18347121197199848620nteger @ Y ) )
% 0.29/0.56        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.29/0.56          = Y ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_eq_of_int_cancel_iff
% 0.29/0.56  thf(fact_4894_dbl__simps_I4_J,axiom,
% 0.29/0.56      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.56      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dbl_simps(4)
% 0.29/0.56  thf(fact_4895_dbl__simps_I4_J,axiom,
% 0.29/0.56      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.56      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dbl_simps(4)
% 0.29/0.56  thf(fact_4896_dbl__simps_I4_J,axiom,
% 0.29/0.56      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.56      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dbl_simps(4)
% 0.29/0.56  thf(fact_4897_dbl__simps_I4_J,axiom,
% 0.29/0.56      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.56      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dbl_simps(4)
% 0.29/0.56  thf(fact_4898_dbl__simps_I4_J,axiom,
% 0.29/0.56      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.56      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dbl_simps(4)
% 0.29/0.56  thf(fact_4899_power__minus1__even,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.56        = one_one_real ) ).
% 0.29/0.56  
% 0.29/0.56  % power_minus1_even
% 0.29/0.56  thf(fact_4900_power__minus1__even,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.56        = one_one_int ) ).
% 0.29/0.56  
% 0.29/0.56  % power_minus1_even
% 0.29/0.56  thf(fact_4901_power__minus1__even,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.56        = one_one_complex ) ).
% 0.29/0.56  
% 0.29/0.56  % power_minus1_even
% 0.29/0.56  thf(fact_4902_power__minus1__even,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.56        = one_one_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % power_minus1_even
% 0.29/0.56  thf(fact_4903_power__minus1__even,axiom,
% 0.29/0.56      ! [N: nat] :
% 0.29/0.56        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.56        = one_one_Code_integer ) ).
% 0.29/0.56  
% 0.29/0.56  % power_minus1_even
% 0.29/0.56  thf(fact_4904_mod2__gr__0,axiom,
% 0.29/0.56      ! [M: nat] :
% 0.29/0.56        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.56        = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.56          = one_one_nat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod2_gr_0
% 0.29/0.56  thf(fact_4905_norm__of__real__add1,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ one_one_real ) )
% 0.29/0.56        = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % norm_of_real_add1
% 0.29/0.56  thf(fact_4906_norm__of__real__add1,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ one_one_complex ) )
% 0.29/0.56        = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % norm_of_real_add1
% 0.29/0.56  thf(fact_4907_norm__of__real__addn,axiom,
% 0.29/0.56      ! [X: real,B: num] :
% 0.29/0.56        ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B ) ) )
% 0.29/0.56        = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % norm_of_real_addn
% 0.29/0.56  thf(fact_4908_norm__of__real__addn,axiom,
% 0.29/0.56      ! [X: real,B: num] :
% 0.29/0.56        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B ) ) )
% 0.29/0.56        = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % norm_of_real_addn
% 0.29/0.56  thf(fact_4909_neg__numeral__less__floor,axiom,
% 0.29/0.56      ! [V: num,X: real] :
% 0.29/0.56        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 0.29/0.56        = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_floor
% 0.29/0.56  thf(fact_4910_neg__numeral__less__floor,axiom,
% 0.29/0.56      ! [V: num,X: rat] :
% 0.29/0.56        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.29/0.56        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_floor
% 0.29/0.56  thf(fact_4911_floor__le__neg__numeral,axiom,
% 0.29/0.56      ! [X: real,V: num] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.29/0.56        = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % floor_le_neg_numeral
% 0.29/0.56  thf(fact_4912_floor__le__neg__numeral,axiom,
% 0.29/0.56      ! [X: rat,V: num] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.29/0.56        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % floor_le_neg_numeral
% 0.29/0.56  thf(fact_4913_ceiling__less__neg__numeral,axiom,
% 0.29/0.56      ! [X: real,V: num] :
% 0.29/0.56        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.29/0.56        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_less_neg_numeral
% 0.29/0.56  thf(fact_4914_ceiling__less__neg__numeral,axiom,
% 0.29/0.56      ! [X: rat,V: num] :
% 0.29/0.56        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.29/0.56        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_less_neg_numeral
% 0.29/0.56  thf(fact_4915_neg__numeral__le__ceiling,axiom,
% 0.29/0.56      ! [V: num,X: real] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 0.29/0.56        = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_ceiling
% 0.29/0.56  thf(fact_4916_neg__numeral__le__ceiling,axiom,
% 0.29/0.56      ! [V: num,X: rat] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.29/0.56        = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_ceiling
% 0.29/0.56  thf(fact_4917_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [A: int,X: num,N: nat] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.29/0.56        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_le_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4918_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [A: int,X: num,N: nat] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.29/0.56        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_le_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4919_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [A: int,X: num,N: nat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.29/0.56        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_le_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4920_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [A: int,X: num,N: nat] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.29/0.56        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_le_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4921_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,A: int] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.29/0.56        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_le_of_int_cancel_iff
% 0.29/0.56  thf(fact_4922_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,A: int] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.29/0.56        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_le_of_int_cancel_iff
% 0.29/0.56  thf(fact_4923_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,A: int] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.29/0.56        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_le_of_int_cancel_iff
% 0.29/0.56  thf(fact_4924_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,A: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.29/0.56        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_le_of_int_cancel_iff
% 0.29/0.56  thf(fact_4925_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [A: int,X: num,N: nat] :
% 0.29/0.56        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.29/0.56        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_less_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4926_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [A: int,X: num,N: nat] :
% 0.29/0.56        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.29/0.56        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_less_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4927_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [A: int,X: num,N: nat] :
% 0.29/0.56        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.29/0.56        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_less_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4928_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.29/0.56      ! [A: int,X: num,N: nat] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.29/0.56        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_less_neg_numeral_power_cancel_iff
% 0.29/0.56  thf(fact_4929_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,A: int] :
% 0.29/0.56        ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.29/0.56        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_less_of_int_cancel_iff
% 0.29/0.56  thf(fact_4930_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,A: int] :
% 0.29/0.56        ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.29/0.56        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_less_of_int_cancel_iff
% 0.29/0.56  thf(fact_4931_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,A: int] :
% 0.29/0.56        ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.29/0.56        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_less_of_int_cancel_iff
% 0.29/0.56  thf(fact_4932_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.29/0.56      ! [X: num,N: nat,A: int] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.29/0.56        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_power_less_of_int_cancel_iff
% 0.29/0.56  thf(fact_4933_square__powr__half,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.56        = ( abs_abs_real @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % square_powr_half
% 0.29/0.56  thf(fact_4934_abs__less__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 0.29/0.56        = ( ( ord_less_real @ A @ B )
% 0.29/0.56          & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_less_iff
% 0.29/0.56  thf(fact_4935_abs__less__iff,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 0.29/0.56        = ( ( ord_less_int @ A @ B )
% 0.29/0.56          & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_less_iff
% 0.29/0.56  thf(fact_4936_abs__less__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 0.29/0.56        = ( ( ord_less_rat @ A @ B )
% 0.29/0.56          & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_less_iff
% 0.29/0.56  thf(fact_4937_abs__less__iff,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.29/0.56        = ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.29/0.56          & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_less_iff
% 0.29/0.56  thf(fact_4938_equation__minus__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( uminus_uminus_real @ B ) )
% 0.29/0.56        = ( B
% 0.29/0.56          = ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % equation_minus_iff
% 0.29/0.56  thf(fact_4939_equation__minus__iff,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( uminus_uminus_int @ B ) )
% 0.29/0.56        = ( B
% 0.29/0.56          = ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % equation_minus_iff
% 0.29/0.56  thf(fact_4940_equation__minus__iff,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( uminus1482373934393186551omplex @ B ) )
% 0.29/0.56        = ( B
% 0.29/0.56          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % equation_minus_iff
% 0.29/0.56  thf(fact_4941_equation__minus__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( uminus_uminus_rat @ B ) )
% 0.29/0.56        = ( B
% 0.29/0.56          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % equation_minus_iff
% 0.29/0.56  thf(fact_4942_equation__minus__iff,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.56        = ( B
% 0.29/0.56          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % equation_minus_iff
% 0.29/0.56  thf(fact_4943_minus__equation__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ( uminus_uminus_real @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( uminus_uminus_real @ B )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_equation_iff
% 0.29/0.56  thf(fact_4944_minus__equation__iff,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ( uminus_uminus_int @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( uminus_uminus_int @ B )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_equation_iff
% 0.29/0.56  thf(fact_4945_minus__equation__iff,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( ( uminus1482373934393186551omplex @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( uminus1482373934393186551omplex @ B )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_equation_iff
% 0.29/0.56  thf(fact_4946_minus__equation__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ( uminus_uminus_rat @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( uminus_uminus_rat @ B )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_equation_iff
% 0.29/0.56  thf(fact_4947_minus__equation__iff,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ( uminus1351360451143612070nteger @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( uminus1351360451143612070nteger @ B )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_equation_iff
% 0.29/0.56  thf(fact_4948_verit__negate__coefficient_I3_J,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( A = B )
% 0.29/0.56       => ( ( uminus_uminus_real @ A )
% 0.29/0.56          = ( uminus_uminus_real @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_negate_coefficient(3)
% 0.29/0.56  thf(fact_4949_verit__negate__coefficient_I3_J,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( A = B )
% 0.29/0.56       => ( ( uminus_uminus_int @ A )
% 0.29/0.56          = ( uminus_uminus_int @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_negate_coefficient(3)
% 0.29/0.56  thf(fact_4950_verit__negate__coefficient_I3_J,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( A = B )
% 0.29/0.56       => ( ( uminus_uminus_rat @ A )
% 0.29/0.56          = ( uminus_uminus_rat @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_negate_coefficient(3)
% 0.29/0.56  thf(fact_4951_verit__negate__coefficient_I3_J,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( A = B )
% 0.29/0.56       => ( ( uminus1351360451143612070nteger @ A )
% 0.29/0.56          = ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_negate_coefficient(3)
% 0.29/0.56  thf(fact_4952_abs__leI,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.56       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.29/0.56         => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_leI
% 0.29/0.56  thf(fact_4953_abs__leI,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.29/0.56       => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.29/0.56         => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_leI
% 0.29/0.56  thf(fact_4954_abs__leI,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.56       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.29/0.56         => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_leI
% 0.29/0.56  thf(fact_4955_abs__leI,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.56       => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.29/0.56         => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_leI
% 0.29/0.56  thf(fact_4956_abs__le__D2,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.29/0.56       => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_D2
% 0.29/0.56  thf(fact_4957_abs__le__D2,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.29/0.56       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_D2
% 0.29/0.56  thf(fact_4958_abs__le__D2,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.29/0.56       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_D2
% 0.29/0.56  thf(fact_4959_abs__le__D2,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.29/0.56       => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_D2
% 0.29/0.56  thf(fact_4960_abs__le__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.29/0.56        = ( ( ord_less_eq_real @ A @ B )
% 0.29/0.56          & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_iff
% 0.29/0.56  thf(fact_4961_abs__le__iff,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.29/0.56        = ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.29/0.56          & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_iff
% 0.29/0.56  thf(fact_4962_abs__le__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.29/0.56        = ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.56          & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_iff
% 0.29/0.56  thf(fact_4963_abs__le__iff,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.29/0.56        = ( ( ord_less_eq_int @ A @ B )
% 0.29/0.56          & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_iff
% 0.29/0.56  thf(fact_4964_abs__ge__minus__self,axiom,
% 0.29/0.56      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_minus_self
% 0.29/0.56  thf(fact_4965_abs__ge__minus__self,axiom,
% 0.29/0.56      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_minus_self
% 0.29/0.56  thf(fact_4966_abs__ge__minus__self,axiom,
% 0.29/0.56      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_minus_self
% 0.29/0.56  thf(fact_4967_abs__ge__minus__self,axiom,
% 0.29/0.56      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_minus_self
% 0.29/0.56  thf(fact_4968_of__nat__mod,axiom,
% 0.29/0.56      ! [M: nat,N: nat] :
% 0.29/0.56        ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 0.29/0.56        = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_nat_mod
% 0.29/0.56  thf(fact_4969_of__nat__mod,axiom,
% 0.29/0.56      ! [M: nat,N: nat] :
% 0.29/0.56        ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.29/0.56        = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_nat_mod
% 0.29/0.56  thf(fact_4970_of__nat__mod,axiom,
% 0.29/0.56      ! [M: nat,N: nat] :
% 0.29/0.56        ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 0.29/0.56        = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_nat_mod
% 0.29/0.56  thf(fact_4971_artanh__tanh__real,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( artanh_real @ ( tanh_real @ X ) )
% 0.29/0.56        = X ) ).
% 0.29/0.56  
% 0.29/0.56  % artanh_tanh_real
% 0.29/0.56  thf(fact_4972_abs__minus__le__zero,axiom,
% 0.29/0.56      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_minus_le_zero
% 0.29/0.56  thf(fact_4973_abs__minus__le__zero,axiom,
% 0.29/0.56      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_minus_le_zero
% 0.29/0.56  thf(fact_4974_abs__minus__le__zero,axiom,
% 0.29/0.56      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_minus_le_zero
% 0.29/0.56  thf(fact_4975_abs__minus__le__zero,axiom,
% 0.29/0.56      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_minus_le_zero
% 0.29/0.56  thf(fact_4976_eq__abs__iff_H,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( abs_abs_real @ B ) )
% 0.29/0.56        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.56          & ( ( B = A )
% 0.29/0.56            | ( B
% 0.29/0.56              = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_abs_iff'
% 0.29/0.56  thf(fact_4977_eq__abs__iff_H,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( abs_abs_Code_integer @ B ) )
% 0.29/0.56        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.56          & ( ( B = A )
% 0.29/0.56            | ( B
% 0.29/0.56              = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_abs_iff'
% 0.29/0.56  thf(fact_4978_eq__abs__iff_H,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( abs_abs_rat @ B ) )
% 0.29/0.56        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.56          & ( ( B = A )
% 0.29/0.56            | ( B
% 0.29/0.56              = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_abs_iff'
% 0.29/0.56  thf(fact_4979_eq__abs__iff_H,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( abs_abs_int @ B ) )
% 0.29/0.56        = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.56          & ( ( B = A )
% 0.29/0.56            | ( B
% 0.29/0.56              = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_abs_iff'
% 0.29/0.56  thf(fact_4980_abs__eq__iff_H,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ( abs_abs_real @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.56          & ( ( A = B )
% 0.29/0.56            | ( A
% 0.29/0.56              = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_iff'
% 0.29/0.56  thf(fact_4981_abs__eq__iff_H,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ( abs_abs_Code_integer @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.29/0.56          & ( ( A = B )
% 0.29/0.56            | ( A
% 0.29/0.56              = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_iff'
% 0.29/0.56  thf(fact_4982_abs__eq__iff_H,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ( abs_abs_rat @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.56          & ( ( A = B )
% 0.29/0.56            | ( A
% 0.29/0.56              = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_iff'
% 0.29/0.56  thf(fact_4983_abs__eq__iff_H,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ( abs_abs_int @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.56          & ( ( A = B )
% 0.29/0.56            | ( A
% 0.29/0.56              = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_iff'
% 0.29/0.56  thf(fact_4984_abs__of__neg,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.56       => ( ( abs_abs_real @ A )
% 0.29/0.56          = ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_neg
% 0.29/0.56  thf(fact_4985_abs__of__neg,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.56       => ( ( abs_abs_int @ A )
% 0.29/0.56          = ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_neg
% 0.29/0.56  thf(fact_4986_abs__of__neg,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.56       => ( ( abs_abs_rat @ A )
% 0.29/0.56          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_neg
% 0.29/0.56  thf(fact_4987_abs__of__neg,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.29/0.56       => ( ( abs_abs_Code_integer @ A )
% 0.29/0.56          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_neg
% 0.29/0.56  thf(fact_4988_abs__if__raw,axiom,
% 0.29/0.56      ( abs_abs_real
% 0.29/0.56      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_if_raw
% 0.29/0.56  thf(fact_4989_abs__if__raw,axiom,
% 0.29/0.56      ( abs_abs_int
% 0.29/0.56      = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_if_raw
% 0.29/0.56  thf(fact_4990_abs__if__raw,axiom,
% 0.29/0.56      ( abs_abs_rat
% 0.29/0.56      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_if_raw
% 0.29/0.56  thf(fact_4991_abs__if__raw,axiom,
% 0.29/0.56      ( abs_abs_Code_integer
% 0.29/0.56      = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_if_raw
% 0.29/0.56  thf(fact_4992_abs__if,axiom,
% 0.29/0.56      ( abs_abs_real
% 0.29/0.56      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_if
% 0.29/0.56  thf(fact_4993_abs__if,axiom,
% 0.29/0.56      ( abs_abs_int
% 0.29/0.56      = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_if
% 0.29/0.56  thf(fact_4994_abs__if,axiom,
% 0.29/0.56      ( abs_abs_rat
% 0.29/0.56      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_if
% 0.29/0.56  thf(fact_4995_abs__if,axiom,
% 0.29/0.56      ( abs_abs_Code_integer
% 0.29/0.56      = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_if
% 0.29/0.56  thf(fact_4996_abs__real__def,axiom,
% 0.29/0.56      ( abs_abs_real
% 0.29/0.56      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_real_def
% 0.29/0.56  thf(fact_4997_abs__ge__self,axiom,
% 0.29/0.56      ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_self
% 0.29/0.56  thf(fact_4998_abs__ge__self,axiom,
% 0.29/0.56      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_self
% 0.29/0.56  thf(fact_4999_abs__ge__self,axiom,
% 0.29/0.56      ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_self
% 0.29/0.56  thf(fact_5000_abs__ge__self,axiom,
% 0.29/0.56      ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_self
% 0.29/0.56  thf(fact_5001_abs__le__D1,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.29/0.56       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_D1
% 0.29/0.56  thf(fact_5002_abs__le__D1,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.29/0.56       => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_D1
% 0.29/0.56  thf(fact_5003_abs__le__D1,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.29/0.56       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_D1
% 0.29/0.56  thf(fact_5004_abs__le__D1,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.29/0.56       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_le_D1
% 0.29/0.56  thf(fact_5005_abs__eq__0__iff,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ( abs_abs_Code_integer @ A )
% 0.29/0.56          = zero_z3403309356797280102nteger )
% 0.29/0.56        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_0_iff
% 0.29/0.56  thf(fact_5006_abs__eq__0__iff,axiom,
% 0.29/0.56      ! [A: complex] :
% 0.29/0.56        ( ( ( abs_abs_complex @ A )
% 0.29/0.56          = zero_zero_complex )
% 0.29/0.56        = ( A = zero_zero_complex ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_0_iff
% 0.29/0.56  thf(fact_5007_abs__eq__0__iff,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ( abs_abs_real @ A )
% 0.29/0.56          = zero_zero_real )
% 0.29/0.56        = ( A = zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_0_iff
% 0.29/0.56  thf(fact_5008_abs__eq__0__iff,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ( abs_abs_rat @ A )
% 0.29/0.56          = zero_zero_rat )
% 0.29/0.56        = ( A = zero_zero_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_0_iff
% 0.29/0.56  thf(fact_5009_abs__eq__0__iff,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ( abs_abs_int @ A )
% 0.29/0.56          = zero_zero_int )
% 0.29/0.56        = ( A = zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_0_iff
% 0.29/0.56  thf(fact_5010_abs__one,axiom,
% 0.29/0.56      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.29/0.56      = one_one_Code_integer ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_one
% 0.29/0.56  thf(fact_5011_abs__one,axiom,
% 0.29/0.56      ( ( abs_abs_real @ one_one_real )
% 0.29/0.56      = one_one_real ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_one
% 0.29/0.56  thf(fact_5012_abs__one,axiom,
% 0.29/0.56      ( ( abs_abs_rat @ one_one_rat )
% 0.29/0.56      = one_one_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_one
% 0.29/0.56  thf(fact_5013_abs__one,axiom,
% 0.29/0.56      ( ( abs_abs_int @ one_one_int )
% 0.29/0.56      = one_one_int ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_one
% 0.29/0.56  thf(fact_5014_abs__minus__commute,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.29/0.56        = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_minus_commute
% 0.29/0.56  thf(fact_5015_abs__minus__commute,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.29/0.56        = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_minus_commute
% 0.29/0.56  thf(fact_5016_abs__minus__commute,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 0.29/0.56        = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_minus_commute
% 0.29/0.56  thf(fact_5017_abs__minus__commute,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 0.29/0.56        = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_minus_commute
% 0.29/0.56  thf(fact_5018_tanh__real__gt__neg1,axiom,
% 0.29/0.56      ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % tanh_real_gt_neg1
% 0.29/0.56  thf(fact_5019_power__abs,axiom,
% 0.29/0.56      ! [A: rat,N: nat] :
% 0.29/0.56        ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
% 0.29/0.56        = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power_abs
% 0.29/0.56  thf(fact_5020_power__abs,axiom,
% 0.29/0.56      ! [A: code_integer,N: nat] :
% 0.29/0.56        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.29/0.56        = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power_abs
% 0.29/0.56  thf(fact_5021_power__abs,axiom,
% 0.29/0.56      ! [A: real,N: nat] :
% 0.29/0.56        ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 0.29/0.56        = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power_abs
% 0.29/0.56  thf(fact_5022_power__abs,axiom,
% 0.29/0.56      ! [A: int,N: nat] :
% 0.29/0.56        ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 0.29/0.56        = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power_abs
% 0.29/0.56  thf(fact_5023_mod__add__eq,axiom,
% 0.29/0.56      ! [A: nat,C: nat,B: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.29/0.56        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_eq
% 0.29/0.56  thf(fact_5024_mod__add__eq,axiom,
% 0.29/0.56      ! [A: int,C: int,B: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.29/0.56        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_eq
% 0.29/0.56  thf(fact_5025_mod__add__eq,axiom,
% 0.29/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.29/0.56        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_eq
% 0.29/0.56  thf(fact_5026_mod__add__cong,axiom,
% 0.29/0.56      ! [A: nat,C: nat,A6: nat,B: nat,B7: nat] :
% 0.29/0.56        ( ( ( modulo_modulo_nat @ A @ C )
% 0.29/0.56          = ( modulo_modulo_nat @ A6 @ C ) )
% 0.29/0.56       => ( ( ( modulo_modulo_nat @ B @ C )
% 0.29/0.56            = ( modulo_modulo_nat @ B7 @ C ) )
% 0.29/0.56         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.56            = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B7 ) @ C ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_cong
% 0.29/0.56  thf(fact_5027_mod__add__cong,axiom,
% 0.29/0.56      ! [A: int,C: int,A6: int,B: int,B7: int] :
% 0.29/0.56        ( ( ( modulo_modulo_int @ A @ C )
% 0.29/0.56          = ( modulo_modulo_int @ A6 @ C ) )
% 0.29/0.56       => ( ( ( modulo_modulo_int @ B @ C )
% 0.29/0.56            = ( modulo_modulo_int @ B7 @ C ) )
% 0.29/0.56         => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.56            = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B7 ) @ C ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_cong
% 0.29/0.56  thf(fact_5028_mod__add__cong,axiom,
% 0.29/0.56      ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B7: code_integer] :
% 0.29/0.56        ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.29/0.56          = ( modulo364778990260209775nteger @ A6 @ C ) )
% 0.29/0.56       => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.29/0.56            = ( modulo364778990260209775nteger @ B7 @ C ) )
% 0.29/0.56         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.29/0.56            = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A6 @ B7 ) @ C ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_cong
% 0.29/0.56  thf(fact_5029_mod__add__left__eq,axiom,
% 0.29/0.56      ! [A: nat,C: nat,B: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.29/0.56        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_left_eq
% 0.29/0.56  thf(fact_5030_mod__add__left__eq,axiom,
% 0.29/0.56      ! [A: int,C: int,B: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.29/0.56        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_left_eq
% 0.29/0.56  thf(fact_5031_mod__add__left__eq,axiom,
% 0.29/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.29/0.56        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_left_eq
% 0.29/0.56  thf(fact_5032_mod__add__right__eq,axiom,
% 0.29/0.56      ! [A: nat,B: nat,C: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.29/0.56        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_right_eq
% 0.29/0.56  thf(fact_5033_mod__add__right__eq,axiom,
% 0.29/0.56      ! [A: int,B: int,C: int] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.29/0.56        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_right_eq
% 0.29/0.56  thf(fact_5034_mod__add__right__eq,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.29/0.56        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_add_right_eq
% 0.29/0.56  thf(fact_5035_ceiling__def,axiom,
% 0.29/0.56      ( archim7802044766580827645g_real
% 0.29/0.56      = ( ^ [X4: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_def
% 0.29/0.56  thf(fact_5036_ceiling__def,axiom,
% 0.29/0.56      ( archim2889992004027027881ng_rat
% 0.29/0.56      = ( ^ [X4: rat] : ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X4 ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_def
% 0.29/0.56  thf(fact_5037_floor__minus,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % floor_minus
% 0.29/0.56  thf(fact_5038_floor__minus,axiom,
% 0.29/0.56      ! [X: rat] :
% 0.29/0.56        ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % floor_minus
% 0.29/0.56  thf(fact_5039_ceiling__minus,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( archim6058952711729229775r_real @ X ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_minus
% 0.29/0.56  thf(fact_5040_ceiling__minus,axiom,
% 0.29/0.56      ! [X: rat] :
% 0.29/0.56        ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ X ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ X ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ceiling_minus
% 0.29/0.56  thf(fact_5041_power__mod,axiom,
% 0.29/0.56      ! [A: nat,B: nat,N: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
% 0.29/0.56        = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power_mod
% 0.29/0.56  thf(fact_5042_power__mod,axiom,
% 0.29/0.56      ! [A: int,B: int,N: nat] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
% 0.29/0.56        = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power_mod
% 0.29/0.56  thf(fact_5043_power__mod,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer,N: nat] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
% 0.29/0.56        = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % power_mod
% 0.29/0.56  thf(fact_5044_VEBT_Osize_I4_J,axiom,
% 0.29/0.56      ! [X21: $o,X222: $o] :
% 0.29/0.56        ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % VEBT.size(4)
% 0.29/0.56  thf(fact_5045_mod__Suc__Suc__eq,axiom,
% 0.29/0.56      ! [M: nat,N: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 0.29/0.56        = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_Suc_Suc_eq
% 0.29/0.56  thf(fact_5046_mod__Suc__eq,axiom,
% 0.29/0.56      ! [M: nat,N: nat] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 0.29/0.56        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_Suc_eq
% 0.29/0.56  thf(fact_5047_le__imp__neg__le,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.56       => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_imp_neg_le
% 0.29/0.56  thf(fact_5048_le__imp__neg__le,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.29/0.56       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_imp_neg_le
% 0.29/0.56  thf(fact_5049_le__imp__neg__le,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.56       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_imp_neg_le
% 0.29/0.56  thf(fact_5050_le__imp__neg__le,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.56       => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_imp_neg_le
% 0.29/0.56  thf(fact_5051_minus__le__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.29/0.56        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_le_iff
% 0.29/0.56  thf(fact_5052_minus__le__iff,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.29/0.56        = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_le_iff
% 0.29/0.56  thf(fact_5053_minus__le__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.29/0.56        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_le_iff
% 0.29/0.56  thf(fact_5054_minus__le__iff,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.29/0.56        = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_le_iff
% 0.29/0.56  thf(fact_5055_le__minus__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.29/0.56        = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_iff
% 0.29/0.56  thf(fact_5056_le__minus__iff,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.56        = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_iff
% 0.29/0.56  thf(fact_5057_le__minus__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.29/0.56        = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_iff
% 0.29/0.56  thf(fact_5058_le__minus__iff,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.29/0.56        = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_iff
% 0.29/0.56  thf(fact_5059_less__minus__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.29/0.56        = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_iff
% 0.29/0.56  thf(fact_5060_less__minus__iff,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 0.29/0.56        = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_iff
% 0.29/0.56  thf(fact_5061_less__minus__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.29/0.56        = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_iff
% 0.29/0.56  thf(fact_5062_less__minus__iff,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.56        = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_iff
% 0.29/0.56  thf(fact_5063_minus__less__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.29/0.56        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_less_iff
% 0.29/0.56  thf(fact_5064_minus__less__iff,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 0.29/0.56        = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_less_iff
% 0.29/0.56  thf(fact_5065_minus__less__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.29/0.56        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_less_iff
% 0.29/0.56  thf(fact_5066_minus__less__iff,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.29/0.56        = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_less_iff
% 0.29/0.56  thf(fact_5067_verit__negate__coefficient_I2_J,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ord_less_real @ A @ B )
% 0.29/0.56       => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_negate_coefficient(2)
% 0.29/0.56  thf(fact_5068_verit__negate__coefficient_I2_J,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_int @ A @ B )
% 0.29/0.56       => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_negate_coefficient(2)
% 0.29/0.56  thf(fact_5069_verit__negate__coefficient_I2_J,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ord_less_rat @ A @ B )
% 0.29/0.56       => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_negate_coefficient(2)
% 0.29/0.56  thf(fact_5070_verit__negate__coefficient_I2_J,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.29/0.56       => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % verit_negate_coefficient(2)
% 0.29/0.56  thf(fact_5071_neg__numeral__neq__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.29/0.56       != ( numeral_numeral_real @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_neq_numeral
% 0.29/0.56  thf(fact_5072_neg__numeral__neq__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.29/0.56       != ( numeral_numeral_int @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_neq_numeral
% 0.29/0.56  thf(fact_5073_neg__numeral__neq__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.29/0.56       != ( numera6690914467698888265omplex @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_neq_numeral
% 0.29/0.56  thf(fact_5074_neg__numeral__neq__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.29/0.56       != ( numeral_numeral_rat @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_neq_numeral
% 0.29/0.56  thf(fact_5075_neg__numeral__neq__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.29/0.56       != ( numera6620942414471956472nteger @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_neq_numeral
% 0.29/0.56  thf(fact_5076_numeral__neq__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( numeral_numeral_real @ M )
% 0.29/0.56       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_neq_neg_numeral
% 0.29/0.56  thf(fact_5077_numeral__neq__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( numeral_numeral_int @ M )
% 0.29/0.56       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_neq_neg_numeral
% 0.29/0.56  thf(fact_5078_numeral__neq__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( numera6690914467698888265omplex @ M )
% 0.29/0.56       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_neq_neg_numeral
% 0.29/0.56  thf(fact_5079_numeral__neq__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( numeral_numeral_rat @ M )
% 0.29/0.56       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_neq_neg_numeral
% 0.29/0.56  thf(fact_5080_numeral__neq__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( numera6620942414471956472nteger @ M )
% 0.29/0.56       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_neq_neg_numeral
% 0.29/0.56  thf(fact_5081_mod__less__eq__dividend,axiom,
% 0.29/0.56      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_less_eq_dividend
% 0.29/0.56  thf(fact_5082_one__neq__neg__one,axiom,
% 0.29/0.56      ( one_one_real
% 0.29/0.56     != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % one_neq_neg_one
% 0.29/0.56  thf(fact_5083_one__neq__neg__one,axiom,
% 0.29/0.56      ( one_one_int
% 0.29/0.56     != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % one_neq_neg_one
% 0.29/0.56  thf(fact_5084_one__neq__neg__one,axiom,
% 0.29/0.56      ( one_one_complex
% 0.29/0.56     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.56  
% 0.29/0.56  % one_neq_neg_one
% 0.29/0.56  thf(fact_5085_one__neq__neg__one,axiom,
% 0.29/0.56      ( one_one_rat
% 0.29/0.56     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % one_neq_neg_one
% 0.29/0.56  thf(fact_5086_one__neq__neg__one,axiom,
% 0.29/0.56      ( one_one_Code_integer
% 0.29/0.56     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % one_neq_neg_one
% 0.29/0.56  thf(fact_5087_is__num__normalize_I8_J,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.29/0.56        = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % is_num_normalize(8)
% 0.29/0.56  thf(fact_5088_is__num__normalize_I8_J,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.29/0.56        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % is_num_normalize(8)
% 0.29/0.56  thf(fact_5089_is__num__normalize_I8_J,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.29/0.56        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % is_num_normalize(8)
% 0.29/0.56  thf(fact_5090_is__num__normalize_I8_J,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.56        = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % is_num_normalize(8)
% 0.29/0.56  thf(fact_5091_is__num__normalize_I8_J,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.29/0.56        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % is_num_normalize(8)
% 0.29/0.56  thf(fact_5092_group__cancel_Oneg1,axiom,
% 0.29/0.56      ! [A2: real,K: real,A: real] :
% 0.29/0.56        ( ( A2
% 0.29/0.56          = ( plus_plus_real @ K @ A ) )
% 0.29/0.56       => ( ( uminus_uminus_real @ A2 )
% 0.29/0.56          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % group_cancel.neg1
% 0.29/0.56  thf(fact_5093_group__cancel_Oneg1,axiom,
% 0.29/0.56      ! [A2: int,K: int,A: int] :
% 0.29/0.56        ( ( A2
% 0.29/0.56          = ( plus_plus_int @ K @ A ) )
% 0.29/0.56       => ( ( uminus_uminus_int @ A2 )
% 0.29/0.56          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % group_cancel.neg1
% 0.29/0.56  thf(fact_5094_group__cancel_Oneg1,axiom,
% 0.29/0.56      ! [A2: complex,K: complex,A: complex] :
% 0.29/0.56        ( ( A2
% 0.29/0.56          = ( plus_plus_complex @ K @ A ) )
% 0.29/0.56       => ( ( uminus1482373934393186551omplex @ A2 )
% 0.29/0.56          = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % group_cancel.neg1
% 0.29/0.56  thf(fact_5095_group__cancel_Oneg1,axiom,
% 0.29/0.56      ! [A2: rat,K: rat,A: rat] :
% 0.29/0.56        ( ( A2
% 0.29/0.56          = ( plus_plus_rat @ K @ A ) )
% 0.29/0.56       => ( ( uminus_uminus_rat @ A2 )
% 0.29/0.56          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % group_cancel.neg1
% 0.29/0.56  thf(fact_5096_group__cancel_Oneg1,axiom,
% 0.29/0.56      ! [A2: code_integer,K: code_integer,A: code_integer] :
% 0.29/0.56        ( ( A2
% 0.29/0.56          = ( plus_p5714425477246183910nteger @ K @ A ) )
% 0.29/0.56       => ( ( uminus1351360451143612070nteger @ A2 )
% 0.29/0.56          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % group_cancel.neg1
% 0.29/0.56  thf(fact_5097_add_Oinverse__distrib__swap,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.29/0.56        = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add.inverse_distrib_swap
% 0.29/0.56  thf(fact_5098_add_Oinverse__distrib__swap,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.29/0.56        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add.inverse_distrib_swap
% 0.29/0.56  thf(fact_5099_add_Oinverse__distrib__swap,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.29/0.56        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add.inverse_distrib_swap
% 0.29/0.56  thf(fact_5100_add_Oinverse__distrib__swap,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.56        = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add.inverse_distrib_swap
% 0.29/0.56  thf(fact_5101_add_Oinverse__distrib__swap,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.29/0.56        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add.inverse_distrib_swap
% 0.29/0.56  thf(fact_5102_minus__diff__commute,axiom,
% 0.29/0.56      ! [B: real,A: real] :
% 0.29/0.56        ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 0.29/0.56        = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_diff_commute
% 0.29/0.56  thf(fact_5103_minus__diff__commute,axiom,
% 0.29/0.56      ! [B: int,A: int] :
% 0.29/0.56        ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.29/0.56        = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_diff_commute
% 0.29/0.56  thf(fact_5104_minus__diff__commute,axiom,
% 0.29/0.56      ! [B: complex,A: complex] :
% 0.29/0.56        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 0.29/0.56        = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_diff_commute
% 0.29/0.56  thf(fact_5105_minus__diff__commute,axiom,
% 0.29/0.56      ! [B: rat,A: rat] :
% 0.29/0.56        ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 0.29/0.56        = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_diff_commute
% 0.29/0.56  thf(fact_5106_minus__diff__commute,axiom,
% 0.29/0.56      ! [B: code_integer,A: code_integer] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 0.29/0.56        = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_diff_commute
% 0.29/0.56  thf(fact_5107_minus__diff__minus,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.29/0.56        = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_diff_minus
% 0.29/0.56  thf(fact_5108_minus__diff__minus,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_diff_minus
% 0.29/0.56  thf(fact_5109_minus__diff__minus,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_diff_minus
% 0.29/0.56  thf(fact_5110_minus__diff__minus,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.29/0.56        = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_diff_minus
% 0.29/0.56  thf(fact_5111_minus__diff__minus,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_diff_minus
% 0.29/0.56  thf(fact_5112_div__minus__right,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.29/0.56        = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % div_minus_right
% 0.29/0.56  thf(fact_5113_div__minus__right,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.56        = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % div_minus_right
% 0.29/0.56  thf(fact_5114_minus__divide__left,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.56        = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_divide_left
% 0.29/0.56  thf(fact_5115_minus__divide__left,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.56        = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_divide_left
% 0.29/0.56  thf(fact_5116_minus__divide__left,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.29/0.56        = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_divide_left
% 0.29/0.56  thf(fact_5117_minus__divide__divide,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.29/0.56        = ( divide_divide_real @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_divide_divide
% 0.29/0.56  thf(fact_5118_minus__divide__divide,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.29/0.56        = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_divide_divide
% 0.29/0.56  thf(fact_5119_minus__divide__divide,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.29/0.56        = ( divide_divide_rat @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_divide_divide
% 0.29/0.56  thf(fact_5120_minus__divide__right,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.56        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_divide_right
% 0.29/0.56  thf(fact_5121_minus__divide__right,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.56        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_divide_right
% 0.29/0.56  thf(fact_5122_minus__divide__right,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.29/0.56        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_divide_right
% 0.29/0.56  thf(fact_5123_VEBT__internal_Oheight_Ocases,axiom,
% 0.29/0.56      ! [X: vEBT_VEBT] :
% 0.29/0.56        ( ! [A5: $o,B6: $o] :
% 0.29/0.56            ( X
% 0.29/0.56           != ( vEBT_Leaf @ A5 @ B6 ) )
% 0.29/0.56       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.29/0.56              ( X
% 0.29/0.56             != ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % VEBT_internal.height.cases
% 0.29/0.56  thf(fact_5124_VEBT_Odistinct_I1_J,axiom,
% 0.29/0.56      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 0.29/0.56        ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.29/0.56       != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 0.29/0.56  
% 0.29/0.56  % VEBT.distinct(1)
% 0.29/0.56  thf(fact_5125_VEBT_Oexhaust,axiom,
% 0.29/0.56      ! [Y: vEBT_VEBT] :
% 0.29/0.56        ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 0.29/0.56            ( Y
% 0.29/0.56           != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 0.29/0.56       => ~ ! [X212: $o,X223: $o] :
% 0.29/0.56              ( Y
% 0.29/0.56             != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % VEBT.exhaust
% 0.29/0.56  thf(fact_5126_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 0.29/0.56      ! [Uu: $o,Uv: $o,Uw: nat] :
% 0.29/0.56        ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 0.29/0.56  
% 0.29/0.56  % VEBT_internal.membermima.simps(1)
% 0.29/0.56  thf(fact_5127_of__int__neg__numeral,axiom,
% 0.29/0.56      ! [K: num] :
% 0.29/0.56        ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.56        = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_neg_numeral
% 0.29/0.56  thf(fact_5128_of__int__neg__numeral,axiom,
% 0.29/0.56      ! [K: num] :
% 0.29/0.56        ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.56        = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_neg_numeral
% 0.29/0.56  thf(fact_5129_of__int__neg__numeral,axiom,
% 0.29/0.56      ! [K: num] :
% 0.29/0.56        ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.56        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_neg_numeral
% 0.29/0.56  thf(fact_5130_of__int__neg__numeral,axiom,
% 0.29/0.56      ! [K: num] :
% 0.29/0.56        ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.56        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_neg_numeral
% 0.29/0.56  thf(fact_5131_of__int__neg__numeral,axiom,
% 0.29/0.56      ! [K: num] :
% 0.29/0.56        ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.56        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % of_int_neg_numeral
% 0.29/0.56  thf(fact_5132_norm__of__real__diff,axiom,
% 0.29/0.56      ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % norm_of_real_diff
% 0.29/0.56  thf(fact_5133_norm__of__real__diff,axiom,
% 0.29/0.56      ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % norm_of_real_diff
% 0.29/0.56  thf(fact_5134_abs__ge__zero,axiom,
% 0.29/0.56      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_zero
% 0.29/0.56  thf(fact_5135_abs__ge__zero,axiom,
% 0.29/0.56      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_zero
% 0.29/0.56  thf(fact_5136_abs__ge__zero,axiom,
% 0.29/0.56      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_zero
% 0.29/0.56  thf(fact_5137_abs__ge__zero,axiom,
% 0.29/0.56      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_ge_zero
% 0.29/0.56  thf(fact_5138_abs__of__pos,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.56       => ( ( abs_abs_Code_integer @ A )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_pos
% 0.29/0.56  thf(fact_5139_abs__of__pos,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.56       => ( ( abs_abs_real @ A )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_pos
% 0.29/0.56  thf(fact_5140_abs__of__pos,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.56       => ( ( abs_abs_rat @ A )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_pos
% 0.29/0.56  thf(fact_5141_abs__of__pos,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.56       => ( ( abs_abs_int @ A )
% 0.29/0.56          = A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_of_pos
% 0.29/0.56  thf(fact_5142_abs__not__less__zero,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_not_less_zero
% 0.29/0.56  thf(fact_5143_abs__not__less__zero,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_not_less_zero
% 0.29/0.56  thf(fact_5144_abs__not__less__zero,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_not_less_zero
% 0.29/0.56  thf(fact_5145_abs__not__less__zero,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_not_less_zero
% 0.29/0.56  thf(fact_5146_abs__triangle__ineq,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq
% 0.29/0.56  thf(fact_5147_abs__triangle__ineq,axiom,
% 0.29/0.56      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq
% 0.29/0.56  thf(fact_5148_abs__triangle__ineq,axiom,
% 0.29/0.56      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq
% 0.29/0.56  thf(fact_5149_abs__triangle__ineq,axiom,
% 0.29/0.56      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq
% 0.29/0.56  thf(fact_5150_abs__mult__less,axiom,
% 0.29/0.56      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 0.29/0.56       => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 0.29/0.56         => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_mult_less
% 0.29/0.56  thf(fact_5151_abs__mult__less,axiom,
% 0.29/0.56      ! [A: real,C: real,B: real,D: real] :
% 0.29/0.56        ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 0.29/0.56       => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 0.29/0.56         => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_mult_less
% 0.29/0.56  thf(fact_5152_abs__mult__less,axiom,
% 0.29/0.56      ! [A: rat,C: rat,B: rat,D: rat] :
% 0.29/0.56        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 0.29/0.56       => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 0.29/0.56         => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_mult_less
% 0.29/0.56  thf(fact_5153_abs__mult__less,axiom,
% 0.29/0.56      ! [A: int,C: int,B: int,D: int] :
% 0.29/0.56        ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 0.29/0.56       => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 0.29/0.56         => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_mult_less
% 0.29/0.56  thf(fact_5154_abs__triangle__ineq2,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq2
% 0.29/0.56  thf(fact_5155_abs__triangle__ineq2,axiom,
% 0.29/0.56      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq2
% 0.29/0.56  thf(fact_5156_abs__triangle__ineq2,axiom,
% 0.29/0.56      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq2
% 0.29/0.56  thf(fact_5157_abs__triangle__ineq2,axiom,
% 0.29/0.56      ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq2
% 0.29/0.56  thf(fact_5158_abs__triangle__ineq3,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq3
% 0.29/0.56  thf(fact_5159_abs__triangle__ineq3,axiom,
% 0.29/0.56      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq3
% 0.29/0.56  thf(fact_5160_abs__triangle__ineq3,axiom,
% 0.29/0.56      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq3
% 0.29/0.56  thf(fact_5161_abs__triangle__ineq3,axiom,
% 0.29/0.56      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq3
% 0.29/0.56  thf(fact_5162_abs__triangle__ineq2__sym,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq2_sym
% 0.29/0.56  thf(fact_5163_abs__triangle__ineq2__sym,axiom,
% 0.29/0.56      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq2_sym
% 0.29/0.56  thf(fact_5164_abs__triangle__ineq2__sym,axiom,
% 0.29/0.56      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq2_sym
% 0.29/0.56  thf(fact_5165_abs__triangle__ineq2__sym,axiom,
% 0.29/0.56      ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq2_sym
% 0.29/0.56  thf(fact_5166_nonzero__abs__divide,axiom,
% 0.29/0.56      ! [B: real,A: real] :
% 0.29/0.56        ( ( B != zero_zero_real )
% 0.29/0.56       => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.56          = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_abs_divide
% 0.29/0.56  thf(fact_5167_nonzero__abs__divide,axiom,
% 0.29/0.56      ! [B: rat,A: rat] :
% 0.29/0.56        ( ( B != zero_zero_rat )
% 0.29/0.56       => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.29/0.56          = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_abs_divide
% 0.29/0.56  thf(fact_5168_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.56       => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.29/0.56  thf(fact_5169_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.56       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.29/0.56  thf(fact_5170_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.56       => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.29/0.56  thf(fact_5171_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.29/0.56      ! [B: nat,A: nat] :
% 0.29/0.56        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.56       => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.29/0.56  thf(fact_5172_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.29/0.56      ! [B: int,A: int] :
% 0.29/0.56        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.56       => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.29/0.56  thf(fact_5173_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.29/0.56      ! [B: code_integer,A: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.29/0.56       => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.29/0.56  thf(fact_5174_mod__eq__self__iff__div__eq__0,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( ( modulo_modulo_nat @ A @ B )
% 0.29/0.56          = A )
% 0.29/0.56        = ( ( divide_divide_nat @ A @ B )
% 0.29/0.56          = zero_zero_nat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_eq_self_iff_div_eq_0
% 0.29/0.56  thf(fact_5175_mod__eq__self__iff__div__eq__0,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.56          = A )
% 0.29/0.56        = ( ( divide_divide_int @ A @ B )
% 0.29/0.56          = zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_eq_self_iff_div_eq_0
% 0.29/0.56  thf(fact_5176_mod__eq__self__iff__div__eq__0,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.29/0.56          = A )
% 0.29/0.56        = ( ( divide6298287555418463151nteger @ A @ B )
% 0.29/0.56          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_eq_self_iff_div_eq_0
% 0.29/0.56  thf(fact_5177_cong__exp__iff__simps_I9_J,axiom,
% 0.29/0.56      ! [M: num,Q3: num,N: num] :
% 0.29/0.56        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.29/0.56          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.56        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.29/0.56          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(9)
% 0.29/0.56  thf(fact_5178_cong__exp__iff__simps_I9_J,axiom,
% 0.29/0.56      ! [M: num,Q3: num,N: num] :
% 0.29/0.56        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.29/0.56          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.56        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 0.29/0.56          = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(9)
% 0.29/0.56  thf(fact_5179_cong__exp__iff__simps_I9_J,axiom,
% 0.29/0.56      ! [M: num,Q3: num,N: num] :
% 0.29/0.56        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.29/0.56          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.56        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.29/0.56          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(9)
% 0.29/0.56  thf(fact_5180_cong__exp__iff__simps_I4_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 0.29/0.56        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(4)
% 0.29/0.56  thf(fact_5181_cong__exp__iff__simps_I4_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 0.29/0.56        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(4)
% 0.29/0.56  thf(fact_5182_cong__exp__iff__simps_I4_J,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 0.29/0.56        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(4)
% 0.29/0.56  thf(fact_5183_mod__eqE,axiom,
% 0.29/0.56      ! [A: int,C: int,B: int] :
% 0.29/0.56        ( ( ( modulo_modulo_int @ A @ C )
% 0.29/0.56          = ( modulo_modulo_int @ B @ C ) )
% 0.29/0.56       => ~ ! [D5: int] :
% 0.29/0.56              ( B
% 0.29/0.56             != ( plus_plus_int @ A @ ( times_times_int @ C @ D5 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_eqE
% 0.29/0.56  thf(fact_5184_mod__eqE,axiom,
% 0.29/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.29/0.56          = ( modulo364778990260209775nteger @ B @ C ) )
% 0.29/0.56       => ~ ! [D5: code_integer] :
% 0.29/0.56              ( B
% 0.29/0.56             != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D5 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_eqE
% 0.29/0.56  thf(fact_5185_div__add1__eq,axiom,
% 0.29/0.56      ! [A: nat,B: nat,C: nat] :
% 0.29/0.56        ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.56        = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % div_add1_eq
% 0.29/0.56  thf(fact_5186_div__add1__eq,axiom,
% 0.29/0.56      ! [A: int,B: int,C: int] :
% 0.29/0.56        ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.56        = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % div_add1_eq
% 0.29/0.56  thf(fact_5187_div__add1__eq,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.56        ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.29/0.56        = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % div_add1_eq
% 0.29/0.56  thf(fact_5188_mod__Suc,axiom,
% 0.29/0.56      ! [M: nat,N: nat] :
% 0.29/0.56        ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.29/0.56            = N )
% 0.29/0.56         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.29/0.56            = zero_zero_nat ) )
% 0.29/0.56        & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.29/0.56           != N )
% 0.29/0.56         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.29/0.56            = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_Suc
% 0.29/0.56  thf(fact_5189_not__numeral__le__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_le_neg_numeral
% 0.29/0.56  thf(fact_5190_not__numeral__le__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_le_neg_numeral
% 0.29/0.56  thf(fact_5191_not__numeral__le__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_le_neg_numeral
% 0.29/0.56  thf(fact_5192_not__numeral__le__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_le_neg_numeral
% 0.29/0.56  thf(fact_5193_neg__numeral__le__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_numeral
% 0.29/0.56  thf(fact_5194_neg__numeral__le__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_numeral
% 0.29/0.56  thf(fact_5195_neg__numeral__le__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_numeral
% 0.29/0.56  thf(fact_5196_neg__numeral__le__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_numeral
% 0.29/0.56  thf(fact_5197_mod__induct,axiom,
% 0.29/0.56      ! [P: nat > $o,N: nat,P5: nat,M: nat] :
% 0.29/0.56        ( ( P @ N )
% 0.29/0.56       => ( ( ord_less_nat @ N @ P5 )
% 0.29/0.56         => ( ( ord_less_nat @ M @ P5 )
% 0.29/0.56           => ( ! [N2: nat] :
% 0.29/0.56                  ( ( ord_less_nat @ N2 @ P5 )
% 0.29/0.56                 => ( ( P @ N2 )
% 0.29/0.56                   => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P5 ) ) ) )
% 0.29/0.56             => ( P @ M ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_induct
% 0.29/0.56  thf(fact_5198_zero__neq__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( zero_zero_real
% 0.29/0.56       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_neq_neg_numeral
% 0.29/0.56  thf(fact_5199_zero__neq__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( zero_zero_int
% 0.29/0.56       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_neq_neg_numeral
% 0.29/0.56  thf(fact_5200_zero__neq__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( zero_zero_complex
% 0.29/0.56       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_neq_neg_numeral
% 0.29/0.56  thf(fact_5201_zero__neq__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( zero_zero_rat
% 0.29/0.56       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_neq_neg_numeral
% 0.29/0.56  thf(fact_5202_zero__neq__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( zero_z3403309356797280102nteger
% 0.29/0.56       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_neq_neg_numeral
% 0.29/0.56  thf(fact_5203_mod__less__divisor,axiom,
% 0.29/0.56      ! [N: nat,M: nat] :
% 0.29/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.56       => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_less_divisor
% 0.29/0.56  thf(fact_5204_not__numeral__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_less_neg_numeral
% 0.29/0.56  thf(fact_5205_not__numeral__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_less_neg_numeral
% 0.29/0.56  thf(fact_5206_not__numeral__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_less_neg_numeral
% 0.29/0.56  thf(fact_5207_not__numeral__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] :
% 0.29/0.56        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_less_neg_numeral
% 0.29/0.56  thf(fact_5208_neg__numeral__less__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_numeral
% 0.29/0.56  thf(fact_5209_neg__numeral__less__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_numeral
% 0.29/0.56  thf(fact_5210_neg__numeral__less__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_numeral
% 0.29/0.56  thf(fact_5211_neg__numeral__less__numeral,axiom,
% 0.29/0.56      ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_numeral
% 0.29/0.56  thf(fact_5212_le__minus__one__simps_I2_J,axiom,
% 0.29/0.56      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(2)
% 0.29/0.56  thf(fact_5213_le__minus__one__simps_I2_J,axiom,
% 0.29/0.56      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(2)
% 0.29/0.56  thf(fact_5214_le__minus__one__simps_I2_J,axiom,
% 0.29/0.56      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(2)
% 0.29/0.56  thf(fact_5215_le__minus__one__simps_I2_J,axiom,
% 0.29/0.56      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(2)
% 0.29/0.56  thf(fact_5216_le__minus__one__simps_I4_J,axiom,
% 0.29/0.56      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(4)
% 0.29/0.56  thf(fact_5217_le__minus__one__simps_I4_J,axiom,
% 0.29/0.56      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(4)
% 0.29/0.56  thf(fact_5218_le__minus__one__simps_I4_J,axiom,
% 0.29/0.56      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(4)
% 0.29/0.56  thf(fact_5219_le__minus__one__simps_I4_J,axiom,
% 0.29/0.56      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(4)
% 0.29/0.56  thf(fact_5220_zero__neq__neg__one,axiom,
% 0.29/0.56      ( zero_zero_real
% 0.29/0.56     != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_neq_neg_one
% 0.29/0.56  thf(fact_5221_zero__neq__neg__one,axiom,
% 0.29/0.56      ( zero_zero_int
% 0.29/0.56     != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_neq_neg_one
% 0.29/0.56  thf(fact_5222_zero__neq__neg__one,axiom,
% 0.29/0.56      ( zero_zero_complex
% 0.29/0.56     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_neq_neg_one
% 0.29/0.56  thf(fact_5223_zero__neq__neg__one,axiom,
% 0.29/0.56      ( zero_zero_rat
% 0.29/0.56     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_neq_neg_one
% 0.29/0.56  thf(fact_5224_zero__neq__neg__one,axiom,
% 0.29/0.56      ( zero_z3403309356797280102nteger
% 0.29/0.56     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_neq_neg_one
% 0.29/0.56  thf(fact_5225_add__eq__0__iff,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ( plus_plus_real @ A @ B )
% 0.29/0.56          = zero_zero_real )
% 0.29/0.56        = ( B
% 0.29/0.56          = ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_eq_0_iff
% 0.29/0.56  thf(fact_5226_add__eq__0__iff,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ( plus_plus_int @ A @ B )
% 0.29/0.56          = zero_zero_int )
% 0.29/0.56        = ( B
% 0.29/0.56          = ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_eq_0_iff
% 0.29/0.56  thf(fact_5227_add__eq__0__iff,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( ( plus_plus_complex @ A @ B )
% 0.29/0.56          = zero_zero_complex )
% 0.29/0.56        = ( B
% 0.29/0.56          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_eq_0_iff
% 0.29/0.56  thf(fact_5228_add__eq__0__iff,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ( plus_plus_rat @ A @ B )
% 0.29/0.56          = zero_zero_rat )
% 0.29/0.56        = ( B
% 0.29/0.56          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_eq_0_iff
% 0.29/0.56  thf(fact_5229_add__eq__0__iff,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.29/0.56          = zero_z3403309356797280102nteger )
% 0.29/0.56        = ( B
% 0.29/0.56          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add_eq_0_iff
% 0.29/0.56  thf(fact_5230_ab__group__add__class_Oab__left__minus,axiom,
% 0.29/0.56      ! [A: real] :
% 0.29/0.56        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.29/0.56        = zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_group_add_class.ab_left_minus
% 0.29/0.56  thf(fact_5231_ab__group__add__class_Oab__left__minus,axiom,
% 0.29/0.56      ! [A: int] :
% 0.29/0.56        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_group_add_class.ab_left_minus
% 0.29/0.56  thf(fact_5232_ab__group__add__class_Oab__left__minus,axiom,
% 0.29/0.56      ! [A: complex] :
% 0.29/0.56        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.29/0.56        = zero_zero_complex ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_group_add_class.ab_left_minus
% 0.29/0.56  thf(fact_5233_ab__group__add__class_Oab__left__minus,axiom,
% 0.29/0.56      ! [A: rat] :
% 0.29/0.56        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.29/0.56        = zero_zero_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_group_add_class.ab_left_minus
% 0.29/0.56  thf(fact_5234_ab__group__add__class_Oab__left__minus,axiom,
% 0.29/0.56      ! [A: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_group_add_class.ab_left_minus
% 0.29/0.56  thf(fact_5235_add_Oinverse__unique,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ( plus_plus_real @ A @ B )
% 0.29/0.56          = zero_zero_real )
% 0.29/0.56       => ( ( uminus_uminus_real @ A )
% 0.29/0.56          = B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add.inverse_unique
% 0.29/0.56  thf(fact_5236_add_Oinverse__unique,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ( plus_plus_int @ A @ B )
% 0.29/0.56          = zero_zero_int )
% 0.29/0.56       => ( ( uminus_uminus_int @ A )
% 0.29/0.56          = B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add.inverse_unique
% 0.29/0.56  thf(fact_5237_add_Oinverse__unique,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( ( plus_plus_complex @ A @ B )
% 0.29/0.56          = zero_zero_complex )
% 0.29/0.56       => ( ( uminus1482373934393186551omplex @ A )
% 0.29/0.56          = B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add.inverse_unique
% 0.29/0.56  thf(fact_5238_add_Oinverse__unique,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ( plus_plus_rat @ A @ B )
% 0.29/0.56          = zero_zero_rat )
% 0.29/0.56       => ( ( uminus_uminus_rat @ A )
% 0.29/0.56          = B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add.inverse_unique
% 0.29/0.56  thf(fact_5239_add_Oinverse__unique,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.29/0.56          = zero_z3403309356797280102nteger )
% 0.29/0.56       => ( ( uminus1351360451143612070nteger @ A )
% 0.29/0.56          = B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % add.inverse_unique
% 0.29/0.56  thf(fact_5240_eq__neg__iff__add__eq__0,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( uminus_uminus_real @ B ) )
% 0.29/0.56        = ( ( plus_plus_real @ A @ B )
% 0.29/0.56          = zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_neg_iff_add_eq_0
% 0.29/0.56  thf(fact_5241_eq__neg__iff__add__eq__0,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( uminus_uminus_int @ B ) )
% 0.29/0.56        = ( ( plus_plus_int @ A @ B )
% 0.29/0.56          = zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_neg_iff_add_eq_0
% 0.29/0.56  thf(fact_5242_eq__neg__iff__add__eq__0,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( uminus1482373934393186551omplex @ B ) )
% 0.29/0.56        = ( ( plus_plus_complex @ A @ B )
% 0.29/0.56          = zero_zero_complex ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_neg_iff_add_eq_0
% 0.29/0.56  thf(fact_5243_eq__neg__iff__add__eq__0,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( uminus_uminus_rat @ B ) )
% 0.29/0.56        = ( ( plus_plus_rat @ A @ B )
% 0.29/0.56          = zero_zero_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_neg_iff_add_eq_0
% 0.29/0.56  thf(fact_5244_eq__neg__iff__add__eq__0,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( A
% 0.29/0.56          = ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.56        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.29/0.56          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % eq_neg_iff_add_eq_0
% 0.29/0.56  thf(fact_5245_neg__eq__iff__add__eq__0,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ( uminus_uminus_real @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( plus_plus_real @ A @ B )
% 0.29/0.56          = zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_eq_iff_add_eq_0
% 0.29/0.56  thf(fact_5246_neg__eq__iff__add__eq__0,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ( uminus_uminus_int @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( plus_plus_int @ A @ B )
% 0.29/0.56          = zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_eq_iff_add_eq_0
% 0.29/0.56  thf(fact_5247_neg__eq__iff__add__eq__0,axiom,
% 0.29/0.56      ! [A: complex,B: complex] :
% 0.29/0.56        ( ( ( uminus1482373934393186551omplex @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( plus_plus_complex @ A @ B )
% 0.29/0.56          = zero_zero_complex ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_eq_iff_add_eq_0
% 0.29/0.56  thf(fact_5248_neg__eq__iff__add__eq__0,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ( uminus_uminus_rat @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( plus_plus_rat @ A @ B )
% 0.29/0.56          = zero_zero_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_eq_iff_add_eq_0
% 0.29/0.56  thf(fact_5249_neg__eq__iff__add__eq__0,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ( uminus1351360451143612070nteger @ A )
% 0.29/0.56          = B )
% 0.29/0.56        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.29/0.56          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_eq_iff_add_eq_0
% 0.29/0.56  thf(fact_5250_mod__Suc__le__divisor,axiom,
% 0.29/0.56      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_Suc_le_divisor
% 0.29/0.56  thf(fact_5251_less__minus__one__simps_I2_J,axiom,
% 0.29/0.56      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(2)
% 0.29/0.56  thf(fact_5252_less__minus__one__simps_I2_J,axiom,
% 0.29/0.56      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(2)
% 0.29/0.56  thf(fact_5253_less__minus__one__simps_I2_J,axiom,
% 0.29/0.56      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(2)
% 0.29/0.56  thf(fact_5254_less__minus__one__simps_I2_J,axiom,
% 0.29/0.56      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(2)
% 0.29/0.56  thf(fact_5255_less__minus__one__simps_I4_J,axiom,
% 0.29/0.56      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(4)
% 0.29/0.56  thf(fact_5256_less__minus__one__simps_I4_J,axiom,
% 0.29/0.56      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(4)
% 0.29/0.56  thf(fact_5257_less__minus__one__simps_I4_J,axiom,
% 0.29/0.56      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(4)
% 0.29/0.56  thf(fact_5258_less__minus__one__simps_I4_J,axiom,
% 0.29/0.56      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(4)
% 0.29/0.56  thf(fact_5259_numeral__times__minus__swap,axiom,
% 0.29/0.56      ! [W: num,X: real] :
% 0.29/0.56        ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
% 0.29/0.56        = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_times_minus_swap
% 0.29/0.56  thf(fact_5260_numeral__times__minus__swap,axiom,
% 0.29/0.56      ! [W: num,X: int] :
% 0.29/0.56        ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
% 0.29/0.56        = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_times_minus_swap
% 0.29/0.56  thf(fact_5261_numeral__times__minus__swap,axiom,
% 0.29/0.56      ! [W: num,X: complex] :
% 0.29/0.56        ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
% 0.29/0.56        = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_times_minus_swap
% 0.29/0.56  thf(fact_5262_numeral__times__minus__swap,axiom,
% 0.29/0.56      ! [W: num,X: rat] :
% 0.29/0.56        ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
% 0.29/0.56        = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_times_minus_swap
% 0.29/0.56  thf(fact_5263_numeral__times__minus__swap,axiom,
% 0.29/0.56      ! [W: num,X: code_integer] :
% 0.29/0.56        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
% 0.29/0.56        = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_times_minus_swap
% 0.29/0.56  thf(fact_5264_nonzero__minus__divide__divide,axiom,
% 0.29/0.56      ! [B: real,A: real] :
% 0.29/0.56        ( ( B != zero_zero_real )
% 0.29/0.56       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.29/0.56          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_minus_divide_divide
% 0.29/0.56  thf(fact_5265_nonzero__minus__divide__divide,axiom,
% 0.29/0.56      ! [B: complex,A: complex] :
% 0.29/0.56        ( ( B != zero_zero_complex )
% 0.29/0.56       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.29/0.56          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_minus_divide_divide
% 0.29/0.56  thf(fact_5266_nonzero__minus__divide__divide,axiom,
% 0.29/0.56      ! [B: rat,A: rat] :
% 0.29/0.56        ( ( B != zero_zero_rat )
% 0.29/0.56       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.29/0.56          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_minus_divide_divide
% 0.29/0.56  thf(fact_5267_nonzero__minus__divide__right,axiom,
% 0.29/0.56      ! [B: real,A: real] :
% 0.29/0.56        ( ( B != zero_zero_real )
% 0.29/0.56       => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.56          = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_minus_divide_right
% 0.29/0.56  thf(fact_5268_nonzero__minus__divide__right,axiom,
% 0.29/0.56      ! [B: complex,A: complex] :
% 0.29/0.56        ( ( B != zero_zero_complex )
% 0.29/0.56       => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.56          = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_minus_divide_right
% 0.29/0.56  thf(fact_5269_nonzero__minus__divide__right,axiom,
% 0.29/0.56      ! [B: rat,A: rat] :
% 0.29/0.56        ( ( B != zero_zero_rat )
% 0.29/0.56       => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.29/0.56          = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_minus_divide_right
% 0.29/0.56  thf(fact_5270_numeral__neq__neg__one,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( numeral_numeral_real @ N )
% 0.29/0.56       != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_neq_neg_one
% 0.29/0.56  thf(fact_5271_numeral__neq__neg__one,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( numeral_numeral_int @ N )
% 0.29/0.56       != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_neq_neg_one
% 0.29/0.56  thf(fact_5272_numeral__neq__neg__one,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( numera6690914467698888265omplex @ N )
% 0.29/0.56       != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_neq_neg_one
% 0.29/0.56  thf(fact_5273_numeral__neq__neg__one,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( numeral_numeral_rat @ N )
% 0.29/0.56       != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_neq_neg_one
% 0.29/0.56  thf(fact_5274_numeral__neq__neg__one,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( numera6620942414471956472nteger @ N )
% 0.29/0.56       != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % numeral_neq_neg_one
% 0.29/0.56  thf(fact_5275_one__neq__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( one_one_real
% 0.29/0.56       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % one_neq_neg_numeral
% 0.29/0.56  thf(fact_5276_one__neq__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( one_one_int
% 0.29/0.56       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % one_neq_neg_numeral
% 0.29/0.56  thf(fact_5277_one__neq__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( one_one_complex
% 0.29/0.56       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % one_neq_neg_numeral
% 0.29/0.56  thf(fact_5278_one__neq__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( one_one_rat
% 0.29/0.56       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % one_neq_neg_numeral
% 0.29/0.56  thf(fact_5279_one__neq__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( one_one_Code_integer
% 0.29/0.56       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % one_neq_neg_numeral
% 0.29/0.56  thf(fact_5280_square__eq__1__iff,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ( ( times_times_real @ X @ X )
% 0.29/0.56          = one_one_real )
% 0.29/0.56        = ( ( X = one_one_real )
% 0.29/0.56          | ( X
% 0.29/0.56            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % square_eq_1_iff
% 0.29/0.56  thf(fact_5281_square__eq__1__iff,axiom,
% 0.29/0.56      ! [X: int] :
% 0.29/0.56        ( ( ( times_times_int @ X @ X )
% 0.29/0.56          = one_one_int )
% 0.29/0.56        = ( ( X = one_one_int )
% 0.29/0.56          | ( X
% 0.29/0.56            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % square_eq_1_iff
% 0.29/0.56  thf(fact_5282_square__eq__1__iff,axiom,
% 0.29/0.56      ! [X: complex] :
% 0.29/0.56        ( ( ( times_times_complex @ X @ X )
% 0.29/0.56          = one_one_complex )
% 0.29/0.56        = ( ( X = one_one_complex )
% 0.29/0.56          | ( X
% 0.29/0.56            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % square_eq_1_iff
% 0.29/0.56  thf(fact_5283_square__eq__1__iff,axiom,
% 0.29/0.56      ! [X: rat] :
% 0.29/0.56        ( ( ( times_times_rat @ X @ X )
% 0.29/0.56          = one_one_rat )
% 0.29/0.56        = ( ( X = one_one_rat )
% 0.29/0.56          | ( X
% 0.29/0.56            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % square_eq_1_iff
% 0.29/0.56  thf(fact_5284_square__eq__1__iff,axiom,
% 0.29/0.56      ! [X: code_integer] :
% 0.29/0.56        ( ( ( times_3573771949741848930nteger @ X @ X )
% 0.29/0.56          = one_one_Code_integer )
% 0.29/0.56        = ( ( X = one_one_Code_integer )
% 0.29/0.56          | ( X
% 0.29/0.56            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % square_eq_1_iff
% 0.29/0.56  thf(fact_5285_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.29/0.56      ( minus_minus_real
% 0.29/0.56      = ( ^ [A3: real,B4: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.29/0.56  thf(fact_5286_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.29/0.56      ( minus_minus_int
% 0.29/0.56      = ( ^ [A3: int,B4: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.29/0.56  thf(fact_5287_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.29/0.56      ( minus_minus_complex
% 0.29/0.56      = ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.29/0.56  thf(fact_5288_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.29/0.56      ( minus_minus_rat
% 0.29/0.56      = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.29/0.56  thf(fact_5289_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.29/0.56      ( minus_8373710615458151222nteger
% 0.29/0.56      = ( ^ [A3: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.29/0.56  thf(fact_5290_diff__conv__add__uminus,axiom,
% 0.29/0.56      ( minus_minus_real
% 0.29/0.56      = ( ^ [A3: real,B4: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_conv_add_uminus
% 0.29/0.56  thf(fact_5291_diff__conv__add__uminus,axiom,
% 0.29/0.56      ( minus_minus_int
% 0.29/0.56      = ( ^ [A3: int,B4: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_conv_add_uminus
% 0.29/0.56  thf(fact_5292_diff__conv__add__uminus,axiom,
% 0.29/0.56      ( minus_minus_complex
% 0.29/0.56      = ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_conv_add_uminus
% 0.29/0.56  thf(fact_5293_diff__conv__add__uminus,axiom,
% 0.29/0.56      ( minus_minus_rat
% 0.29/0.56      = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_conv_add_uminus
% 0.29/0.56  thf(fact_5294_diff__conv__add__uminus,axiom,
% 0.29/0.56      ( minus_8373710615458151222nteger
% 0.29/0.56      = ( ^ [A3: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % diff_conv_add_uminus
% 0.29/0.56  thf(fact_5295_group__cancel_Osub2,axiom,
% 0.29/0.56      ! [B2: real,K: real,B: real,A: real] :
% 0.29/0.56        ( ( B2
% 0.29/0.56          = ( plus_plus_real @ K @ B ) )
% 0.29/0.56       => ( ( minus_minus_real @ A @ B2 )
% 0.29/0.56          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % group_cancel.sub2
% 0.29/0.56  thf(fact_5296_group__cancel_Osub2,axiom,
% 0.29/0.56      ! [B2: int,K: int,B: int,A: int] :
% 0.29/0.56        ( ( B2
% 0.29/0.56          = ( plus_plus_int @ K @ B ) )
% 0.29/0.56       => ( ( minus_minus_int @ A @ B2 )
% 0.29/0.56          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % group_cancel.sub2
% 0.29/0.56  thf(fact_5297_group__cancel_Osub2,axiom,
% 0.29/0.56      ! [B2: complex,K: complex,B: complex,A: complex] :
% 0.29/0.56        ( ( B2
% 0.29/0.56          = ( plus_plus_complex @ K @ B ) )
% 0.29/0.56       => ( ( minus_minus_complex @ A @ B2 )
% 0.29/0.56          = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % group_cancel.sub2
% 0.29/0.56  thf(fact_5298_group__cancel_Osub2,axiom,
% 0.29/0.56      ! [B2: rat,K: rat,B: rat,A: rat] :
% 0.29/0.56        ( ( B2
% 0.29/0.56          = ( plus_plus_rat @ K @ B ) )
% 0.29/0.56       => ( ( minus_minus_rat @ A @ B2 )
% 0.29/0.56          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % group_cancel.sub2
% 0.29/0.56  thf(fact_5299_group__cancel_Osub2,axiom,
% 0.29/0.56      ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 0.29/0.56        ( ( B2
% 0.29/0.56          = ( plus_p5714425477246183910nteger @ K @ B ) )
% 0.29/0.56       => ( ( minus_8373710615458151222nteger @ A @ B2 )
% 0.29/0.56          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % group_cancel.sub2
% 0.29/0.56  thf(fact_5300_mod__eq__0D,axiom,
% 0.29/0.56      ! [M: nat,D: nat] :
% 0.29/0.56        ( ( ( modulo_modulo_nat @ M @ D )
% 0.29/0.56          = zero_zero_nat )
% 0.29/0.56       => ? [Q2: nat] :
% 0.29/0.56            ( M
% 0.29/0.56            = ( times_times_nat @ D @ Q2 ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_eq_0D
% 0.29/0.56  thf(fact_5301_mod__geq,axiom,
% 0.29/0.56      ! [M: nat,N: nat] :
% 0.29/0.56        ( ~ ( ord_less_nat @ M @ N )
% 0.29/0.56       => ( ( modulo_modulo_nat @ M @ N )
% 0.29/0.56          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_geq
% 0.29/0.56  thf(fact_5302_mod__if,axiom,
% 0.29/0.56      ( modulo_modulo_nat
% 0.29/0.56      = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N3 ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_if
% 0.29/0.56  thf(fact_5303_le__mod__geq,axiom,
% 0.29/0.56      ! [N: nat,M: nat] :
% 0.29/0.56        ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.56       => ( ( modulo_modulo_nat @ M @ N )
% 0.29/0.56          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_mod_geq
% 0.29/0.56  thf(fact_5304_nat__mod__eq__iff,axiom,
% 0.29/0.56      ! [X: nat,N: nat,Y: nat] :
% 0.29/0.56        ( ( ( modulo_modulo_nat @ X @ N )
% 0.29/0.56          = ( modulo_modulo_nat @ Y @ N ) )
% 0.29/0.56        = ( ? [Q1: nat,Q22: nat] :
% 0.29/0.56              ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
% 0.29/0.56              = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nat_mod_eq_iff
% 0.29/0.56  thf(fact_5305_real__minus__mult__self__le,axiom,
% 0.29/0.56      ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% 0.29/0.56  
% 0.29/0.56  % real_minus_mult_self_le
% 0.29/0.56  thf(fact_5306_vebt__buildup_Osimps_I1_J,axiom,
% 0.29/0.56      ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 0.29/0.56      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.29/0.56  
% 0.29/0.56  % vebt_buildup.simps(1)
% 0.29/0.56  thf(fact_5307_minus__real__def,axiom,
% 0.29/0.56      ( minus_minus_real
% 0.29/0.56      = ( ^ [X4: real,Y6: real] : ( plus_plus_real @ X4 @ ( uminus_uminus_real @ Y6 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_real_def
% 0.29/0.56  thf(fact_5308_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 0.29/0.56      ! [Uu: $o,Uv: $o,D: nat] :
% 0.29/0.56        ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 0.29/0.56        = ( D = one_one_nat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % VEBT_internal.valid'.simps(1)
% 0.29/0.56  thf(fact_5309_VEBT__internal_Oheight_Osimps_I1_J,axiom,
% 0.29/0.56      ! [A: $o,B: $o] :
% 0.29/0.56        ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A @ B ) )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % VEBT_internal.height.simps(1)
% 0.29/0.56  thf(fact_5310_tanh__real__lt__1,axiom,
% 0.29/0.56      ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% 0.29/0.56  
% 0.29/0.56  % tanh_real_lt_1
% 0.29/0.56  thf(fact_5311_dense__eq0__I,axiom,
% 0.29/0.56      ! [X: real] :
% 0.29/0.56        ( ! [E: real] :
% 0.29/0.56            ( ( ord_less_real @ zero_zero_real @ E )
% 0.29/0.56           => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
% 0.29/0.56       => ( X = zero_zero_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dense_eq0_I
% 0.29/0.56  thf(fact_5312_dense__eq0__I,axiom,
% 0.29/0.56      ! [X: rat] :
% 0.29/0.56        ( ! [E: rat] :
% 0.29/0.56            ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.29/0.56           => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
% 0.29/0.56       => ( X = zero_zero_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % dense_eq0_I
% 0.29/0.56  thf(fact_5313_abs__eq__mult,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.56            | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 0.29/0.56          & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.29/0.56            | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 0.29/0.56       => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.29/0.56          = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_mult
% 0.29/0.56  thf(fact_5314_abs__eq__mult,axiom,
% 0.29/0.56      ! [A: real,B: real] :
% 0.29/0.56        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.56            | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.29/0.56          & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.56            | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.29/0.56       => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.29/0.56          = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_mult
% 0.29/0.56  thf(fact_5315_abs__eq__mult,axiom,
% 0.29/0.56      ! [A: rat,B: rat] :
% 0.29/0.56        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.56            | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.29/0.56          & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.56            | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.29/0.56       => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.56          = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_mult
% 0.29/0.56  thf(fact_5316_abs__eq__mult,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.56            | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.29/0.56          & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.56            | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.29/0.56       => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.29/0.56          = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_eq_mult
% 0.29/0.56  thf(fact_5317_abs__mult__pos,axiom,
% 0.29/0.56      ! [X: code_integer,Y: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.29/0.56       => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
% 0.29/0.56          = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_mult_pos
% 0.29/0.56  thf(fact_5318_abs__mult__pos,axiom,
% 0.29/0.56      ! [X: real,Y: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.56       => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
% 0.29/0.56          = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_mult_pos
% 0.29/0.56  thf(fact_5319_abs__mult__pos,axiom,
% 0.29/0.56      ! [X: rat,Y: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.29/0.56       => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
% 0.29/0.56          = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_mult_pos
% 0.29/0.56  thf(fact_5320_abs__mult__pos,axiom,
% 0.29/0.56      ! [X: int,Y: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.56       => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
% 0.29/0.56          = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_mult_pos
% 0.29/0.56  thf(fact_5321_zero__le__power__abs,axiom,
% 0.29/0.56      ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_le_power_abs
% 0.29/0.56  thf(fact_5322_zero__le__power__abs,axiom,
% 0.29/0.56      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_le_power_abs
% 0.29/0.56  thf(fact_5323_zero__le__power__abs,axiom,
% 0.29/0.56      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_le_power_abs
% 0.29/0.56  thf(fact_5324_zero__le__power__abs,axiom,
% 0.29/0.56      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % zero_le_power_abs
% 0.29/0.56  thf(fact_5325_abs__div__pos,axiom,
% 0.29/0.56      ! [Y: real,X: real] :
% 0.29/0.56        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.56       => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
% 0.29/0.56          = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_div_pos
% 0.29/0.56  thf(fact_5326_abs__div__pos,axiom,
% 0.29/0.56      ! [Y: rat,X: rat] :
% 0.29/0.56        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.29/0.56       => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
% 0.29/0.56          = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_div_pos
% 0.29/0.56  thf(fact_5327_abs__triangle__ineq4,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq4
% 0.29/0.56  thf(fact_5328_abs__triangle__ineq4,axiom,
% 0.29/0.56      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq4
% 0.29/0.56  thf(fact_5329_abs__triangle__ineq4,axiom,
% 0.29/0.56      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq4
% 0.29/0.56  thf(fact_5330_abs__triangle__ineq4,axiom,
% 0.29/0.56      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_triangle_ineq4
% 0.29/0.56  thf(fact_5331_abs__diff__triangle__ineq,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_triangle_ineq
% 0.29/0.56  thf(fact_5332_abs__diff__triangle__ineq,axiom,
% 0.29/0.56      ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_triangle_ineq
% 0.29/0.56  thf(fact_5333_abs__diff__triangle__ineq,axiom,
% 0.29/0.56      ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_triangle_ineq
% 0.29/0.56  thf(fact_5334_abs__diff__triangle__ineq,axiom,
% 0.29/0.56      ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_triangle_ineq
% 0.29/0.56  thf(fact_5335_abs__diff__le__iff,axiom,
% 0.29/0.56      ! [X: code_integer,A: code_integer,R2: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 0.29/0.56        = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 0.29/0.56          & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_le_iff
% 0.29/0.56  thf(fact_5336_abs__diff__le__iff,axiom,
% 0.29/0.56      ! [X: real,A: real,R2: real] :
% 0.29/0.56        ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 0.29/0.56        = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 0.29/0.56          & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_le_iff
% 0.29/0.56  thf(fact_5337_abs__diff__le__iff,axiom,
% 0.29/0.56      ! [X: rat,A: rat,R2: rat] :
% 0.29/0.56        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 0.29/0.56        = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 0.29/0.56          & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_le_iff
% 0.29/0.56  thf(fact_5338_abs__diff__le__iff,axiom,
% 0.29/0.56      ! [X: int,A: int,R2: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 0.29/0.56        = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 0.29/0.56          & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_le_iff
% 0.29/0.56  thf(fact_5339_abs__diff__less__iff,axiom,
% 0.29/0.56      ! [X: code_integer,A: code_integer,R2: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 0.29/0.56        = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 0.29/0.56          & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_less_iff
% 0.29/0.56  thf(fact_5340_abs__diff__less__iff,axiom,
% 0.29/0.56      ! [X: real,A: real,R2: real] :
% 0.29/0.56        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 0.29/0.56        = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 0.29/0.56          & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_less_iff
% 0.29/0.56  thf(fact_5341_abs__diff__less__iff,axiom,
% 0.29/0.56      ! [X: rat,A: rat,R2: rat] :
% 0.29/0.56        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 0.29/0.56        = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 0.29/0.56          & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_less_iff
% 0.29/0.56  thf(fact_5342_abs__diff__less__iff,axiom,
% 0.29/0.56      ! [X: int,A: int,R2: int] :
% 0.29/0.56        ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 0.29/0.56        = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 0.29/0.56          & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % abs_diff_less_iff
% 0.29/0.56  thf(fact_5343_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.29/0.56      ! [B: code_integer,A: code_integer] :
% 0.29/0.56        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.29/0.56       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.29/0.56  thf(fact_5344_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.29/0.56      ! [B: nat,A: nat] :
% 0.29/0.56        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.56       => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.29/0.56  thf(fact_5345_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.29/0.56      ! [B: int,A: int] :
% 0.29/0.56        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.56       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.29/0.56  thf(fact_5346_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.56       => ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.29/0.56         => ( ( modulo364778990260209775nteger @ A @ B )
% 0.29/0.56            = A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.mod_less
% 0.29/0.56  thf(fact_5347_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.56       => ( ( ord_less_nat @ A @ B )
% 0.29/0.56         => ( ( modulo_modulo_nat @ A @ B )
% 0.29/0.56            = A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.mod_less
% 0.29/0.56  thf(fact_5348_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.56       => ( ( ord_less_int @ A @ B )
% 0.29/0.56         => ( ( modulo_modulo_int @ A @ B )
% 0.29/0.56            = A ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % unique_euclidean_semiring_numeral_class.mod_less
% 0.29/0.56  thf(fact_5349_nonzero__of__real__divide,axiom,
% 0.29/0.56      ! [Y: real,X: real] :
% 0.29/0.56        ( ( Y != zero_zero_real )
% 0.29/0.56       => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
% 0.29/0.56          = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_of_real_divide
% 0.29/0.56  thf(fact_5350_nonzero__of__real__divide,axiom,
% 0.29/0.56      ! [Y: real,X: real] :
% 0.29/0.56        ( ( Y != zero_zero_real )
% 0.29/0.56       => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
% 0.29/0.56          = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_of_real_divide
% 0.29/0.56  thf(fact_5351_cong__exp__iff__simps_I2_J,axiom,
% 0.29/0.56      ! [N: num,Q3: num] :
% 0.29/0.56        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.29/0.56          = zero_zero_nat )
% 0.29/0.56        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.29/0.56          = zero_zero_nat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(2)
% 0.29/0.56  thf(fact_5352_cong__exp__iff__simps_I2_J,axiom,
% 0.29/0.56      ! [N: num,Q3: num] :
% 0.29/0.56        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.29/0.56          = zero_zero_int )
% 0.29/0.56        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 0.29/0.56          = zero_zero_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(2)
% 0.29/0.56  thf(fact_5353_cong__exp__iff__simps_I2_J,axiom,
% 0.29/0.56      ! [N: num,Q3: num] :
% 0.29/0.56        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.29/0.56          = zero_z3403309356797280102nteger )
% 0.29/0.56        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.29/0.56          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(2)
% 0.29/0.56  thf(fact_5354_cong__exp__iff__simps_I1_J,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 0.29/0.56        = zero_zero_nat ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(1)
% 0.29/0.56  thf(fact_5355_cong__exp__iff__simps_I1_J,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 0.29/0.56        = zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(1)
% 0.29/0.56  thf(fact_5356_cong__exp__iff__simps_I1_J,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 0.29/0.56        = zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(1)
% 0.29/0.56  thf(fact_5357_cong__exp__iff__simps_I8_J,axiom,
% 0.29/0.56      ! [M: num,Q3: num] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.29/0.56       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(8)
% 0.29/0.56  thf(fact_5358_cong__exp__iff__simps_I8_J,axiom,
% 0.29/0.56      ! [M: num,Q3: num] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.29/0.56       != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(8)
% 0.29/0.56  thf(fact_5359_cong__exp__iff__simps_I8_J,axiom,
% 0.29/0.56      ! [M: num,Q3: num] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.29/0.56       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(8)
% 0.29/0.56  thf(fact_5360_cong__exp__iff__simps_I6_J,axiom,
% 0.29/0.56      ! [Q3: num,N: num] :
% 0.29/0.56        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.29/0.56       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(6)
% 0.29/0.56  thf(fact_5361_cong__exp__iff__simps_I6_J,axiom,
% 0.29/0.56      ! [Q3: num,N: num] :
% 0.29/0.56        ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.29/0.56       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(6)
% 0.29/0.56  thf(fact_5362_cong__exp__iff__simps_I6_J,axiom,
% 0.29/0.56      ! [Q3: num,N: num] :
% 0.29/0.56        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.29/0.56       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cong_exp_iff_simps(6)
% 0.29/0.56  thf(fact_5363_div__mult1__eq,axiom,
% 0.29/0.56      ! [A: nat,B: nat,C: nat] :
% 0.29/0.56        ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.29/0.56        = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % div_mult1_eq
% 0.29/0.56  thf(fact_5364_div__mult1__eq,axiom,
% 0.29/0.56      ! [A: int,B: int,C: int] :
% 0.29/0.56        ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 0.29/0.56        = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % div_mult1_eq
% 0.29/0.56  thf(fact_5365_div__mult1__eq,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.56        ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.29/0.56        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % div_mult1_eq
% 0.29/0.56  thf(fact_5366_cancel__div__mod__rules_I2_J,axiom,
% 0.29/0.56      ! [B: nat,A: nat,C: nat] :
% 0.29/0.56        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.29/0.56        = ( plus_plus_nat @ A @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cancel_div_mod_rules(2)
% 0.29/0.56  thf(fact_5367_cancel__div__mod__rules_I2_J,axiom,
% 0.29/0.56      ! [B: int,A: int,C: int] :
% 0.29/0.56        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.29/0.56        = ( plus_plus_int @ A @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cancel_div_mod_rules(2)
% 0.29/0.56  thf(fact_5368_cancel__div__mod__rules_I2_J,axiom,
% 0.29/0.56      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.29/0.56        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cancel_div_mod_rules(2)
% 0.29/0.56  thf(fact_5369_cancel__div__mod__rules_I1_J,axiom,
% 0.29/0.56      ! [A: nat,B: nat,C: nat] :
% 0.29/0.56        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.29/0.56        = ( plus_plus_nat @ A @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cancel_div_mod_rules(1)
% 0.29/0.56  thf(fact_5370_cancel__div__mod__rules_I1_J,axiom,
% 0.29/0.56      ! [A: int,B: int,C: int] :
% 0.29/0.56        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.29/0.56        = ( plus_plus_int @ A @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cancel_div_mod_rules(1)
% 0.29/0.56  thf(fact_5371_cancel__div__mod__rules_I1_J,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.29/0.56        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.29/0.56  
% 0.29/0.56  % cancel_div_mod_rules(1)
% 0.29/0.56  thf(fact_5372_mod__div__decomp,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( A
% 0.29/0.56        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_div_decomp
% 0.29/0.56  thf(fact_5373_mod__div__decomp,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( A
% 0.29/0.56        = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_div_decomp
% 0.29/0.56  thf(fact_5374_mod__div__decomp,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( A
% 0.29/0.56        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_div_decomp
% 0.29/0.56  thf(fact_5375_div__mult__mod__eq,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % div_mult_mod_eq
% 0.29/0.56  thf(fact_5376_div__mult__mod__eq,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % div_mult_mod_eq
% 0.29/0.56  thf(fact_5377_div__mult__mod__eq,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % div_mult_mod_eq
% 0.29/0.56  thf(fact_5378_mod__div__mult__eq,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_div_mult_eq
% 0.29/0.56  thf(fact_5379_mod__div__mult__eq,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_div_mult_eq
% 0.29/0.56  thf(fact_5380_mod__div__mult__eq,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_div_mult_eq
% 0.29/0.56  thf(fact_5381_mod__mult__div__eq,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_div_eq
% 0.29/0.56  thf(fact_5382_mod__mult__div__eq,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_div_eq
% 0.29/0.56  thf(fact_5383_mod__mult__div__eq,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_mult_div_eq
% 0.29/0.56  thf(fact_5384_mult__div__mod__eq,axiom,
% 0.29/0.56      ! [B: nat,A: nat] :
% 0.29/0.56        ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_div_mod_eq
% 0.29/0.56  thf(fact_5385_mult__div__mod__eq,axiom,
% 0.29/0.56      ! [B: int,A: int] :
% 0.29/0.56        ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_div_mod_eq
% 0.29/0.56  thf(fact_5386_mult__div__mod__eq,axiom,
% 0.29/0.56      ! [B: code_integer,A: code_integer] :
% 0.29/0.56        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.29/0.56        = A ) ).
% 0.29/0.56  
% 0.29/0.56  % mult_div_mod_eq
% 0.29/0.56  thf(fact_5387_minus__div__mult__eq__mod,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.29/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_div_mult_eq_mod
% 0.29/0.56  thf(fact_5388_minus__div__mult__eq__mod,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.29/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_div_mult_eq_mod
% 0.29/0.56  thf(fact_5389_minus__div__mult__eq__mod,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.29/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_div_mult_eq_mod
% 0.29/0.56  thf(fact_5390_minus__mod__eq__div__mult,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.29/0.56        = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_mod_eq_div_mult
% 0.29/0.56  thf(fact_5391_minus__mod__eq__div__mult,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.29/0.56        = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_mod_eq_div_mult
% 0.29/0.56  thf(fact_5392_minus__mod__eq__div__mult,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.29/0.56        = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_mod_eq_div_mult
% 0.29/0.56  thf(fact_5393_minus__mod__eq__mult__div,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.29/0.56        = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_mod_eq_mult_div
% 0.29/0.56  thf(fact_5394_minus__mod__eq__mult__div,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.29/0.56        = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_mod_eq_mult_div
% 0.29/0.56  thf(fact_5395_minus__mod__eq__mult__div,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.29/0.56        = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_mod_eq_mult_div
% 0.29/0.56  thf(fact_5396_minus__mult__div__eq__mod,axiom,
% 0.29/0.56      ! [A: nat,B: nat] :
% 0.29/0.56        ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.29/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_mult_div_eq_mod
% 0.29/0.56  thf(fact_5397_minus__mult__div__eq__mod,axiom,
% 0.29/0.56      ! [A: int,B: int] :
% 0.29/0.56        ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.29/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_mult_div_eq_mod
% 0.29/0.56  thf(fact_5398_minus__mult__div__eq__mod,axiom,
% 0.29/0.56      ! [A: code_integer,B: code_integer] :
% 0.29/0.56        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.29/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.29/0.56  
% 0.29/0.56  % minus_mult_div_eq_mod
% 0.29/0.56  thf(fact_5399_not__zero__le__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_zero_le_neg_numeral
% 0.29/0.56  thf(fact_5400_not__zero__le__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_zero_le_neg_numeral
% 0.29/0.56  thf(fact_5401_not__zero__le__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_zero_le_neg_numeral
% 0.29/0.56  thf(fact_5402_not__zero__le__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_zero_le_neg_numeral
% 0.29/0.56  thf(fact_5403_neg__numeral__le__zero,axiom,
% 0.29/0.56      ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_zero
% 0.29/0.56  thf(fact_5404_neg__numeral__le__zero,axiom,
% 0.29/0.56      ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_zero
% 0.29/0.56  thf(fact_5405_neg__numeral__le__zero,axiom,
% 0.29/0.56      ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_zero
% 0.29/0.56  thf(fact_5406_neg__numeral__le__zero,axiom,
% 0.29/0.56      ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_zero
% 0.29/0.56  thf(fact_5407_not__zero__less__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_zero_less_neg_numeral
% 0.29/0.56  thf(fact_5408_not__zero__less__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_zero_less_neg_numeral
% 0.29/0.56  thf(fact_5409_not__zero__less__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_zero_less_neg_numeral
% 0.29/0.56  thf(fact_5410_not__zero__less__neg__numeral,axiom,
% 0.29/0.56      ! [N: num] :
% 0.29/0.56        ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_zero_less_neg_numeral
% 0.29/0.56  thf(fact_5411_neg__numeral__less__zero,axiom,
% 0.29/0.56      ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_zero
% 0.29/0.56  thf(fact_5412_neg__numeral__less__zero,axiom,
% 0.29/0.56      ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_zero
% 0.29/0.56  thf(fact_5413_neg__numeral__less__zero,axiom,
% 0.29/0.56      ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_zero
% 0.29/0.56  thf(fact_5414_neg__numeral__less__zero,axiom,
% 0.29/0.56      ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_zero
% 0.29/0.56  thf(fact_5415_le__minus__one__simps_I3_J,axiom,
% 0.29/0.56      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(3)
% 0.29/0.56  thf(fact_5416_le__minus__one__simps_I3_J,axiom,
% 0.29/0.56      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(3)
% 0.29/0.56  thf(fact_5417_le__minus__one__simps_I3_J,axiom,
% 0.29/0.56      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(3)
% 0.29/0.56  thf(fact_5418_le__minus__one__simps_I3_J,axiom,
% 0.29/0.56      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(3)
% 0.29/0.56  thf(fact_5419_le__minus__one__simps_I1_J,axiom,
% 0.29/0.56      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(1)
% 0.29/0.56  thf(fact_5420_le__minus__one__simps_I1_J,axiom,
% 0.29/0.56      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(1)
% 0.29/0.56  thf(fact_5421_le__minus__one__simps_I1_J,axiom,
% 0.29/0.56      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(1)
% 0.29/0.56  thf(fact_5422_le__minus__one__simps_I1_J,axiom,
% 0.29/0.56      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.29/0.56  
% 0.29/0.56  % le_minus_one_simps(1)
% 0.29/0.56  thf(fact_5423_less__minus__one__simps_I3_J,axiom,
% 0.29/0.56      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(3)
% 0.29/0.56  thf(fact_5424_less__minus__one__simps_I3_J,axiom,
% 0.29/0.56      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(3)
% 0.29/0.56  thf(fact_5425_less__minus__one__simps_I3_J,axiom,
% 0.29/0.56      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(3)
% 0.29/0.56  thf(fact_5426_less__minus__one__simps_I3_J,axiom,
% 0.29/0.56      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(3)
% 0.29/0.56  thf(fact_5427_less__minus__one__simps_I1_J,axiom,
% 0.29/0.56      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(1)
% 0.29/0.56  thf(fact_5428_less__minus__one__simps_I1_J,axiom,
% 0.29/0.56      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(1)
% 0.29/0.56  thf(fact_5429_less__minus__one__simps_I1_J,axiom,
% 0.29/0.56      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(1)
% 0.29/0.56  thf(fact_5430_less__minus__one__simps_I1_J,axiom,
% 0.29/0.56      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.29/0.56  
% 0.29/0.56  % less_minus_one_simps(1)
% 0.29/0.56  thf(fact_5431_neg__numeral__le__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_one
% 0.29/0.56  thf(fact_5432_neg__numeral__le__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_one
% 0.29/0.56  thf(fact_5433_neg__numeral__le__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_one
% 0.29/0.56  thf(fact_5434_neg__numeral__le__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_one
% 0.29/0.56  thf(fact_5435_neg__one__le__numeral,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_le_numeral
% 0.29/0.56  thf(fact_5436_neg__one__le__numeral,axiom,
% 0.29/0.56      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_le_numeral
% 0.29/0.56  thf(fact_5437_neg__one__le__numeral,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_le_numeral
% 0.29/0.56  thf(fact_5438_neg__one__le__numeral,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_le_numeral
% 0.29/0.56  thf(fact_5439_neg__numeral__le__neg__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_neg_one
% 0.29/0.56  thf(fact_5440_neg__numeral__le__neg__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_neg_one
% 0.29/0.56  thf(fact_5441_neg__numeral__le__neg__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_neg_one
% 0.29/0.56  thf(fact_5442_neg__numeral__le__neg__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_le_neg_one
% 0.29/0.56  thf(fact_5443_not__numeral__le__neg__one,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_le_neg_one
% 0.29/0.56  thf(fact_5444_not__numeral__le__neg__one,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_le_neg_one
% 0.29/0.56  thf(fact_5445_not__numeral__le__neg__one,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_le_neg_one
% 0.29/0.56  thf(fact_5446_not__numeral__le__neg__one,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_le_neg_one
% 0.29/0.56  thf(fact_5447_not__one__le__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_one_le_neg_numeral
% 0.29/0.56  thf(fact_5448_not__one__le__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_one_le_neg_numeral
% 0.29/0.56  thf(fact_5449_not__one__le__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_one_le_neg_numeral
% 0.29/0.56  thf(fact_5450_not__one__le__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_one_le_neg_numeral
% 0.29/0.56  thf(fact_5451_neg__numeral__less__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_one
% 0.29/0.56  thf(fact_5452_neg__numeral__less__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_one
% 0.29/0.56  thf(fact_5453_neg__numeral__less__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_one
% 0.29/0.56  thf(fact_5454_neg__numeral__less__one,axiom,
% 0.29/0.56      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_numeral_less_one
% 0.29/0.56  thf(fact_5455_neg__one__less__numeral,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_less_numeral
% 0.29/0.56  thf(fact_5456_neg__one__less__numeral,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_less_numeral
% 0.29/0.56  thf(fact_5457_neg__one__less__numeral,axiom,
% 0.29/0.56      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_less_numeral
% 0.29/0.56  thf(fact_5458_neg__one__less__numeral,axiom,
% 0.29/0.56      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.29/0.56  
% 0.29/0.56  % neg_one_less_numeral
% 0.29/0.56  thf(fact_5459_not__numeral__less__neg__one,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_less_neg_one
% 0.29/0.56  thf(fact_5460_not__numeral__less__neg__one,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_less_neg_one
% 0.29/0.56  thf(fact_5461_not__numeral__less__neg__one,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_less_neg_one
% 0.29/0.56  thf(fact_5462_not__numeral__less__neg__one,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_numeral_less_neg_one
% 0.29/0.56  thf(fact_5463_not__one__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_one_less_neg_numeral
% 0.29/0.56  thf(fact_5464_not__one__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_one_less_neg_numeral
% 0.29/0.56  thf(fact_5465_not__one__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_one_less_neg_numeral
% 0.29/0.56  thf(fact_5466_not__one__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_one_less_neg_numeral
% 0.29/0.56  thf(fact_5467_not__neg__one__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_neg_one_less_neg_numeral
% 0.29/0.56  thf(fact_5468_not__neg__one__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_neg_one_less_neg_numeral
% 0.29/0.56  thf(fact_5469_not__neg__one__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_neg_one_less_neg_numeral
% 0.29/0.56  thf(fact_5470_not__neg__one__less__neg__numeral,axiom,
% 0.29/0.56      ! [M: num] :
% 0.29/0.56        ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % not_neg_one_less_neg_numeral
% 0.29/0.56  thf(fact_5471_mod__le__divisor,axiom,
% 0.29/0.56      ! [N: nat,M: nat] :
% 0.29/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.56       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.29/0.56  
% 0.29/0.56  % mod_le_divisor
% 0.29/0.56  thf(fact_5472_nonzero__neg__divide__eq__eq2,axiom,
% 0.29/0.56      ! [B: real,C: real,A: real] :
% 0.29/0.56        ( ( B != zero_zero_real )
% 0.29/0.56       => ( ( C
% 0.29/0.56            = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 0.29/0.56          = ( ( times_times_real @ C @ B )
% 0.29/0.56            = ( uminus_uminus_real @ A ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_neg_divide_eq_eq2
% 0.29/0.56  thf(fact_5473_nonzero__neg__divide__eq__eq2,axiom,
% 0.29/0.56      ! [B: complex,C: complex,A: complex] :
% 0.29/0.56        ( ( B != zero_zero_complex )
% 0.29/0.56       => ( ( C
% 0.29/0.56            = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.29/0.56          = ( ( times_times_complex @ C @ B )
% 0.29/0.56            = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_neg_divide_eq_eq2
% 0.29/0.56  thf(fact_5474_nonzero__neg__divide__eq__eq2,axiom,
% 0.29/0.56      ! [B: rat,C: rat,A: rat] :
% 0.29/0.56        ( ( B != zero_zero_rat )
% 0.29/0.56       => ( ( C
% 0.29/0.56            = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 0.29/0.56          = ( ( times_times_rat @ C @ B )
% 0.29/0.56            = ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_neg_divide_eq_eq2
% 0.29/0.56  thf(fact_5475_nonzero__neg__divide__eq__eq,axiom,
% 0.29/0.56      ! [B: real,A: real,C: real] :
% 0.29/0.56        ( ( B != zero_zero_real )
% 0.29/0.56       => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.56            = C )
% 0.29/0.56          = ( ( uminus_uminus_real @ A )
% 0.29/0.56            = ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_neg_divide_eq_eq
% 0.29/0.56  thf(fact_5476_nonzero__neg__divide__eq__eq,axiom,
% 0.29/0.56      ! [B: complex,A: complex,C: complex] :
% 0.29/0.56        ( ( B != zero_zero_complex )
% 0.29/0.56       => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.56            = C )
% 0.29/0.56          = ( ( uminus1482373934393186551omplex @ A )
% 0.29/0.56            = ( times_times_complex @ C @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_neg_divide_eq_eq
% 0.29/0.56  thf(fact_5477_nonzero__neg__divide__eq__eq,axiom,
% 0.29/0.56      ! [B: rat,A: rat,C: rat] :
% 0.29/0.56        ( ( B != zero_zero_rat )
% 0.29/0.56       => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.29/0.56            = C )
% 0.29/0.56          = ( ( uminus_uminus_rat @ A )
% 0.29/0.56            = ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.56  
% 0.29/0.56  % nonzero_neg_divide_eq_eq
% 0.29/0.56  thf(fact_5478_minus__divide__eq__eq,axiom,
% 0.29/0.56      ! [B: real,C: real,A: real] :
% 0.29/0.56        ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 0.29/0.56          = A )
% 0.29/0.56        = ( ( ( C != zero_zero_real )
% 0.29/0.56           => ( ( uminus_uminus_real @ B )
% 0.29/0.56              = ( times_times_real @ A @ C ) ) )
% 0.29/0.56          & ( ( C = zero_zero_real )
% 0.29/0.56           => ( A = zero_zero_real ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_eq_eq
% 0.29/0.57  thf(fact_5479_minus__divide__eq__eq,axiom,
% 0.29/0.57      ! [B: complex,C: complex,A: complex] :
% 0.29/0.57        ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.57          = A )
% 0.29/0.57        = ( ( ( C != zero_zero_complex )
% 0.29/0.57           => ( ( uminus1482373934393186551omplex @ B )
% 0.29/0.57              = ( times_times_complex @ A @ C ) ) )
% 0.29/0.57          & ( ( C = zero_zero_complex )
% 0.29/0.57           => ( A = zero_zero_complex ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_eq_eq
% 0.29/0.57  thf(fact_5480_minus__divide__eq__eq,axiom,
% 0.29/0.57      ! [B: rat,C: rat,A: rat] :
% 0.29/0.57        ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.57          = A )
% 0.29/0.57        = ( ( ( C != zero_zero_rat )
% 0.29/0.57           => ( ( uminus_uminus_rat @ B )
% 0.29/0.57              = ( times_times_rat @ A @ C ) ) )
% 0.29/0.57          & ( ( C = zero_zero_rat )
% 0.29/0.57           => ( A = zero_zero_rat ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_eq_eq
% 0.29/0.57  thf(fact_5481_eq__minus__divide__eq,axiom,
% 0.29/0.57      ! [A: real,B: real,C: real] :
% 0.29/0.57        ( ( A
% 0.29/0.57          = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.29/0.57        = ( ( ( C != zero_zero_real )
% 0.29/0.57           => ( ( times_times_real @ A @ C )
% 0.29/0.57              = ( uminus_uminus_real @ B ) ) )
% 0.29/0.57          & ( ( C = zero_zero_real )
% 0.29/0.57           => ( A = zero_zero_real ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % eq_minus_divide_eq
% 0.29/0.57  thf(fact_5482_eq__minus__divide__eq,axiom,
% 0.29/0.57      ! [A: complex,B: complex,C: complex] :
% 0.29/0.57        ( ( A
% 0.29/0.57          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 0.29/0.57        = ( ( ( C != zero_zero_complex )
% 0.29/0.57           => ( ( times_times_complex @ A @ C )
% 0.29/0.57              = ( uminus1482373934393186551omplex @ B ) ) )
% 0.29/0.57          & ( ( C = zero_zero_complex )
% 0.29/0.57           => ( A = zero_zero_complex ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % eq_minus_divide_eq
% 0.29/0.57  thf(fact_5483_eq__minus__divide__eq,axiom,
% 0.29/0.57      ! [A: rat,B: rat,C: rat] :
% 0.29/0.57        ( ( A
% 0.29/0.57          = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.29/0.57        = ( ( ( C != zero_zero_rat )
% 0.29/0.57           => ( ( times_times_rat @ A @ C )
% 0.29/0.57              = ( uminus_uminus_rat @ B ) ) )
% 0.29/0.57          & ( ( C = zero_zero_rat )
% 0.29/0.57           => ( A = zero_zero_rat ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % eq_minus_divide_eq
% 0.29/0.57  thf(fact_5484_divide__eq__minus__1__iff,axiom,
% 0.29/0.57      ! [A: real,B: real] :
% 0.29/0.57        ( ( ( divide_divide_real @ A @ B )
% 0.29/0.57          = ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.57        = ( ( B != zero_zero_real )
% 0.29/0.57          & ( A
% 0.29/0.57            = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_eq_minus_1_iff
% 0.29/0.57  thf(fact_5485_divide__eq__minus__1__iff,axiom,
% 0.29/0.57      ! [A: complex,B: complex] :
% 0.29/0.57        ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.29/0.57          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.57        = ( ( B != zero_zero_complex )
% 0.29/0.57          & ( A
% 0.29/0.57            = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_eq_minus_1_iff
% 0.29/0.57  thf(fact_5486_divide__eq__minus__1__iff,axiom,
% 0.29/0.57      ! [A: rat,B: rat] :
% 0.29/0.57        ( ( ( divide_divide_rat @ A @ B )
% 0.29/0.57          = ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.57        = ( ( B != zero_zero_rat )
% 0.29/0.57          & ( A
% 0.29/0.57            = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_eq_minus_1_iff
% 0.29/0.57  thf(fact_5487_mult__1s__ring__1_I2_J,axiom,
% 0.29/0.57      ! [B: real] :
% 0.29/0.57        ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 0.29/0.57        = ( uminus_uminus_real @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_1s_ring_1(2)
% 0.29/0.57  thf(fact_5488_mult__1s__ring__1_I2_J,axiom,
% 0.29/0.57      ! [B: int] :
% 0.29/0.57        ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 0.29/0.57        = ( uminus_uminus_int @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_1s_ring_1(2)
% 0.29/0.57  thf(fact_5489_mult__1s__ring__1_I2_J,axiom,
% 0.29/0.57      ! [B: complex] :
% 0.29/0.57        ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 0.29/0.57        = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_1s_ring_1(2)
% 0.29/0.57  thf(fact_5490_mult__1s__ring__1_I2_J,axiom,
% 0.29/0.57      ! [B: rat] :
% 0.29/0.57        ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 0.29/0.57        = ( uminus_uminus_rat @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_1s_ring_1(2)
% 0.29/0.57  thf(fact_5491_mult__1s__ring__1_I2_J,axiom,
% 0.29/0.57      ! [B: code_integer] :
% 0.29/0.57        ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 0.29/0.57        = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_1s_ring_1(2)
% 0.29/0.57  thf(fact_5492_mult__1s__ring__1_I1_J,axiom,
% 0.29/0.57      ! [B: real] :
% 0.29/0.57        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 0.29/0.57        = ( uminus_uminus_real @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_1s_ring_1(1)
% 0.29/0.57  thf(fact_5493_mult__1s__ring__1_I1_J,axiom,
% 0.29/0.57      ! [B: int] :
% 0.29/0.57        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 0.29/0.57        = ( uminus_uminus_int @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_1s_ring_1(1)
% 0.29/0.57  thf(fact_5494_mult__1s__ring__1_I1_J,axiom,
% 0.29/0.57      ! [B: complex] :
% 0.29/0.57        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 0.29/0.57        = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_1s_ring_1(1)
% 0.29/0.57  thf(fact_5495_mult__1s__ring__1_I1_J,axiom,
% 0.29/0.57      ! [B: rat] :
% 0.29/0.57        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 0.29/0.57        = ( uminus_uminus_rat @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_1s_ring_1(1)
% 0.29/0.57  thf(fact_5496_mult__1s__ring__1_I1_J,axiom,
% 0.29/0.57      ! [B: code_integer] :
% 0.29/0.57        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 0.29/0.57        = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_1s_ring_1(1)
% 0.29/0.57  thf(fact_5497_uminus__numeral__One,axiom,
% 0.29/0.57      ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 0.29/0.57      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % uminus_numeral_One
% 0.29/0.57  thf(fact_5498_uminus__numeral__One,axiom,
% 0.29/0.57      ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 0.29/0.57      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % uminus_numeral_One
% 0.29/0.57  thf(fact_5499_uminus__numeral__One,axiom,
% 0.29/0.57      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.29/0.57      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.57  
% 0.29/0.57  % uminus_numeral_One
% 0.29/0.57  thf(fact_5500_uminus__numeral__One,axiom,
% 0.29/0.57      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 0.29/0.57      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % uminus_numeral_One
% 0.29/0.57  thf(fact_5501_uminus__numeral__One,axiom,
% 0.29/0.57      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 0.29/0.57      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.57  
% 0.29/0.57  % uminus_numeral_One
% 0.29/0.57  thf(fact_5502_power__minus,axiom,
% 0.29/0.57      ! [A: real,N: nat] :
% 0.29/0.57        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.29/0.57        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus
% 0.29/0.57  thf(fact_5503_power__minus,axiom,
% 0.29/0.57      ! [A: int,N: nat] :
% 0.29/0.57        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.29/0.57        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus
% 0.29/0.57  thf(fact_5504_power__minus,axiom,
% 0.29/0.57      ! [A: complex,N: nat] :
% 0.29/0.57        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.29/0.57        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus
% 0.29/0.57  thf(fact_5505_power__minus,axiom,
% 0.29/0.57      ! [A: rat,N: nat] :
% 0.29/0.57        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.29/0.57        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus
% 0.29/0.57  thf(fact_5506_power__minus,axiom,
% 0.29/0.57      ! [A: code_integer,N: nat] :
% 0.29/0.57        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.29/0.57        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus
% 0.29/0.57  thf(fact_5507_div__less__mono,axiom,
% 0.29/0.57      ! [A2: nat,B2: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ A2 @ B2 )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57         => ( ( ( modulo_modulo_nat @ A2 @ N )
% 0.29/0.57              = zero_zero_nat )
% 0.29/0.57           => ( ( ( modulo_modulo_nat @ B2 @ N )
% 0.29/0.57                = zero_zero_nat )
% 0.29/0.57             => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_less_mono
% 0.29/0.57  thf(fact_5508_mod__eq__nat1E,axiom,
% 0.29/0.57      ! [M: nat,Q3: nat,N: nat] :
% 0.29/0.57        ( ( ( modulo_modulo_nat @ M @ Q3 )
% 0.29/0.57          = ( modulo_modulo_nat @ N @ Q3 ) )
% 0.29/0.57       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.57         => ~ ! [S3: nat] :
% 0.29/0.57                ( M
% 0.29/0.57               != ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_eq_nat1E
% 0.29/0.57  thf(fact_5509_mod__eq__nat2E,axiom,
% 0.29/0.57      ! [M: nat,Q3: nat,N: nat] :
% 0.29/0.57        ( ( ( modulo_modulo_nat @ M @ Q3 )
% 0.29/0.57          = ( modulo_modulo_nat @ N @ Q3 ) )
% 0.29/0.57       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57         => ~ ! [S3: nat] :
% 0.29/0.57                ( N
% 0.29/0.57               != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_eq_nat2E
% 0.29/0.57  thf(fact_5510_nat__mod__eq__lemma,axiom,
% 0.29/0.57      ! [X: nat,N: nat,Y: nat] :
% 0.29/0.57        ( ( ( modulo_modulo_nat @ X @ N )
% 0.29/0.57          = ( modulo_modulo_nat @ Y @ N ) )
% 0.29/0.57       => ( ( ord_less_eq_nat @ Y @ X )
% 0.29/0.57         => ? [Q2: nat] :
% 0.29/0.57              ( X
% 0.29/0.57              = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q2 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % nat_mod_eq_lemma
% 0.29/0.57  thf(fact_5511_lemma__interval__lt,axiom,
% 0.29/0.57      ! [A: real,X: real,B: real] :
% 0.29/0.57        ( ( ord_less_real @ A @ X )
% 0.29/0.57       => ( ( ord_less_real @ X @ B )
% 0.29/0.57         => ? [D5: real] :
% 0.29/0.57              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.29/0.57              & ! [Y5: real] :
% 0.29/0.57                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D5 )
% 0.29/0.57                 => ( ( ord_less_real @ A @ Y5 )
% 0.29/0.57                    & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % lemma_interval_lt
% 0.29/0.57  thf(fact_5512_power__minus__Bit0,axiom,
% 0.29/0.57      ! [X: real,K: num] :
% 0.29/0.57        ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.29/0.57        = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_Bit0
% 0.29/0.57  thf(fact_5513_power__minus__Bit0,axiom,
% 0.29/0.57      ! [X: int,K: num] :
% 0.29/0.57        ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.29/0.57        = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_Bit0
% 0.29/0.57  thf(fact_5514_power__minus__Bit0,axiom,
% 0.29/0.57      ! [X: complex,K: num] :
% 0.29/0.57        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.29/0.57        = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_Bit0
% 0.29/0.57  thf(fact_5515_power__minus__Bit0,axiom,
% 0.29/0.57      ! [X: rat,K: num] :
% 0.29/0.57        ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.29/0.57        = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_Bit0
% 0.29/0.57  thf(fact_5516_power__minus__Bit0,axiom,
% 0.29/0.57      ! [X: code_integer,K: num] :
% 0.29/0.57        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.29/0.57        = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_Bit0
% 0.29/0.57  thf(fact_5517_norm__uminus__minus,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
% 0.29/0.57        = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % norm_uminus_minus
% 0.29/0.57  thf(fact_5518_norm__uminus__minus,axiom,
% 0.29/0.57      ! [X: complex,Y: complex] :
% 0.29/0.57        ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
% 0.29/0.57        = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % norm_uminus_minus
% 0.29/0.57  thf(fact_5519_invar__vebt_Ointros_I1_J,axiom,
% 0.29/0.57      ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % invar_vebt.intros(1)
% 0.29/0.57  thf(fact_5520_mod__mult2__eq,axiom,
% 0.29/0.57      ! [M: nat,N: nat,Q3: nat] :
% 0.29/0.57        ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q3 ) )
% 0.29/0.57        = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_mult2_eq
% 0.29/0.57  thf(fact_5521_div__mod__decomp,axiom,
% 0.29/0.57      ! [A2: nat,N: nat] :
% 0.29/0.57        ( A2
% 0.29/0.57        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_mod_decomp
% 0.29/0.57  thf(fact_5522_modulo__nat__def,axiom,
% 0.29/0.57      ( modulo_modulo_nat
% 0.29/0.57      = ( ^ [M6: nat,N3: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N3 ) @ N3 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % modulo_nat_def
% 0.29/0.57  thf(fact_5523_sin__bound__lemma,axiom,
% 0.29/0.57      ! [X: real,Y: real,U: real,V: real] :
% 0.29/0.57        ( ( X = Y )
% 0.29/0.57       => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.29/0.57         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % sin_bound_lemma
% 0.29/0.57  thf(fact_5524_powr__minus__divide,axiom,
% 0.29/0.57      ! [X: real,A: real] :
% 0.29/0.57        ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
% 0.29/0.57        = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % powr_minus_divide
% 0.29/0.57  thf(fact_5525_real__0__less__add__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.57        = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 0.29/0.57  
% 0.29/0.57  % real_0_less_add_iff
% 0.29/0.57  thf(fact_5526_real__add__less__0__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.29/0.57        = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % real_add_less_0_iff
% 0.29/0.57  thf(fact_5527_real__add__le__0__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.29/0.57        = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % real_add_le_0_iff
% 0.29/0.57  thf(fact_5528_real__0__le__add__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.57        = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 0.29/0.57  
% 0.29/0.57  % real_0_le_add_iff
% 0.29/0.57  thf(fact_5529_vebt__buildup_Osimps_I2_J,axiom,
% 0.29/0.57      ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 0.29/0.57      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.29/0.57  
% 0.29/0.57  % vebt_buildup.simps(2)
% 0.29/0.57  thf(fact_5530_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 0.29/0.57      ! [A: $o,B: $o,X: nat] :
% 0.29/0.57        ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.29/0.57        = ( ( ( X = zero_zero_nat )
% 0.29/0.57           => A )
% 0.29/0.57          & ( ( X != zero_zero_nat )
% 0.29/0.57           => ( ( ( X = one_one_nat )
% 0.29/0.57               => B )
% 0.29/0.57              & ( X = one_one_nat ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % VEBT_internal.naive_member.simps(1)
% 0.29/0.57  thf(fact_5531_divide__powr__uminus,axiom,
% 0.29/0.57      ! [A: real,B: real,C: real] :
% 0.29/0.57        ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 0.29/0.57        = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_powr_uminus
% 0.29/0.57  thf(fact_5532_abs__add__one__gt__zero,axiom,
% 0.29/0.57      ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_add_one_gt_zero
% 0.29/0.57  thf(fact_5533_abs__add__one__gt__zero,axiom,
% 0.29/0.57      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_add_one_gt_zero
% 0.29/0.57  thf(fact_5534_abs__add__one__gt__zero,axiom,
% 0.29/0.57      ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_add_one_gt_zero
% 0.29/0.57  thf(fact_5535_abs__add__one__gt__zero,axiom,
% 0.29/0.57      ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_add_one_gt_zero
% 0.29/0.57  thf(fact_5536_norm__less__p1,axiom,
% 0.29/0.57      ! [X: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X ) ) @ one_one_real ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % norm_less_p1
% 0.29/0.57  thf(fact_5537_norm__less__p1,axiom,
% 0.29/0.57      ! [X: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X ) ) @ one_one_complex ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % norm_less_p1
% 0.29/0.57  thf(fact_5538_of__int__leD,axiom,
% 0.29/0.57      ! [N: int,X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.29/0.57       => ( ( N = zero_zero_int )
% 0.29/0.57          | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % of_int_leD
% 0.29/0.57  thf(fact_5539_of__int__leD,axiom,
% 0.29/0.57      ! [N: int,X: code_integer] :
% 0.29/0.57        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 0.29/0.57       => ( ( N = zero_zero_int )
% 0.29/0.57          | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % of_int_leD
% 0.29/0.57  thf(fact_5540_of__int__leD,axiom,
% 0.29/0.57      ! [N: int,X: rat] :
% 0.29/0.57        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 0.29/0.57       => ( ( N = zero_zero_int )
% 0.29/0.57          | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % of_int_leD
% 0.29/0.57  thf(fact_5541_of__int__leD,axiom,
% 0.29/0.57      ! [N: int,X: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.29/0.57       => ( ( N = zero_zero_int )
% 0.29/0.57          | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % of_int_leD
% 0.29/0.57  thf(fact_5542_of__int__lessD,axiom,
% 0.29/0.57      ! [N: int,X: code_integer] :
% 0.29/0.57        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 0.29/0.57       => ( ( N = zero_zero_int )
% 0.29/0.57          | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % of_int_lessD
% 0.29/0.57  thf(fact_5543_of__int__lessD,axiom,
% 0.29/0.57      ! [N: int,X: real] :
% 0.29/0.57        ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.29/0.57       => ( ( N = zero_zero_int )
% 0.29/0.57          | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % of_int_lessD
% 0.29/0.57  thf(fact_5544_of__int__lessD,axiom,
% 0.29/0.57      ! [N: int,X: rat] :
% 0.29/0.57        ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 0.29/0.57       => ( ( N = zero_zero_int )
% 0.29/0.57          | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % of_int_lessD
% 0.29/0.57  thf(fact_5545_of__int__lessD,axiom,
% 0.29/0.57      ! [N: int,X: int] :
% 0.29/0.57        ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.29/0.57       => ( ( N = zero_zero_int )
% 0.29/0.57          | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % of_int_lessD
% 0.29/0.57  thf(fact_5546_mod__mult2__eq_H,axiom,
% 0.29/0.57      ! [A: code_integer,M: nat,N: nat] :
% 0.29/0.57        ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 0.29/0.57        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_mult2_eq'
% 0.29/0.57  thf(fact_5547_mod__mult2__eq_H,axiom,
% 0.29/0.57      ! [A: nat,M: nat,N: nat] :
% 0.29/0.57        ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.29/0.57        = ( 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 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_mult2_eq'
% 0.29/0.57  thf(fact_5548_mod__mult2__eq_H,axiom,
% 0.29/0.57      ! [A: int,M: nat,N: nat] :
% 0.29/0.57        ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.29/0.57        = ( 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 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_mult2_eq'
% 0.29/0.57  thf(fact_5549_pos__minus__divide__less__eq,axiom,
% 0.29/0.57      ! [C: real,B: real,A: real] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.29/0.57          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_minus_divide_less_eq
% 0.29/0.57  thf(fact_5550_pos__minus__divide__less__eq,axiom,
% 0.29/0.57      ! [C: rat,B: rat,A: rat] :
% 0.29/0.57        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.29/0.57          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_minus_divide_less_eq
% 0.29/0.57  thf(fact_5551_pos__less__minus__divide__eq,axiom,
% 0.29/0.57      ! [C: real,A: real,B: real] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.29/0.57          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_less_minus_divide_eq
% 0.29/0.57  thf(fact_5552_pos__less__minus__divide__eq,axiom,
% 0.29/0.57      ! [C: rat,A: rat,B: rat] :
% 0.29/0.57        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.29/0.57          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_less_minus_divide_eq
% 0.29/0.57  thf(fact_5553_neg__minus__divide__less__eq,axiom,
% 0.29/0.57      ! [C: real,B: real,A: real] :
% 0.29/0.57        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.29/0.57          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_minus_divide_less_eq
% 0.29/0.57  thf(fact_5554_neg__minus__divide__less__eq,axiom,
% 0.29/0.57      ! [C: rat,B: rat,A: rat] :
% 0.29/0.57        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.29/0.57          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_minus_divide_less_eq
% 0.29/0.57  thf(fact_5555_neg__less__minus__divide__eq,axiom,
% 0.29/0.57      ! [C: real,A: real,B: real] :
% 0.29/0.57        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.29/0.57          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_less_minus_divide_eq
% 0.29/0.57  thf(fact_5556_neg__less__minus__divide__eq,axiom,
% 0.29/0.57      ! [C: rat,A: rat,B: rat] :
% 0.29/0.57        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.29/0.57          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_less_minus_divide_eq
% 0.29/0.57  thf(fact_5557_minus__divide__less__eq,axiom,
% 0.29/0.57      ! [B: real,C: real,A: real] :
% 0.29/0.57        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.29/0.57        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.29/0.57          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.29/0.57              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_less_eq
% 0.29/0.57  thf(fact_5558_minus__divide__less__eq,axiom,
% 0.29/0.57      ! [B: rat,C: rat,A: rat] :
% 0.29/0.57        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.29/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.57          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.29/0.57              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_less_eq
% 0.29/0.57  thf(fact_5559_less__minus__divide__eq,axiom,
% 0.29/0.57      ! [A: real,B: real,C: real] :
% 0.29/0.57        ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.29/0.57        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.29/0.57          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.29/0.57              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % less_minus_divide_eq
% 0.29/0.57  thf(fact_5560_less__minus__divide__eq,axiom,
% 0.29/0.57      ! [A: rat,B: rat,C: rat] :
% 0.29/0.57        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.29/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.29/0.57          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.57              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % less_minus_divide_eq
% 0.29/0.57  thf(fact_5561_eq__divide__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [W: num,B: real,C: real] :
% 0.29/0.57        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.29/0.57          = ( divide_divide_real @ B @ C ) )
% 0.29/0.57        = ( ( ( C != zero_zero_real )
% 0.29/0.57           => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 0.29/0.57              = B ) )
% 0.29/0.57          & ( ( C = zero_zero_real )
% 0.29/0.57           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.29/0.57              = zero_zero_real ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % eq_divide_eq_numeral(2)
% 0.29/0.57  thf(fact_5562_eq__divide__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [W: num,B: complex,C: complex] :
% 0.29/0.57        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.29/0.57          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.57        = ( ( ( C != zero_zero_complex )
% 0.29/0.57           => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 0.29/0.57              = B ) )
% 0.29/0.57          & ( ( C = zero_zero_complex )
% 0.29/0.57           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.29/0.57              = zero_zero_complex ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % eq_divide_eq_numeral(2)
% 0.29/0.57  thf(fact_5563_eq__divide__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [W: num,B: rat,C: rat] :
% 0.29/0.57        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.29/0.57          = ( divide_divide_rat @ B @ C ) )
% 0.29/0.57        = ( ( ( C != zero_zero_rat )
% 0.29/0.57           => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 0.29/0.57              = B ) )
% 0.29/0.57          & ( ( C = zero_zero_rat )
% 0.29/0.57           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.29/0.57              = zero_zero_rat ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % eq_divide_eq_numeral(2)
% 0.29/0.57  thf(fact_5564_divide__eq__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [B: real,C: real,W: num] :
% 0.29/0.57        ( ( ( divide_divide_real @ B @ C )
% 0.29/0.57          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.57        = ( ( ( C != zero_zero_real )
% 0.29/0.57           => ( B
% 0.29/0.57              = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.29/0.57          & ( ( C = zero_zero_real )
% 0.29/0.57           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.29/0.57              = zero_zero_real ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_eq_eq_numeral(2)
% 0.29/0.57  thf(fact_5565_divide__eq__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [B: complex,C: complex,W: num] :
% 0.29/0.57        ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.29/0.57          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.57        = ( ( ( C != zero_zero_complex )
% 0.29/0.57           => ( B
% 0.29/0.57              = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 0.29/0.57          & ( ( C = zero_zero_complex )
% 0.29/0.57           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.29/0.57              = zero_zero_complex ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_eq_eq_numeral(2)
% 0.29/0.57  thf(fact_5566_divide__eq__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [B: rat,C: rat,W: num] :
% 0.29/0.57        ( ( ( divide_divide_rat @ B @ C )
% 0.29/0.57          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.29/0.57        = ( ( ( C != zero_zero_rat )
% 0.29/0.57           => ( B
% 0.29/0.57              = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.29/0.57          & ( ( C = zero_zero_rat )
% 0.29/0.57           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.29/0.57              = zero_zero_rat ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_eq_eq_numeral(2)
% 0.29/0.57  thf(fact_5567_add__divide__eq__if__simps_I3_J,axiom,
% 0.29/0.57      ! [Z: real,A: real,B: real] :
% 0.29/0.57        ( ( ( Z = zero_zero_real )
% 0.29/0.57         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.29/0.57            = B ) )
% 0.29/0.57        & ( ( Z != zero_zero_real )
% 0.29/0.57         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.29/0.57            = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_divide_eq_if_simps(3)
% 0.29/0.57  thf(fact_5568_add__divide__eq__if__simps_I3_J,axiom,
% 0.29/0.57      ! [Z: complex,A: complex,B: complex] :
% 0.29/0.57        ( ( ( Z = zero_zero_complex )
% 0.29/0.57         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.29/0.57            = B ) )
% 0.29/0.57        & ( ( Z != zero_zero_complex )
% 0.29/0.57         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.29/0.57            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_divide_eq_if_simps(3)
% 0.29/0.57  thf(fact_5569_add__divide__eq__if__simps_I3_J,axiom,
% 0.29/0.57      ! [Z: rat,A: rat,B: rat] :
% 0.29/0.57        ( ( ( Z = zero_zero_rat )
% 0.29/0.57         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.29/0.57            = B ) )
% 0.29/0.57        & ( ( Z != zero_zero_rat )
% 0.29/0.57         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.29/0.57            = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_divide_eq_if_simps(3)
% 0.29/0.57  thf(fact_5570_minus__divide__add__eq__iff,axiom,
% 0.29/0.57      ! [Z: real,X: real,Y: real] :
% 0.29/0.57        ( ( Z != zero_zero_real )
% 0.29/0.57       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
% 0.29/0.57          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_add_eq_iff
% 0.29/0.57  thf(fact_5571_minus__divide__add__eq__iff,axiom,
% 0.29/0.57      ! [Z: complex,X: complex,Y: complex] :
% 0.29/0.57        ( ( Z != zero_zero_complex )
% 0.29/0.57       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
% 0.29/0.57          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_add_eq_iff
% 0.29/0.57  thf(fact_5572_minus__divide__add__eq__iff,axiom,
% 0.29/0.57      ! [Z: rat,X: rat,Y: rat] :
% 0.29/0.57        ( ( Z != zero_zero_rat )
% 0.29/0.57       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
% 0.29/0.57          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_add_eq_iff
% 0.29/0.57  thf(fact_5573_add__divide__eq__if__simps_I6_J,axiom,
% 0.29/0.57      ! [Z: real,A: real,B: real] :
% 0.29/0.57        ( ( ( Z = zero_zero_real )
% 0.29/0.57         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.29/0.57            = ( uminus_uminus_real @ B ) ) )
% 0.29/0.57        & ( ( Z != zero_zero_real )
% 0.29/0.57         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.29/0.57            = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_divide_eq_if_simps(6)
% 0.29/0.57  thf(fact_5574_add__divide__eq__if__simps_I6_J,axiom,
% 0.29/0.57      ! [Z: complex,A: complex,B: complex] :
% 0.29/0.57        ( ( ( Z = zero_zero_complex )
% 0.29/0.57         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.29/0.57            = ( uminus1482373934393186551omplex @ B ) ) )
% 0.29/0.57        & ( ( Z != zero_zero_complex )
% 0.29/0.57         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.29/0.57            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_divide_eq_if_simps(6)
% 0.29/0.57  thf(fact_5575_add__divide__eq__if__simps_I6_J,axiom,
% 0.29/0.57      ! [Z: rat,A: rat,B: rat] :
% 0.29/0.57        ( ( ( Z = zero_zero_rat )
% 0.29/0.57         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.29/0.57            = ( uminus_uminus_rat @ B ) ) )
% 0.29/0.57        & ( ( Z != zero_zero_rat )
% 0.29/0.57         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.29/0.57            = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_divide_eq_if_simps(6)
% 0.29/0.57  thf(fact_5576_add__divide__eq__if__simps_I5_J,axiom,
% 0.29/0.57      ! [Z: real,A: real,B: real] :
% 0.29/0.57        ( ( ( Z = zero_zero_real )
% 0.29/0.57         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.29/0.57            = ( uminus_uminus_real @ B ) ) )
% 0.29/0.57        & ( ( Z != zero_zero_real )
% 0.29/0.57         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.29/0.57            = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_divide_eq_if_simps(5)
% 0.29/0.57  thf(fact_5577_add__divide__eq__if__simps_I5_J,axiom,
% 0.29/0.57      ! [Z: complex,A: complex,B: complex] :
% 0.29/0.57        ( ( ( Z = zero_zero_complex )
% 0.29/0.57         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.29/0.57            = ( uminus1482373934393186551omplex @ B ) ) )
% 0.29/0.57        & ( ( Z != zero_zero_complex )
% 0.29/0.57         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.29/0.57            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_divide_eq_if_simps(5)
% 0.29/0.57  thf(fact_5578_add__divide__eq__if__simps_I5_J,axiom,
% 0.29/0.57      ! [Z: rat,A: rat,B: rat] :
% 0.29/0.57        ( ( ( Z = zero_zero_rat )
% 0.29/0.57         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.29/0.57            = ( uminus_uminus_rat @ B ) ) )
% 0.29/0.57        & ( ( Z != zero_zero_rat )
% 0.29/0.57         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.29/0.57            = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_divide_eq_if_simps(5)
% 0.29/0.57  thf(fact_5579_minus__divide__diff__eq__iff,axiom,
% 0.29/0.57      ! [Z: real,X: real,Y: real] :
% 0.29/0.57        ( ( Z != zero_zero_real )
% 0.29/0.57       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
% 0.29/0.57          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_diff_eq_iff
% 0.29/0.57  thf(fact_5580_minus__divide__diff__eq__iff,axiom,
% 0.29/0.57      ! [Z: complex,X: complex,Y: complex] :
% 0.29/0.57        ( ( Z != zero_zero_complex )
% 0.29/0.57       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
% 0.29/0.57          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_diff_eq_iff
% 0.29/0.57  thf(fact_5581_minus__divide__diff__eq__iff,axiom,
% 0.29/0.57      ! [Z: rat,X: rat,Y: rat] :
% 0.29/0.57        ( ( Z != zero_zero_rat )
% 0.29/0.57       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
% 0.29/0.57          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_diff_eq_iff
% 0.29/0.57  thf(fact_5582_field__char__0__class_Oof__nat__div,axiom,
% 0.29/0.57      ! [M: nat,N: nat] :
% 0.29/0.57        ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 0.29/0.57        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % field_char_0_class.of_nat_div
% 0.29/0.57  thf(fact_5583_field__char__0__class_Oof__nat__div,axiom,
% 0.29/0.57      ! [M: nat,N: nat] :
% 0.29/0.57        ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 0.29/0.57        = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % field_char_0_class.of_nat_div
% 0.29/0.57  thf(fact_5584_field__char__0__class_Oof__nat__div,axiom,
% 0.29/0.57      ! [M: nat,N: nat] :
% 0.29/0.57        ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 0.29/0.57        = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % field_char_0_class.of_nat_div
% 0.29/0.57  thf(fact_5585_split__mod,axiom,
% 0.29/0.57      ! [P: nat > $o,M: nat,N: nat] :
% 0.29/0.57        ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 0.29/0.57        = ( ( ( N = zero_zero_nat )
% 0.29/0.57           => ( P @ M ) )
% 0.29/0.57          & ( ( N != zero_zero_nat )
% 0.29/0.57           => ! [I3: nat,J2: nat] :
% 0.29/0.57                ( ( ord_less_nat @ J2 @ N )
% 0.29/0.57               => ( ( M
% 0.29/0.57                    = ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J2 ) )
% 0.29/0.57                 => ( P @ J2 ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % split_mod
% 0.29/0.57  thf(fact_5586_lemma__interval,axiom,
% 0.29/0.57      ! [A: real,X: real,B: real] :
% 0.29/0.57        ( ( ord_less_real @ A @ X )
% 0.29/0.57       => ( ( ord_less_real @ X @ B )
% 0.29/0.57         => ? [D5: real] :
% 0.29/0.57              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.29/0.57              & ! [Y5: real] :
% 0.29/0.57                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D5 )
% 0.29/0.57                 => ( ( ord_less_eq_real @ A @ Y5 )
% 0.29/0.57                    & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % lemma_interval
% 0.29/0.57  thf(fact_5587_power2__eq__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57        = ( ( X = Y )
% 0.29/0.57          | ( X
% 0.29/0.57            = ( uminus_uminus_real @ Y ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_eq_iff
% 0.29/0.57  thf(fact_5588_power2__eq__iff,axiom,
% 0.29/0.57      ! [X: int,Y: int] :
% 0.29/0.57        ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57        = ( ( X = Y )
% 0.29/0.57          | ( X
% 0.29/0.57            = ( uminus_uminus_int @ Y ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_eq_iff
% 0.29/0.57  thf(fact_5589_power2__eq__iff,axiom,
% 0.29/0.57      ! [X: complex,Y: complex] :
% 0.29/0.57        ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57        = ( ( X = Y )
% 0.29/0.57          | ( X
% 0.29/0.57            = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_eq_iff
% 0.29/0.57  thf(fact_5590_power2__eq__iff,axiom,
% 0.29/0.57      ! [X: rat,Y: rat] :
% 0.29/0.57        ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57        = ( ( X = Y )
% 0.29/0.57          | ( X
% 0.29/0.57            = ( uminus_uminus_rat @ Y ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_eq_iff
% 0.29/0.57  thf(fact_5591_power2__eq__iff,axiom,
% 0.29/0.57      ! [X: code_integer,Y: code_integer] :
% 0.29/0.57        ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57        = ( ( X = Y )
% 0.29/0.57          | ( X
% 0.29/0.57            = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_eq_iff
% 0.29/0.57  thf(fact_5592_norm__triangle__ineq3,axiom,
% 0.29/0.57      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % norm_triangle_ineq3
% 0.29/0.57  thf(fact_5593_norm__triangle__ineq3,axiom,
% 0.29/0.57      ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % norm_triangle_ineq3
% 0.29/0.57  thf(fact_5594_round__diff__minimal,axiom,
% 0.29/0.57      ! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % round_diff_minimal
% 0.29/0.57  thf(fact_5595_round__diff__minimal,axiom,
% 0.29/0.57      ! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % round_diff_minimal
% 0.29/0.57  thf(fact_5596_real__of__nat__div__aux,axiom,
% 0.29/0.57      ! [X: nat,D: nat] :
% 0.29/0.57        ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
% 0.29/0.57        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % real_of_nat_div_aux
% 0.29/0.57  thf(fact_5597_abs__le__square__iff,axiom,
% 0.29/0.57      ! [X: code_integer,Y: code_integer] :
% 0.29/0.57        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
% 0.29/0.57        = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_le_square_iff
% 0.29/0.57  thf(fact_5598_abs__le__square__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
% 0.29/0.57        = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_le_square_iff
% 0.29/0.57  thf(fact_5599_abs__le__square__iff,axiom,
% 0.29/0.57      ! [X: rat,Y: rat] :
% 0.29/0.57        ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
% 0.29/0.57        = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_le_square_iff
% 0.29/0.57  thf(fact_5600_abs__le__square__iff,axiom,
% 0.29/0.57      ! [X: int,Y: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
% 0.29/0.57        = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_le_square_iff
% 0.29/0.57  thf(fact_5601_abs__square__eq__1,axiom,
% 0.29/0.57      ! [X: code_integer] :
% 0.29/0.57        ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = one_one_Code_integer )
% 0.29/0.57        = ( ( abs_abs_Code_integer @ X )
% 0.29/0.57          = one_one_Code_integer ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_eq_1
% 0.29/0.57  thf(fact_5602_abs__square__eq__1,axiom,
% 0.29/0.57      ! [X: rat] :
% 0.29/0.57        ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = one_one_rat )
% 0.29/0.57        = ( ( abs_abs_rat @ X )
% 0.29/0.57          = one_one_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_eq_1
% 0.29/0.57  thf(fact_5603_abs__square__eq__1,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = one_one_real )
% 0.29/0.57        = ( ( abs_abs_real @ X )
% 0.29/0.57          = one_one_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_eq_1
% 0.29/0.57  thf(fact_5604_abs__square__eq__1,axiom,
% 0.29/0.57      ! [X: int] :
% 0.29/0.57        ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = one_one_int )
% 0.29/0.57        = ( ( abs_abs_int @ X )
% 0.29/0.57          = one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_eq_1
% 0.29/0.57  thf(fact_5605_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.29/0.57      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.29/0.57       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.29/0.57          = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.29/0.57  thf(fact_5606_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.29/0.57      ! [C: nat,A: nat,B: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.57       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.29/0.57          = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.29/0.57  thf(fact_5607_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.29/0.57      ! [C: int,A: int,B: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.57       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.29/0.57          = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.29/0.57  thf(fact_5608_pos__minus__divide__le__eq,axiom,
% 0.29/0.57      ! [C: real,B: real,A: real] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.29/0.57          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_minus_divide_le_eq
% 0.29/0.57  thf(fact_5609_pos__minus__divide__le__eq,axiom,
% 0.29/0.57      ! [C: rat,B: rat,A: rat] :
% 0.29/0.57        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.29/0.57          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_minus_divide_le_eq
% 0.29/0.57  thf(fact_5610_pos__le__minus__divide__eq,axiom,
% 0.29/0.57      ! [C: real,A: real,B: real] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.29/0.57          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_le_minus_divide_eq
% 0.29/0.57  thf(fact_5611_pos__le__minus__divide__eq,axiom,
% 0.29/0.57      ! [C: rat,A: rat,B: rat] :
% 0.29/0.57        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.29/0.57          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_le_minus_divide_eq
% 0.29/0.57  thf(fact_5612_neg__minus__divide__le__eq,axiom,
% 0.29/0.57      ! [C: real,B: real,A: real] :
% 0.29/0.57        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.29/0.57          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_minus_divide_le_eq
% 0.29/0.57  thf(fact_5613_neg__minus__divide__le__eq,axiom,
% 0.29/0.57      ! [C: rat,B: rat,A: rat] :
% 0.29/0.57        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.29/0.57          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_minus_divide_le_eq
% 0.29/0.57  thf(fact_5614_neg__le__minus__divide__eq,axiom,
% 0.29/0.57      ! [C: real,A: real,B: real] :
% 0.29/0.57        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.29/0.57          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_le_minus_divide_eq
% 0.29/0.57  thf(fact_5615_neg__le__minus__divide__eq,axiom,
% 0.29/0.57      ! [C: rat,A: rat,B: rat] :
% 0.29/0.57        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.29/0.57          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_le_minus_divide_eq
% 0.29/0.57  thf(fact_5616_minus__divide__le__eq,axiom,
% 0.29/0.57      ! [B: real,C: real,A: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.29/0.57        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.29/0.57          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.29/0.57              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_le_eq
% 0.29/0.57  thf(fact_5617_minus__divide__le__eq,axiom,
% 0.29/0.57      ! [B: rat,C: rat,A: rat] :
% 0.29/0.57        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.29/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.57          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.29/0.57              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_divide_le_eq
% 0.29/0.57  thf(fact_5618_le__minus__divide__eq,axiom,
% 0.29/0.57      ! [A: real,B: real,C: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.29/0.57        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.29/0.57          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.29/0.57              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % le_minus_divide_eq
% 0.29/0.57  thf(fact_5619_le__minus__divide__eq,axiom,
% 0.29/0.57      ! [A: rat,B: rat,C: rat] :
% 0.29/0.57        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.29/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.29/0.57          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.57              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % le_minus_divide_eq
% 0.29/0.57  thf(fact_5620_less__divide__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [W: num,B: real,C: real] :
% 0.29/0.57        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.29/0.57        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.29/0.57          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.29/0.57              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % less_divide_eq_numeral(2)
% 0.29/0.57  thf(fact_5621_less__divide__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [W: num,B: rat,C: rat] :
% 0.29/0.57        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.29/0.57          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.29/0.57              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % less_divide_eq_numeral(2)
% 0.29/0.57  thf(fact_5622_divide__less__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [B: real,C: real,W: num] :
% 0.29/0.57        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.57        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.29/0.57          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.29/0.57              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_less_eq_numeral(2)
% 0.29/0.57  thf(fact_5623_divide__less__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [B: rat,C: rat,W: num] :
% 0.29/0.57        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.29/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.29/0.57          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.29/0.57              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_less_eq_numeral(2)
% 0.29/0.57  thf(fact_5624_Suc__times__mod__eq,axiom,
% 0.29/0.57      ! [M: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.57       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 0.29/0.57          = one_one_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Suc_times_mod_eq
% 0.29/0.57  thf(fact_5625_power2__eq__1__iff,axiom,
% 0.29/0.57      ! [A: real] :
% 0.29/0.57        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = one_one_real )
% 0.29/0.57        = ( ( A = one_one_real )
% 0.29/0.57          | ( A
% 0.29/0.57            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_eq_1_iff
% 0.29/0.57  thf(fact_5626_power2__eq__1__iff,axiom,
% 0.29/0.57      ! [A: int] :
% 0.29/0.57        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = one_one_int )
% 0.29/0.57        = ( ( A = one_one_int )
% 0.29/0.57          | ( A
% 0.29/0.57            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_eq_1_iff
% 0.29/0.57  thf(fact_5627_power2__eq__1__iff,axiom,
% 0.29/0.57      ! [A: complex] :
% 0.29/0.57        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = one_one_complex )
% 0.29/0.57        = ( ( A = one_one_complex )
% 0.29/0.57          | ( A
% 0.29/0.57            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_eq_1_iff
% 0.29/0.57  thf(fact_5628_power2__eq__1__iff,axiom,
% 0.29/0.57      ! [A: rat] :
% 0.29/0.57        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = one_one_rat )
% 0.29/0.57        = ( ( A = one_one_rat )
% 0.29/0.57          | ( A
% 0.29/0.57            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_eq_1_iff
% 0.29/0.57  thf(fact_5629_power2__eq__1__iff,axiom,
% 0.29/0.57      ! [A: code_integer] :
% 0.29/0.57        ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = one_one_Code_integer )
% 0.29/0.57        = ( ( A = one_one_Code_integer )
% 0.29/0.57          | ( A
% 0.29/0.57            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_eq_1_iff
% 0.29/0.57  thf(fact_5630_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 0.29/0.57          = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_one_power_add_eq_neg_one_power_diff
% 0.29/0.57  thf(fact_5631_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 0.29/0.57          = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_one_power_add_eq_neg_one_power_diff
% 0.29/0.57  thf(fact_5632_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 0.29/0.57          = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_one_power_add_eq_neg_one_power_diff
% 0.29/0.57  thf(fact_5633_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 0.29/0.57          = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_one_power_add_eq_neg_one_power_diff
% 0.29/0.57  thf(fact_5634_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 0.29/0.57          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_one_power_add_eq_neg_one_power_diff
% 0.29/0.57  thf(fact_5635_realpow__square__minus__le,axiom,
% 0.29/0.57      ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % realpow_square_minus_le
% 0.29/0.57  thf(fact_5636_powr__neg__one,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.57       => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.57          = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % powr_neg_one
% 0.29/0.57  thf(fact_5637_ln__add__one__self__le__self2,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.57       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ln_add_one_self_le_self2
% 0.29/0.57  thf(fact_5638_power2__le__iff__abs__le,axiom,
% 0.29/0.57      ! [Y: code_integer,X: code_integer] :
% 0.29/0.57        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 0.29/0.57       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_le_iff_abs_le
% 0.29/0.57  thf(fact_5639_power2__le__iff__abs__le,axiom,
% 0.29/0.57      ! [Y: real,X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.57       => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_le_iff_abs_le
% 0.29/0.57  thf(fact_5640_power2__le__iff__abs__le,axiom,
% 0.29/0.57      ! [Y: rat,X: rat] :
% 0.29/0.57        ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.29/0.57       => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_le_iff_abs_le
% 0.29/0.57  thf(fact_5641_power2__le__iff__abs__le,axiom,
% 0.29/0.57      ! [Y: int,X: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.57       => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power2_le_iff_abs_le
% 0.29/0.57  thf(fact_5642_abs__square__le__1,axiom,
% 0.29/0.57      ! [X: code_integer] :
% 0.29/0.57        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.29/0.57        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_le_1
% 0.29/0.57  thf(fact_5643_abs__square__le__1,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.29/0.57        = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_le_1
% 0.29/0.57  thf(fact_5644_abs__square__le__1,axiom,
% 0.29/0.57      ! [X: rat] :
% 0.29/0.57        ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.29/0.57        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_le_1
% 0.29/0.57  thf(fact_5645_abs__square__le__1,axiom,
% 0.29/0.57      ! [X: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.29/0.57        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_le_1
% 0.29/0.57  thf(fact_5646_abs__square__less__1,axiom,
% 0.29/0.57      ! [X: code_integer] :
% 0.29/0.57        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.29/0.57        = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_less_1
% 0.29/0.57  thf(fact_5647_abs__square__less__1,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.29/0.57        = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_less_1
% 0.29/0.57  thf(fact_5648_abs__square__less__1,axiom,
% 0.29/0.57      ! [X: rat] :
% 0.29/0.57        ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.29/0.57        = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_less_1
% 0.29/0.57  thf(fact_5649_abs__square__less__1,axiom,
% 0.29/0.57      ! [X: int] :
% 0.29/0.57        ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.29/0.57        = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_square_less_1
% 0.29/0.57  thf(fact_5650_divmod__digit__0_I2_J,axiom,
% 0.29/0.57      ! [B: nat,A: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.57       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.29/0.57         => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 0.29/0.57            = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_0(2)
% 0.29/0.57  thf(fact_5651_divmod__digit__0_I2_J,axiom,
% 0.29/0.57      ! [B: int,A: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.57       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.29/0.57         => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 0.29/0.57            = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_0(2)
% 0.29/0.57  thf(fact_5652_divmod__digit__0_I2_J,axiom,
% 0.29/0.57      ! [B: code_integer,A: code_integer] :
% 0.29/0.57        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.29/0.57       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.29/0.57         => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 0.29/0.57            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_0(2)
% 0.29/0.57  thf(fact_5653_bits__stable__imp__add__self,axiom,
% 0.29/0.57      ! [A: nat] :
% 0.29/0.57        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = A )
% 0.29/0.57       => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = zero_zero_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % bits_stable_imp_add_self
% 0.29/0.57  thf(fact_5654_bits__stable__imp__add__self,axiom,
% 0.29/0.57      ! [A: int] :
% 0.29/0.57        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.57          = A )
% 0.29/0.57       => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % bits_stable_imp_add_self
% 0.29/0.57  thf(fact_5655_bits__stable__imp__add__self,axiom,
% 0.29/0.57      ! [A: code_integer] :
% 0.29/0.57        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.57          = A )
% 0.29/0.57       => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.57  
% 0.29/0.57  % bits_stable_imp_add_self
% 0.29/0.57  thf(fact_5656_div__exp__mod__exp__eq,axiom,
% 0.29/0.57      ! [A: nat,N: nat,M: nat] :
% 0.29/0.57        ( ( 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 ) )
% 0.29/0.57        = ( 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 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_exp_mod_exp_eq
% 0.29/0.57  thf(fact_5657_div__exp__mod__exp__eq,axiom,
% 0.29/0.57      ! [A: int,N: nat,M: nat] :
% 0.29/0.57        ( ( 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 ) )
% 0.29/0.57        = ( 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 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_exp_mod_exp_eq
% 0.29/0.57  thf(fact_5658_div__exp__mod__exp__eq,axiom,
% 0.29/0.57      ! [A: code_integer,N: nat,M: nat] :
% 0.29/0.57        ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.57        = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_exp_mod_exp_eq
% 0.29/0.57  thf(fact_5659_le__divide__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [W: num,B: real,C: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.29/0.57        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.29/0.57          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.29/0.57              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % le_divide_eq_numeral(2)
% 0.29/0.57  thf(fact_5660_le__divide__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [W: num,B: rat,C: rat] :
% 0.29/0.57        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.29/0.57          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.29/0.57              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % le_divide_eq_numeral(2)
% 0.29/0.57  thf(fact_5661_divide__le__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [B: real,C: real,W: num] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.57        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.29/0.57          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.57           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.29/0.57              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.57               => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_le_eq_numeral(2)
% 0.29/0.57  thf(fact_5662_divide__le__eq__numeral_I2_J,axiom,
% 0.29/0.57      ! [B: rat,C: rat,W: num] :
% 0.29/0.57        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.29/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.29/0.57          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.57           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.29/0.57              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.57               => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divide_le_eq_numeral(2)
% 0.29/0.57  thf(fact_5663_square__le__1,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.57       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.57         => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % square_le_1
% 0.29/0.57  thf(fact_5664_square__le__1,axiom,
% 0.29/0.57      ! [X: code_integer] :
% 0.29/0.57        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.29/0.57       => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 0.29/0.57         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % square_le_1
% 0.29/0.57  thf(fact_5665_square__le__1,axiom,
% 0.29/0.57      ! [X: rat] :
% 0.29/0.57        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 0.29/0.57       => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 0.29/0.57         => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % square_le_1
% 0.29/0.57  thf(fact_5666_square__le__1,axiom,
% 0.29/0.57      ! [X: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.29/0.57       => ( ( ord_less_eq_int @ X @ one_one_int )
% 0.29/0.57         => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % square_le_1
% 0.29/0.57  thf(fact_5667_verit__le__mono__div,axiom,
% 0.29/0.57      ! [A2: nat,B2: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ A2 @ B2 )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57         => ( ord_less_eq_nat
% 0.29/0.57            @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
% 0.29/0.57              @ ( if_nat
% 0.29/0.57                @ ( ( modulo_modulo_nat @ B2 @ N )
% 0.29/0.57                  = zero_zero_nat )
% 0.29/0.57                @ one_one_nat
% 0.29/0.57                @ zero_zero_nat ) )
% 0.29/0.57            @ ( divide_divide_nat @ B2 @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % verit_le_mono_div
% 0.29/0.57  thf(fact_5668_minus__power__mult__self,axiom,
% 0.29/0.57      ! [A: real,N: nat] :
% 0.29/0.57        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.29/0.57        = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_power_mult_self
% 0.29/0.57  thf(fact_5669_minus__power__mult__self,axiom,
% 0.29/0.57      ! [A: int,N: nat] :
% 0.29/0.57        ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.29/0.57        = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_power_mult_self
% 0.29/0.57  thf(fact_5670_minus__power__mult__self,axiom,
% 0.29/0.57      ! [A: complex,N: nat] :
% 0.29/0.57        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 0.29/0.57        = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_power_mult_self
% 0.29/0.57  thf(fact_5671_minus__power__mult__self,axiom,
% 0.29/0.57      ! [A: rat,N: nat] :
% 0.29/0.57        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.29/0.57        = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_power_mult_self
% 0.29/0.57  thf(fact_5672_minus__power__mult__self,axiom,
% 0.29/0.57      ! [A: code_integer,N: nat] :
% 0.29/0.57        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.29/0.57        = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_power_mult_self
% 0.29/0.57  thf(fact_5673_ln__one__minus__pos__upper__bound,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.57       => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.57         => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ln_one_minus_pos_upper_bound
% 0.29/0.57  thf(fact_5674_Bernoulli__inequality,axiom,
% 0.29/0.57      ! [X: real,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.57       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Bernoulli_inequality
% 0.29/0.57  thf(fact_5675_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.57       => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.57         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_ln_one_plus_x_minus_x_bound_nonpos
% 0.29/0.57  thf(fact_5676_divmod__digit__0_I1_J,axiom,
% 0.29/0.57      ! [B: nat,A: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.57       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.29/0.57         => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.29/0.57            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_0(1)
% 0.29/0.57  thf(fact_5677_divmod__digit__0_I1_J,axiom,
% 0.29/0.57      ! [B: int,A: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.57       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.29/0.57         => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.29/0.57            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_0(1)
% 0.29/0.57  thf(fact_5678_divmod__digit__0_I1_J,axiom,
% 0.29/0.57      ! [B: code_integer,A: code_integer] :
% 0.29/0.57        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.29/0.57       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.29/0.57         => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.29/0.57            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_0(1)
% 0.29/0.57  thf(fact_5679_mult__exp__mod__exp__eq,axiom,
% 0.29/0.57      ! [M: nat,N: nat,A: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57       => ( ( 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 ) )
% 0.29/0.57          = ( 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 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_exp_mod_exp_eq
% 0.29/0.57  thf(fact_5680_mult__exp__mod__exp__eq,axiom,
% 0.29/0.57      ! [M: nat,N: nat,A: int] :
% 0.29/0.57        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57       => ( ( 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 ) )
% 0.29/0.57          = ( 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 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_exp_mod_exp_eq
% 0.29/0.57  thf(fact_5681_mult__exp__mod__exp__eq,axiom,
% 0.29/0.57      ! [M: nat,N: nat,A: code_integer] :
% 0.29/0.57        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57          = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mult_exp_mod_exp_eq
% 0.29/0.57  thf(fact_5682_power__minus1__odd,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.57        = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus1_odd
% 0.29/0.57  thf(fact_5683_power__minus1__odd,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.57        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus1_odd
% 0.29/0.57  thf(fact_5684_power__minus1__odd,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.57        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus1_odd
% 0.29/0.57  thf(fact_5685_power__minus1__odd,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.57        = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus1_odd
% 0.29/0.57  thf(fact_5686_power__minus1__odd,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.57        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus1_odd
% 0.29/0.57  thf(fact_5687_mod__double__modulus,axiom,
% 0.29/0.57      ! [M: code_integer,X: code_integer] :
% 0.29/0.57        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.29/0.57       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.29/0.57         => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.57              = ( modulo364778990260209775nteger @ X @ M ) )
% 0.29/0.57            | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.57              = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_double_modulus
% 0.29/0.57  thf(fact_5688_mod__double__modulus,axiom,
% 0.29/0.57      ! [M: nat,X: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.57       => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.29/0.57         => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.57              = ( modulo_modulo_nat @ X @ M ) )
% 0.29/0.57            | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.57              = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_double_modulus
% 0.29/0.57  thf(fact_5689_mod__double__modulus,axiom,
% 0.29/0.57      ! [M: int,X: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ M )
% 0.29/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.57         => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.57              = ( modulo_modulo_int @ X @ M ) )
% 0.29/0.57            | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.57              = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_double_modulus
% 0.29/0.57  thf(fact_5690_divmod__digit__1_I2_J,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.57       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.29/0.57         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.29/0.57           => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.29/0.57              = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_1(2)
% 0.29/0.57  thf(fact_5691_divmod__digit__1_I2_J,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.57         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.29/0.57           => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.29/0.57              = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_1(2)
% 0.29/0.57  thf(fact_5692_divmod__digit__1_I2_J,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.57       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.57         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.29/0.57           => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.29/0.57              = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_1(2)
% 0.29/0.57  thf(fact_5693_log__minus__eq__powr,axiom,
% 0.29/0.57      ! [B: real,X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.57       => ( ( B != one_one_real )
% 0.29/0.57         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.57           => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
% 0.29/0.57              = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % log_minus_eq_powr
% 0.29/0.57  thf(fact_5694_unset__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: code_integer] :
% 0.29/0.57        ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unset_bit_Suc
% 0.29/0.57  thf(fact_5695_unset__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: int] :
% 0.29/0.57        ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unset_bit_Suc
% 0.29/0.57  thf(fact_5696_unset__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: nat] :
% 0.29/0.57        ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unset_bit_Suc
% 0.29/0.57  thf(fact_5697_set__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: code_integer] :
% 0.29/0.57        ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % set_bit_Suc
% 0.29/0.57  thf(fact_5698_set__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: int] :
% 0.29/0.57        ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % set_bit_Suc
% 0.29/0.57  thf(fact_5699_set__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: nat] :
% 0.29/0.57        ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % set_bit_Suc
% 0.29/0.57  thf(fact_5700_powr__neg__numeral,axiom,
% 0.29/0.57      ! [X: real,N: num] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.57       => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.57          = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % powr_neg_numeral
% 0.29/0.57  thf(fact_5701_of__int__round__abs__le,axiom,
% 0.29/0.57      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % of_int_round_abs_le
% 0.29/0.57  thf(fact_5702_of__int__round__abs__le,axiom,
% 0.29/0.57      ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % of_int_round_abs_le
% 0.29/0.57  thf(fact_5703_round__unique_H,axiom,
% 0.29/0.57      ! [X: real,N: int] :
% 0.29/0.57        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.57       => ( ( archim8280529875227126926d_real @ X )
% 0.29/0.57          = N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % round_unique'
% 0.29/0.57  thf(fact_5704_round__unique_H,axiom,
% 0.29/0.57      ! [X: rat,N: int] :
% 0.29/0.57        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.29/0.57       => ( ( archim7778729529865785530nd_rat @ X )
% 0.29/0.57          = N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % round_unique'
% 0.29/0.57  thf(fact_5705_divmod__digit__1_I1_J,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.57       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.29/0.57         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.29/0.57           => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 0.29/0.57              = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_1(1)
% 0.29/0.57  thf(fact_5706_divmod__digit__1_I1_J,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.57         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.29/0.57           => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
% 0.29/0.57              = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_1(1)
% 0.29/0.57  thf(fact_5707_divmod__digit__1_I1_J,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.57       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.57         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.29/0.57           => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
% 0.29/0.57              = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % divmod_digit_1(1)
% 0.29/0.57  thf(fact_5708_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.57       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.57         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_ln_one_plus_x_minus_x_bound_nonneg
% 0.29/0.57  thf(fact_5709_VEBT__internal_Oheight_Oelims,axiom,
% 0.29/0.57      ! [X: vEBT_VEBT,Y: nat] :
% 0.29/0.57        ( ( ( vEBT_VEBT_height @ X )
% 0.29/0.57          = Y )
% 0.29/0.57       => ( ( ? [A5: $o,B6: $o] :
% 0.29/0.57                ( X
% 0.29/0.57                = ( vEBT_Leaf @ A5 @ B6 ) )
% 0.29/0.57           => ( Y != zero_zero_nat ) )
% 0.29/0.57         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.29/0.57                ( ( X
% 0.29/0.57                  = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.57               => ( Y
% 0.29/0.57                 != ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % VEBT_internal.height.elims
% 0.29/0.57  thf(fact_5710_arsinh__def,axiom,
% 0.29/0.57      ( arsinh_real
% 0.29/0.57      = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arsinh_def
% 0.29/0.57  thf(fact_5711_abs__sqrt__wlog,axiom,
% 0.29/0.57      ! [P: code_integer > code_integer > $o,X: code_integer] :
% 0.29/0.57        ( ! [X3: code_integer] :
% 0.29/0.57            ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 0.29/0.57           => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.57       => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_sqrt_wlog
% 0.29/0.57  thf(fact_5712_abs__sqrt__wlog,axiom,
% 0.29/0.57      ! [P: real > real > $o,X: real] :
% 0.29/0.57        ( ! [X3: real] :
% 0.29/0.57            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.57           => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.57       => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_sqrt_wlog
% 0.29/0.57  thf(fact_5713_abs__sqrt__wlog,axiom,
% 0.29/0.57      ! [P: rat > rat > $o,X: rat] :
% 0.29/0.57        ( ! [X3: rat] :
% 0.29/0.57            ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.57           => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.57       => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_sqrt_wlog
% 0.29/0.57  thf(fact_5714_abs__sqrt__wlog,axiom,
% 0.29/0.57      ! [P: int > int > $o,X: int] :
% 0.29/0.57        ( ! [X3: int] :
% 0.29/0.57            ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.29/0.57           => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.57       => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_sqrt_wlog
% 0.29/0.57  thf(fact_5715_compl__le__compl__iff,axiom,
% 0.29/0.57      ! [X: set_int,Y: set_int] :
% 0.29/0.57        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
% 0.29/0.57        = ( ord_less_eq_set_int @ Y @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % compl_le_compl_iff
% 0.29/0.57  thf(fact_5716_flip__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: code_integer] :
% 0.29/0.57        ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % flip_bit_Suc
% 0.29/0.57  thf(fact_5717_flip__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: int] :
% 0.29/0.57        ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % flip_bit_Suc
% 0.29/0.57  thf(fact_5718_flip__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: nat] :
% 0.29/0.57        ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % flip_bit_Suc
% 0.29/0.57  thf(fact_5719_signed__take__bit__rec,axiom,
% 0.29/0.57      ( bit_ri6519982836138164636nteger
% 0.29/0.57      = ( ^ [N3: nat,A3: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_rec
% 0.29/0.57  thf(fact_5720_signed__take__bit__rec,axiom,
% 0.29/0.57      ( bit_ri631733984087533419it_int
% 0.29/0.57      = ( ^ [N3: nat,A3: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_rec
% 0.29/0.57  thf(fact_5721_round__altdef,axiom,
% 0.29/0.57      ( archim8280529875227126926d_real
% 0.29/0.57      = ( ^ [X4: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X4 ) ) @ ( archim7802044766580827645g_real @ X4 ) @ ( archim6058952711729229775r_real @ X4 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % round_altdef
% 0.29/0.57  thf(fact_5722_round__altdef,axiom,
% 0.29/0.57      ( archim7778729529865785530nd_rat
% 0.29/0.57      = ( ^ [X4: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X4 ) ) @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim3151403230148437115or_rat @ X4 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % round_altdef
% 0.29/0.57  thf(fact_5723_arctan__double,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.57       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 0.29/0.57          = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_double
% 0.29/0.57  thf(fact_5724_pochhammer__double,axiom,
% 0.29/0.57      ! [Z: complex,N: nat] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57        = ( 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 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_double
% 0.29/0.57  thf(fact_5725_pochhammer__double,axiom,
% 0.29/0.57      ! [Z: real,N: nat] :
% 0.29/0.57        ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57        = ( 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 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_double
% 0.29/0.57  thf(fact_5726_pochhammer__double,axiom,
% 0.29/0.57      ! [Z: rat,N: nat] :
% 0.29/0.57        ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57        = ( 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 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_double
% 0.29/0.57  thf(fact_5727_Compl__subset__Compl__iff,axiom,
% 0.29/0.57      ! [A2: set_int,B2: set_int] :
% 0.29/0.57        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B2 ) )
% 0.29/0.57        = ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Compl_subset_Compl_iff
% 0.29/0.57  thf(fact_5728_Compl__anti__mono,axiom,
% 0.29/0.57      ! [A2: set_int,B2: set_int] :
% 0.29/0.57        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.57       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Compl_anti_mono
% 0.29/0.57  thf(fact_5729_flip__bit__nonnegative__int__iff,axiom,
% 0.29/0.57      ! [N: nat,K: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 0.29/0.57        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.29/0.57  
% 0.29/0.57  % flip_bit_nonnegative_int_iff
% 0.29/0.57  thf(fact_5730_flip__bit__negative__int__iff,axiom,
% 0.29/0.57      ! [N: nat,K: int] :
% 0.29/0.57        ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 0.29/0.57        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % flip_bit_negative_int_iff
% 0.29/0.57  thf(fact_5731_signed__take__bit__of__0,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 0.29/0.57        = zero_zero_int ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_of_0
% 0.29/0.57  thf(fact_5732_arctan__eq__zero__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ( arctan @ X )
% 0.29/0.57          = zero_zero_real )
% 0.29/0.57        = ( X = zero_zero_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_eq_zero_iff
% 0.29/0.57  thf(fact_5733_arctan__zero__zero,axiom,
% 0.29/0.57      ( ( arctan @ zero_zero_real )
% 0.29/0.57      = zero_zero_real ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_zero_zero
% 0.29/0.57  thf(fact_5734_mod__pos__pos__trivial,axiom,
% 0.29/0.57      ! [K: int,L: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.57       => ( ( ord_less_int @ K @ L )
% 0.29/0.57         => ( ( modulo_modulo_int @ K @ L )
% 0.29/0.57            = K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_pos_pos_trivial
% 0.29/0.57  thf(fact_5735_mod__neg__neg__trivial,axiom,
% 0.29/0.57      ! [K: int,L: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.29/0.57       => ( ( ord_less_int @ L @ K )
% 0.29/0.57         => ( ( modulo_modulo_int @ K @ L )
% 0.29/0.57            = K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_neg_neg_trivial
% 0.29/0.57  thf(fact_5736_negative__eq__positive,axiom,
% 0.29/0.57      ! [N: nat,M: nat] :
% 0.29/0.57        ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.29/0.57          = ( semiri1314217659103216013at_int @ M ) )
% 0.29/0.57        = ( ( N = zero_zero_nat )
% 0.29/0.57          & ( M = zero_zero_nat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % negative_eq_positive
% 0.29/0.57  thf(fact_5737_signed__take__bit__of__minus__1,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.57        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_of_minus_1
% 0.29/0.57  thf(fact_5738_signed__take__bit__of__minus__1,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.57        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_of_minus_1
% 0.29/0.57  thf(fact_5739_signed__take__bit__Suc__1,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 0.29/0.57        = one_one_int ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_Suc_1
% 0.29/0.57  thf(fact_5740_signed__take__bit__numeral__of__1,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 0.29/0.57        = one_one_int ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_numeral_of_1
% 0.29/0.57  thf(fact_5741_pochhammer__0,axiom,
% 0.29/0.57      ! [A: complex] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 0.29/0.57        = one_one_complex ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_0
% 0.29/0.57  thf(fact_5742_pochhammer__0,axiom,
% 0.29/0.57      ! [A: real] :
% 0.29/0.57        ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 0.29/0.57        = one_one_real ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_0
% 0.29/0.57  thf(fact_5743_pochhammer__0,axiom,
% 0.29/0.57      ! [A: rat] :
% 0.29/0.57        ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 0.29/0.57        = one_one_rat ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_0
% 0.29/0.57  thf(fact_5744_pochhammer__0,axiom,
% 0.29/0.57      ! [A: nat] :
% 0.29/0.57        ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 0.29/0.57        = one_one_nat ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_0
% 0.29/0.57  thf(fact_5745_pochhammer__0,axiom,
% 0.29/0.57      ! [A: int] :
% 0.29/0.57        ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 0.29/0.57        = one_one_int ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_0
% 0.29/0.57  thf(fact_5746_negative__zle,axiom,
% 0.29/0.57      ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.29/0.57  
% 0.29/0.57  % negative_zle
% 0.29/0.57  thf(fact_5747_zero__less__arctan__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
% 0.29/0.57        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zero_less_arctan_iff
% 0.29/0.57  thf(fact_5748_arctan__less__zero__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
% 0.29/0.57        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_less_zero_iff
% 0.29/0.57  thf(fact_5749_zero__le__arctan__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
% 0.29/0.57        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zero_le_arctan_iff
% 0.29/0.57  thf(fact_5750_arctan__le__zero__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
% 0.29/0.57        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_le_zero_iff
% 0.29/0.57  thf(fact_5751_frac__of__int,axiom,
% 0.29/0.57      ! [Z: int] :
% 0.29/0.57        ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
% 0.29/0.57        = zero_zero_real ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_of_int
% 0.29/0.57  thf(fact_5752_frac__of__int,axiom,
% 0.29/0.57      ! [Z: int] :
% 0.29/0.57        ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.29/0.57        = zero_zero_rat ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_of_int
% 0.29/0.57  thf(fact_5753_zmod__numeral__Bit0,axiom,
% 0.29/0.57      ! [V: num,W: num] :
% 0.29/0.57        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.29/0.57        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_numeral_Bit0
% 0.29/0.57  thf(fact_5754_negative__zless,axiom,
% 0.29/0.57      ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.29/0.57  
% 0.29/0.57  % negative_zless
% 0.29/0.57  thf(fact_5755_zabs__less__one__iff,axiom,
% 0.29/0.57      ! [Z: int] :
% 0.29/0.57        ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 0.29/0.57        = ( Z = zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zabs_less_one_iff
% 0.29/0.57  thf(fact_5756_signed__take__bit__Suc__bit0,axiom,
% 0.29/0.57      ! [N: nat,K: num] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.29/0.57        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_Suc_bit0
% 0.29/0.57  thf(fact_5757_signed__take__bit__Suc__minus__bit0,axiom,
% 0.29/0.57      ! [N: nat,K: num] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.29/0.57        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_Suc_minus_bit0
% 0.29/0.57  thf(fact_5758_ceiling__divide__eq__div__numeral,axiom,
% 0.29/0.57      ! [A: num,B: num] :
% 0.29/0.57        ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.29/0.57        = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ceiling_divide_eq_div_numeral
% 0.29/0.57  thf(fact_5759_signed__take__bit__0,axiom,
% 0.29/0.57      ! [A: code_integer] :
% 0.29/0.57        ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 0.29/0.57        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_0
% 0.29/0.57  thf(fact_5760_signed__take__bit__0,axiom,
% 0.29/0.57      ! [A: int] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 0.29/0.57        = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_0
% 0.29/0.57  thf(fact_5761_floor__minus__divide__eq__div__numeral,axiom,
% 0.29/0.57      ! [A: num,B: num] :
% 0.29/0.57        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.29/0.57        = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % floor_minus_divide_eq_div_numeral
% 0.29/0.57  thf(fact_5762_ceiling__minus__divide__eq__div__numeral,axiom,
% 0.29/0.57      ! [A: num,B: num] :
% 0.29/0.57        ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.29/0.57        = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ceiling_minus_divide_eq_div_numeral
% 0.29/0.57  thf(fact_5763_floor__minus__one__divide__eq__div__numeral,axiom,
% 0.29/0.57      ! [B: num] :
% 0.29/0.57        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 0.29/0.57        = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % floor_minus_one_divide_eq_div_numeral
% 0.29/0.57  thf(fact_5764_zmod__zminus1__not__zero,axiom,
% 0.29/0.57      ! [K: int,L: int] :
% 0.29/0.57        ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 0.29/0.57         != zero_zero_int )
% 0.29/0.57       => ( ( modulo_modulo_int @ K @ L )
% 0.29/0.57         != zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_zminus1_not_zero
% 0.29/0.57  thf(fact_5765_zmod__zminus2__not__zero,axiom,
% 0.29/0.57      ! [K: int,L: int] :
% 0.29/0.57        ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
% 0.29/0.57         != zero_zero_int )
% 0.29/0.57       => ( ( modulo_modulo_int @ K @ L )
% 0.29/0.57         != zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_zminus2_not_zero
% 0.29/0.57  thf(fact_5766_pochhammer__of__real,axiom,
% 0.29/0.57      ! [X: real,N: nat] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ ( real_V4546457046886955230omplex @ X ) @ N )
% 0.29/0.57        = ( real_V4546457046886955230omplex @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_real
% 0.29/0.57  thf(fact_5767_arctan__eq__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ( arctan @ X )
% 0.29/0.57          = ( arctan @ Y ) )
% 0.29/0.57        = ( X = Y ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_eq_iff
% 0.29/0.57  thf(fact_5768_pochhammer__of__nat,axiom,
% 0.29/0.57      ! [X: nat,N: nat] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ ( semiri8010041392384452111omplex @ X ) @ N )
% 0.29/0.57        = ( semiri8010041392384452111omplex @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat
% 0.29/0.57  thf(fact_5769_pochhammer__of__nat,axiom,
% 0.29/0.57      ! [X: nat,N: nat] :
% 0.29/0.57        ( ( comm_s7457072308508201937r_real @ ( semiri5074537144036343181t_real @ X ) @ N )
% 0.29/0.57        = ( semiri5074537144036343181t_real @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat
% 0.29/0.57  thf(fact_5770_pochhammer__of__nat,axiom,
% 0.29/0.57      ! [X: nat,N: nat] :
% 0.29/0.57        ( ( comm_s4028243227959126397er_rat @ ( semiri681578069525770553at_rat @ X ) @ N )
% 0.29/0.57        = ( semiri681578069525770553at_rat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat
% 0.29/0.57  thf(fact_5771_pochhammer__of__nat,axiom,
% 0.29/0.57      ! [X: nat,N: nat] :
% 0.29/0.57        ( ( comm_s4663373288045622133er_nat @ ( semiri1316708129612266289at_nat @ X ) @ N )
% 0.29/0.57        = ( semiri1316708129612266289at_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat
% 0.29/0.57  thf(fact_5772_pochhammer__of__nat,axiom,
% 0.29/0.57      ! [X: nat,N: nat] :
% 0.29/0.57        ( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X ) @ N )
% 0.29/0.57        = ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat
% 0.29/0.57  thf(fact_5773_pochhammer__of__int,axiom,
% 0.29/0.57      ! [X: int,N: nat] :
% 0.29/0.57        ( ( comm_s7457072308508201937r_real @ ( ring_1_of_int_real @ X ) @ N )
% 0.29/0.57        = ( ring_1_of_int_real @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_int
% 0.29/0.57  thf(fact_5774_pochhammer__of__int,axiom,
% 0.29/0.57      ! [X: int,N: nat] :
% 0.29/0.57        ( ( comm_s4028243227959126397er_rat @ ( ring_1_of_int_rat @ X ) @ N )
% 0.29/0.57        = ( ring_1_of_int_rat @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_int
% 0.29/0.57  thf(fact_5775_pochhammer__of__int,axiom,
% 0.29/0.57      ! [X: int,N: nat] :
% 0.29/0.57        ( ( comm_s8582702949713902594nteger @ ( ring_18347121197199848620nteger @ X ) @ N )
% 0.29/0.57        = ( ring_18347121197199848620nteger @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_int
% 0.29/0.57  thf(fact_5776_pochhammer__of__int,axiom,
% 0.29/0.57      ! [X: int,N: nat] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ ( ring_17405671764205052669omplex @ X ) @ N )
% 0.29/0.57        = ( ring_17405671764205052669omplex @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_int
% 0.29/0.57  thf(fact_5777_abs__mod__less,axiom,
% 0.29/0.57      ! [L: int,K: int] :
% 0.29/0.57        ( ( L != zero_zero_int )
% 0.29/0.57       => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_mod_less
% 0.29/0.57  thf(fact_5778_zabs__def,axiom,
% 0.29/0.57      ( abs_abs_int
% 0.29/0.57      = ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zabs_def
% 0.29/0.57  thf(fact_5779_zmod__zminus1__eq__if,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.57            = zero_zero_int )
% 0.29/0.57         => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.29/0.57            = zero_zero_int ) )
% 0.29/0.57        & ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.57           != zero_zero_int )
% 0.29/0.57         => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.29/0.57            = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_zminus1_eq_if
% 0.29/0.57  thf(fact_5780_zmod__zminus2__eq__if,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.57            = zero_zero_int )
% 0.29/0.57         => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.29/0.57            = zero_zero_int ) )
% 0.29/0.57        & ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.57           != zero_zero_int )
% 0.29/0.57         => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.29/0.57            = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_zminus2_eq_if
% 0.29/0.57  thf(fact_5781_signed__take__bit__add,axiom,
% 0.29/0.57      ! [N: nat,K: int,L: int] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.29/0.57        = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_add
% 0.29/0.57  thf(fact_5782_uminus__int__code_I1_J,axiom,
% 0.29/0.57      ( ( uminus_uminus_int @ zero_zero_int )
% 0.29/0.57      = zero_zero_int ) ).
% 0.29/0.57  
% 0.29/0.57  % uminus_int_code(1)
% 0.29/0.57  thf(fact_5783_arctan__monotone,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_real @ X @ Y )
% 0.29/0.57       => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_monotone
% 0.29/0.57  thf(fact_5784_arctan__less__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 0.29/0.57        = ( ord_less_real @ X @ Y ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_less_iff
% 0.29/0.57  thf(fact_5785_arctan__le__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 0.29/0.57        = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_le_iff
% 0.29/0.57  thf(fact_5786_arctan__monotone_H,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.57       => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_monotone'
% 0.29/0.57  thf(fact_5787_int__cases2,axiom,
% 0.29/0.57      ! [Z: int] :
% 0.29/0.57        ( ! [N2: nat] :
% 0.29/0.57            ( Z
% 0.29/0.57           != ( semiri1314217659103216013at_int @ N2 ) )
% 0.29/0.57       => ~ ! [N2: nat] :
% 0.29/0.57              ( Z
% 0.29/0.57             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % int_cases2
% 0.29/0.57  thf(fact_5788_arctan__minus,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( arctan @ ( uminus_uminus_real @ X ) )
% 0.29/0.57        = ( uminus_uminus_real @ ( arctan @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_minus
% 0.29/0.57  thf(fact_5789_minus__mod__int__eq,axiom,
% 0.29/0.57      ! [L: int,K: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.29/0.57       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 0.29/0.57          = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_mod_int_eq
% 0.29/0.57  thf(fact_5790_zmod__minus1,axiom,
% 0.29/0.57      ! [B: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.57       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.29/0.57          = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_minus1
% 0.29/0.57  thf(fact_5791_zdiv__zminus2__eq__if,axiom,
% 0.29/0.57      ! [B: int,A: int] :
% 0.29/0.57        ( ( B != zero_zero_int )
% 0.29/0.57       => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.57              = zero_zero_int )
% 0.29/0.57           => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.29/0.57              = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.29/0.57          & ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.57             != zero_zero_int )
% 0.29/0.57           => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.29/0.57              = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zdiv_zminus2_eq_if
% 0.29/0.57  thf(fact_5792_zdiv__zminus1__eq__if,axiom,
% 0.29/0.57      ! [B: int,A: int] :
% 0.29/0.57        ( ( B != zero_zero_int )
% 0.29/0.57       => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.57              = zero_zero_int )
% 0.29/0.57           => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.29/0.57              = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.29/0.57          & ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.57             != zero_zero_int )
% 0.29/0.57           => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.29/0.57              = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zdiv_zminus1_eq_if
% 0.29/0.57  thf(fact_5793_zmod__le__nonneg__dividend,axiom,
% 0.29/0.57      ! [M: int,K: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.29/0.57       => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_le_nonneg_dividend
% 0.29/0.57  thf(fact_5794_Euclidean__Division_Opos__mod__bound,axiom,
% 0.29/0.57      ! [L: int,K: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ L )
% 0.29/0.57       => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Euclidean_Division.pos_mod_bound
% 0.29/0.57  thf(fact_5795_neg__mod__bound,axiom,
% 0.29/0.57      ! [L: int,K: int] :
% 0.29/0.57        ( ( ord_less_int @ L @ zero_zero_int )
% 0.29/0.57       => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_mod_bound
% 0.29/0.57  thf(fact_5796_zmod__eq__0D,axiom,
% 0.29/0.57      ! [M: int,D: int] :
% 0.29/0.57        ( ( ( modulo_modulo_int @ M @ D )
% 0.29/0.57          = zero_zero_int )
% 0.29/0.57       => ? [Q2: int] :
% 0.29/0.57            ( M
% 0.29/0.57            = ( times_times_int @ D @ Q2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_eq_0D
% 0.29/0.57  thf(fact_5797_zmod__eq__0__iff,axiom,
% 0.29/0.57      ! [M: int,D: int] :
% 0.29/0.57        ( ( ( modulo_modulo_int @ M @ D )
% 0.29/0.57          = zero_zero_int )
% 0.29/0.57        = ( ? [Q4: int] :
% 0.29/0.57              ( M
% 0.29/0.57              = ( times_times_int @ D @ Q4 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_eq_0_iff
% 0.29/0.57  thf(fact_5798_pochhammer__pos,axiom,
% 0.29/0.57      ! [X: real,N: nat] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.57       => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_pos
% 0.29/0.57  thf(fact_5799_pochhammer__pos,axiom,
% 0.29/0.57      ! [X: rat,N: nat] :
% 0.29/0.57        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.29/0.57       => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_pos
% 0.29/0.57  thf(fact_5800_pochhammer__pos,axiom,
% 0.29/0.57      ! [X: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.29/0.57       => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_pos
% 0.29/0.57  thf(fact_5801_pochhammer__pos,axiom,
% 0.29/0.57      ! [X: int,N: nat] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ X )
% 0.29/0.57       => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_pos
% 0.29/0.57  thf(fact_5802_abs__zmult__eq__1,axiom,
% 0.29/0.57      ! [M: int,N: int] :
% 0.29/0.57        ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 0.29/0.57          = one_one_int )
% 0.29/0.57       => ( ( abs_abs_int @ M )
% 0.29/0.57          = one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_zmult_eq_1
% 0.29/0.57  thf(fact_5803_pochhammer__eq__0__mono,axiom,
% 0.29/0.57      ! [A: complex,N: nat,M: nat] :
% 0.29/0.57        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.29/0.57          = zero_zero_complex )
% 0.29/0.57       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.57         => ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.29/0.57            = zero_zero_complex ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_eq_0_mono
% 0.29/0.57  thf(fact_5804_pochhammer__eq__0__mono,axiom,
% 0.29/0.57      ! [A: real,N: nat,M: nat] :
% 0.29/0.57        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.29/0.57          = zero_zero_real )
% 0.29/0.57       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.57         => ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.29/0.57            = zero_zero_real ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_eq_0_mono
% 0.29/0.57  thf(fact_5805_pochhammer__eq__0__mono,axiom,
% 0.29/0.57      ! [A: rat,N: nat,M: nat] :
% 0.29/0.57        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.29/0.57          = zero_zero_rat )
% 0.29/0.57       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.57         => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.29/0.57            = zero_zero_rat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_eq_0_mono
% 0.29/0.57  thf(fact_5806_pochhammer__neq__0__mono,axiom,
% 0.29/0.57      ! [A: complex,M: nat,N: nat] :
% 0.29/0.57        ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.29/0.57         != zero_zero_complex )
% 0.29/0.57       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.57         => ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.29/0.57           != zero_zero_complex ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_neq_0_mono
% 0.29/0.57  thf(fact_5807_pochhammer__neq__0__mono,axiom,
% 0.29/0.57      ! [A: real,M: nat,N: nat] :
% 0.29/0.57        ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.29/0.57         != zero_zero_real )
% 0.29/0.57       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.57         => ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.29/0.57           != zero_zero_real ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_neq_0_mono
% 0.29/0.57  thf(fact_5808_pochhammer__neq__0__mono,axiom,
% 0.29/0.57      ! [A: rat,M: nat,N: nat] :
% 0.29/0.57        ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.29/0.57         != zero_zero_rat )
% 0.29/0.57       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.57         => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.29/0.57           != zero_zero_rat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_neq_0_mono
% 0.29/0.57  thf(fact_5809_zmod__int,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
% 0.29/0.57        = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_int
% 0.29/0.57  thf(fact_5810_int__of__nat__induct,axiom,
% 0.29/0.57      ! [P: int > $o,Z: int] :
% 0.29/0.57        ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.29/0.57       => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
% 0.29/0.57         => ( P @ Z ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % int_of_nat_induct
% 0.29/0.57  thf(fact_5811_int__cases,axiom,
% 0.29/0.57      ! [Z: int] :
% 0.29/0.57        ( ! [N2: nat] :
% 0.29/0.57            ( Z
% 0.29/0.57           != ( semiri1314217659103216013at_int @ N2 ) )
% 0.29/0.57       => ~ ! [N2: nat] :
% 0.29/0.57              ( Z
% 0.29/0.57             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % int_cases
% 0.29/0.57  thf(fact_5812_pos__zmult__eq__1__iff__lemma,axiom,
% 0.29/0.57      ! [M: int,N: int] :
% 0.29/0.57        ( ( ( times_times_int @ M @ N )
% 0.29/0.57          = one_one_int )
% 0.29/0.57       => ( ( M = one_one_int )
% 0.29/0.57          | ( M
% 0.29/0.57            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_zmult_eq_1_iff_lemma
% 0.29/0.57  thf(fact_5813_zmult__eq__1__iff,axiom,
% 0.29/0.57      ! [M: int,N: int] :
% 0.29/0.57        ( ( ( times_times_int @ M @ N )
% 0.29/0.57          = one_one_int )
% 0.29/0.57        = ( ( ( M = one_one_int )
% 0.29/0.57            & ( N = one_one_int ) )
% 0.29/0.57          | ( ( M
% 0.29/0.57              = ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.57            & ( N
% 0.29/0.57              = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmult_eq_1_iff
% 0.29/0.57  thf(fact_5814_minus__int__code_I2_J,axiom,
% 0.29/0.57      ! [L: int] :
% 0.29/0.57        ( ( minus_minus_int @ zero_zero_int @ L )
% 0.29/0.57        = ( uminus_uminus_int @ L ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_int_code(2)
% 0.29/0.57  thf(fact_5815_not__int__zless__negative,axiom,
% 0.29/0.57      ! [N: nat,M: nat] :
% 0.29/0.57        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % not_int_zless_negative
% 0.29/0.57  thf(fact_5816_signed__take__bit__int__less__eq__self__iff,axiom,
% 0.29/0.57      ! [N: nat,K: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.29/0.57        = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_int_less_eq_self_iff
% 0.29/0.57  thf(fact_5817_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 0.29/0.57      ! [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 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_int_greater_eq_minus_exp
% 0.29/0.57  thf(fact_5818_signed__take__bit__int__greater__self__iff,axiom,
% 0.29/0.57      ! [K: int,N: nat] :
% 0.29/0.57        ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.29/0.57        = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_int_greater_self_iff
% 0.29/0.57  thf(fact_5819_frac__ge__0,axiom,
% 0.29/0.57      ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_ge_0
% 0.29/0.57  thf(fact_5820_frac__ge__0,axiom,
% 0.29/0.57      ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_ge_0
% 0.29/0.57  thf(fact_5821_frac__lt__1,axiom,
% 0.29/0.57      ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_lt_1
% 0.29/0.57  thf(fact_5822_frac__lt__1,axiom,
% 0.29/0.57      ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_lt_1
% 0.29/0.57  thf(fact_5823_frac__1__eq,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
% 0.29/0.57        = ( archim2898591450579166408c_real @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_1_eq
% 0.29/0.57  thf(fact_5824_frac__1__eq,axiom,
% 0.29/0.57      ! [X: rat] :
% 0.29/0.57        ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
% 0.29/0.57        = ( archimedean_frac_rat @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_1_eq
% 0.29/0.57  thf(fact_5825_signed__take__bit__int__eq__self,axiom,
% 0.29/0.57      ! [N: nat,K: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.29/0.57       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57         => ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.29/0.57            = K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_int_eq_self
% 0.29/0.57  thf(fact_5826_signed__take__bit__int__eq__self__iff,axiom,
% 0.29/0.57      ! [N: nat,K: int] :
% 0.29/0.57        ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.29/0.57          = K )
% 0.29/0.57        = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.29/0.57          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_int_eq_self_iff
% 0.29/0.57  thf(fact_5827_zmod__trivial__iff,axiom,
% 0.29/0.57      ! [I: int,K: int] :
% 0.29/0.57        ( ( ( modulo_modulo_int @ I @ K )
% 0.29/0.57          = I )
% 0.29/0.57        = ( ( K = zero_zero_int )
% 0.29/0.57          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.29/0.57            & ( ord_less_int @ I @ K ) )
% 0.29/0.57          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.29/0.57            & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_trivial_iff
% 0.29/0.57  thf(fact_5828_pos__mod__conj,axiom,
% 0.29/0.57      ! [B: int,A: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 0.29/0.57          & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_mod_conj
% 0.29/0.57  thf(fact_5829_neg__mod__conj,axiom,
% 0.29/0.57      ! [B: int,A: int] :
% 0.29/0.57        ( ( ord_less_int @ B @ zero_zero_int )
% 0.29/0.57       => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 0.29/0.57          & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_mod_conj
% 0.29/0.57  thf(fact_5830_Euclidean__Division_Opos__mod__sign,axiom,
% 0.29/0.57      ! [L: int,K: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ L )
% 0.29/0.57       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Euclidean_Division.pos_mod_sign
% 0.29/0.57  thf(fact_5831_neg__mod__sign,axiom,
% 0.29/0.57      ! [L: int,K: int] :
% 0.29/0.57        ( ( ord_less_int @ L @ zero_zero_int )
% 0.29/0.57       => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_mod_sign
% 0.29/0.57  thf(fact_5832_pochhammer__nonneg,axiom,
% 0.29/0.57      ! [X: real,N: nat] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.57       => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_nonneg
% 0.29/0.57  thf(fact_5833_pochhammer__nonneg,axiom,
% 0.29/0.57      ! [X: rat,N: nat] :
% 0.29/0.57        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.29/0.57       => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_nonneg
% 0.29/0.57  thf(fact_5834_pochhammer__nonneg,axiom,
% 0.29/0.57      ! [X: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.29/0.57       => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_nonneg
% 0.29/0.57  thf(fact_5835_pochhammer__nonneg,axiom,
% 0.29/0.57      ! [X: int,N: nat] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ X )
% 0.29/0.57       => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_nonneg
% 0.29/0.57  thf(fact_5836_zdiv__mono__strict,axiom,
% 0.29/0.57      ! [A2: int,B2: int,N: int] :
% 0.29/0.57        ( ( ord_less_int @ A2 @ B2 )
% 0.29/0.57       => ( ( ord_less_int @ zero_zero_int @ N )
% 0.29/0.57         => ( ( ( modulo_modulo_int @ A2 @ N )
% 0.29/0.57              = zero_zero_int )
% 0.29/0.57           => ( ( ( modulo_modulo_int @ B2 @ N )
% 0.29/0.57                = zero_zero_int )
% 0.29/0.57             => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B2 @ N ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zdiv_mono_strict
% 0.29/0.57  thf(fact_5837_pochhammer__0__left,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( ( N = zero_zero_nat )
% 0.29/0.57         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.29/0.57            = one_one_complex ) )
% 0.29/0.57        & ( ( N != zero_zero_nat )
% 0.29/0.57         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.29/0.57            = zero_zero_complex ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_0_left
% 0.29/0.57  thf(fact_5838_pochhammer__0__left,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( ( N = zero_zero_nat )
% 0.29/0.57         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.29/0.57            = one_one_real ) )
% 0.29/0.57        & ( ( N != zero_zero_nat )
% 0.29/0.57         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.29/0.57            = zero_zero_real ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_0_left
% 0.29/0.57  thf(fact_5839_pochhammer__0__left,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( ( N = zero_zero_nat )
% 0.29/0.57         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.29/0.57            = one_one_rat ) )
% 0.29/0.57        & ( ( N != zero_zero_nat )
% 0.29/0.57         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.29/0.57            = zero_zero_rat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_0_left
% 0.29/0.57  thf(fact_5840_pochhammer__0__left,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( ( N = zero_zero_nat )
% 0.29/0.57         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.29/0.57            = one_one_nat ) )
% 0.29/0.57        & ( ( N != zero_zero_nat )
% 0.29/0.57         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.29/0.57            = zero_zero_nat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_0_left
% 0.29/0.57  thf(fact_5841_pochhammer__0__left,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( ( N = zero_zero_nat )
% 0.29/0.57         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.29/0.57            = one_one_int ) )
% 0.29/0.57        & ( ( N != zero_zero_nat )
% 0.29/0.57         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.29/0.57            = zero_zero_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_0_left
% 0.29/0.57  thf(fact_5842_div__mod__decomp__int,axiom,
% 0.29/0.57      ! [A2: int,N: int] :
% 0.29/0.57        ( A2
% 0.29/0.57        = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_mod_decomp_int
% 0.29/0.57  thf(fact_5843_int__cases4,axiom,
% 0.29/0.57      ! [M: int] :
% 0.29/0.57        ( ! [N2: nat] :
% 0.29/0.57            ( M
% 0.29/0.57           != ( semiri1314217659103216013at_int @ N2 ) )
% 0.29/0.57       => ~ ! [N2: nat] :
% 0.29/0.57              ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.57             => ( M
% 0.29/0.57               != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % int_cases4
% 0.29/0.57  thf(fact_5844_int__zle__neg,axiom,
% 0.29/0.57      ! [N: nat,M: nat] :
% 0.29/0.57        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.29/0.57        = ( ( N = zero_zero_nat )
% 0.29/0.57          & ( M = zero_zero_nat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % int_zle_neg
% 0.29/0.57  thf(fact_5845_negative__zle__0,axiom,
% 0.29/0.57      ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 0.29/0.57  
% 0.29/0.57  % negative_zle_0
% 0.29/0.57  thf(fact_5846_nonpos__int__cases,axiom,
% 0.29/0.57      ! [K: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.29/0.57       => ~ ! [N2: nat] :
% 0.29/0.57              ( K
% 0.29/0.57             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % nonpos_int_cases
% 0.29/0.57  thf(fact_5847_signed__take__bit__int__greater__eq,axiom,
% 0.29/0.57      ! [K: int,N: nat] :
% 0.29/0.57        ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.57       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_int_greater_eq
% 0.29/0.57  thf(fact_5848_mod__pos__neg__trivial,axiom,
% 0.29/0.57      ! [K: int,L: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.57       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.29/0.57         => ( ( modulo_modulo_int @ K @ L )
% 0.29/0.57            = ( plus_plus_int @ K @ L ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_pos_neg_trivial
% 0.29/0.57  thf(fact_5849_mod__pos__geq,axiom,
% 0.29/0.57      ! [L: int,K: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ L )
% 0.29/0.57       => ( ( ord_less_eq_int @ L @ K )
% 0.29/0.57         => ( ( modulo_modulo_int @ K @ L )
% 0.29/0.57            = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_pos_geq
% 0.29/0.57  thf(fact_5850_frac__def,axiom,
% 0.29/0.57      ( archim2898591450579166408c_real
% 0.29/0.57      = ( ^ [X4: real] : ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_def
% 0.29/0.57  thf(fact_5851_frac__def,axiom,
% 0.29/0.57      ( archimedean_frac_rat
% 0.29/0.57      = ( ^ [X4: rat] : ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_def
% 0.29/0.57  thf(fact_5852_pochhammer__rec,axiom,
% 0.29/0.57      ! [A: complex,N: nat] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_rec
% 0.29/0.57  thf(fact_5853_pochhammer__rec,axiom,
% 0.29/0.57      ! [A: real,N: nat] :
% 0.29/0.57        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_rec
% 0.29/0.57  thf(fact_5854_pochhammer__rec,axiom,
% 0.29/0.57      ! [A: rat,N: nat] :
% 0.29/0.57        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_rec
% 0.29/0.57  thf(fact_5855_pochhammer__rec,axiom,
% 0.29/0.57      ! [A: nat,N: nat] :
% 0.29/0.57        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_rec
% 0.29/0.57  thf(fact_5856_pochhammer__rec,axiom,
% 0.29/0.57      ! [A: int,N: nat] :
% 0.29/0.57        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_rec
% 0.29/0.57  thf(fact_5857_pochhammer__rec_H,axiom,
% 0.29/0.57      ! [Z: complex,N: nat] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_rec'
% 0.29/0.57  thf(fact_5858_pochhammer__rec_H,axiom,
% 0.29/0.57      ! [Z: real,N: nat] :
% 0.29/0.57        ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_rec'
% 0.29/0.57  thf(fact_5859_pochhammer__rec_H,axiom,
% 0.29/0.57      ! [Z: rat,N: nat] :
% 0.29/0.57        ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_rec'
% 0.29/0.57  thf(fact_5860_pochhammer__rec_H,axiom,
% 0.29/0.57      ! [Z: nat,N: nat] :
% 0.29/0.57        ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_rec'
% 0.29/0.57  thf(fact_5861_pochhammer__rec_H,axiom,
% 0.29/0.57      ! [Z: int,N: nat] :
% 0.29/0.57        ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_rec'
% 0.29/0.57  thf(fact_5862_pochhammer__Suc,axiom,
% 0.29/0.57      ! [A: complex,N: nat] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_Suc
% 0.29/0.57  thf(fact_5863_pochhammer__Suc,axiom,
% 0.29/0.57      ! [A: real,N: nat] :
% 0.29/0.57        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_Suc
% 0.29/0.57  thf(fact_5864_pochhammer__Suc,axiom,
% 0.29/0.57      ! [A: rat,N: nat] :
% 0.29/0.57        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_Suc
% 0.29/0.57  thf(fact_5865_pochhammer__Suc,axiom,
% 0.29/0.57      ! [A: nat,N: nat] :
% 0.29/0.57        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_Suc
% 0.29/0.57  thf(fact_5866_pochhammer__Suc,axiom,
% 0.29/0.57      ! [A: int,N: nat] :
% 0.29/0.57        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.29/0.57        = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_Suc
% 0.29/0.57  thf(fact_5867_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ord_less_nat @ N @ K )
% 0.29/0.57       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.29/0.57          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_lemma
% 0.29/0.57  thf(fact_5868_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ord_less_nat @ N @ K )
% 0.29/0.57       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.29/0.57          = zero_zero_complex ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_lemma
% 0.29/0.57  thf(fact_5869_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ord_less_nat @ N @ K )
% 0.29/0.57       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.29/0.57          = zero_zero_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_lemma
% 0.29/0.57  thf(fact_5870_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ord_less_nat @ N @ K )
% 0.29/0.57       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.29/0.57          = zero_zero_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_lemma
% 0.29/0.57  thf(fact_5871_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ord_less_nat @ N @ K )
% 0.29/0.57       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.29/0.57          = zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_lemma
% 0.29/0.57  thf(fact_5872_pochhammer__of__nat__eq__0__iff,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.29/0.57          = zero_z3403309356797280102nteger )
% 0.29/0.57        = ( ord_less_nat @ N @ K ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_iff
% 0.29/0.57  thf(fact_5873_pochhammer__of__nat__eq__0__iff,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.29/0.57          = zero_zero_complex )
% 0.29/0.57        = ( ord_less_nat @ N @ K ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_iff
% 0.29/0.57  thf(fact_5874_pochhammer__of__nat__eq__0__iff,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.29/0.57          = zero_zero_real )
% 0.29/0.57        = ( ord_less_nat @ N @ K ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_iff
% 0.29/0.57  thf(fact_5875_pochhammer__of__nat__eq__0__iff,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.29/0.57          = zero_zero_rat )
% 0.29/0.57        = ( ord_less_nat @ N @ K ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_iff
% 0.29/0.57  thf(fact_5876_pochhammer__of__nat__eq__0__iff,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.29/0.57          = zero_zero_int )
% 0.29/0.57        = ( ord_less_nat @ N @ K ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_iff
% 0.29/0.57  thf(fact_5877_pochhammer__eq__0__iff,axiom,
% 0.29/0.57      ! [A: complex,N: nat] :
% 0.29/0.57        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.29/0.57          = zero_zero_complex )
% 0.29/0.57        = ( ? [K2: nat] :
% 0.29/0.57              ( ( ord_less_nat @ K2 @ N )
% 0.29/0.57              & ( A
% 0.29/0.57                = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_eq_0_iff
% 0.29/0.57  thf(fact_5878_pochhammer__eq__0__iff,axiom,
% 0.29/0.57      ! [A: real,N: nat] :
% 0.29/0.57        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.29/0.57          = zero_zero_real )
% 0.29/0.57        = ( ? [K2: nat] :
% 0.29/0.57              ( ( ord_less_nat @ K2 @ N )
% 0.29/0.57              & ( A
% 0.29/0.57                = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_eq_0_iff
% 0.29/0.57  thf(fact_5879_pochhammer__eq__0__iff,axiom,
% 0.29/0.57      ! [A: rat,N: nat] :
% 0.29/0.57        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.29/0.57          = zero_zero_rat )
% 0.29/0.57        = ( ? [K2: nat] :
% 0.29/0.57              ( ( ord_less_nat @ K2 @ N )
% 0.29/0.57              & ( A
% 0.29/0.57                = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_eq_0_iff
% 0.29/0.57  thf(fact_5880_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.29/0.57         != zero_z3403309356797280102nteger ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_lemma'
% 0.29/0.57  thf(fact_5881_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.29/0.57         != zero_zero_complex ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_lemma'
% 0.29/0.57  thf(fact_5882_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.29/0.57         != zero_zero_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_lemma'
% 0.29/0.57  thf(fact_5883_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.29/0.57         != zero_zero_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_lemma'
% 0.29/0.57  thf(fact_5884_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.29/0.57         != zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_of_nat_eq_0_lemma'
% 0.29/0.57  thf(fact_5885_pochhammer__product_H,axiom,
% 0.29/0.57      ! [Z: complex,N: nat,M: nat] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.29/0.57        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_product'
% 0.29/0.57  thf(fact_5886_pochhammer__product_H,axiom,
% 0.29/0.57      ! [Z: real,N: nat,M: nat] :
% 0.29/0.57        ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.29/0.57        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_product'
% 0.29/0.57  thf(fact_5887_pochhammer__product_H,axiom,
% 0.29/0.57      ! [Z: rat,N: nat,M: nat] :
% 0.29/0.57        ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.29/0.57        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_product'
% 0.29/0.57  thf(fact_5888_pochhammer__product_H,axiom,
% 0.29/0.57      ! [Z: nat,N: nat,M: nat] :
% 0.29/0.57        ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.29/0.57        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_product'
% 0.29/0.57  thf(fact_5889_pochhammer__product_H,axiom,
% 0.29/0.57      ! [Z: int,N: nat,M: nat] :
% 0.29/0.57        ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.29/0.57        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_product'
% 0.29/0.57  thf(fact_5890_int__cases3,axiom,
% 0.29/0.57      ! [K: int] :
% 0.29/0.57        ( ( K != zero_zero_int )
% 0.29/0.57       => ( ! [N2: nat] :
% 0.29/0.57              ( ( K
% 0.29/0.57                = ( semiri1314217659103216013at_int @ N2 ) )
% 0.29/0.57             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.29/0.57         => ~ ! [N2: nat] :
% 0.29/0.57                ( ( K
% 0.29/0.57                  = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.29/0.57               => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % int_cases3
% 0.29/0.57  thf(fact_5891_not__zle__0__negative,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % not_zle_0_negative
% 0.29/0.57  thf(fact_5892_negative__zless__0,axiom,
% 0.29/0.57      ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 0.29/0.57  
% 0.29/0.57  % negative_zless_0
% 0.29/0.57  thf(fact_5893_negD,axiom,
% 0.29/0.57      ! [X: int] :
% 0.29/0.57        ( ( ord_less_int @ X @ zero_zero_int )
% 0.29/0.57       => ? [N2: nat] :
% 0.29/0.57            ( X
% 0.29/0.57            = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % negD
% 0.29/0.57  thf(fact_5894_real__of__int__div__aux,axiom,
% 0.29/0.57      ! [X: int,D: int] :
% 0.29/0.57        ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
% 0.29/0.57        = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % real_of_int_div_aux
% 0.29/0.57  thf(fact_5895_verit__less__mono__div__int2,axiom,
% 0.29/0.57      ! [A2: int,B2: int,N: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ A2 @ B2 )
% 0.29/0.57       => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 0.29/0.57         => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % verit_less_mono_div_int2
% 0.29/0.57  thf(fact_5896_div__eq__minus1,axiom,
% 0.29/0.57      ! [B: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.57       => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.29/0.57          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_eq_minus1
% 0.29/0.57  thf(fact_5897_ceiling__divide__eq__div,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
% 0.29/0.57        = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ceiling_divide_eq_div
% 0.29/0.57  thf(fact_5898_ceiling__divide__eq__div,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
% 0.29/0.57        = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ceiling_divide_eq_div
% 0.29/0.57  thf(fact_5899_frac__eq,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ( archim2898591450579166408c_real @ X )
% 0.29/0.57          = X )
% 0.29/0.57        = ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.57          & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_eq
% 0.29/0.57  thf(fact_5900_frac__eq,axiom,
% 0.29/0.57      ! [X: rat] :
% 0.29/0.57        ( ( ( archimedean_frac_rat @ X )
% 0.29/0.57          = X )
% 0.29/0.57        = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.29/0.57          & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_eq
% 0.29/0.57  thf(fact_5901_split__zmod,axiom,
% 0.29/0.57      ! [P: int > $o,N: int,K: int] :
% 0.29/0.57        ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 0.29/0.57        = ( ( ( K = zero_zero_int )
% 0.29/0.57           => ( P @ N ) )
% 0.29/0.57          & ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.57           => ! [I3: int,J2: int] :
% 0.29/0.57                ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
% 0.29/0.57                  & ( ord_less_int @ J2 @ K )
% 0.29/0.57                  & ( N
% 0.29/0.57                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 0.29/0.57               => ( P @ J2 ) ) )
% 0.29/0.57          & ( ( ord_less_int @ K @ zero_zero_int )
% 0.29/0.57           => ! [I3: int,J2: int] :
% 0.29/0.57                ( ( ( ord_less_int @ K @ J2 )
% 0.29/0.57                  & ( ord_less_eq_int @ J2 @ zero_zero_int )
% 0.29/0.57                  & ( N
% 0.29/0.57                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 0.29/0.57               => ( P @ J2 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % split_zmod
% 0.29/0.57  thf(fact_5902_int__mod__neg__eq,axiom,
% 0.29/0.57      ! [A: int,B: int,Q3: int,R2: int] :
% 0.29/0.57        ( ( A
% 0.29/0.57          = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.29/0.57       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.29/0.57         => ( ( ord_less_int @ B @ R2 )
% 0.29/0.57           => ( ( modulo_modulo_int @ A @ B )
% 0.29/0.57              = R2 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % int_mod_neg_eq
% 0.29/0.57  thf(fact_5903_int__mod__pos__eq,axiom,
% 0.29/0.57      ! [A: int,B: int,Q3: int,R2: int] :
% 0.29/0.57        ( ( A
% 0.29/0.57          = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.29/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.29/0.57         => ( ( ord_less_int @ R2 @ B )
% 0.29/0.57           => ( ( modulo_modulo_int @ A @ B )
% 0.29/0.57              = R2 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % int_mod_pos_eq
% 0.29/0.57  thf(fact_5904_frac__add,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.29/0.57         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.57            = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
% 0.29/0.57        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.29/0.57         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.57            = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_add
% 0.29/0.57  thf(fact_5905_frac__add,axiom,
% 0.29/0.57      ! [X: rat,Y: rat] :
% 0.29/0.57        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.29/0.57         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.29/0.57            = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
% 0.29/0.57        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.29/0.57         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.29/0.57            = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % frac_add
% 0.29/0.57  thf(fact_5906_signed__take__bit__int__less__exp,axiom,
% 0.29/0.57      ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_int_less_exp
% 0.29/0.57  thf(fact_5907_zmod__zmult2__eq,axiom,
% 0.29/0.57      ! [C: int,A: int,B: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.57       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.29/0.57          = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_zmult2_eq
% 0.29/0.57  thf(fact_5908_pochhammer__product,axiom,
% 0.29/0.57      ! [M: nat,N: nat,Z: complex] :
% 0.29/0.57        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57       => ( ( comm_s2602460028002588243omplex @ Z @ N )
% 0.29/0.57          = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_product
% 0.29/0.57  thf(fact_5909_pochhammer__product,axiom,
% 0.29/0.57      ! [M: nat,N: nat,Z: real] :
% 0.29/0.57        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57       => ( ( comm_s7457072308508201937r_real @ Z @ N )
% 0.29/0.57          = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_product
% 0.29/0.57  thf(fact_5910_pochhammer__product,axiom,
% 0.29/0.57      ! [M: nat,N: nat,Z: rat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57       => ( ( comm_s4028243227959126397er_rat @ Z @ N )
% 0.29/0.57          = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_product
% 0.29/0.57  thf(fact_5911_pochhammer__product,axiom,
% 0.29/0.57      ! [M: nat,N: nat,Z: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57       => ( ( comm_s4663373288045622133er_nat @ Z @ N )
% 0.29/0.57          = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_product
% 0.29/0.57  thf(fact_5912_pochhammer__product,axiom,
% 0.29/0.57      ! [M: nat,N: nat,Z: int] :
% 0.29/0.57        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57       => ( ( comm_s4660882817536571857er_int @ Z @ N )
% 0.29/0.57          = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_product
% 0.29/0.57  thf(fact_5913_neg__int__cases,axiom,
% 0.29/0.57      ! [K: int] :
% 0.29/0.57        ( ( ord_less_int @ K @ zero_zero_int )
% 0.29/0.57       => ~ ! [N2: nat] :
% 0.29/0.57              ( ( K
% 0.29/0.57                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.29/0.57             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_int_cases
% 0.29/0.57  thf(fact_5914_nat__intermed__int__val,axiom,
% 0.29/0.57      ! [M: nat,N: nat,F: nat > int,K: int] :
% 0.29/0.57        ( ! [I2: nat] :
% 0.29/0.57            ( ( ( ord_less_eq_nat @ M @ I2 )
% 0.29/0.57              & ( ord_less_nat @ I2 @ N ) )
% 0.29/0.57           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.29/0.57       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57         => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.29/0.57           => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.29/0.57             => ? [I2: nat] :
% 0.29/0.57                  ( ( ord_less_eq_nat @ M @ I2 )
% 0.29/0.57                  & ( ord_less_eq_nat @ I2 @ N )
% 0.29/0.57                  & ( ( F @ I2 )
% 0.29/0.57                    = K ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % nat_intermed_int_val
% 0.29/0.57  thf(fact_5915_decr__lemma,axiom,
% 0.29/0.57      ! [D: int,X: int,Z: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.57       => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 0.29/0.57  
% 0.29/0.57  % decr_lemma
% 0.29/0.57  thf(fact_5916_incr__lemma,axiom,
% 0.29/0.57      ! [D: int,Z: int,X: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.57       => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % incr_lemma
% 0.29/0.57  thf(fact_5917_verit__le__mono__div__int,axiom,
% 0.29/0.57      ! [A2: int,B2: int,N: int] :
% 0.29/0.57        ( ( ord_less_int @ A2 @ B2 )
% 0.29/0.57       => ( ( ord_less_int @ zero_zero_int @ N )
% 0.29/0.57         => ( ord_less_eq_int
% 0.29/0.57            @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
% 0.29/0.57              @ ( if_int
% 0.29/0.57                @ ( ( modulo_modulo_int @ B2 @ N )
% 0.29/0.57                  = zero_zero_int )
% 0.29/0.57                @ one_one_int
% 0.29/0.57                @ zero_zero_int ) )
% 0.29/0.57            @ ( divide_divide_int @ B2 @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % verit_le_mono_div_int
% 0.29/0.57  thf(fact_5918_split__neg__lemma,axiom,
% 0.29/0.57      ! [K: int,P: int > int > $o,N: int] :
% 0.29/0.57        ( ( ord_less_int @ K @ zero_zero_int )
% 0.29/0.57       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.29/0.57          = ( ! [I3: int,J2: int] :
% 0.29/0.57                ( ( ( ord_less_int @ K @ J2 )
% 0.29/0.57                  & ( ord_less_eq_int @ J2 @ zero_zero_int )
% 0.29/0.57                  & ( N
% 0.29/0.57                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 0.29/0.57               => ( P @ I3 @ J2 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % split_neg_lemma
% 0.29/0.57  thf(fact_5919_split__pos__lemma,axiom,
% 0.29/0.57      ! [K: int,P: int > int > $o,N: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.57       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.29/0.57          = ( ! [I3: int,J2: int] :
% 0.29/0.57                ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
% 0.29/0.57                  & ( ord_less_int @ J2 @ K )
% 0.29/0.57                  & ( N
% 0.29/0.57                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 0.29/0.57               => ( P @ I3 @ J2 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % split_pos_lemma
% 0.29/0.57  thf(fact_5920_signed__take__bit__int__less__self__iff,axiom,
% 0.29/0.57      ! [N: nat,K: int] :
% 0.29/0.57        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.29/0.57        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_int_less_self_iff
% 0.29/0.57  thf(fact_5921_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.29/0.57      ! [K: int,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.29/0.57        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_int_greater_eq_self_iff
% 0.29/0.57  thf(fact_5922_pochhammer__absorb__comp,axiom,
% 0.29/0.57      ! [R2: code_integer,K: nat] :
% 0.29/0.57        ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
% 0.29/0.57        = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_absorb_comp
% 0.29/0.57  thf(fact_5923_pochhammer__absorb__comp,axiom,
% 0.29/0.57      ! [R2: complex,K: nat] :
% 0.29/0.57        ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
% 0.29/0.57        = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_absorb_comp
% 0.29/0.57  thf(fact_5924_pochhammer__absorb__comp,axiom,
% 0.29/0.57      ! [R2: real,K: nat] :
% 0.29/0.57        ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
% 0.29/0.57        = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_absorb_comp
% 0.29/0.57  thf(fact_5925_pochhammer__absorb__comp,axiom,
% 0.29/0.57      ! [R2: rat,K: nat] :
% 0.29/0.57        ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
% 0.29/0.57        = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_absorb_comp
% 0.29/0.57  thf(fact_5926_pochhammer__absorb__comp,axiom,
% 0.29/0.57      ! [R2: int,K: nat] :
% 0.29/0.57        ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
% 0.29/0.57        = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_absorb_comp
% 0.29/0.57  thf(fact_5927_minus__1__div__exp__eq__int,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % minus_1_div_exp_eq_int
% 0.29/0.57  thf(fact_5928_div__pos__neg__trivial,axiom,
% 0.29/0.57      ! [K: int,L: int] :
% 0.29/0.57        ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.57       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.29/0.57         => ( ( divide_divide_int @ K @ L )
% 0.29/0.57            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_pos_neg_trivial
% 0.29/0.57  thf(fact_5929_compl__le__swap2,axiom,
% 0.29/0.57      ! [Y: set_int,X: set_int] :
% 0.29/0.57        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
% 0.29/0.57       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).
% 0.29/0.57  
% 0.29/0.57  % compl_le_swap2
% 0.29/0.57  thf(fact_5930_compl__le__swap1,axiom,
% 0.29/0.57      ! [Y: set_int,X: set_int] :
% 0.29/0.57        ( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
% 0.29/0.57       => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % compl_le_swap1
% 0.29/0.57  thf(fact_5931_compl__mono,axiom,
% 0.29/0.57      ! [X: set_int,Y: set_int] :
% 0.29/0.57        ( ( ord_less_eq_set_int @ X @ Y )
% 0.29/0.57       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % compl_mono
% 0.29/0.57  thf(fact_5932_nat__ivt__aux,axiom,
% 0.29/0.57      ! [N: nat,F: nat > int,K: int] :
% 0.29/0.57        ( ! [I2: nat] :
% 0.29/0.57            ( ( ord_less_nat @ I2 @ N )
% 0.29/0.57           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.29/0.57       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.29/0.57         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.29/0.57           => ? [I2: nat] :
% 0.29/0.57                ( ( ord_less_eq_nat @ I2 @ N )
% 0.29/0.57                & ( ( F @ I2 )
% 0.29/0.57                  = K ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % nat_ivt_aux
% 0.29/0.57  thf(fact_5933_complex__mod__minus__le__complex__mod,axiom,
% 0.29/0.57      ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % complex_mod_minus_le_complex_mod
% 0.29/0.57  thf(fact_5934_signed__take__bit__int__less__eq,axiom,
% 0.29/0.57      ! [N: nat,K: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.29/0.57       => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_int_less_eq
% 0.29/0.57  thf(fact_5935_pochhammer__minus_H,axiom,
% 0.29/0.57      ! [B: code_integer,K: nat] :
% 0.29/0.57        ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 0.29/0.57        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_minus'
% 0.29/0.57  thf(fact_5936_pochhammer__minus_H,axiom,
% 0.29/0.57      ! [B: complex,K: nat] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 0.29/0.57        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_minus'
% 0.29/0.57  thf(fact_5937_pochhammer__minus_H,axiom,
% 0.29/0.57      ! [B: real,K: nat] :
% 0.29/0.57        ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 0.29/0.57        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_minus'
% 0.29/0.57  thf(fact_5938_pochhammer__minus_H,axiom,
% 0.29/0.57      ! [B: rat,K: nat] :
% 0.29/0.57        ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 0.29/0.57        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_minus'
% 0.29/0.57  thf(fact_5939_pochhammer__minus_H,axiom,
% 0.29/0.57      ! [B: int,K: nat] :
% 0.29/0.57        ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 0.29/0.57        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_minus'
% 0.29/0.57  thf(fact_5940_pochhammer__minus,axiom,
% 0.29/0.57      ! [B: code_integer,K: nat] :
% 0.29/0.57        ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 0.29/0.57        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_minus
% 0.29/0.57  thf(fact_5941_pochhammer__minus,axiom,
% 0.29/0.57      ! [B: complex,K: nat] :
% 0.29/0.57        ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 0.29/0.57        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_minus
% 0.29/0.57  thf(fact_5942_pochhammer__minus,axiom,
% 0.29/0.57      ! [B: real,K: nat] :
% 0.29/0.57        ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 0.29/0.57        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_minus
% 0.29/0.57  thf(fact_5943_pochhammer__minus,axiom,
% 0.29/0.57      ! [B: rat,K: nat] :
% 0.29/0.57        ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 0.29/0.57        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_minus
% 0.29/0.57  thf(fact_5944_pochhammer__minus,axiom,
% 0.29/0.57      ! [B: int,K: nat] :
% 0.29/0.57        ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 0.29/0.57        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pochhammer_minus
% 0.29/0.57  thf(fact_5945_complex__mod__triangle__ineq2,axiom,
% 0.29/0.57      ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % complex_mod_triangle_ineq2
% 0.29/0.57  thf(fact_5946_int__bit__induct,axiom,
% 0.29/0.57      ! [P: int > $o,K: int] :
% 0.29/0.57        ( ( P @ zero_zero_int )
% 0.29/0.57       => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.57         => ( ! [K3: int] :
% 0.29/0.57                ( ( P @ K3 )
% 0.29/0.57               => ( ( K3 != zero_zero_int )
% 0.29/0.57                 => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.57           => ( ! [K3: int] :
% 0.29/0.57                  ( ( P @ K3 )
% 0.29/0.57                 => ( ( K3
% 0.29/0.57                     != ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.57                   => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 0.29/0.57             => ( P @ K ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % int_bit_induct
% 0.29/0.57  thf(fact_5947_floor__add,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.29/0.57         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.57            = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
% 0.29/0.57        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.29/0.57         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.57            = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % floor_add
% 0.29/0.57  thf(fact_5948_floor__add,axiom,
% 0.29/0.57      ! [X: rat,Y: rat] :
% 0.29/0.57        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.29/0.57         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.29/0.57            = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
% 0.29/0.57        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.29/0.57         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.29/0.57            = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % floor_add
% 0.29/0.57  thf(fact_5949_pos__zmod__mult__2,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.57       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.57          = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pos_zmod_mult_2
% 0.29/0.57  thf(fact_5950_nat0__intermed__int__val,axiom,
% 0.29/0.57      ! [N: nat,F: nat > int,K: int] :
% 0.29/0.57        ( ! [I2: nat] :
% 0.29/0.57            ( ( ord_less_nat @ I2 @ N )
% 0.29/0.57           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.29/0.57       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.29/0.57         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.29/0.57           => ? [I2: nat] :
% 0.29/0.57                ( ( ord_less_eq_nat @ I2 @ N )
% 0.29/0.57                & ( ( F @ I2 )
% 0.29/0.57                  = K ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % nat0_intermed_int_val
% 0.29/0.57  thf(fact_5951_arctan__add,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.57       => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.29/0.57         => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 0.29/0.57            = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % arctan_add
% 0.29/0.57  thf(fact_5952_neg__zmod__mult__2,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.57       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.57          = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % neg_zmod_mult_2
% 0.29/0.57  thf(fact_5953_signed__take__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: code_integer] :
% 0.29/0.57        ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_Suc
% 0.29/0.57  thf(fact_5954_signed__take__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,A: int] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 0.29/0.57        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_Suc
% 0.29/0.57  thf(fact_5955_central__binomial__lower__bound,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57       => ( 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 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % central_binomial_lower_bound
% 0.29/0.57  thf(fact_5956_signed__take__bit__Suc__minus__bit1,axiom,
% 0.29/0.57      ! [N: nat,K: num] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.29/0.57        = ( 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 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_Suc_minus_bit1
% 0.29/0.57  thf(fact_5957_gcd__nat__induct,axiom,
% 0.29/0.57      ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.29/0.57        ( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
% 0.29/0.57       => ( ! [M5: nat,N2: nat] :
% 0.29/0.57              ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.57             => ( ( P @ N2 @ ( modulo_modulo_nat @ M5 @ N2 ) )
% 0.29/0.57               => ( P @ M5 @ N2 ) ) )
% 0.29/0.57         => ( P @ M @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % gcd_nat_induct
% 0.29/0.57  thf(fact_5958_option_Osize_I3_J,axiom,
% 0.29/0.57      ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 0.29/0.57      = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % option.size(3)
% 0.29/0.57  thf(fact_5959_option_Osize_I3_J,axiom,
% 0.29/0.57      ( ( size_size_option_num @ none_num )
% 0.29/0.57      = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % option.size(3)
% 0.29/0.57  thf(fact_5960_verit__eq__simplify_I9_J,axiom,
% 0.29/0.57      ! [X32: num,Y32: num] :
% 0.29/0.57        ( ( ( bit1 @ X32 )
% 0.29/0.57          = ( bit1 @ Y32 ) )
% 0.29/0.57        = ( X32 = Y32 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % verit_eq_simplify(9)
% 0.29/0.57  thf(fact_5961_semiring__norm_I90_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( ( bit1 @ M )
% 0.29/0.57          = ( bit1 @ N ) )
% 0.29/0.57        = ( M = N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(90)
% 0.29/0.57  thf(fact_5962_ComplI,axiom,
% 0.29/0.57      ! [C: complex,A2: set_complex] :
% 0.29/0.57        ( ~ ( member_complex @ C @ A2 )
% 0.29/0.57       => ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ComplI
% 0.29/0.57  thf(fact_5963_ComplI,axiom,
% 0.29/0.57      ! [C: real,A2: set_real] :
% 0.29/0.57        ( ~ ( member_real @ C @ A2 )
% 0.29/0.57       => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ComplI
% 0.29/0.57  thf(fact_5964_ComplI,axiom,
% 0.29/0.57      ! [C: set_nat,A2: set_set_nat] :
% 0.29/0.57        ( ~ ( member_set_nat @ C @ A2 )
% 0.29/0.57       => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ComplI
% 0.29/0.57  thf(fact_5965_ComplI,axiom,
% 0.29/0.57      ! [C: nat,A2: set_nat] :
% 0.29/0.57        ( ~ ( member_nat @ C @ A2 )
% 0.29/0.57       => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ComplI
% 0.29/0.57  thf(fact_5966_ComplI,axiom,
% 0.29/0.57      ! [C: int,A2: set_int] :
% 0.29/0.57        ( ~ ( member_int @ C @ A2 )
% 0.29/0.57       => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ComplI
% 0.29/0.57  thf(fact_5967_Compl__iff,axiom,
% 0.29/0.57      ! [C: complex,A2: set_complex] :
% 0.29/0.57        ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
% 0.29/0.57        = ( ~ ( member_complex @ C @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Compl_iff
% 0.29/0.57  thf(fact_5968_Compl__iff,axiom,
% 0.29/0.57      ! [C: real,A2: set_real] :
% 0.29/0.57        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 0.29/0.57        = ( ~ ( member_real @ C @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Compl_iff
% 0.29/0.57  thf(fact_5969_Compl__iff,axiom,
% 0.29/0.57      ! [C: set_nat,A2: set_set_nat] :
% 0.29/0.57        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 0.29/0.57        = ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Compl_iff
% 0.29/0.57  thf(fact_5970_Compl__iff,axiom,
% 0.29/0.57      ! [C: nat,A2: set_nat] :
% 0.29/0.57        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.29/0.57        = ( ~ ( member_nat @ C @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Compl_iff
% 0.29/0.57  thf(fact_5971_Compl__iff,axiom,
% 0.29/0.57      ! [C: int,A2: set_int] :
% 0.29/0.57        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.29/0.57        = ( ~ ( member_int @ C @ A2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Compl_iff
% 0.29/0.57  thf(fact_5972_semiring__norm_I89_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( bit1 @ M )
% 0.29/0.57       != ( bit0 @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(89)
% 0.29/0.57  thf(fact_5973_semiring__norm_I88_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( bit0 @ M )
% 0.29/0.57       != ( bit1 @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(88)
% 0.29/0.57  thf(fact_5974_semiring__norm_I86_J,axiom,
% 0.29/0.57      ! [M: num] :
% 0.29/0.57        ( ( bit1 @ M )
% 0.29/0.57       != one ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(86)
% 0.29/0.57  thf(fact_5975_semiring__norm_I84_J,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( one
% 0.29/0.57       != ( bit1 @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(84)
% 0.29/0.57  thf(fact_5976_semiring__norm_I80_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.57        = ( ord_less_num @ M @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(80)
% 0.29/0.57  thf(fact_5977_semiring__norm_I73_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.57        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(73)
% 0.29/0.57  thf(fact_5978_semiring__norm_I7_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.57        = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(7)
% 0.29/0.57  thf(fact_5979_semiring__norm_I9_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.29/0.57        = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(9)
% 0.29/0.57  thf(fact_5980_semiring__norm_I14_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.57        = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(14)
% 0.29/0.57  thf(fact_5981_semiring__norm_I15_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.29/0.57        = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(15)
% 0.29/0.57  thf(fact_5982_semiring__norm_I81_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.29/0.57        = ( ord_less_num @ M @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(81)
% 0.29/0.57  thf(fact_5983_semiring__norm_I72_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.57        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(72)
% 0.29/0.57  thf(fact_5984_semiring__norm_I77_J,axiom,
% 0.29/0.57      ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(77)
% 0.29/0.57  thf(fact_5985_semiring__norm_I70_J,axiom,
% 0.29/0.57      ! [M: num] :
% 0.29/0.57        ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(70)
% 0.29/0.57  thf(fact_5986_zdiv__numeral__Bit1,axiom,
% 0.29/0.57      ! [V: num,W: num] :
% 0.29/0.57        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.29/0.57        = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zdiv_numeral_Bit1
% 0.29/0.57  thf(fact_5987_semiring__norm_I10_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.57        = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(10)
% 0.29/0.57  thf(fact_5988_semiring__norm_I8_J,axiom,
% 0.29/0.57      ! [M: num] :
% 0.29/0.57        ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 0.29/0.57        = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(8)
% 0.29/0.57  thf(fact_5989_semiring__norm_I5_J,axiom,
% 0.29/0.57      ! [M: num] :
% 0.29/0.57        ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 0.29/0.57        = ( bit1 @ M ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(5)
% 0.29/0.57  thf(fact_5990_semiring__norm_I4_J,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 0.29/0.57        = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(4)
% 0.29/0.57  thf(fact_5991_semiring__norm_I3_J,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 0.29/0.57        = ( bit1 @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(3)
% 0.29/0.57  thf(fact_5992_semiring__norm_I16_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.57        = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(16)
% 0.29/0.57  thf(fact_5993_semiring__norm_I74_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.29/0.57        = ( ord_less_num @ M @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(74)
% 0.29/0.57  thf(fact_5994_semiring__norm_I79_J,axiom,
% 0.29/0.57      ! [M: num,N: num] :
% 0.29/0.57        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.57        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % semiring_norm(79)
% 0.29/0.57  thf(fact_5995_Suc__div__eq__add3__div__numeral,axiom,
% 0.29/0.57      ! [M: nat,V: num] :
% 0.29/0.57        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.29/0.57        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Suc_div_eq_add3_div_numeral
% 0.29/0.57  thf(fact_5996_div__Suc__eq__div__add3,axiom,
% 0.29/0.57      ! [M: nat,N: nat] :
% 0.29/0.57        ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.29/0.57        = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_Suc_eq_div_add3
% 0.29/0.57  thf(fact_5997_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.29/0.57      ! [M: nat,V: num] :
% 0.29/0.57        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.29/0.57        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Suc_mod_eq_add3_mod_numeral
% 0.29/0.57  thf(fact_5998_mod__Suc__eq__mod__add3,axiom,
% 0.29/0.57      ! [M: nat,N: nat] :
% 0.29/0.57        ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.29/0.57        = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_Suc_eq_mod_add3
% 0.29/0.57  thf(fact_5999_zmod__numeral__Bit1,axiom,
% 0.29/0.57      ! [V: num,W: num] :
% 0.29/0.57        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.29/0.57        = ( 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 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zmod_numeral_Bit1
% 0.29/0.57  thf(fact_6000_signed__take__bit__Suc__bit1,axiom,
% 0.29/0.57      ! [N: nat,K: num] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.29/0.57        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_Suc_bit1
% 0.29/0.57  thf(fact_6001_ComplD,axiom,
% 0.29/0.57      ! [C: complex,A2: set_complex] :
% 0.29/0.57        ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
% 0.29/0.57       => ~ ( member_complex @ C @ A2 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ComplD
% 0.29/0.57  thf(fact_6002_ComplD,axiom,
% 0.29/0.57      ! [C: real,A2: set_real] :
% 0.29/0.57        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 0.29/0.57       => ~ ( member_real @ C @ A2 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ComplD
% 0.29/0.57  thf(fact_6003_ComplD,axiom,
% 0.29/0.57      ! [C: set_nat,A2: set_set_nat] :
% 0.29/0.57        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 0.29/0.57       => ~ ( member_set_nat @ C @ A2 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ComplD
% 0.29/0.57  thf(fact_6004_ComplD,axiom,
% 0.29/0.57      ! [C: nat,A2: set_nat] :
% 0.29/0.57        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.29/0.57       => ~ ( member_nat @ C @ A2 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ComplD
% 0.29/0.57  thf(fact_6005_ComplD,axiom,
% 0.29/0.57      ! [C: int,A2: set_int] :
% 0.29/0.57        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.29/0.57       => ~ ( member_int @ C @ A2 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % ComplD
% 0.29/0.57  thf(fact_6006_verit__eq__simplify_I14_J,axiom,
% 0.29/0.57      ! [X22: num,X32: num] :
% 0.29/0.57        ( ( bit0 @ X22 )
% 0.29/0.57       != ( bit1 @ X32 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % verit_eq_simplify(14)
% 0.29/0.57  thf(fact_6007_verit__eq__simplify_I12_J,axiom,
% 0.29/0.57      ! [X32: num] :
% 0.29/0.57        ( one
% 0.29/0.57       != ( bit1 @ X32 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % verit_eq_simplify(12)
% 0.29/0.57  thf(fact_6008_num_Oexhaust,axiom,
% 0.29/0.57      ! [Y: num] :
% 0.29/0.57        ( ( Y != one )
% 0.29/0.57       => ( ! [X23: num] :
% 0.29/0.57              ( Y
% 0.29/0.57             != ( bit0 @ X23 ) )
% 0.29/0.57         => ~ ! [X33: num] :
% 0.29/0.57                ( Y
% 0.29/0.57               != ( bit1 @ X33 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % num.exhaust
% 0.29/0.57  thf(fact_6009_numeral__Bit1,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.29/0.57        = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.29/0.57  
% 0.29/0.57  % numeral_Bit1
% 0.29/0.57  thf(fact_6010_numeral__Bit1,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.29/0.57        = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % numeral_Bit1
% 0.29/0.57  thf(fact_6011_numeral__Bit1,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.29/0.57        = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % numeral_Bit1
% 0.29/0.57  thf(fact_6012_numeral__Bit1,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.29/0.57        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % numeral_Bit1
% 0.29/0.57  thf(fact_6013_numeral__Bit1,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.29/0.57        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % numeral_Bit1
% 0.29/0.57  thf(fact_6014_eval__nat__numeral_I3_J,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.29/0.57        = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % eval_nat_numeral(3)
% 0.29/0.57  thf(fact_6015_cong__exp__iff__simps_I13_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num,N: num] :
% 0.29/0.57        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.29/0.57          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.57        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.29/0.57          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(13)
% 0.29/0.57  thf(fact_6016_cong__exp__iff__simps_I13_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num,N: num] :
% 0.29/0.57        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.29/0.57          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.57        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 0.29/0.57          = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(13)
% 0.29/0.57  thf(fact_6017_cong__exp__iff__simps_I13_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num,N: num] :
% 0.29/0.57        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.29/0.57          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.57        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.29/0.57          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(13)
% 0.29/0.57  thf(fact_6018_cong__exp__iff__simps_I12_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num,N: num] :
% 0.29/0.57        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.29/0.57       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(12)
% 0.29/0.57  thf(fact_6019_cong__exp__iff__simps_I12_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num,N: num] :
% 0.29/0.57        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.29/0.57       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(12)
% 0.29/0.57  thf(fact_6020_cong__exp__iff__simps_I12_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num,N: num] :
% 0.29/0.57        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.29/0.57       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(12)
% 0.29/0.57  thf(fact_6021_cong__exp__iff__simps_I10_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num,N: num] :
% 0.29/0.57        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.29/0.57       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(10)
% 0.29/0.57  thf(fact_6022_cong__exp__iff__simps_I10_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num,N: num] :
% 0.29/0.57        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.29/0.57       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(10)
% 0.29/0.57  thf(fact_6023_cong__exp__iff__simps_I10_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num,N: num] :
% 0.29/0.57        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.29/0.57       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(10)
% 0.29/0.57  thf(fact_6024_power__minus__Bit1,axiom,
% 0.29/0.57      ! [X: real,K: num] :
% 0.29/0.57        ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.29/0.57        = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_Bit1
% 0.29/0.57  thf(fact_6025_power__minus__Bit1,axiom,
% 0.29/0.57      ! [X: int,K: num] :
% 0.29/0.57        ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.29/0.57        = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_Bit1
% 0.29/0.57  thf(fact_6026_power__minus__Bit1,axiom,
% 0.29/0.57      ! [X: complex,K: num] :
% 0.29/0.57        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.29/0.57        = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_Bit1
% 0.29/0.57  thf(fact_6027_power__minus__Bit1,axiom,
% 0.29/0.57      ! [X: rat,K: num] :
% 0.29/0.57        ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.29/0.57        = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_Bit1
% 0.29/0.57  thf(fact_6028_power__minus__Bit1,axiom,
% 0.29/0.57      ! [X: code_integer,K: num] :
% 0.29/0.57        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.29/0.57        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_Bit1
% 0.29/0.57  thf(fact_6029_bounded__Max__nat,axiom,
% 0.29/0.57      ! [P: nat > $o,X: nat,M2: nat] :
% 0.29/0.57        ( ( P @ X )
% 0.29/0.57       => ( ! [X3: nat] :
% 0.29/0.57              ( ( P @ X3 )
% 0.29/0.57             => ( ord_less_eq_nat @ X3 @ M2 ) )
% 0.29/0.57         => ~ ! [M5: nat] :
% 0.29/0.57                ( ( P @ M5 )
% 0.29/0.57               => ~ ! [X2: nat] :
% 0.29/0.57                      ( ( P @ X2 )
% 0.29/0.57                     => ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % bounded_Max_nat
% 0.29/0.57  thf(fact_6030_numeral__Bit1__div__2,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57        = ( numeral_numeral_nat @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % numeral_Bit1_div_2
% 0.29/0.57  thf(fact_6031_numeral__Bit1__div__2,axiom,
% 0.29/0.57      ! [N: num] :
% 0.29/0.57        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.57        = ( numeral_numeral_int @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % numeral_Bit1_div_2
% 0.29/0.57  thf(fact_6032_cong__exp__iff__simps_I3_J,axiom,
% 0.29/0.57      ! [N: num,Q3: num] :
% 0.29/0.57        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.29/0.57       != zero_zero_nat ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(3)
% 0.29/0.57  thf(fact_6033_cong__exp__iff__simps_I3_J,axiom,
% 0.29/0.57      ! [N: num,Q3: num] :
% 0.29/0.57        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.29/0.57       != zero_zero_int ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(3)
% 0.29/0.57  thf(fact_6034_cong__exp__iff__simps_I3_J,axiom,
% 0.29/0.57      ! [N: num,Q3: num] :
% 0.29/0.57        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.29/0.57       != zero_z3403309356797280102nteger ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(3)
% 0.29/0.57  thf(fact_6035_power3__eq__cube,axiom,
% 0.29/0.57      ! [A: complex] :
% 0.29/0.57        ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.29/0.57        = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power3_eq_cube
% 0.29/0.57  thf(fact_6036_power3__eq__cube,axiom,
% 0.29/0.57      ! [A: real] :
% 0.29/0.57        ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.29/0.57        = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power3_eq_cube
% 0.29/0.57  thf(fact_6037_power3__eq__cube,axiom,
% 0.29/0.57      ! [A: rat] :
% 0.29/0.57        ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.29/0.57        = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power3_eq_cube
% 0.29/0.57  thf(fact_6038_power3__eq__cube,axiom,
% 0.29/0.57      ! [A: nat] :
% 0.29/0.57        ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.29/0.57        = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power3_eq_cube
% 0.29/0.57  thf(fact_6039_power3__eq__cube,axiom,
% 0.29/0.57      ! [A: int] :
% 0.29/0.57        ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.29/0.57        = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power3_eq_cube
% 0.29/0.57  thf(fact_6040_numeral__3__eq__3,axiom,
% 0.29/0.57      ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.29/0.57      = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % numeral_3_eq_3
% 0.29/0.57  thf(fact_6041_Suc3__eq__add__3,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( suc @ ( suc @ ( suc @ N ) ) )
% 0.29/0.57        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Suc3_eq_add_3
% 0.29/0.57  thf(fact_6042_binomial__maximum_H,axiom,
% 0.29/0.57      ! [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 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_maximum'
% 0.29/0.57  thf(fact_6043_binomial__mono,axiom,
% 0.29/0.57      ! [K: nat,K5: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ K5 )
% 0.29/0.57       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 0.29/0.57         => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_mono
% 0.29/0.57  thf(fact_6044_binomial__antimono,axiom,
% 0.29/0.57      ! [K: nat,K5: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ K5 )
% 0.29/0.57       => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.29/0.57         => ( ( ord_less_eq_nat @ K5 @ N )
% 0.29/0.57           => ( ord_less_eq_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_antimono
% 0.29/0.57  thf(fact_6045_binomial__maximum,axiom,
% 0.29/0.57      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_maximum
% 0.29/0.57  thf(fact_6046_num_Osize_I6_J,axiom,
% 0.29/0.57      ! [X32: num] :
% 0.29/0.57        ( ( size_size_num @ ( bit1 @ X32 ) )
% 0.29/0.57        = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % num.size(6)
% 0.29/0.57  thf(fact_6047_cong__exp__iff__simps_I11_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num] :
% 0.29/0.57        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.29/0.57          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.57        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.29/0.57          = zero_zero_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(11)
% 0.29/0.57  thf(fact_6048_cong__exp__iff__simps_I11_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num] :
% 0.29/0.57        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.29/0.57          = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.57        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 0.29/0.57          = zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(11)
% 0.29/0.57  thf(fact_6049_cong__exp__iff__simps_I11_J,axiom,
% 0.29/0.57      ! [M: num,Q3: num] :
% 0.29/0.57        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.29/0.57          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.57        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.29/0.57          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(11)
% 0.29/0.57  thf(fact_6050_cong__exp__iff__simps_I7_J,axiom,
% 0.29/0.57      ! [Q3: num,N: num] :
% 0.29/0.57        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.29/0.57          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.57        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.29/0.57          = zero_zero_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(7)
% 0.29/0.57  thf(fact_6051_cong__exp__iff__simps_I7_J,axiom,
% 0.29/0.57      ! [Q3: num,N: num] :
% 0.29/0.57        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.29/0.57          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.57        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 0.29/0.57          = zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(7)
% 0.29/0.57  thf(fact_6052_cong__exp__iff__simps_I7_J,axiom,
% 0.29/0.57      ! [Q3: num,N: num] :
% 0.29/0.57        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.29/0.57          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 0.29/0.57        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.29/0.57          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.57  
% 0.29/0.57  % cong_exp_iff_simps(7)
% 0.29/0.57  thf(fact_6053_Suc__div__eq__add3__div,axiom,
% 0.29/0.57      ! [M: nat,N: nat] :
% 0.29/0.57        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.29/0.57        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Suc_div_eq_add3_div
% 0.29/0.57  thf(fact_6054_Suc__mod__eq__add3__mod,axiom,
% 0.29/0.57      ! [M: nat,N: nat] :
% 0.29/0.57        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.29/0.57        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Suc_mod_eq_add3_mod
% 0.29/0.57  thf(fact_6055_binomial__strict__mono,axiom,
% 0.29/0.57      ! [K: nat,K5: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ K @ K5 )
% 0.29/0.57       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 0.29/0.57         => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_strict_mono
% 0.29/0.57  thf(fact_6056_binomial__strict__antimono,axiom,
% 0.29/0.57      ! [K: nat,K5: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ K @ K5 )
% 0.29/0.57       => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.29/0.57         => ( ( ord_less_eq_nat @ K5 @ N )
% 0.29/0.57           => ( ord_less_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_strict_antimono
% 0.29/0.57  thf(fact_6057_binomial__less__binomial__Suc,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_less_binomial_Suc
% 0.29/0.57  thf(fact_6058_mod__exhaust__less__4,axiom,
% 0.29/0.57      ! [M: nat] :
% 0.29/0.57        ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = zero_zero_nat )
% 0.29/0.57        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = one_one_nat )
% 0.29/0.57        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.57          = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % mod_exhaust_less_4
% 0.29/0.57  thf(fact_6059_zero__notin__Suc__image,axiom,
% 0.29/0.57      ! [A2: set_nat] :
% 0.29/0.57        ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zero_notin_Suc_image
% 0.29/0.57  thf(fact_6060_Euclid__induct,axiom,
% 0.29/0.57      ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.29/0.57        ( ! [A5: nat,B6: nat] :
% 0.29/0.57            ( ( P @ A5 @ B6 )
% 0.29/0.57            = ( P @ B6 @ A5 ) )
% 0.29/0.57       => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
% 0.29/0.57         => ( ! [A5: nat,B6: nat] :
% 0.29/0.57                ( ( P @ A5 @ B6 )
% 0.29/0.57               => ( P @ A5 @ ( plus_plus_nat @ A5 @ B6 ) ) )
% 0.29/0.57           => ( P @ A @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Euclid_induct
% 0.29/0.57  thf(fact_6061_zero__less__binomial__iff,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 0.29/0.57        = ( ord_less_eq_nat @ K @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zero_less_binomial_iff
% 0.29/0.57  thf(fact_6062_choose__two,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57        = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % choose_two
% 0.29/0.57  thf(fact_6063_binomial__n__0,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( binomial @ N @ zero_zero_nat )
% 0.29/0.57        = one_one_nat ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_n_0
% 0.29/0.57  thf(fact_6064_binomial__Suc__Suc,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.29/0.57        = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_Suc_Suc
% 0.29/0.57  thf(fact_6065_binomial__eq__0__iff,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ( binomial @ N @ K )
% 0.29/0.57          = zero_zero_nat )
% 0.29/0.57        = ( ord_less_nat @ N @ K ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_eq_0_iff
% 0.29/0.57  thf(fact_6066_binomial__1,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.57        = N ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_1
% 0.29/0.57  thf(fact_6067_binomial__Suc__n,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( binomial @ ( suc @ N ) @ N )
% 0.29/0.57        = ( suc @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_Suc_n
% 0.29/0.57  thf(fact_6068_binomial__n__n,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( binomial @ N @ N )
% 0.29/0.57        = one_one_nat ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_n_n
% 0.29/0.57  thf(fact_6069_binomial__0__Suc,axiom,
% 0.29/0.57      ! [K: nat] :
% 0.29/0.57        ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 0.29/0.57        = zero_zero_nat ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_0_Suc
% 0.29/0.57  thf(fact_6070_choose__one,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( binomial @ N @ one_one_nat )
% 0.29/0.57        = N ) ).
% 0.29/0.57  
% 0.29/0.57  % choose_one
% 0.29/0.57  thf(fact_6071_binomial__eq__0,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ord_less_nat @ N @ K )
% 0.29/0.57       => ( ( binomial @ N @ K )
% 0.29/0.57          = zero_zero_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_eq_0
% 0.29/0.57  thf(fact_6072_Suc__times__binomial__eq,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 0.29/0.57        = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Suc_times_binomial_eq
% 0.29/0.57  thf(fact_6073_Suc__times__binomial,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 0.29/0.57        = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Suc_times_binomial
% 0.29/0.57  thf(fact_6074_binomial__symmetric,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ( binomial @ N @ K )
% 0.29/0.57          = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_symmetric
% 0.29/0.57  thf(fact_6075_choose__mult__lemma,axiom,
% 0.29/0.57      ! [M: nat,R2: nat,K: nat] :
% 0.29/0.57        ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 0.29/0.57        = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % choose_mult_lemma
% 0.29/0.57  thf(fact_6076_binomial__le__pow,axiom,
% 0.29/0.57      ! [R2: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ R2 @ N )
% 0.29/0.57       => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_le_pow
% 0.29/0.57  thf(fact_6077_zero__less__binomial,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zero_less_binomial
% 0.29/0.57  thf(fact_6078_Suc__times__binomial__add,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 0.29/0.57        = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Suc_times_binomial_add
% 0.29/0.57  thf(fact_6079_choose__mult,axiom,
% 0.29/0.57      ! [K: nat,M: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.57       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.57         => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 0.29/0.57            = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % choose_mult
% 0.29/0.57  thf(fact_6080_binomial__Suc__Suc__eq__times,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.29/0.57        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_Suc_Suc_eq_times
% 0.29/0.57  thf(fact_6081_binomial__absorb__comp,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 0.29/0.57        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_absorb_comp
% 0.29/0.57  thf(fact_6082_binomial__absorption,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 0.29/0.57        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_absorption
% 0.29/0.57  thf(fact_6083_binomial__ge__n__over__k__pow__k,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_ge_n_over_k_pow_k
% 0.29/0.57  thf(fact_6084_binomial__ge__n__over__k__pow__k,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.57       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_ge_n_over_k_pow_k
% 0.29/0.57  thf(fact_6085_binomial__le__pow2,axiom,
% 0.29/0.57      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_le_pow2
% 0.29/0.57  thf(fact_6086_choose__reduce__nat,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.57         => ( ( binomial @ N @ K )
% 0.29/0.57            = ( 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 ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % choose_reduce_nat
% 0.29/0.57  thf(fact_6087_times__binomial__minus1__eq,axiom,
% 0.29/0.57      ! [K: nat,N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.57       => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 0.29/0.57          = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % times_binomial_minus1_eq
% 0.29/0.57  thf(fact_6088_binomial__addition__formula,axiom,
% 0.29/0.57      ! [N: nat,K: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57       => ( ( binomial @ N @ ( suc @ K ) )
% 0.29/0.57          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % binomial_addition_formula
% 0.29/0.57  thf(fact_6089_signed__take__bit__numeral__minus__bit1,axiom,
% 0.29/0.57      ! [L: num,K: num] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.29/0.57        = ( 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 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_numeral_minus_bit1
% 0.29/0.57  thf(fact_6090_dbl__dec__simps_I4_J,axiom,
% 0.29/0.57      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.57      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(4)
% 0.29/0.57  thf(fact_6091_dbl__dec__simps_I4_J,axiom,
% 0.29/0.57      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.57      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(4)
% 0.29/0.57  thf(fact_6092_dbl__dec__simps_I4_J,axiom,
% 0.29/0.57      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.57      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(4)
% 0.29/0.57  thf(fact_6093_dbl__dec__simps_I4_J,axiom,
% 0.29/0.57      ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.57      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(4)
% 0.29/0.57  thf(fact_6094_dbl__dec__simps_I4_J,axiom,
% 0.29/0.57      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.57      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(4)
% 0.29/0.57  thf(fact_6095_signed__take__bit__numeral__bit1,axiom,
% 0.29/0.57      ! [L: num,K: num] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.29/0.57        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_numeral_bit1
% 0.29/0.57  thf(fact_6096_dbl__inc__simps_I3_J,axiom,
% 0.29/0.57      ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 0.29/0.57      = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(3)
% 0.29/0.57  thf(fact_6097_dbl__inc__simps_I3_J,axiom,
% 0.29/0.57      ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 0.29/0.57      = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(3)
% 0.29/0.57  thf(fact_6098_dbl__inc__simps_I3_J,axiom,
% 0.29/0.57      ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 0.29/0.57      = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(3)
% 0.29/0.57  thf(fact_6099_dbl__inc__simps_I3_J,axiom,
% 0.29/0.57      ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 0.29/0.57      = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(3)
% 0.29/0.57  thf(fact_6100_translation__Compl,axiom,
% 0.29/0.57      ! [A: real,T: set_real] :
% 0.29/0.57        ( ( image_real_real @ ( plus_plus_real @ A ) @ ( uminus612125837232591019t_real @ T ) )
% 0.29/0.57        = ( uminus612125837232591019t_real @ ( image_real_real @ ( plus_plus_real @ A ) @ T ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % translation_Compl
% 0.29/0.57  thf(fact_6101_translation__Compl,axiom,
% 0.29/0.57      ! [A: rat,T: set_rat] :
% 0.29/0.57        ( ( image_rat_rat @ ( plus_plus_rat @ A ) @ ( uminus2201863774496077783et_rat @ T ) )
% 0.29/0.57        = ( uminus2201863774496077783et_rat @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ T ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % translation_Compl
% 0.29/0.57  thf(fact_6102_translation__Compl,axiom,
% 0.29/0.57      ! [A: int,T: set_int] :
% 0.29/0.57        ( ( image_int_int @ ( plus_plus_int @ A ) @ ( uminus1532241313380277803et_int @ T ) )
% 0.29/0.57        = ( uminus1532241313380277803et_int @ ( image_int_int @ ( plus_plus_int @ A ) @ T ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % translation_Compl
% 0.29/0.57  thf(fact_6103_translation__Compl,axiom,
% 0.29/0.57      ! [A: complex,T: set_complex] :
% 0.29/0.57        ( ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ ( uminus8566677241136511917omplex @ T ) )
% 0.29/0.57        = ( uminus8566677241136511917omplex @ ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ T ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % translation_Compl
% 0.29/0.57  thf(fact_6104_concat__bit__Suc,axiom,
% 0.29/0.57      ! [N: nat,K: int,L: int] :
% 0.29/0.57        ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 0.29/0.57        = ( 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 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % concat_bit_Suc
% 0.29/0.57  thf(fact_6105_concat__bit__0,axiom,
% 0.29/0.57      ! [K: int,L: int] :
% 0.29/0.57        ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 0.29/0.57        = L ) ).
% 0.29/0.57  
% 0.29/0.57  % concat_bit_0
% 0.29/0.57  thf(fact_6106_dbl__dec__simps_I3_J,axiom,
% 0.29/0.57      ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 0.29/0.57      = one_one_complex ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(3)
% 0.29/0.57  thf(fact_6107_dbl__dec__simps_I3_J,axiom,
% 0.29/0.57      ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 0.29/0.57      = one_one_real ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(3)
% 0.29/0.57  thf(fact_6108_dbl__dec__simps_I3_J,axiom,
% 0.29/0.57      ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 0.29/0.57      = one_one_rat ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(3)
% 0.29/0.57  thf(fact_6109_dbl__dec__simps_I3_J,axiom,
% 0.29/0.57      ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 0.29/0.57      = one_one_int ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(3)
% 0.29/0.57  thf(fact_6110_pred__numeral__simps_I1_J,axiom,
% 0.29/0.57      ( ( pred_numeral @ one )
% 0.29/0.57      = zero_zero_nat ) ).
% 0.29/0.57  
% 0.29/0.57  % pred_numeral_simps(1)
% 0.29/0.57  thf(fact_6111_Suc__eq__numeral,axiom,
% 0.29/0.57      ! [N: nat,K: num] :
% 0.29/0.57        ( ( ( suc @ N )
% 0.29/0.57          = ( numeral_numeral_nat @ K ) )
% 0.29/0.57        = ( N
% 0.29/0.57          = ( pred_numeral @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Suc_eq_numeral
% 0.29/0.57  thf(fact_6112_eq__numeral__Suc,axiom,
% 0.29/0.57      ! [K: num,N: nat] :
% 0.29/0.57        ( ( ( numeral_numeral_nat @ K )
% 0.29/0.57          = ( suc @ N ) )
% 0.29/0.57        = ( ( pred_numeral @ K )
% 0.29/0.57          = N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % eq_numeral_Suc
% 0.29/0.57  thf(fact_6113_concat__bit__nonnegative__iff,axiom,
% 0.29/0.57      ! [N: nat,K: int,L: int] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 0.29/0.57        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 0.29/0.57  
% 0.29/0.57  % concat_bit_nonnegative_iff
% 0.29/0.57  thf(fact_6114_concat__bit__negative__iff,axiom,
% 0.29/0.57      ! [N: nat,K: int,L: int] :
% 0.29/0.57        ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 0.29/0.57        = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % concat_bit_negative_iff
% 0.29/0.57  thf(fact_6115_dbl__inc__simps_I2_J,axiom,
% 0.29/0.57      ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 0.29/0.57      = one_one_complex ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(2)
% 0.29/0.57  thf(fact_6116_dbl__inc__simps_I2_J,axiom,
% 0.29/0.57      ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 0.29/0.57      = one_one_real ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(2)
% 0.29/0.57  thf(fact_6117_dbl__inc__simps_I2_J,axiom,
% 0.29/0.57      ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 0.29/0.57      = one_one_rat ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(2)
% 0.29/0.57  thf(fact_6118_dbl__inc__simps_I2_J,axiom,
% 0.29/0.57      ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 0.29/0.57      = one_one_int ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(2)
% 0.29/0.57  thf(fact_6119_dbl__inc__simps_I4_J,axiom,
% 0.29/0.57      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.57      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(4)
% 0.29/0.57  thf(fact_6120_dbl__inc__simps_I4_J,axiom,
% 0.29/0.57      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.57      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(4)
% 0.29/0.57  thf(fact_6121_dbl__inc__simps_I4_J,axiom,
% 0.29/0.57      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.57      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(4)
% 0.29/0.57  thf(fact_6122_dbl__inc__simps_I4_J,axiom,
% 0.29/0.57      ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.57      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(4)
% 0.29/0.57  thf(fact_6123_dbl__inc__simps_I4_J,axiom,
% 0.29/0.57      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.57      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(4)
% 0.29/0.57  thf(fact_6124_dbl__inc__simps_I5_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.29/0.57        = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(5)
% 0.29/0.57  thf(fact_6125_dbl__inc__simps_I5_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 0.29/0.57        = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(5)
% 0.29/0.57  thf(fact_6126_dbl__inc__simps_I5_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 0.29/0.57        = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(5)
% 0.29/0.57  thf(fact_6127_dbl__inc__simps_I5_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 0.29/0.57        = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(5)
% 0.29/0.57  thf(fact_6128_pred__numeral__simps_I3_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( pred_numeral @ ( bit1 @ K ) )
% 0.29/0.57        = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pred_numeral_simps(3)
% 0.29/0.57  thf(fact_6129_less__Suc__numeral,axiom,
% 0.29/0.57      ! [N: nat,K: num] :
% 0.29/0.57        ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.29/0.57        = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % less_Suc_numeral
% 0.29/0.57  thf(fact_6130_less__numeral__Suc,axiom,
% 0.29/0.57      ! [K: num,N: nat] :
% 0.29/0.57        ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.29/0.57        = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % less_numeral_Suc
% 0.29/0.57  thf(fact_6131_le__Suc__numeral,axiom,
% 0.29/0.57      ! [N: nat,K: num] :
% 0.29/0.57        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.29/0.57        = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % le_Suc_numeral
% 0.29/0.57  thf(fact_6132_le__numeral__Suc,axiom,
% 0.29/0.57      ! [K: num,N: nat] :
% 0.29/0.57        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.29/0.57        = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % le_numeral_Suc
% 0.29/0.57  thf(fact_6133_diff__Suc__numeral,axiom,
% 0.29/0.57      ! [N: nat,K: num] :
% 0.29/0.57        ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.29/0.57        = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % diff_Suc_numeral
% 0.29/0.57  thf(fact_6134_diff__numeral__Suc,axiom,
% 0.29/0.57      ! [K: num,N: nat] :
% 0.29/0.57        ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.29/0.57        = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % diff_numeral_Suc
% 0.29/0.57  thf(fact_6135_dbl__dec__simps_I2_J,axiom,
% 0.29/0.57      ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.29/0.57      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(2)
% 0.29/0.57  thf(fact_6136_dbl__dec__simps_I2_J,axiom,
% 0.29/0.57      ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.29/0.57      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(2)
% 0.29/0.57  thf(fact_6137_dbl__dec__simps_I2_J,axiom,
% 0.29/0.57      ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.29/0.57      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(2)
% 0.29/0.57  thf(fact_6138_dbl__dec__simps_I2_J,axiom,
% 0.29/0.57      ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.29/0.57      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(2)
% 0.29/0.57  thf(fact_6139_dbl__dec__simps_I2_J,axiom,
% 0.29/0.57      ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.29/0.57      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(2)
% 0.29/0.57  thf(fact_6140_dbl__dec__simps_I1_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.29/0.57        = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(1)
% 0.29/0.57  thf(fact_6141_dbl__dec__simps_I1_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.57        = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(1)
% 0.29/0.57  thf(fact_6142_dbl__dec__simps_I1_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.29/0.57        = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(1)
% 0.29/0.57  thf(fact_6143_dbl__dec__simps_I1_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.29/0.57        = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(1)
% 0.29/0.57  thf(fact_6144_dbl__dec__simps_I1_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.29/0.57        = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_simps(1)
% 0.29/0.57  thf(fact_6145_dbl__inc__simps_I1_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.29/0.57        = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(1)
% 0.29/0.57  thf(fact_6146_dbl__inc__simps_I1_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.57        = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(1)
% 0.29/0.57  thf(fact_6147_dbl__inc__simps_I1_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.29/0.57        = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(1)
% 0.29/0.57  thf(fact_6148_dbl__inc__simps_I1_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.29/0.57        = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(1)
% 0.29/0.57  thf(fact_6149_dbl__inc__simps_I1_J,axiom,
% 0.29/0.57      ! [K: num] :
% 0.29/0.57        ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.29/0.57        = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_simps(1)
% 0.29/0.57  thf(fact_6150_signed__take__bit__numeral__bit0,axiom,
% 0.29/0.57      ! [L: num,K: num] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.29/0.57        = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_numeral_bit0
% 0.29/0.57  thf(fact_6151_signed__take__bit__numeral__minus__bit0,axiom,
% 0.29/0.57      ! [L: num,K: num] :
% 0.29/0.57        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.29/0.57        = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % signed_take_bit_numeral_minus_bit0
% 0.29/0.57  thf(fact_6152_concat__bit__assoc,axiom,
% 0.29/0.57      ! [N: nat,K: int,M: nat,L: int,R2: int] :
% 0.29/0.57        ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
% 0.29/0.57        = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).
% 0.29/0.57  
% 0.29/0.57  % concat_bit_assoc
% 0.29/0.57  thf(fact_6153_numeral__eq__Suc,axiom,
% 0.29/0.57      ( numeral_numeral_nat
% 0.29/0.57      = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % numeral_eq_Suc
% 0.29/0.57  thf(fact_6154_pred__numeral__def,axiom,
% 0.29/0.57      ( pred_numeral
% 0.29/0.57      = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pred_numeral_def
% 0.29/0.57  thf(fact_6155_dbl__inc__def,axiom,
% 0.29/0.57      ( neg_nu8295874005876285629c_real
% 0.29/0.57      = ( ^ [X4: real] : ( plus_plus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_def
% 0.29/0.57  thf(fact_6156_dbl__inc__def,axiom,
% 0.29/0.57      ( neg_nu5219082963157363817nc_rat
% 0.29/0.57      = ( ^ [X4: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_def
% 0.29/0.57  thf(fact_6157_dbl__inc__def,axiom,
% 0.29/0.57      ( neg_nu5851722552734809277nc_int
% 0.29/0.57      = ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_def
% 0.29/0.57  thf(fact_6158_dbl__inc__def,axiom,
% 0.29/0.57      ( neg_nu8557863876264182079omplex
% 0.29/0.57      = ( ^ [X4: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_inc_def
% 0.29/0.57  thf(fact_6159_dbl__dec__def,axiom,
% 0.29/0.57      ( neg_nu6511756317524482435omplex
% 0.29/0.57      = ( ^ [X4: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_def
% 0.29/0.57  thf(fact_6160_dbl__dec__def,axiom,
% 0.29/0.57      ( neg_nu6075765906172075777c_real
% 0.29/0.57      = ( ^ [X4: real] : ( minus_minus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_def
% 0.29/0.57  thf(fact_6161_dbl__dec__def,axiom,
% 0.29/0.57      ( neg_nu3179335615603231917ec_rat
% 0.29/0.57      = ( ^ [X4: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_def
% 0.29/0.57  thf(fact_6162_dbl__dec__def,axiom,
% 0.29/0.57      ( neg_nu3811975205180677377ec_int
% 0.29/0.57      = ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dbl_dec_def
% 0.29/0.57  thf(fact_6163_translation__diff,axiom,
% 0.29/0.57      ! [A: real,S: set_real,T: set_real] :
% 0.29/0.57        ( ( image_real_real @ ( plus_plus_real @ A ) @ ( minus_minus_set_real @ S @ T ) )
% 0.29/0.57        = ( minus_minus_set_real @ ( image_real_real @ ( plus_plus_real @ A ) @ S ) @ ( image_real_real @ ( plus_plus_real @ A ) @ T ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % translation_diff
% 0.29/0.57  thf(fact_6164_translation__diff,axiom,
% 0.29/0.57      ! [A: rat,S: set_rat,T: set_rat] :
% 0.29/0.57        ( ( image_rat_rat @ ( plus_plus_rat @ A ) @ ( minus_minus_set_rat @ S @ T ) )
% 0.29/0.57        = ( minus_minus_set_rat @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ S ) @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ T ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % translation_diff
% 0.29/0.57  thf(fact_6165_translation__diff,axiom,
% 0.29/0.57      ! [A: int,S: set_int,T: set_int] :
% 0.29/0.57        ( ( image_int_int @ ( plus_plus_int @ A ) @ ( minus_minus_set_int @ S @ T ) )
% 0.29/0.57        = ( minus_minus_set_int @ ( image_int_int @ ( plus_plus_int @ A ) @ S ) @ ( image_int_int @ ( plus_plus_int @ A ) @ T ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % translation_diff
% 0.29/0.57  thf(fact_6166_translation__diff,axiom,
% 0.29/0.57      ! [A: complex,S: set_complex,T: set_complex] :
% 0.29/0.57        ( ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ ( minus_811609699411566653omplex @ S @ T ) )
% 0.29/0.57        = ( minus_811609699411566653omplex @ ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ S ) @ ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ T ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % translation_diff
% 0.29/0.57  thf(fact_6167_even__succ__mod__exp,axiom,
% 0.29/0.57      ! [A: nat,N: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57            = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_succ_mod_exp
% 0.29/0.57  thf(fact_6168_even__succ__mod__exp,axiom,
% 0.29/0.57      ! [A: int,N: nat] :
% 0.29/0.57        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_succ_mod_exp
% 0.29/0.57  thf(fact_6169_even__succ__mod__exp,axiom,
% 0.29/0.57      ! [A: code_integer,N: nat] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57            = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_succ_mod_exp
% 0.29/0.57  thf(fact_6170_even__succ__div__exp,axiom,
% 0.29/0.57      ! [A: code_integer,N: nat] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57            = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_succ_div_exp
% 0.29/0.57  thf(fact_6171_even__succ__div__exp,axiom,
% 0.29/0.57      ! [A: nat,N: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57         => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57            = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_succ_div_exp
% 0.29/0.57  thf(fact_6172_even__succ__div__exp,axiom,
% 0.29/0.57      ! [A: int,N: nat] :
% 0.29/0.57        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57         => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.57            = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_succ_div_exp
% 0.29/0.57  thf(fact_6173_log__base__10__eq1,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.57       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.57          = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % log_base_10_eq1
% 0.29/0.57  thf(fact_6174_add__scale__eq__noteq,axiom,
% 0.29/0.57      ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
% 0.29/0.57        ( ( R2 != zero_zero_complex )
% 0.29/0.57       => ( ( ( A = B )
% 0.29/0.57            & ( C != D ) )
% 0.29/0.57         => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
% 0.29/0.57           != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_scale_eq_noteq
% 0.29/0.57  thf(fact_6175_add__scale__eq__noteq,axiom,
% 0.29/0.57      ! [R2: real,A: real,B: real,C: real,D: real] :
% 0.29/0.57        ( ( R2 != zero_zero_real )
% 0.29/0.57       => ( ( ( A = B )
% 0.29/0.57            & ( C != D ) )
% 0.29/0.57         => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 0.29/0.57           != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_scale_eq_noteq
% 0.29/0.57  thf(fact_6176_add__scale__eq__noteq,axiom,
% 0.29/0.57      ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.57        ( ( R2 != zero_zero_rat )
% 0.29/0.57       => ( ( ( A = B )
% 0.29/0.57            & ( C != D ) )
% 0.29/0.57         => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 0.29/0.57           != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_scale_eq_noteq
% 0.29/0.57  thf(fact_6177_add__scale__eq__noteq,axiom,
% 0.29/0.57      ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.57        ( ( R2 != zero_zero_nat )
% 0.29/0.57       => ( ( ( A = B )
% 0.29/0.57            & ( C != D ) )
% 0.29/0.57         => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 0.29/0.57           != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_scale_eq_noteq
% 0.29/0.57  thf(fact_6178_add__scale__eq__noteq,axiom,
% 0.29/0.57      ! [R2: int,A: int,B: int,C: int,D: int] :
% 0.29/0.57        ( ( R2 != zero_zero_int )
% 0.29/0.57       => ( ( ( A = B )
% 0.29/0.57            & ( C != D ) )
% 0.29/0.57         => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 0.29/0.57           != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % add_scale_eq_noteq
% 0.29/0.57  thf(fact_6179_nat__dvd__1__iff__1,axiom,
% 0.29/0.57      ! [M: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.29/0.57        = ( M = one_one_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % nat_dvd_1_iff_1
% 0.29/0.57  thf(fact_6180_int__dvd__int__iff,axiom,
% 0.29/0.57      ! [M: nat,N: nat] :
% 0.29/0.57        ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.29/0.57        = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % int_dvd_int_iff
% 0.29/0.57  thf(fact_6181_exp__inj__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ( exp_real @ X )
% 0.29/0.57          = ( exp_real @ Y ) )
% 0.29/0.57        = ( X = Y ) ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_inj_iff
% 0.29/0.57  thf(fact_6182_dvd__0__right,axiom,
% 0.29/0.57      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_right
% 0.29/0.57  thf(fact_6183_dvd__0__right,axiom,
% 0.29/0.57      ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_right
% 0.29/0.57  thf(fact_6184_dvd__0__right,axiom,
% 0.29/0.57      ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_right
% 0.29/0.57  thf(fact_6185_dvd__0__right,axiom,
% 0.29/0.57      ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_right
% 0.29/0.57  thf(fact_6186_dvd__0__right,axiom,
% 0.29/0.57      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_right
% 0.29/0.57  thf(fact_6187_dvd__0__right,axiom,
% 0.29/0.57      ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_right
% 0.29/0.57  thf(fact_6188_dvd__0__left__iff,axiom,
% 0.29/0.57      ! [A: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.29/0.57        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_left_iff
% 0.29/0.57  thf(fact_6189_dvd__0__left__iff,axiom,
% 0.29/0.57      ! [A: complex] :
% 0.29/0.57        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.29/0.57        = ( A = zero_zero_complex ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_left_iff
% 0.29/0.57  thf(fact_6190_dvd__0__left__iff,axiom,
% 0.29/0.57      ! [A: real] :
% 0.29/0.57        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.29/0.57        = ( A = zero_zero_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_left_iff
% 0.29/0.57  thf(fact_6191_dvd__0__left__iff,axiom,
% 0.29/0.57      ! [A: rat] :
% 0.29/0.57        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.29/0.57        = ( A = zero_zero_rat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_left_iff
% 0.29/0.57  thf(fact_6192_dvd__0__left__iff,axiom,
% 0.29/0.57      ! [A: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.29/0.57        = ( A = zero_zero_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_left_iff
% 0.29/0.57  thf(fact_6193_dvd__0__left__iff,axiom,
% 0.29/0.57      ! [A: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.29/0.57        = ( A = zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_0_left_iff
% 0.29/0.57  thf(fact_6194_dvd__add__triv__right__iff,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 0.29/0.57        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_right_iff
% 0.29/0.57  thf(fact_6195_dvd__add__triv__right__iff,axiom,
% 0.29/0.57      ! [A: real,B: real] :
% 0.29/0.57        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.29/0.57        = ( dvd_dvd_real @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_right_iff
% 0.29/0.57  thf(fact_6196_dvd__add__triv__right__iff,axiom,
% 0.29/0.57      ! [A: rat,B: rat] :
% 0.29/0.57        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.29/0.57        = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_right_iff
% 0.29/0.57  thf(fact_6197_dvd__add__triv__right__iff,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.29/0.57        = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_right_iff
% 0.29/0.57  thf(fact_6198_dvd__add__triv__right__iff,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.29/0.57        = ( dvd_dvd_int @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_right_iff
% 0.29/0.57  thf(fact_6199_dvd__add__triv__right__iff,axiom,
% 0.29/0.57      ! [A: complex,B: complex] :
% 0.29/0.57        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ A ) )
% 0.29/0.57        = ( dvd_dvd_complex @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_right_iff
% 0.29/0.57  thf(fact_6200_dvd__add__triv__left__iff,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.29/0.57        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_left_iff
% 0.29/0.57  thf(fact_6201_dvd__add__triv__left__iff,axiom,
% 0.29/0.57      ! [A: real,B: real] :
% 0.29/0.57        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.29/0.57        = ( dvd_dvd_real @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_left_iff
% 0.29/0.57  thf(fact_6202_dvd__add__triv__left__iff,axiom,
% 0.29/0.57      ! [A: rat,B: rat] :
% 0.29/0.57        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.57        = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_left_iff
% 0.29/0.57  thf(fact_6203_dvd__add__triv__left__iff,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.57        = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_left_iff
% 0.29/0.57  thf(fact_6204_dvd__add__triv__left__iff,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.29/0.57        = ( dvd_dvd_int @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_left_iff
% 0.29/0.57  thf(fact_6205_dvd__add__triv__left__iff,axiom,
% 0.29/0.57      ! [A: complex,B: complex] :
% 0.29/0.57        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 0.29/0.57        = ( dvd_dvd_complex @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_triv_left_iff
% 0.29/0.57  thf(fact_6206_dvd__1__iff__1,axiom,
% 0.29/0.57      ! [M: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.29/0.57        = ( M
% 0.29/0.57          = ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_1_iff_1
% 0.29/0.57  thf(fact_6207_dvd__1__left,axiom,
% 0.29/0.57      ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_1_left
% 0.29/0.57  thf(fact_6208_div__dvd__div,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.29/0.57       => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.29/0.57         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 0.29/0.57            = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_dvd_div
% 0.29/0.57  thf(fact_6209_div__dvd__div,axiom,
% 0.29/0.57      ! [A: nat,B: nat,C: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ B )
% 0.29/0.57       => ( ( dvd_dvd_nat @ A @ C )
% 0.29/0.57         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 0.29/0.57            = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_dvd_div
% 0.29/0.57  thf(fact_6210_div__dvd__div,axiom,
% 0.29/0.57      ! [A: int,B: int,C: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ B )
% 0.29/0.57       => ( ( dvd_dvd_int @ A @ C )
% 0.29/0.57         => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 0.29/0.57            = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_dvd_div
% 0.29/0.57  thf(fact_6211_nat__mult__dvd__cancel__disj,axiom,
% 0.29/0.57      ! [K: nat,M: nat,N: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.57        = ( ( K = zero_zero_nat )
% 0.29/0.57          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % nat_mult_dvd_cancel_disj
% 0.29/0.57  thf(fact_6212_zdvd1__eq,axiom,
% 0.29/0.57      ! [X: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ X @ one_one_int )
% 0.29/0.57        = ( ( abs_abs_int @ X )
% 0.29/0.57          = one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zdvd1_eq
% 0.29/0.57  thf(fact_6213_exp__less__cancel__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.29/0.57        = ( ord_less_real @ X @ Y ) ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_less_cancel_iff
% 0.29/0.57  thf(fact_6214_exp__less__mono,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_real @ X @ Y )
% 0.29/0.57       => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_less_mono
% 0.29/0.57  thf(fact_6215_exp__le__cancel__iff,axiom,
% 0.29/0.57      ! [X: real,Y: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.29/0.57        = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_le_cancel_iff
% 0.29/0.57  thf(fact_6216_abs__exp__cancel,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( abs_abs_real @ ( exp_real @ X ) )
% 0.29/0.57        = ( exp_real @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % abs_exp_cancel
% 0.29/0.57  thf(fact_6217_ln__exp,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ln_ln_real @ ( exp_real @ X ) )
% 0.29/0.57        = X ) ).
% 0.29/0.57  
% 0.29/0.57  % ln_exp
% 0.29/0.57  thf(fact_6218_dvd__times__right__cancel__iff,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.57        ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.57       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 0.29/0.57          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_times_right_cancel_iff
% 0.29/0.57  thf(fact_6219_dvd__times__right__cancel__iff,axiom,
% 0.29/0.57      ! [A: nat,B: nat,C: nat] :
% 0.29/0.57        ( ( A != zero_zero_nat )
% 0.29/0.57       => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 0.29/0.57          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_times_right_cancel_iff
% 0.29/0.57  thf(fact_6220_dvd__times__right__cancel__iff,axiom,
% 0.29/0.57      ! [A: int,B: int,C: int] :
% 0.29/0.57        ( ( A != zero_zero_int )
% 0.29/0.57       => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 0.29/0.57          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_times_right_cancel_iff
% 0.29/0.57  thf(fact_6221_dvd__times__left__cancel__iff,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.57        ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.57       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 0.29/0.57          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_times_left_cancel_iff
% 0.29/0.57  thf(fact_6222_dvd__times__left__cancel__iff,axiom,
% 0.29/0.57      ! [A: nat,B: nat,C: nat] :
% 0.29/0.57        ( ( A != zero_zero_nat )
% 0.29/0.57       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 0.29/0.57          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_times_left_cancel_iff
% 0.29/0.57  thf(fact_6223_dvd__times__left__cancel__iff,axiom,
% 0.29/0.57      ! [A: int,B: int,C: int] :
% 0.29/0.57        ( ( A != zero_zero_int )
% 0.29/0.57       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 0.29/0.57          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_times_left_cancel_iff
% 0.29/0.57  thf(fact_6224_dvd__mult__cancel__right,axiom,
% 0.29/0.57      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.29/0.57        = ( ( C = zero_z3403309356797280102nteger )
% 0.29/0.57          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_cancel_right
% 0.29/0.57  thf(fact_6225_dvd__mult__cancel__right,axiom,
% 0.29/0.57      ! [A: complex,C: complex,B: complex] :
% 0.29/0.57        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.29/0.57        = ( ( C = zero_zero_complex )
% 0.29/0.57          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_cancel_right
% 0.29/0.57  thf(fact_6226_dvd__mult__cancel__right,axiom,
% 0.29/0.57      ! [A: real,C: real,B: real] :
% 0.29/0.57        ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.57        = ( ( C = zero_zero_real )
% 0.29/0.57          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_cancel_right
% 0.29/0.57  thf(fact_6227_dvd__mult__cancel__right,axiom,
% 0.29/0.57      ! [A: rat,C: rat,B: rat] :
% 0.29/0.57        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.57        = ( ( C = zero_zero_rat )
% 0.29/0.57          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_cancel_right
% 0.29/0.57  thf(fact_6228_dvd__mult__cancel__right,axiom,
% 0.29/0.57      ! [A: int,C: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.57        = ( ( C = zero_zero_int )
% 0.29/0.57          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_cancel_right
% 0.29/0.57  thf(fact_6229_dvd__mult__cancel__left,axiom,
% 0.29/0.57      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.29/0.57        = ( ( C = zero_z3403309356797280102nteger )
% 0.29/0.57          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_cancel_left
% 0.29/0.57  thf(fact_6230_dvd__mult__cancel__left,axiom,
% 0.29/0.57      ! [C: complex,A: complex,B: complex] :
% 0.29/0.57        ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.29/0.57        = ( ( C = zero_zero_complex )
% 0.29/0.57          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_cancel_left
% 0.29/0.57  thf(fact_6231_dvd__mult__cancel__left,axiom,
% 0.29/0.57      ! [C: real,A: real,B: real] :
% 0.29/0.57        ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.57        = ( ( C = zero_zero_real )
% 0.29/0.57          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_cancel_left
% 0.29/0.57  thf(fact_6232_dvd__mult__cancel__left,axiom,
% 0.29/0.57      ! [C: rat,A: rat,B: rat] :
% 0.29/0.57        ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.57        = ( ( C = zero_zero_rat )
% 0.29/0.57          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_cancel_left
% 0.29/0.57  thf(fact_6233_dvd__mult__cancel__left,axiom,
% 0.29/0.57      ! [C: int,A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.57        = ( ( C = zero_zero_int )
% 0.29/0.57          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_cancel_left
% 0.29/0.57  thf(fact_6234_unit__prod,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.57       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.57         => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_prod
% 0.29/0.57  thf(fact_6235_unit__prod,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.57       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.57         => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_prod
% 0.29/0.57  thf(fact_6236_unit__prod,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.57       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.57         => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_prod
% 0.29/0.57  thf(fact_6237_dvd__add__times__triv__right__iff,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 0.29/0.57        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_right_iff
% 0.29/0.57  thf(fact_6238_dvd__add__times__triv__right__iff,axiom,
% 0.29/0.57      ! [A: complex,B: complex,C: complex] :
% 0.29/0.57        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ ( times_times_complex @ C @ A ) ) )
% 0.29/0.57        = ( dvd_dvd_complex @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_right_iff
% 0.29/0.57  thf(fact_6239_dvd__add__times__triv__right__iff,axiom,
% 0.29/0.57      ! [A: real,B: real,C: real] :
% 0.29/0.57        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 0.29/0.57        = ( dvd_dvd_real @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_right_iff
% 0.29/0.57  thf(fact_6240_dvd__add__times__triv__right__iff,axiom,
% 0.29/0.57      ! [A: rat,B: rat,C: rat] :
% 0.29/0.57        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 0.29/0.57        = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_right_iff
% 0.29/0.57  thf(fact_6241_dvd__add__times__triv__right__iff,axiom,
% 0.29/0.57      ! [A: nat,B: nat,C: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 0.29/0.57        = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_right_iff
% 0.29/0.57  thf(fact_6242_dvd__add__times__triv__right__iff,axiom,
% 0.29/0.57      ! [A: int,B: int,C: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 0.29/0.57        = ( dvd_dvd_int @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_right_iff
% 0.29/0.57  thf(fact_6243_dvd__add__times__triv__left__iff,axiom,
% 0.29/0.57      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 0.29/0.57        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_left_iff
% 0.29/0.57  thf(fact_6244_dvd__add__times__triv__left__iff,axiom,
% 0.29/0.57      ! [A: complex,C: complex,B: complex] :
% 0.29/0.57        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ ( times_times_complex @ C @ A ) @ B ) )
% 0.29/0.57        = ( dvd_dvd_complex @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_left_iff
% 0.29/0.57  thf(fact_6245_dvd__add__times__triv__left__iff,axiom,
% 0.29/0.57      ! [A: real,C: real,B: real] :
% 0.29/0.57        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 0.29/0.57        = ( dvd_dvd_real @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_left_iff
% 0.29/0.57  thf(fact_6246_dvd__add__times__triv__left__iff,axiom,
% 0.29/0.57      ! [A: rat,C: rat,B: rat] :
% 0.29/0.57        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 0.29/0.57        = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_left_iff
% 0.29/0.57  thf(fact_6247_dvd__add__times__triv__left__iff,axiom,
% 0.29/0.57      ! [A: nat,C: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 0.29/0.57        = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_left_iff
% 0.29/0.57  thf(fact_6248_dvd__add__times__triv__left__iff,axiom,
% 0.29/0.57      ! [A: int,C: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 0.29/0.57        = ( dvd_dvd_int @ A @ B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_add_times_triv_left_iff
% 0.29/0.57  thf(fact_6249_dvd__mult__div__cancel,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.29/0.57       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 0.29/0.57          = B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_div_cancel
% 0.29/0.57  thf(fact_6250_dvd__mult__div__cancel,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ B )
% 0.29/0.57       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 0.29/0.57          = B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_div_cancel
% 0.29/0.57  thf(fact_6251_dvd__mult__div__cancel,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ B )
% 0.29/0.57       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 0.29/0.57          = B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_mult_div_cancel
% 0.29/0.57  thf(fact_6252_dvd__div__mult__self,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.29/0.57       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.29/0.57          = B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_div_mult_self
% 0.29/0.57  thf(fact_6253_dvd__div__mult__self,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ B )
% 0.29/0.57       => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.29/0.57          = B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_div_mult_self
% 0.29/0.57  thf(fact_6254_dvd__div__mult__self,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ B )
% 0.29/0.57       => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.29/0.57          = B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_div_mult_self
% 0.29/0.57  thf(fact_6255_unit__div__1__div__1,axiom,
% 0.29/0.57      ! [A: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.57       => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.29/0.57          = A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div_1_div_1
% 0.29/0.57  thf(fact_6256_unit__div__1__div__1,axiom,
% 0.29/0.57      ! [A: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.57       => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.29/0.57          = A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div_1_div_1
% 0.29/0.57  thf(fact_6257_unit__div__1__div__1,axiom,
% 0.29/0.57      ! [A: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.57       => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 0.29/0.57          = A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div_1_div_1
% 0.29/0.57  thf(fact_6258_unit__div__1__unit,axiom,
% 0.29/0.57      ! [A: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.57       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div_1_unit
% 0.29/0.57  thf(fact_6259_unit__div__1__unit,axiom,
% 0.29/0.57      ! [A: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.57       => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div_1_unit
% 0.29/0.57  thf(fact_6260_unit__div__1__unit,axiom,
% 0.29/0.57      ! [A: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.57       => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div_1_unit
% 0.29/0.57  thf(fact_6261_unit__div,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.57       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.57         => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div
% 0.29/0.57  thf(fact_6262_unit__div,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.57       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.57         => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div
% 0.29/0.57  thf(fact_6263_unit__div,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.57       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.57         => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div
% 0.29/0.57  thf(fact_6264_div__add,axiom,
% 0.29/0.57      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.29/0.57       => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.29/0.57         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.29/0.57            = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_add
% 0.29/0.57  thf(fact_6265_div__add,axiom,
% 0.29/0.57      ! [C: nat,A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ C @ A )
% 0.29/0.57       => ( ( dvd_dvd_nat @ C @ B )
% 0.29/0.57         => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.57            = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_add
% 0.29/0.57  thf(fact_6266_div__add,axiom,
% 0.29/0.57      ! [C: int,A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ C @ A )
% 0.29/0.57       => ( ( dvd_dvd_int @ C @ B )
% 0.29/0.57         => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.57            = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_add
% 0.29/0.57  thf(fact_6267_div__diff,axiom,
% 0.29/0.57      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.29/0.57       => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.29/0.57         => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.29/0.57            = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_diff
% 0.29/0.57  thf(fact_6268_div__diff,axiom,
% 0.29/0.57      ! [C: int,A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ C @ A )
% 0.29/0.57       => ( ( dvd_dvd_int @ C @ B )
% 0.29/0.57         => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.29/0.57            = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % div_diff
% 0.29/0.57  thf(fact_6269_dvd__imp__mod__0,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ B )
% 0.29/0.57       => ( ( modulo_modulo_nat @ B @ A )
% 0.29/0.57          = zero_zero_nat ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_imp_mod_0
% 0.29/0.57  thf(fact_6270_dvd__imp__mod__0,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ B )
% 0.29/0.57       => ( ( modulo_modulo_int @ B @ A )
% 0.29/0.57          = zero_zero_int ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_imp_mod_0
% 0.29/0.57  thf(fact_6271_dvd__imp__mod__0,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.29/0.57       => ( ( modulo364778990260209775nteger @ B @ A )
% 0.29/0.57          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.57  
% 0.29/0.57  % dvd_imp_mod_0
% 0.29/0.57  thf(fact_6272_exp__zero,axiom,
% 0.29/0.57      ( ( exp_complex @ zero_zero_complex )
% 0.29/0.57      = one_one_complex ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_zero
% 0.29/0.57  thf(fact_6273_exp__zero,axiom,
% 0.29/0.57      ( ( exp_real @ zero_zero_real )
% 0.29/0.57      = one_one_real ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_zero
% 0.29/0.57  thf(fact_6274_exp__eq__one__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ( exp_real @ X )
% 0.29/0.57          = one_one_real )
% 0.29/0.57        = ( X = zero_zero_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_eq_one_iff
% 0.29/0.57  thf(fact_6275_unit__div__mult__self,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.57       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.29/0.57          = B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div_mult_self
% 0.29/0.57  thf(fact_6276_unit__div__mult__self,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.57       => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.29/0.57          = B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div_mult_self
% 0.29/0.57  thf(fact_6277_unit__div__mult__self,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.57       => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.29/0.57          = B ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_div_mult_self
% 0.29/0.57  thf(fact_6278_unit__mult__div__div,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.57       => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.29/0.57          = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_mult_div_div
% 0.29/0.57  thf(fact_6279_unit__mult__div__div,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.57       => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.29/0.57          = ( divide_divide_nat @ B @ A ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_mult_div_div
% 0.29/0.57  thf(fact_6280_unit__mult__div__div,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.57       => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 0.29/0.57          = ( divide_divide_int @ B @ A ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % unit_mult_div_div
% 0.29/0.57  thf(fact_6281_even__Suc__Suc__iff,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 0.29/0.57        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_Suc_Suc_iff
% 0.29/0.57  thf(fact_6282_even__Suc,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 0.29/0.57        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_Suc
% 0.29/0.57  thf(fact_6283_pow__divides__pow__iff,axiom,
% 0.29/0.57      ! [N: nat,A: nat,B: nat] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57       => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.29/0.57          = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pow_divides_pow_iff
% 0.29/0.57  thf(fact_6284_pow__divides__pow__iff,axiom,
% 0.29/0.57      ! [N: nat,A: int,B: int] :
% 0.29/0.57        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.57       => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.29/0.57          = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % pow_divides_pow_iff
% 0.29/0.57  thf(fact_6285_one__less__exp__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 0.29/0.57        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % one_less_exp_iff
% 0.29/0.57  thf(fact_6286_exp__less__one__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 0.29/0.57        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_less_one_iff
% 0.29/0.57  thf(fact_6287_exp__le__one__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
% 0.29/0.57        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_le_one_iff
% 0.29/0.57  thf(fact_6288_one__le__exp__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
% 0.29/0.57        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % one_le_exp_iff
% 0.29/0.57  thf(fact_6289_exp__ln__iff,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ( exp_real @ ( ln_ln_real @ X ) )
% 0.29/0.57          = X )
% 0.29/0.57        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_ln_iff
% 0.29/0.57  thf(fact_6290_exp__ln,axiom,
% 0.29/0.57      ! [X: real] :
% 0.29/0.57        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.57       => ( ( exp_real @ ( ln_ln_real @ X ) )
% 0.29/0.57          = X ) ) ).
% 0.29/0.57  
% 0.29/0.57  % exp_ln
% 0.29/0.57  thf(fact_6291_even__mult__iff,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.29/0.57        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.57          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_mult_iff
% 0.29/0.57  thf(fact_6292_even__mult__iff,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 0.29/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.57          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_mult_iff
% 0.29/0.57  thf(fact_6293_even__mult__iff,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 0.29/0.57        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.57          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_mult_iff
% 0.29/0.57  thf(fact_6294_even__add,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.29/0.57        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.57          = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_add
% 0.29/0.57  thf(fact_6295_even__add,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.57          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_add
% 0.29/0.57  thf(fact_6296_even__add,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 0.29/0.57        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.57          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_add
% 0.29/0.57  thf(fact_6297_odd__add,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 0.29/0.57        = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.57         != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % odd_add
% 0.29/0.57  thf(fact_6298_odd__add,axiom,
% 0.29/0.57      ! [A: nat,B: nat] :
% 0.29/0.57        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 0.29/0.57        = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.57         != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % odd_add
% 0.29/0.57  thf(fact_6299_odd__add,axiom,
% 0.29/0.57      ! [A: int,B: int] :
% 0.29/0.57        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 0.29/0.57        = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.57         != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % odd_add
% 0.29/0.57  thf(fact_6300_power__minus__odd,axiom,
% 0.29/0.57      ! [N: nat,A: real] :
% 0.29/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.29/0.57          = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_odd
% 0.29/0.57  thf(fact_6301_power__minus__odd,axiom,
% 0.29/0.57      ! [N: nat,A: int] :
% 0.29/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.29/0.57          = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_odd
% 0.29/0.57  thf(fact_6302_power__minus__odd,axiom,
% 0.29/0.57      ! [N: nat,A: complex] :
% 0.29/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.29/0.57          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_odd
% 0.29/0.57  thf(fact_6303_power__minus__odd,axiom,
% 0.29/0.57      ! [N: nat,A: rat] :
% 0.29/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.29/0.57          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_odd
% 0.29/0.57  thf(fact_6304_power__minus__odd,axiom,
% 0.29/0.57      ! [N: nat,A: code_integer] :
% 0.29/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.29/0.57          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_minus_odd
% 0.29/0.57  thf(fact_6305_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.29/0.57      ! [N: nat,A: real] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.29/0.57          = ( power_power_real @ A @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Parity.ring_1_class.power_minus_even
% 0.29/0.57  thf(fact_6306_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.29/0.57      ! [N: nat,A: int] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.29/0.57          = ( power_power_int @ A @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Parity.ring_1_class.power_minus_even
% 0.29/0.57  thf(fact_6307_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.29/0.57      ! [N: nat,A: complex] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.29/0.57          = ( power_power_complex @ A @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Parity.ring_1_class.power_minus_even
% 0.29/0.57  thf(fact_6308_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.29/0.57      ! [N: nat,A: rat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.29/0.57          = ( power_power_rat @ A @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Parity.ring_1_class.power_minus_even
% 0.29/0.57  thf(fact_6309_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.29/0.57      ! [N: nat,A: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.29/0.57          = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % Parity.ring_1_class.power_minus_even
% 0.29/0.57  thf(fact_6310_even__mod__2__iff,axiom,
% 0.29/0.57      ! [A: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.57        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_mod_2_iff
% 0.29/0.57  thf(fact_6311_even__mod__2__iff,axiom,
% 0.29/0.57      ! [A: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.29/0.57        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_mod_2_iff
% 0.29/0.57  thf(fact_6312_even__mod__2__iff,axiom,
% 0.29/0.57      ! [A: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.29/0.57        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_mod_2_iff
% 0.29/0.57  thf(fact_6313_power__even__abs__numeral,axiom,
% 0.29/0.57      ! [W: num,A: rat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57       => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57          = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_even_abs_numeral
% 0.29/0.57  thf(fact_6314_power__even__abs__numeral,axiom,
% 0.29/0.57      ! [W: num,A: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57       => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57          = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_even_abs_numeral
% 0.29/0.57  thf(fact_6315_power__even__abs__numeral,axiom,
% 0.29/0.57      ! [W: num,A: real] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57       => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57          = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_even_abs_numeral
% 0.29/0.57  thf(fact_6316_power__even__abs__numeral,axiom,
% 0.29/0.57      ! [W: num,A: int] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57       => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57          = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_even_abs_numeral
% 0.29/0.57  thf(fact_6317_odd__Suc__div__two,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % odd_Suc_div_two
% 0.29/0.57  thf(fact_6318_even__Suc__div__two,axiom,
% 0.29/0.57      ! [N: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57       => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.57          = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_Suc_div_two
% 0.29/0.57  thf(fact_6319_zero__le__power__eq__numeral,axiom,
% 0.29/0.57      ! [A: real,W: num] :
% 0.29/0.57        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.29/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zero_le_power_eq_numeral
% 0.29/0.57  thf(fact_6320_zero__le__power__eq__numeral,axiom,
% 0.29/0.57      ! [A: rat,W: num] :
% 0.29/0.57        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.29/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zero_le_power_eq_numeral
% 0.29/0.57  thf(fact_6321_zero__le__power__eq__numeral,axiom,
% 0.29/0.57      ! [A: int,W: num] :
% 0.29/0.57        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.29/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % zero_le_power_eq_numeral
% 0.29/0.57  thf(fact_6322_power__less__zero__eq__numeral,axiom,
% 0.29/0.57      ! [A: real,W: num] :
% 0.29/0.57        ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.29/0.57        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_less_zero_eq_numeral
% 0.29/0.57  thf(fact_6323_power__less__zero__eq__numeral,axiom,
% 0.29/0.57      ! [A: rat,W: num] :
% 0.29/0.57        ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.29/0.57        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_less_zero_eq_numeral
% 0.29/0.57  thf(fact_6324_power__less__zero__eq__numeral,axiom,
% 0.29/0.57      ! [A: int,W: num] :
% 0.29/0.57        ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.29/0.57        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.57          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_less_zero_eq_numeral
% 0.29/0.57  thf(fact_6325_power__less__zero__eq,axiom,
% 0.29/0.57      ! [A: real,N: nat] :
% 0.29/0.57        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.29/0.57        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_less_zero_eq
% 0.29/0.57  thf(fact_6326_power__less__zero__eq,axiom,
% 0.29/0.57      ! [A: rat,N: nat] :
% 0.29/0.57        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.29/0.57        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_less_zero_eq
% 0.29/0.57  thf(fact_6327_power__less__zero__eq,axiom,
% 0.29/0.57      ! [A: int,N: nat] :
% 0.29/0.57        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.29/0.57        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.57          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % power_less_zero_eq
% 0.29/0.57  thf(fact_6328_even__plus__one__iff,axiom,
% 0.29/0.57      ! [A: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 0.29/0.57        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_plus_one_iff
% 0.29/0.57  thf(fact_6329_even__plus__one__iff,axiom,
% 0.29/0.57      ! [A: nat] :
% 0.29/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 0.29/0.57        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_plus_one_iff
% 0.29/0.57  thf(fact_6330_even__plus__one__iff,axiom,
% 0.29/0.57      ! [A: int] :
% 0.29/0.57        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 0.29/0.57        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.57  
% 0.29/0.57  % even_plus_one_iff
% 0.29/0.57  thf(fact_6331_even__diff,axiom,
% 0.29/0.57      ! [A: code_integer,B: code_integer] :
% 0.29/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.29/0.57        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_diff
% 0.29/0.58  thf(fact_6332_even__diff,axiom,
% 0.29/0.58      ! [A: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 0.29/0.58        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_diff
% 0.29/0.58  thf(fact_6333_neg__one__even__power,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.29/0.58          = one_one_real ) ) ).
% 0.29/0.58  
% 0.29/0.58  % neg_one_even_power
% 0.29/0.58  thf(fact_6334_neg__one__even__power,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.29/0.58          = one_one_int ) ) ).
% 0.29/0.58  
% 0.29/0.58  % neg_one_even_power
% 0.29/0.58  thf(fact_6335_neg__one__even__power,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.29/0.58          = one_one_complex ) ) ).
% 0.29/0.58  
% 0.29/0.58  % neg_one_even_power
% 0.29/0.58  thf(fact_6336_neg__one__even__power,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.29/0.58          = one_one_rat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % neg_one_even_power
% 0.29/0.58  thf(fact_6337_neg__one__even__power,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.29/0.58          = one_one_Code_integer ) ) ).
% 0.29/0.58  
% 0.29/0.58  % neg_one_even_power
% 0.29/0.58  thf(fact_6338_neg__one__odd__power,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.29/0.58          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % neg_one_odd_power
% 0.29/0.58  thf(fact_6339_neg__one__odd__power,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.29/0.58          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % neg_one_odd_power
% 0.29/0.58  thf(fact_6340_neg__one__odd__power,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.29/0.58          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % neg_one_odd_power
% 0.29/0.58  thf(fact_6341_neg__one__odd__power,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.29/0.58          = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % neg_one_odd_power
% 0.29/0.58  thf(fact_6342_neg__one__odd__power,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.29/0.58          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % neg_one_odd_power
% 0.29/0.58  thf(fact_6343_even__of__nat,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.29/0.58        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_of_nat
% 0.29/0.58  thf(fact_6344_even__of__nat,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.29/0.58        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_of_nat
% 0.29/0.58  thf(fact_6345_even__of__nat,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.29/0.58        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_of_nat
% 0.29/0.58  thf(fact_6346_odd__Suc__minus__one,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.29/0.58          = N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_Suc_minus_one
% 0.29/0.58  thf(fact_6347_even__diff__nat,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.58        = ( ( ord_less_nat @ M @ N )
% 0.29/0.58          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_diff_nat
% 0.29/0.58  thf(fact_6348_zero__less__power__eq__numeral,axiom,
% 0.29/0.58      ! [A: real,W: num] :
% 0.29/0.58        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.29/0.58        = ( ( ( numeral_numeral_nat @ W )
% 0.29/0.58            = zero_zero_nat )
% 0.29/0.58          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58            & ( A != zero_zero_real ) )
% 0.29/0.58          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_power_eq_numeral
% 0.29/0.58  thf(fact_6349_zero__less__power__eq__numeral,axiom,
% 0.29/0.58      ! [A: rat,W: num] :
% 0.29/0.58        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.29/0.58        = ( ( ( numeral_numeral_nat @ W )
% 0.29/0.58            = zero_zero_nat )
% 0.29/0.58          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58            & ( A != zero_zero_rat ) )
% 0.29/0.58          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_power_eq_numeral
% 0.29/0.58  thf(fact_6350_zero__less__power__eq__numeral,axiom,
% 0.29/0.58      ! [A: int,W: num] :
% 0.29/0.58        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.29/0.58        = ( ( ( numeral_numeral_nat @ W )
% 0.29/0.58            = zero_zero_nat )
% 0.29/0.58          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58            & ( A != zero_zero_int ) )
% 0.29/0.58          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_power_eq_numeral
% 0.29/0.58  thf(fact_6351_even__succ__div__two,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_succ_div_two
% 0.29/0.58  thf(fact_6352_even__succ__div__two,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_succ_div_two
% 0.29/0.58  thf(fact_6353_even__succ__div__two,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_succ_div_two
% 0.29/0.58  thf(fact_6354_odd__succ__div__two,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_succ_div_two
% 0.29/0.58  thf(fact_6355_odd__succ__div__two,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58          = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_succ_div_two
% 0.29/0.58  thf(fact_6356_odd__succ__div__two,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58          = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_succ_div_two
% 0.29/0.58  thf(fact_6357_even__succ__div__2,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_succ_div_2
% 0.29/0.58  thf(fact_6358_even__succ__div__2,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_succ_div_2
% 0.29/0.58  thf(fact_6359_even__succ__div__2,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_succ_div_2
% 0.29/0.58  thf(fact_6360_even__power,axiom,
% 0.29/0.58      ! [A: code_integer,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.29/0.58        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_power
% 0.29/0.58  thf(fact_6361_even__power,axiom,
% 0.29/0.58      ! [A: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 0.29/0.58        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_power
% 0.29/0.58  thf(fact_6362_even__power,axiom,
% 0.29/0.58      ! [A: int,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 0.29/0.58        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_power
% 0.29/0.58  thf(fact_6363_odd__two__times__div__two__nat,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.58          = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_two_times_div_two_nat
% 0.29/0.58  thf(fact_6364_odd__two__times__div__two__succ,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 0.29/0.58          = A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_two_times_div_two_succ
% 0.29/0.58  thf(fact_6365_odd__two__times__div__two__succ,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 0.29/0.58          = A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_two_times_div_two_succ
% 0.29/0.58  thf(fact_6366_odd__two__times__div__two__succ,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 0.29/0.58          = A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_two_times_div_two_succ
% 0.29/0.58  thf(fact_6367_power__le__zero__eq__numeral,axiom,
% 0.29/0.58      ! [A: real,W: num] :
% 0.29/0.58        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.29/0.58        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.29/0.58            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58              & ( A = zero_zero_real ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_zero_eq_numeral
% 0.29/0.58  thf(fact_6368_power__le__zero__eq__numeral,axiom,
% 0.29/0.58      ! [A: rat,W: num] :
% 0.29/0.58        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.29/0.58        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.29/0.58            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58              & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_zero_eq_numeral
% 0.29/0.58  thf(fact_6369_power__le__zero__eq__numeral,axiom,
% 0.29/0.58      ! [A: int,W: num] :
% 0.29/0.58        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.29/0.58        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.29/0.58            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.29/0.58              & ( A = zero_zero_int ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_zero_eq_numeral
% 0.29/0.58  thf(fact_6370_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 0.29/0.58        = ( N = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % semiring_parity_class.even_mask_iff
% 0.29/0.58  thf(fact_6371_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 0.29/0.58        = ( N = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % semiring_parity_class.even_mask_iff
% 0.29/0.58  thf(fact_6372_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.29/0.58        = ( N = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % semiring_parity_class.even_mask_iff
% 0.29/0.58  thf(fact_6373_of__real__exp,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ( ( real_V1803761363581548252l_real @ ( exp_real @ X ) )
% 0.29/0.58        = ( exp_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_real_exp
% 0.29/0.58  thf(fact_6374_of__real__exp,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ( ( real_V4546457046886955230omplex @ ( exp_real @ X ) )
% 0.29/0.58        = ( exp_complex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_real_exp
% 0.29/0.58  thf(fact_6375_of__nat__dvd__iff,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.29/0.58        = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_dvd_iff
% 0.29/0.58  thf(fact_6376_of__nat__dvd__iff,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.29/0.58        = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_dvd_iff
% 0.29/0.58  thf(fact_6377_of__nat__dvd__iff,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.29/0.58        = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_dvd_iff
% 0.29/0.58  thf(fact_6378_norm__exp,axiom,
% 0.29/0.58      ! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % norm_exp
% 0.29/0.58  thf(fact_6379_norm__exp,axiom,
% 0.29/0.58      ! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % norm_exp
% 0.29/0.58  thf(fact_6380_exp__less__cancel,axiom,
% 0.29/0.58      ! [X: real,Y: real] :
% 0.29/0.58        ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.29/0.58       => ( ord_less_real @ X @ Y ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_less_cancel
% 0.29/0.58  thf(fact_6381_dvd__0__left,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.29/0.58       => ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_0_left
% 0.29/0.58  thf(fact_6382_dvd__0__left,axiom,
% 0.29/0.58      ! [A: complex] :
% 0.29/0.58        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.29/0.58       => ( A = zero_zero_complex ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_0_left
% 0.29/0.58  thf(fact_6383_dvd__0__left,axiom,
% 0.29/0.58      ! [A: real] :
% 0.29/0.58        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.29/0.58       => ( A = zero_zero_real ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_0_left
% 0.29/0.58  thf(fact_6384_dvd__0__left,axiom,
% 0.29/0.58      ! [A: rat] :
% 0.29/0.58        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.29/0.58       => ( A = zero_zero_rat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_0_left
% 0.29/0.58  thf(fact_6385_dvd__0__left,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.29/0.58       => ( A = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_0_left
% 0.29/0.58  thf(fact_6386_dvd__0__left,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.29/0.58       => ( A = zero_zero_int ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_0_left
% 0.29/0.58  thf(fact_6387_dvd__field__iff,axiom,
% 0.29/0.58      ( dvd_dvd_complex
% 0.29/0.58      = ( ^ [A3: complex,B4: complex] :
% 0.29/0.58            ( ( A3 = zero_zero_complex )
% 0.29/0.58           => ( B4 = zero_zero_complex ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_field_iff
% 0.29/0.58  thf(fact_6388_dvd__field__iff,axiom,
% 0.29/0.58      ( dvd_dvd_real
% 0.29/0.58      = ( ^ [A3: real,B4: real] :
% 0.29/0.58            ( ( A3 = zero_zero_real )
% 0.29/0.58           => ( B4 = zero_zero_real ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_field_iff
% 0.29/0.58  thf(fact_6389_dvd__field__iff,axiom,
% 0.29/0.58      ( dvd_dvd_rat
% 0.29/0.58      = ( ^ [A3: rat,B4: rat] :
% 0.29/0.58            ( ( A3 = zero_zero_rat )
% 0.29/0.58           => ( B4 = zero_zero_rat ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_field_iff
% 0.29/0.58  thf(fact_6390_exp__not__eq__zero,axiom,
% 0.29/0.58      ! [X: complex] :
% 0.29/0.58        ( ( exp_complex @ X )
% 0.29/0.58       != zero_zero_complex ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_not_eq_zero
% 0.29/0.58  thf(fact_6391_exp__not__eq__zero,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ( ( exp_real @ X )
% 0.29/0.58       != zero_zero_real ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_not_eq_zero
% 0.29/0.58  thf(fact_6392_exp__times__arg__commute,axiom,
% 0.29/0.58      ! [A2: complex] :
% 0.29/0.58        ( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
% 0.29/0.58        = ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_times_arg_commute
% 0.29/0.58  thf(fact_6393_exp__times__arg__commute,axiom,
% 0.29/0.58      ! [A2: real] :
% 0.29/0.58        ( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
% 0.29/0.58        = ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_times_arg_commute
% 0.29/0.58  thf(fact_6394_dvd__unit__imp__unit,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58         => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_unit_imp_unit
% 0.29/0.58  thf(fact_6395_dvd__unit__imp__unit,axiom,
% 0.29/0.58      ! [A: nat,B: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58         => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_unit_imp_unit
% 0.29/0.58  thf(fact_6396_dvd__unit__imp__unit,axiom,
% 0.29/0.58      ! [A: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58         => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_unit_imp_unit
% 0.29/0.58  thf(fact_6397_unit__imp__dvd,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_imp_dvd
% 0.29/0.58  thf(fact_6398_unit__imp__dvd,axiom,
% 0.29/0.58      ! [B: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_imp_dvd
% 0.29/0.58  thf(fact_6399_unit__imp__dvd,axiom,
% 0.29/0.58      ! [B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( dvd_dvd_int @ B @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_imp_dvd
% 0.29/0.58  thf(fact_6400_one__dvd,axiom,
% 0.29/0.58      ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 0.29/0.58  
% 0.29/0.58  % one_dvd
% 0.29/0.58  thf(fact_6401_one__dvd,axiom,
% 0.29/0.58      ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 0.29/0.58  
% 0.29/0.58  % one_dvd
% 0.29/0.58  thf(fact_6402_one__dvd,axiom,
% 0.29/0.58      ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 0.29/0.58  
% 0.29/0.58  % one_dvd
% 0.29/0.58  thf(fact_6403_one__dvd,axiom,
% 0.29/0.58      ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 0.29/0.58  
% 0.29/0.58  % one_dvd
% 0.29/0.58  thf(fact_6404_one__dvd,axiom,
% 0.29/0.58      ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 0.29/0.58  
% 0.29/0.58  % one_dvd
% 0.29/0.58  thf(fact_6405_one__dvd,axiom,
% 0.29/0.58      ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 0.29/0.58  
% 0.29/0.58  % one_dvd
% 0.29/0.58  thf(fact_6406_dvd__add__right__iff,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_right_iff
% 0.29/0.58  thf(fact_6407_dvd__add__right__iff,axiom,
% 0.29/0.58      ! [A: real,B: real,C: real] :
% 0.29/0.58        ( ( dvd_dvd_real @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_real @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_right_iff
% 0.29/0.58  thf(fact_6408_dvd__add__right__iff,axiom,
% 0.29/0.58      ! [A: rat,B: rat,C: rat] :
% 0.29/0.58        ( ( dvd_dvd_rat @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_right_iff
% 0.29/0.58  thf(fact_6409_dvd__add__right__iff,axiom,
% 0.29/0.58      ! [A: nat,B: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_right_iff
% 0.29/0.58  thf(fact_6410_dvd__add__right__iff,axiom,
% 0.29/0.58      ! [A: int,B: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_right_iff
% 0.29/0.58  thf(fact_6411_dvd__add__right__iff,axiom,
% 0.29/0.58      ! [A: complex,B: complex,C: complex] :
% 0.29/0.58        ( ( dvd_dvd_complex @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_complex @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_right_iff
% 0.29/0.58  thf(fact_6412_dvd__add__left__iff,axiom,
% 0.29/0.58      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ A @ C )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_left_iff
% 0.29/0.58  thf(fact_6413_dvd__add__left__iff,axiom,
% 0.29/0.58      ! [A: real,C: real,B: real] :
% 0.29/0.58        ( ( dvd_dvd_real @ A @ C )
% 0.29/0.58       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_left_iff
% 0.29/0.58  thf(fact_6414_dvd__add__left__iff,axiom,
% 0.29/0.58      ! [A: rat,C: rat,B: rat] :
% 0.29/0.58        ( ( dvd_dvd_rat @ A @ C )
% 0.29/0.58       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_left_iff
% 0.29/0.58  thf(fact_6415_dvd__add__left__iff,axiom,
% 0.29/0.58      ! [A: nat,C: nat,B: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ A @ C )
% 0.29/0.58       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_left_iff
% 0.29/0.58  thf(fact_6416_dvd__add__left__iff,axiom,
% 0.29/0.58      ! [A: int,C: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ C )
% 0.29/0.58       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_left_iff
% 0.29/0.58  thf(fact_6417_dvd__add__left__iff,axiom,
% 0.29/0.58      ! [A: complex,C: complex,B: complex] :
% 0.29/0.58        ( ( dvd_dvd_complex @ A @ C )
% 0.29/0.58       => ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add_left_iff
% 0.29/0.58  thf(fact_6418_dvd__add,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.29/0.58         => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add
% 0.29/0.58  thf(fact_6419_dvd__add,axiom,
% 0.29/0.58      ! [A: real,B: real,C: real] :
% 0.29/0.58        ( ( dvd_dvd_real @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_real @ A @ C )
% 0.29/0.58         => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add
% 0.29/0.58  thf(fact_6420_dvd__add,axiom,
% 0.29/0.58      ! [A: rat,B: rat,C: rat] :
% 0.29/0.58        ( ( dvd_dvd_rat @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_rat @ A @ C )
% 0.29/0.58         => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add
% 0.29/0.58  thf(fact_6421_dvd__add,axiom,
% 0.29/0.58      ! [A: nat,B: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_nat @ A @ C )
% 0.29/0.58         => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add
% 0.29/0.58  thf(fact_6422_dvd__add,axiom,
% 0.29/0.58      ! [A: int,B: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_int @ A @ C )
% 0.29/0.58         => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add
% 0.29/0.58  thf(fact_6423_dvd__add,axiom,
% 0.29/0.58      ! [A: complex,B: complex,C: complex] :
% 0.29/0.58        ( ( dvd_dvd_complex @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_complex @ A @ C )
% 0.29/0.58         => ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_add
% 0.29/0.58  thf(fact_6424_dvd__div__eq__iff,axiom,
% 0.29/0.58      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.29/0.58         => ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.29/0.58              = ( divide6298287555418463151nteger @ B @ C ) )
% 0.29/0.58            = ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_iff
% 0.29/0.58  thf(fact_6425_dvd__div__eq__iff,axiom,
% 0.29/0.58      ! [C: complex,A: complex,B: complex] :
% 0.29/0.58        ( ( dvd_dvd_complex @ C @ A )
% 0.29/0.58       => ( ( dvd_dvd_complex @ C @ B )
% 0.29/0.58         => ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.29/0.58              = ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.58            = ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_iff
% 0.29/0.58  thf(fact_6426_dvd__div__eq__iff,axiom,
% 0.29/0.58      ! [C: real,A: real,B: real] :
% 0.29/0.58        ( ( dvd_dvd_real @ C @ A )
% 0.29/0.58       => ( ( dvd_dvd_real @ C @ B )
% 0.29/0.58         => ( ( ( divide_divide_real @ A @ C )
% 0.29/0.58              = ( divide_divide_real @ B @ C ) )
% 0.29/0.58            = ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_iff
% 0.29/0.58  thf(fact_6427_dvd__div__eq__iff,axiom,
% 0.29/0.58      ! [C: rat,A: rat,B: rat] :
% 0.29/0.58        ( ( dvd_dvd_rat @ C @ A )
% 0.29/0.58       => ( ( dvd_dvd_rat @ C @ B )
% 0.29/0.58         => ( ( ( divide_divide_rat @ A @ C )
% 0.29/0.58              = ( divide_divide_rat @ B @ C ) )
% 0.29/0.58            = ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_iff
% 0.29/0.58  thf(fact_6428_dvd__div__eq__iff,axiom,
% 0.29/0.58      ! [C: nat,A: nat,B: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ C @ A )
% 0.29/0.58       => ( ( dvd_dvd_nat @ C @ B )
% 0.29/0.58         => ( ( ( divide_divide_nat @ A @ C )
% 0.29/0.58              = ( divide_divide_nat @ B @ C ) )
% 0.29/0.58            = ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_iff
% 0.29/0.58  thf(fact_6429_dvd__div__eq__iff,axiom,
% 0.29/0.58      ! [C: int,A: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ C @ A )
% 0.29/0.58       => ( ( dvd_dvd_int @ C @ B )
% 0.29/0.58         => ( ( ( divide_divide_int @ A @ C )
% 0.29/0.58              = ( divide_divide_int @ B @ C ) )
% 0.29/0.58            = ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_iff
% 0.29/0.58  thf(fact_6430_dvd__div__eq__cancel,axiom,
% 0.29/0.58      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.29/0.58        ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.29/0.58          = ( divide6298287555418463151nteger @ B @ C ) )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ C @ A )
% 0.29/0.58         => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.29/0.58           => ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_cancel
% 0.29/0.58  thf(fact_6431_dvd__div__eq__cancel,axiom,
% 0.29/0.58      ! [A: complex,C: complex,B: complex] :
% 0.29/0.58        ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.29/0.58          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.58       => ( ( dvd_dvd_complex @ C @ A )
% 0.29/0.58         => ( ( dvd_dvd_complex @ C @ B )
% 0.29/0.58           => ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_cancel
% 0.29/0.58  thf(fact_6432_dvd__div__eq__cancel,axiom,
% 0.29/0.58      ! [A: real,C: real,B: real] :
% 0.29/0.58        ( ( ( divide_divide_real @ A @ C )
% 0.29/0.58          = ( divide_divide_real @ B @ C ) )
% 0.29/0.58       => ( ( dvd_dvd_real @ C @ A )
% 0.29/0.58         => ( ( dvd_dvd_real @ C @ B )
% 0.29/0.58           => ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_cancel
% 0.29/0.58  thf(fact_6433_dvd__div__eq__cancel,axiom,
% 0.29/0.58      ! [A: rat,C: rat,B: rat] :
% 0.29/0.58        ( ( ( divide_divide_rat @ A @ C )
% 0.29/0.58          = ( divide_divide_rat @ B @ C ) )
% 0.29/0.58       => ( ( dvd_dvd_rat @ C @ A )
% 0.29/0.58         => ( ( dvd_dvd_rat @ C @ B )
% 0.29/0.58           => ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_cancel
% 0.29/0.58  thf(fact_6434_dvd__div__eq__cancel,axiom,
% 0.29/0.58      ! [A: nat,C: nat,B: nat] :
% 0.29/0.58        ( ( ( divide_divide_nat @ A @ C )
% 0.29/0.58          = ( divide_divide_nat @ B @ C ) )
% 0.29/0.58       => ( ( dvd_dvd_nat @ C @ A )
% 0.29/0.58         => ( ( dvd_dvd_nat @ C @ B )
% 0.29/0.58           => ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_cancel
% 0.29/0.58  thf(fact_6435_dvd__div__eq__cancel,axiom,
% 0.29/0.58      ! [A: int,C: int,B: int] :
% 0.29/0.58        ( ( ( divide_divide_int @ A @ C )
% 0.29/0.58          = ( divide_divide_int @ B @ C ) )
% 0.29/0.58       => ( ( dvd_dvd_int @ C @ A )
% 0.29/0.58         => ( ( dvd_dvd_int @ C @ B )
% 0.29/0.58           => ( A = B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_cancel
% 0.29/0.58  thf(fact_6436_div__div__div__same,axiom,
% 0.29/0.58      ! [D: code_integer,B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ D @ B )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.29/0.58         => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 0.29/0.58            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_div_div_same
% 0.29/0.58  thf(fact_6437_div__div__div__same,axiom,
% 0.29/0.58      ! [D: nat,B: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ D @ B )
% 0.29/0.58       => ( ( dvd_dvd_nat @ B @ A )
% 0.29/0.58         => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 0.29/0.58            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_div_div_same
% 0.29/0.58  thf(fact_6438_div__div__div__same,axiom,
% 0.29/0.58      ! [D: int,B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ D @ B )
% 0.29/0.58       => ( ( dvd_dvd_int @ B @ A )
% 0.29/0.58         => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 0.29/0.58            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_div_div_same
% 0.29/0.58  thf(fact_6439_dvd__power__same,axiom,
% 0.29/0.58      ! [X: code_integer,Y: code_integer,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ X @ Y )
% 0.29/0.58       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_same
% 0.29/0.58  thf(fact_6440_dvd__power__same,axiom,
% 0.29/0.58      ! [X: nat,Y: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ X @ Y )
% 0.29/0.58       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_same
% 0.29/0.58  thf(fact_6441_dvd__power__same,axiom,
% 0.29/0.58      ! [X: real,Y: real,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_real @ X @ Y )
% 0.29/0.58       => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_same
% 0.29/0.58  thf(fact_6442_dvd__power__same,axiom,
% 0.29/0.58      ! [X: int,Y: int,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_int @ X @ Y )
% 0.29/0.58       => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_same
% 0.29/0.58  thf(fact_6443_dvd__power__same,axiom,
% 0.29/0.58      ! [X: complex,Y: complex,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_complex @ X @ Y )
% 0.29/0.58       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_same
% 0.29/0.58  thf(fact_6444_gcd__nat_Oextremum__uniqueI,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.29/0.58       => ( A = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % gcd_nat.extremum_uniqueI
% 0.29/0.58  thf(fact_6445_gcd__nat_Onot__eq__extremum,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( A != zero_zero_nat )
% 0.29/0.58        = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
% 0.29/0.58          & ( A != zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % gcd_nat.not_eq_extremum
% 0.29/0.58  thf(fact_6446_gcd__nat_Oextremum__unique,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.29/0.58        = ( A = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % gcd_nat.extremum_unique
% 0.29/0.58  thf(fact_6447_gcd__nat_Oextremum__strict,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.29/0.58          & ( zero_zero_nat != A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % gcd_nat.extremum_strict
% 0.29/0.58  thf(fact_6448_gcd__nat_Oextremum,axiom,
% 0.29/0.58      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % gcd_nat.extremum
% 0.29/0.58  thf(fact_6449_ln__unique,axiom,
% 0.29/0.58      ! [Y: real,X: real] :
% 0.29/0.58        ( ( ( exp_real @ Y )
% 0.29/0.58          = X )
% 0.29/0.58       => ( ( ln_ln_real @ X )
% 0.29/0.58          = Y ) ) ).
% 0.29/0.58  
% 0.29/0.58  % ln_unique
% 0.29/0.58  thf(fact_6450_dvd__diff__nat,axiom,
% 0.29/0.58      ! [K: nat,M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ K @ M )
% 0.29/0.58       => ( ( dvd_dvd_nat @ K @ N )
% 0.29/0.58         => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_diff_nat
% 0.29/0.58  thf(fact_6451_zdvd__zdiffD,axiom,
% 0.29/0.58      ! [K: int,M: int,N: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 0.29/0.58       => ( ( dvd_dvd_int @ K @ N )
% 0.29/0.58         => ( dvd_dvd_int @ K @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zdvd_zdiffD
% 0.29/0.58  thf(fact_6452_zdvd__antisym__abs,axiom,
% 0.29/0.58      ! [A: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ B )
% 0.29/0.58       => ( ( dvd_dvd_int @ B @ A )
% 0.29/0.58         => ( ( abs_abs_int @ A )
% 0.29/0.58            = ( abs_abs_int @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zdvd_antisym_abs
% 0.29/0.58  thf(fact_6453_not__exp__less__zero,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 0.29/0.58  
% 0.29/0.58  % not_exp_less_zero
% 0.29/0.58  thf(fact_6454_exp__gt__zero,axiom,
% 0.29/0.58      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_gt_zero
% 0.29/0.58  thf(fact_6455_exp__total,axiom,
% 0.29/0.58      ! [Y: real] :
% 0.29/0.58        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.58       => ? [X3: real] :
% 0.29/0.58            ( ( exp_real @ X3 )
% 0.29/0.58            = Y ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_total
% 0.29/0.58  thf(fact_6456_not__exp__le__zero,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 0.29/0.58  
% 0.29/0.58  % not_exp_le_zero
% 0.29/0.58  thf(fact_6457_exp__ge__zero,axiom,
% 0.29/0.58      ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_ge_zero
% 0.29/0.58  thf(fact_6458_not__is__unit__0,axiom,
% 0.29/0.58      ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.29/0.58  
% 0.29/0.58  % not_is_unit_0
% 0.29/0.58  thf(fact_6459_not__is__unit__0,axiom,
% 0.29/0.58      ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % not_is_unit_0
% 0.29/0.58  thf(fact_6460_not__is__unit__0,axiom,
% 0.29/0.58      ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.29/0.58  
% 0.29/0.58  % not_is_unit_0
% 0.29/0.58  thf(fact_6461_pinf_I9_J,axiom,
% 0.29/0.58      ! [D: code_integer,S: code_integer] :
% 0.29/0.58      ? [Z3: code_integer] :
% 0.29/0.58      ! [X2: code_integer] :
% 0.29/0.58        ( ( ord_le6747313008572928689nteger @ Z3 @ X2 )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 0.29/0.58          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pinf(9)
% 0.29/0.58  thf(fact_6462_pinf_I9_J,axiom,
% 0.29/0.58      ! [D: real,S: real] :
% 0.29/0.58      ? [Z3: real] :
% 0.29/0.58      ! [X2: real] :
% 0.29/0.58        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.58       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 0.29/0.58          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pinf(9)
% 0.29/0.58  thf(fact_6463_pinf_I9_J,axiom,
% 0.29/0.58      ! [D: rat,S: rat] :
% 0.29/0.58      ? [Z3: rat] :
% 0.29/0.58      ! [X2: rat] :
% 0.29/0.58        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.58       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 0.29/0.58          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pinf(9)
% 0.29/0.58  thf(fact_6464_pinf_I9_J,axiom,
% 0.29/0.58      ! [D: nat,S: nat] :
% 0.29/0.58      ? [Z3: nat] :
% 0.29/0.58      ! [X2: nat] :
% 0.29/0.58        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.58       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 0.29/0.58          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pinf(9)
% 0.29/0.58  thf(fact_6465_pinf_I9_J,axiom,
% 0.29/0.58      ! [D: int,S: int] :
% 0.29/0.58      ? [Z3: int] :
% 0.29/0.58      ! [X2: int] :
% 0.29/0.58        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.58       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 0.29/0.58          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pinf(9)
% 0.29/0.58  thf(fact_6466_pinf_I10_J,axiom,
% 0.29/0.58      ! [D: code_integer,S: code_integer] :
% 0.29/0.58      ? [Z3: code_integer] :
% 0.29/0.58      ! [X2: code_integer] :
% 0.29/0.58        ( ( ord_le6747313008572928689nteger @ Z3 @ X2 )
% 0.29/0.58       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 0.29/0.58          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pinf(10)
% 0.29/0.58  thf(fact_6467_pinf_I10_J,axiom,
% 0.29/0.58      ! [D: real,S: real] :
% 0.29/0.58      ? [Z3: real] :
% 0.29/0.58      ! [X2: real] :
% 0.29/0.58        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.58       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 0.29/0.58          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pinf(10)
% 0.29/0.58  thf(fact_6468_pinf_I10_J,axiom,
% 0.29/0.58      ! [D: rat,S: rat] :
% 0.29/0.58      ? [Z3: rat] :
% 0.29/0.58      ! [X2: rat] :
% 0.29/0.58        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.58       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 0.29/0.58          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pinf(10)
% 0.29/0.58  thf(fact_6469_pinf_I10_J,axiom,
% 0.29/0.58      ! [D: nat,S: nat] :
% 0.29/0.58      ? [Z3: nat] :
% 0.29/0.58      ! [X2: nat] :
% 0.29/0.58        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.58       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 0.29/0.58          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pinf(10)
% 0.29/0.58  thf(fact_6470_pinf_I10_J,axiom,
% 0.29/0.58      ! [D: int,S: int] :
% 0.29/0.58      ? [Z3: int] :
% 0.29/0.58      ! [X2: int] :
% 0.29/0.58        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.58       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 0.29/0.58          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pinf(10)
% 0.29/0.58  thf(fact_6471_minf_I9_J,axiom,
% 0.29/0.58      ! [D: code_integer,S: code_integer] :
% 0.29/0.58      ? [Z3: code_integer] :
% 0.29/0.58      ! [X2: code_integer] :
% 0.29/0.58        ( ( ord_le6747313008572928689nteger @ X2 @ Z3 )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 0.29/0.58          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minf(9)
% 0.29/0.58  thf(fact_6472_minf_I9_J,axiom,
% 0.29/0.58      ! [D: real,S: real] :
% 0.29/0.58      ? [Z3: real] :
% 0.29/0.58      ! [X2: real] :
% 0.29/0.58        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.58       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 0.29/0.58          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minf(9)
% 0.29/0.58  thf(fact_6473_minf_I9_J,axiom,
% 0.29/0.58      ! [D: rat,S: rat] :
% 0.29/0.58      ? [Z3: rat] :
% 0.29/0.58      ! [X2: rat] :
% 0.29/0.58        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.58       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 0.29/0.58          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minf(9)
% 0.29/0.58  thf(fact_6474_minf_I9_J,axiom,
% 0.29/0.58      ! [D: nat,S: nat] :
% 0.29/0.58      ? [Z3: nat] :
% 0.29/0.58      ! [X2: nat] :
% 0.29/0.58        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.58       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 0.29/0.58          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minf(9)
% 0.29/0.58  thf(fact_6475_minf_I9_J,axiom,
% 0.29/0.58      ! [D: int,S: int] :
% 0.29/0.58      ? [Z3: int] :
% 0.29/0.58      ! [X2: int] :
% 0.29/0.58        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.58       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 0.29/0.58          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minf(9)
% 0.29/0.58  thf(fact_6476_minf_I10_J,axiom,
% 0.29/0.58      ! [D: code_integer,S: code_integer] :
% 0.29/0.58      ? [Z3: code_integer] :
% 0.29/0.58      ! [X2: code_integer] :
% 0.29/0.58        ( ( ord_le6747313008572928689nteger @ X2 @ Z3 )
% 0.29/0.58       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 0.29/0.58          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minf(10)
% 0.29/0.58  thf(fact_6477_minf_I10_J,axiom,
% 0.29/0.58      ! [D: real,S: real] :
% 0.29/0.58      ? [Z3: real] :
% 0.29/0.58      ! [X2: real] :
% 0.29/0.58        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.58       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 0.29/0.58          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minf(10)
% 0.29/0.58  thf(fact_6478_minf_I10_J,axiom,
% 0.29/0.58      ! [D: rat,S: rat] :
% 0.29/0.58      ? [Z3: rat] :
% 0.29/0.58      ! [X2: rat] :
% 0.29/0.58        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.58       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 0.29/0.58          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minf(10)
% 0.29/0.58  thf(fact_6479_minf_I10_J,axiom,
% 0.29/0.58      ! [D: nat,S: nat] :
% 0.29/0.58      ? [Z3: nat] :
% 0.29/0.58      ! [X2: nat] :
% 0.29/0.58        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.58       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 0.29/0.58          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minf(10)
% 0.29/0.58  thf(fact_6480_minf_I10_J,axiom,
% 0.29/0.58      ! [D: int,S: int] :
% 0.29/0.58      ? [Z3: int] :
% 0.29/0.58      ! [X2: int] :
% 0.29/0.58        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.58       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 0.29/0.58          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minf(10)
% 0.29/0.58  thf(fact_6481_dvd__div__eq__0__iff,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.29/0.58       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.29/0.58            = zero_z3403309356797280102nteger )
% 0.29/0.58          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_0_iff
% 0.29/0.58  thf(fact_6482_dvd__div__eq__0__iff,axiom,
% 0.29/0.58      ! [B: complex,A: complex] :
% 0.29/0.58        ( ( dvd_dvd_complex @ B @ A )
% 0.29/0.58       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.29/0.58            = zero_zero_complex )
% 0.29/0.58          = ( A = zero_zero_complex ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_0_iff
% 0.29/0.58  thf(fact_6483_dvd__div__eq__0__iff,axiom,
% 0.29/0.58      ! [B: real,A: real] :
% 0.29/0.58        ( ( dvd_dvd_real @ B @ A )
% 0.29/0.58       => ( ( ( divide_divide_real @ A @ B )
% 0.29/0.58            = zero_zero_real )
% 0.29/0.58          = ( A = zero_zero_real ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_0_iff
% 0.29/0.58  thf(fact_6484_dvd__div__eq__0__iff,axiom,
% 0.29/0.58      ! [B: rat,A: rat] :
% 0.29/0.58        ( ( dvd_dvd_rat @ B @ A )
% 0.29/0.58       => ( ( ( divide_divide_rat @ A @ B )
% 0.29/0.58            = zero_zero_rat )
% 0.29/0.58          = ( A = zero_zero_rat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_0_iff
% 0.29/0.58  thf(fact_6485_dvd__div__eq__0__iff,axiom,
% 0.29/0.58      ! [B: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ A )
% 0.29/0.58       => ( ( ( divide_divide_nat @ A @ B )
% 0.29/0.58            = zero_zero_nat )
% 0.29/0.58          = ( A = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_0_iff
% 0.29/0.58  thf(fact_6486_dvd__div__eq__0__iff,axiom,
% 0.29/0.58      ! [B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ A )
% 0.29/0.58       => ( ( ( divide_divide_int @ A @ B )
% 0.29/0.58            = zero_zero_int )
% 0.29/0.58          = ( A = zero_zero_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_0_iff
% 0.29/0.58  thf(fact_6487_unit__mult__right__cancel,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.58       => ( ( ( times_3573771949741848930nteger @ B @ A )
% 0.29/0.58            = ( times_3573771949741848930nteger @ C @ A ) )
% 0.29/0.58          = ( B = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_mult_right_cancel
% 0.29/0.58  thf(fact_6488_unit__mult__right__cancel,axiom,
% 0.29/0.58      ! [A: nat,B: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.58       => ( ( ( times_times_nat @ B @ A )
% 0.29/0.58            = ( times_times_nat @ C @ A ) )
% 0.29/0.58          = ( B = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_mult_right_cancel
% 0.29/0.58  thf(fact_6489_unit__mult__right__cancel,axiom,
% 0.29/0.58      ! [A: int,B: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.58       => ( ( ( times_times_int @ B @ A )
% 0.29/0.58            = ( times_times_int @ C @ A ) )
% 0.29/0.58          = ( B = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_mult_right_cancel
% 0.29/0.58  thf(fact_6490_unit__mult__left__cancel,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.58       => ( ( ( times_3573771949741848930nteger @ A @ B )
% 0.29/0.58            = ( times_3573771949741848930nteger @ A @ C ) )
% 0.29/0.58          = ( B = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_mult_left_cancel
% 0.29/0.58  thf(fact_6491_unit__mult__left__cancel,axiom,
% 0.29/0.58      ! [A: nat,B: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.58       => ( ( ( times_times_nat @ A @ B )
% 0.29/0.58            = ( times_times_nat @ A @ C ) )
% 0.29/0.58          = ( B = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_mult_left_cancel
% 0.29/0.58  thf(fact_6492_unit__mult__left__cancel,axiom,
% 0.29/0.58      ! [A: int,B: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.58       => ( ( ( times_times_int @ A @ B )
% 0.29/0.58            = ( times_times_int @ A @ C ) )
% 0.29/0.58          = ( B = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_mult_left_cancel
% 0.29/0.58  thf(fact_6493_mult__unit__dvd__iff_H,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.29/0.58          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mult_unit_dvd_iff'
% 0.29/0.58  thf(fact_6494_mult__unit__dvd__iff_H,axiom,
% 0.29/0.58      ! [A: nat,B: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.29/0.58          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mult_unit_dvd_iff'
% 0.29/0.58  thf(fact_6495_mult__unit__dvd__iff_H,axiom,
% 0.29/0.58      ! [A: int,B: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.29/0.58          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mult_unit_dvd_iff'
% 0.29/0.58  thf(fact_6496_dvd__mult__unit__iff_H,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_unit_iff'
% 0.29/0.58  thf(fact_6497_dvd__mult__unit__iff_H,axiom,
% 0.29/0.58      ! [B: nat,A: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_unit_iff'
% 0.29/0.58  thf(fact_6498_dvd__mult__unit__iff_H,axiom,
% 0.29/0.58      ! [B: int,A: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.29/0.58          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_unit_iff'
% 0.29/0.58  thf(fact_6499_mult__unit__dvd__iff,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.29/0.58          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mult_unit_dvd_iff
% 0.29/0.58  thf(fact_6500_mult__unit__dvd__iff,axiom,
% 0.29/0.58      ! [B: nat,A: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.29/0.58          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mult_unit_dvd_iff
% 0.29/0.58  thf(fact_6501_mult__unit__dvd__iff,axiom,
% 0.29/0.58      ! [B: int,A: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.29/0.58          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mult_unit_dvd_iff
% 0.29/0.58  thf(fact_6502_dvd__mult__unit__iff,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.29/0.58          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_unit_iff
% 0.29/0.58  thf(fact_6503_dvd__mult__unit__iff,axiom,
% 0.29/0.58      ! [B: nat,A: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 0.29/0.58          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_unit_iff
% 0.29/0.58  thf(fact_6504_dvd__mult__unit__iff,axiom,
% 0.29/0.58      ! [B: int,A: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 0.29/0.58          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_unit_iff
% 0.29/0.58  thf(fact_6505_is__unit__mult__iff,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 0.29/0.58        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.58          & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_mult_iff
% 0.29/0.58  thf(fact_6506_is__unit__mult__iff,axiom,
% 0.29/0.58      ! [A: nat,B: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 0.29/0.58        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.58          & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_mult_iff
% 0.29/0.58  thf(fact_6507_is__unit__mult__iff,axiom,
% 0.29/0.58      ! [A: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 0.29/0.58        = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.58          & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_mult_iff
% 0.29/0.58  thf(fact_6508_mult__exp__exp,axiom,
% 0.29/0.58      ! [X: complex,Y: complex] :
% 0.29/0.58        ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
% 0.29/0.58        = ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mult_exp_exp
% 0.29/0.58  thf(fact_6509_mult__exp__exp,axiom,
% 0.29/0.58      ! [X: real,Y: real] :
% 0.29/0.58        ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.29/0.58        = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mult_exp_exp
% 0.29/0.58  thf(fact_6510_exp__add__commuting,axiom,
% 0.29/0.58      ! [X: complex,Y: complex] :
% 0.29/0.58        ( ( ( times_times_complex @ X @ Y )
% 0.29/0.58          = ( times_times_complex @ Y @ X ) )
% 0.29/0.58       => ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.29/0.58          = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_add_commuting
% 0.29/0.58  thf(fact_6511_exp__add__commuting,axiom,
% 0.29/0.58      ! [X: real,Y: real] :
% 0.29/0.58        ( ( ( times_times_real @ X @ Y )
% 0.29/0.58          = ( times_times_real @ Y @ X ) )
% 0.29/0.58       => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.58          = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_add_commuting
% 0.29/0.58  thf(fact_6512_div__mult__div__if__dvd,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ D @ C )
% 0.29/0.58         => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 0.29/0.58            = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_mult_div_if_dvd
% 0.29/0.58  thf(fact_6513_div__mult__div__if__dvd,axiom,
% 0.29/0.58      ! [B: nat,A: nat,D: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ A )
% 0.29/0.58       => ( ( dvd_dvd_nat @ D @ C )
% 0.29/0.58         => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 0.29/0.58            = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_mult_div_if_dvd
% 0.29/0.58  thf(fact_6514_div__mult__div__if__dvd,axiom,
% 0.29/0.58      ! [B: int,A: int,D: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ A )
% 0.29/0.58       => ( ( dvd_dvd_int @ D @ C )
% 0.29/0.58         => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 0.29/0.58            = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_mult_div_if_dvd
% 0.29/0.58  thf(fact_6515_dvd__mult__imp__div,axiom,
% 0.29/0.58      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 0.29/0.58       => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_imp_div
% 0.29/0.58  thf(fact_6516_dvd__mult__imp__div,axiom,
% 0.29/0.58      ! [A: nat,C: nat,B: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 0.29/0.58       => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_imp_div
% 0.29/0.58  thf(fact_6517_dvd__mult__imp__div,axiom,
% 0.29/0.58      ! [A: int,C: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 0.29/0.58       => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_imp_div
% 0.29/0.58  thf(fact_6518_dvd__div__mult2__eq,axiom,
% 0.29/0.58      ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 0.29/0.58       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.29/0.58          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_mult2_eq
% 0.29/0.58  thf(fact_6519_dvd__div__mult2__eq,axiom,
% 0.29/0.58      ! [B: nat,C: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 0.29/0.58       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.29/0.58          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_mult2_eq
% 0.29/0.58  thf(fact_6520_dvd__div__mult2__eq,axiom,
% 0.29/0.58      ! [B: int,C: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 0.29/0.58       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.29/0.58          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_mult2_eq
% 0.29/0.58  thf(fact_6521_div__div__eq__right,axiom,
% 0.29/0.58      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ C @ B )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.29/0.58         => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.29/0.58            = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_div_eq_right
% 0.29/0.58  thf(fact_6522_div__div__eq__right,axiom,
% 0.29/0.58      ! [C: nat,B: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ C @ B )
% 0.29/0.58       => ( ( dvd_dvd_nat @ B @ A )
% 0.29/0.58         => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.29/0.58            = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_div_eq_right
% 0.29/0.58  thf(fact_6523_div__div__eq__right,axiom,
% 0.29/0.58      ! [C: int,B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ C @ B )
% 0.29/0.58       => ( ( dvd_dvd_int @ B @ A )
% 0.29/0.58         => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.29/0.58            = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_div_eq_right
% 0.29/0.58  thf(fact_6524_div__mult__swap,axiom,
% 0.29/0.58      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ C @ B )
% 0.29/0.58       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.29/0.58          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_mult_swap
% 0.29/0.58  thf(fact_6525_div__mult__swap,axiom,
% 0.29/0.58      ! [C: nat,B: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ C @ B )
% 0.29/0.58       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.29/0.58          = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_mult_swap
% 0.29/0.58  thf(fact_6526_div__mult__swap,axiom,
% 0.29/0.58      ! [C: int,B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ C @ B )
% 0.29/0.58       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.29/0.58          = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_mult_swap
% 0.29/0.58  thf(fact_6527_dvd__div__mult,axiom,
% 0.29/0.58      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ C @ B )
% 0.29/0.58       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 0.29/0.58          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_mult
% 0.29/0.58  thf(fact_6528_dvd__div__mult,axiom,
% 0.29/0.58      ! [C: nat,B: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ C @ B )
% 0.29/0.58       => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 0.29/0.58          = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_mult
% 0.29/0.58  thf(fact_6529_dvd__div__mult,axiom,
% 0.29/0.58      ! [C: int,B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ C @ B )
% 0.29/0.58       => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 0.29/0.58          = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_mult
% 0.29/0.58  thf(fact_6530_dvd__div__unit__iff,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 0.29/0.58          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_unit_iff
% 0.29/0.58  thf(fact_6531_dvd__div__unit__iff,axiom,
% 0.29/0.58      ! [B: nat,A: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 0.29/0.58          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_unit_iff
% 0.29/0.58  thf(fact_6532_dvd__div__unit__iff,axiom,
% 0.29/0.58      ! [B: int,A: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 0.29/0.58          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_unit_iff
% 0.29/0.58  thf(fact_6533_div__unit__dvd__iff,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.29/0.58          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_unit_dvd_iff
% 0.29/0.58  thf(fact_6534_div__unit__dvd__iff,axiom,
% 0.29/0.58      ! [B: nat,A: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.29/0.58          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_unit_dvd_iff
% 0.29/0.58  thf(fact_6535_div__unit__dvd__iff,axiom,
% 0.29/0.58      ! [B: int,A: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.29/0.58          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_unit_dvd_iff
% 0.29/0.58  thf(fact_6536_unit__div__cancel,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.58       => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.29/0.58            = ( divide6298287555418463151nteger @ C @ A ) )
% 0.29/0.58          = ( B = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_cancel
% 0.29/0.58  thf(fact_6537_unit__div__cancel,axiom,
% 0.29/0.58      ! [A: nat,B: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.58       => ( ( ( divide_divide_nat @ B @ A )
% 0.29/0.58            = ( divide_divide_nat @ C @ A ) )
% 0.29/0.58          = ( B = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_cancel
% 0.29/0.58  thf(fact_6538_unit__div__cancel,axiom,
% 0.29/0.58      ! [A: int,B: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.58       => ( ( ( divide_divide_int @ B @ A )
% 0.29/0.58            = ( divide_divide_int @ C @ A ) )
% 0.29/0.58          = ( B = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_cancel
% 0.29/0.58  thf(fact_6539_div__plus__div__distrib__dvd__right,axiom,
% 0.29/0.58      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ C @ B )
% 0.29/0.58       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.29/0.58          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_plus_div_distrib_dvd_right
% 0.29/0.58  thf(fact_6540_div__plus__div__distrib__dvd__right,axiom,
% 0.29/0.58      ! [C: nat,B: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ C @ B )
% 0.29/0.58       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.58          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_plus_div_distrib_dvd_right
% 0.29/0.58  thf(fact_6541_div__plus__div__distrib__dvd__right,axiom,
% 0.29/0.58      ! [C: int,B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ C @ B )
% 0.29/0.58       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.58          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_plus_div_distrib_dvd_right
% 0.29/0.58  thf(fact_6542_div__plus__div__distrib__dvd__left,axiom,
% 0.29/0.58      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.29/0.58       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.29/0.58          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_plus_div_distrib_dvd_left
% 0.29/0.58  thf(fact_6543_div__plus__div__distrib__dvd__left,axiom,
% 0.29/0.58      ! [C: nat,A: nat,B: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ C @ A )
% 0.29/0.58       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.58          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_plus_div_distrib_dvd_left
% 0.29/0.58  thf(fact_6544_div__plus__div__distrib__dvd__left,axiom,
% 0.29/0.58      ! [C: int,A: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ C @ A )
% 0.29/0.58       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.58          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_plus_div_distrib_dvd_left
% 0.29/0.58  thf(fact_6545_exp__diff,axiom,
% 0.29/0.58      ! [X: complex,Y: complex] :
% 0.29/0.58        ( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.29/0.58        = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_diff
% 0.29/0.58  thf(fact_6546_exp__diff,axiom,
% 0.29/0.58      ! [X: real,Y: real] :
% 0.29/0.58        ( ( exp_real @ ( minus_minus_real @ X @ Y ) )
% 0.29/0.58        = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_diff
% 0.29/0.58  thf(fact_6547_dvd__neg__div,axiom,
% 0.29/0.58      ! [B: real,A: real] :
% 0.29/0.58        ( ( dvd_dvd_real @ B @ A )
% 0.29/0.58       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 0.29/0.58          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_neg_div
% 0.29/0.58  thf(fact_6548_dvd__neg__div,axiom,
% 0.29/0.58      ! [B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ A )
% 0.29/0.58       => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.29/0.58          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_neg_div
% 0.29/0.58  thf(fact_6549_dvd__neg__div,axiom,
% 0.29/0.58      ! [B: complex,A: complex] :
% 0.29/0.58        ( ( dvd_dvd_complex @ B @ A )
% 0.29/0.58       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.29/0.58          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_neg_div
% 0.29/0.58  thf(fact_6550_dvd__neg__div,axiom,
% 0.29/0.58      ! [B: rat,A: rat] :
% 0.29/0.58        ( ( dvd_dvd_rat @ B @ A )
% 0.29/0.58       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.29/0.58          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_neg_div
% 0.29/0.58  thf(fact_6551_dvd__neg__div,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.29/0.58       => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.29/0.58          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_neg_div
% 0.29/0.58  thf(fact_6552_dvd__div__neg,axiom,
% 0.29/0.58      ! [B: real,A: real] :
% 0.29/0.58        ( ( dvd_dvd_real @ B @ A )
% 0.29/0.58       => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 0.29/0.58          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_neg
% 0.29/0.58  thf(fact_6553_dvd__div__neg,axiom,
% 0.29/0.58      ! [B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ A )
% 0.29/0.58       => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.29/0.58          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_neg
% 0.29/0.58  thf(fact_6554_dvd__div__neg,axiom,
% 0.29/0.58      ! [B: complex,A: complex] :
% 0.29/0.58        ( ( dvd_dvd_complex @ B @ A )
% 0.29/0.58       => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.29/0.58          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_neg
% 0.29/0.58  thf(fact_6555_dvd__div__neg,axiom,
% 0.29/0.58      ! [B: rat,A: rat] :
% 0.29/0.58        ( ( dvd_dvd_rat @ B @ A )
% 0.29/0.58       => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.29/0.58          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_neg
% 0.29/0.58  thf(fact_6556_dvd__div__neg,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.29/0.58       => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.29/0.58          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_neg
% 0.29/0.58  thf(fact_6557_div__power,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.29/0.58       => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
% 0.29/0.58          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_power
% 0.29/0.58  thf(fact_6558_div__power,axiom,
% 0.29/0.58      ! [B: nat,A: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ A )
% 0.29/0.58       => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
% 0.29/0.58          = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_power
% 0.29/0.58  thf(fact_6559_div__power,axiom,
% 0.29/0.58      ! [B: int,A: int,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ A )
% 0.29/0.58       => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
% 0.29/0.58          = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_power
% 0.29/0.58  thf(fact_6560_mod__eq__0__iff__dvd,axiom,
% 0.29/0.58      ! [A: nat,B: nat] :
% 0.29/0.58        ( ( ( modulo_modulo_nat @ A @ B )
% 0.29/0.58          = zero_zero_nat )
% 0.29/0.58        = ( dvd_dvd_nat @ B @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod_eq_0_iff_dvd
% 0.29/0.58  thf(fact_6561_mod__eq__0__iff__dvd,axiom,
% 0.29/0.58      ! [A: int,B: int] :
% 0.29/0.58        ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.58          = zero_zero_int )
% 0.29/0.58        = ( dvd_dvd_int @ B @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod_eq_0_iff_dvd
% 0.29/0.58  thf(fact_6562_mod__eq__0__iff__dvd,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.29/0.58          = zero_z3403309356797280102nteger )
% 0.29/0.58        = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod_eq_0_iff_dvd
% 0.29/0.58  thf(fact_6563_dvd__eq__mod__eq__0,axiom,
% 0.29/0.58      ( dvd_dvd_nat
% 0.29/0.58      = ( ^ [A3: nat,B4: nat] :
% 0.29/0.58            ( ( modulo_modulo_nat @ B4 @ A3 )
% 0.29/0.58            = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_eq_mod_eq_0
% 0.29/0.58  thf(fact_6564_dvd__eq__mod__eq__0,axiom,
% 0.29/0.58      ( dvd_dvd_int
% 0.29/0.58      = ( ^ [A3: int,B4: int] :
% 0.29/0.58            ( ( modulo_modulo_int @ B4 @ A3 )
% 0.29/0.58            = zero_zero_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_eq_mod_eq_0
% 0.29/0.58  thf(fact_6565_dvd__eq__mod__eq__0,axiom,
% 0.29/0.58      ( dvd_dvd_Code_integer
% 0.29/0.58      = ( ^ [A3: code_integer,B4: code_integer] :
% 0.29/0.58            ( ( modulo364778990260209775nteger @ B4 @ A3 )
% 0.29/0.58            = zero_z3403309356797280102nteger ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_eq_mod_eq_0
% 0.29/0.58  thf(fact_6566_mod__0__imp__dvd,axiom,
% 0.29/0.58      ! [A: nat,B: nat] :
% 0.29/0.58        ( ( ( modulo_modulo_nat @ A @ B )
% 0.29/0.58          = zero_zero_nat )
% 0.29/0.58       => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod_0_imp_dvd
% 0.29/0.58  thf(fact_6567_mod__0__imp__dvd,axiom,
% 0.29/0.58      ! [A: int,B: int] :
% 0.29/0.58        ( ( ( modulo_modulo_int @ A @ B )
% 0.29/0.58          = zero_zero_int )
% 0.29/0.58       => ( dvd_dvd_int @ B @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod_0_imp_dvd
% 0.29/0.58  thf(fact_6568_mod__0__imp__dvd,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.29/0.58          = zero_z3403309356797280102nteger )
% 0.29/0.58       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod_0_imp_dvd
% 0.29/0.58  thf(fact_6569_dvd__power__le,axiom,
% 0.29/0.58      ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ X @ Y )
% 0.29/0.58       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_le
% 0.29/0.58  thf(fact_6570_dvd__power__le,axiom,
% 0.29/0.58      ! [X: nat,Y: nat,N: nat,M: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ X @ Y )
% 0.29/0.58       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_le
% 0.29/0.58  thf(fact_6571_dvd__power__le,axiom,
% 0.29/0.58      ! [X: real,Y: real,N: nat,M: nat] :
% 0.29/0.58        ( ( dvd_dvd_real @ X @ Y )
% 0.29/0.58       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_le
% 0.29/0.58  thf(fact_6572_dvd__power__le,axiom,
% 0.29/0.58      ! [X: int,Y: int,N: nat,M: nat] :
% 0.29/0.58        ( ( dvd_dvd_int @ X @ Y )
% 0.29/0.58       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_le
% 0.29/0.58  thf(fact_6573_dvd__power__le,axiom,
% 0.29/0.58      ! [X: complex,Y: complex,N: nat,M: nat] :
% 0.29/0.58        ( ( dvd_dvd_complex @ X @ Y )
% 0.29/0.58       => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_le
% 0.29/0.58  thf(fact_6574_power__le__dvd,axiom,
% 0.29/0.58      ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 0.29/0.58       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_dvd
% 0.29/0.58  thf(fact_6575_power__le__dvd,axiom,
% 0.29/0.58      ! [A: nat,N: nat,B: nat,M: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 0.29/0.58       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58         => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_dvd
% 0.29/0.58  thf(fact_6576_power__le__dvd,axiom,
% 0.29/0.58      ! [A: real,N: nat,B: real,M: nat] :
% 0.29/0.58        ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 0.29/0.58       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58         => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_dvd
% 0.29/0.58  thf(fact_6577_power__le__dvd,axiom,
% 0.29/0.58      ! [A: int,N: nat,B: int,M: nat] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 0.29/0.58       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58         => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_dvd
% 0.29/0.58  thf(fact_6578_power__le__dvd,axiom,
% 0.29/0.58      ! [A: complex,N: nat,B: complex,M: nat] :
% 0.29/0.58        ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 0.29/0.58       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58         => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_dvd
% 0.29/0.58  thf(fact_6579_le__imp__power__dvd,axiom,
% 0.29/0.58      ! [M: nat,N: nat,A: code_integer] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % le_imp_power_dvd
% 0.29/0.58  thf(fact_6580_le__imp__power__dvd,axiom,
% 0.29/0.58      ! [M: nat,N: nat,A: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % le_imp_power_dvd
% 0.29/0.58  thf(fact_6581_le__imp__power__dvd,axiom,
% 0.29/0.58      ! [M: nat,N: nat,A: real] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % le_imp_power_dvd
% 0.29/0.58  thf(fact_6582_le__imp__power__dvd,axiom,
% 0.29/0.58      ! [M: nat,N: nat,A: int] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % le_imp_power_dvd
% 0.29/0.58  thf(fact_6583_le__imp__power__dvd,axiom,
% 0.29/0.58      ! [M: nat,N: nat,A: complex] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % le_imp_power_dvd
% 0.29/0.58  thf(fact_6584_nat__dvd__not__less,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.58       => ( ( ord_less_nat @ M @ N )
% 0.29/0.58         => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % nat_dvd_not_less
% 0.29/0.58  thf(fact_6585_dvd__pos__nat,axiom,
% 0.29/0.58      ! [N: nat,M: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58       => ( ( dvd_dvd_nat @ M @ N )
% 0.29/0.58         => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_pos_nat
% 0.29/0.58  thf(fact_6586_dvd__minus__self,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 0.29/0.58        = ( ( ord_less_nat @ N @ M )
% 0.29/0.58          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_minus_self
% 0.29/0.58  thf(fact_6587_zdvd__antisym__nonneg,axiom,
% 0.29/0.58      ! [M: int,N: int] :
% 0.29/0.58        ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.29/0.58       => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.29/0.58         => ( ( dvd_dvd_int @ M @ N )
% 0.29/0.58           => ( ( dvd_dvd_int @ N @ M )
% 0.29/0.58             => ( M = N ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zdvd_antisym_nonneg
% 0.29/0.58  thf(fact_6588_zdvd__not__zless,axiom,
% 0.29/0.58      ! [M: int,N: int] :
% 0.29/0.58        ( ( ord_less_int @ zero_zero_int @ M )
% 0.29/0.58       => ( ( ord_less_int @ M @ N )
% 0.29/0.58         => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zdvd_not_zless
% 0.29/0.58  thf(fact_6589_dvd__diffD,axiom,
% 0.29/0.58      ! [K: nat,M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.58       => ( ( dvd_dvd_nat @ K @ N )
% 0.29/0.58         => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58           => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_diffD
% 0.29/0.58  thf(fact_6590_dvd__diffD1,axiom,
% 0.29/0.58      ! [K: nat,M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.58       => ( ( dvd_dvd_nat @ K @ M )
% 0.29/0.58         => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58           => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_diffD1
% 0.29/0.58  thf(fact_6591_less__eq__dvd__minus,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ( dvd_dvd_nat @ M @ N )
% 0.29/0.58          = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % less_eq_dvd_minus
% 0.29/0.58  thf(fact_6592_zdvd__mono,axiom,
% 0.29/0.58      ! [K: int,M: int,T: int] :
% 0.29/0.58        ( ( K != zero_zero_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ M @ T )
% 0.29/0.58          = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zdvd_mono
% 0.29/0.58  thf(fact_6593_zdvd__mult__cancel,axiom,
% 0.29/0.58      ! [K: int,M: int,N: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 0.29/0.58       => ( ( K != zero_zero_int )
% 0.29/0.58         => ( dvd_dvd_int @ M @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zdvd_mult_cancel
% 0.29/0.58  thf(fact_6594_bezout__add__nat,axiom,
% 0.29/0.58      ! [A: nat,B: nat] :
% 0.29/0.58      ? [D5: nat,X3: nat,Y4: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ D5 @ A )
% 0.29/0.58        & ( dvd_dvd_nat @ D5 @ B )
% 0.29/0.58        & ( ( ( times_times_nat @ A @ X3 )
% 0.29/0.58            = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D5 ) )
% 0.29/0.58          | ( ( times_times_nat @ B @ X3 )
% 0.29/0.58            = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D5 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bezout_add_nat
% 0.29/0.58  thf(fact_6595_bezout__lemma__nat,axiom,
% 0.29/0.58      ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ D @ A )
% 0.29/0.58       => ( ( dvd_dvd_nat @ D @ B )
% 0.29/0.58         => ( ( ( ( times_times_nat @ A @ X )
% 0.29/0.58                = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
% 0.29/0.58              | ( ( times_times_nat @ B @ X )
% 0.29/0.58                = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
% 0.29/0.58           => ? [X3: nat,Y4: nat] :
% 0.29/0.58                ( ( dvd_dvd_nat @ D @ A )
% 0.29/0.58                & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.58                & ( ( ( times_times_nat @ A @ X3 )
% 0.29/0.58                    = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
% 0.29/0.58                  | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
% 0.29/0.58                    = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bezout_lemma_nat
% 0.29/0.58  thf(fact_6596_zdvd__reduce,axiom,
% 0.29/0.58      ! [K: int,N: int,M: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 0.29/0.58        = ( dvd_dvd_int @ K @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zdvd_reduce
% 0.29/0.58  thf(fact_6597_zdvd__period,axiom,
% 0.29/0.58      ! [A: int,D: int,X: int,T: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ D )
% 0.29/0.58       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
% 0.29/0.58          = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zdvd_period
% 0.29/0.58  thf(fact_6598_abs__div,axiom,
% 0.29/0.58      ! [Y: int,X: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ Y @ X )
% 0.29/0.58       => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
% 0.29/0.58          = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % abs_div
% 0.29/0.58  thf(fact_6599_even__of__int__iff,axiom,
% 0.29/0.58      ! [K: int] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 0.29/0.58        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_of_int_iff
% 0.29/0.58  thf(fact_6600_even__of__int__iff,axiom,
% 0.29/0.58      ! [K: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 0.29/0.58        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_of_int_iff
% 0.29/0.58  thf(fact_6601_div2__even__ext__nat,axiom,
% 0.29/0.58      ! [X: nat,Y: nat] :
% 0.29/0.58        ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58          = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.58       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
% 0.29/0.58            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
% 0.29/0.58         => ( X = Y ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div2_even_ext_nat
% 0.29/0.58  thf(fact_6602_exp__gt__one,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.58       => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_gt_one
% 0.29/0.58  thf(fact_6603_exp__ge__add__one__self,axiom,
% 0.29/0.58      ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_ge_add_one_self
% 0.29/0.58  thf(fact_6604_unit__dvdE,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.58       => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.58           => ! [C4: code_integer] :
% 0.29/0.58                ( B
% 0.29/0.58               != ( times_3573771949741848930nteger @ A @ C4 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_dvdE
% 0.29/0.58  thf(fact_6605_unit__dvdE,axiom,
% 0.29/0.58      ! [A: nat,B: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.58       => ~ ( ( A != zero_zero_nat )
% 0.29/0.58           => ! [C4: nat] :
% 0.29/0.58                ( B
% 0.29/0.58               != ( times_times_nat @ A @ C4 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_dvdE
% 0.29/0.58  thf(fact_6606_unit__dvdE,axiom,
% 0.29/0.58      ! [A: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.58       => ~ ( ( A != zero_zero_int )
% 0.29/0.58           => ! [C4: int] :
% 0.29/0.58                ( B
% 0.29/0.58               != ( times_times_int @ A @ C4 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_dvdE
% 0.29/0.58  thf(fact_6607_unity__coeff__ex,axiom,
% 0.29/0.58      ! [P: code_integer > $o,L: code_integer] :
% 0.29/0.58        ( ( ? [X4: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X4 ) ) )
% 0.29/0.58        = ( ? [X4: code_integer] :
% 0.29/0.58              ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X4 @ zero_z3403309356797280102nteger ) )
% 0.29/0.58              & ( P @ X4 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unity_coeff_ex
% 0.29/0.58  thf(fact_6608_unity__coeff__ex,axiom,
% 0.29/0.58      ! [P: complex > $o,L: complex] :
% 0.29/0.58        ( ( ? [X4: complex] : ( P @ ( times_times_complex @ L @ X4 ) ) )
% 0.29/0.58        = ( ? [X4: complex] :
% 0.29/0.58              ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X4 @ zero_zero_complex ) )
% 0.29/0.58              & ( P @ X4 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unity_coeff_ex
% 0.29/0.58  thf(fact_6609_unity__coeff__ex,axiom,
% 0.29/0.58      ! [P: real > $o,L: real] :
% 0.29/0.58        ( ( ? [X4: real] : ( P @ ( times_times_real @ L @ X4 ) ) )
% 0.29/0.58        = ( ? [X4: real] :
% 0.29/0.58              ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X4 @ zero_zero_real ) )
% 0.29/0.58              & ( P @ X4 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unity_coeff_ex
% 0.29/0.58  thf(fact_6610_unity__coeff__ex,axiom,
% 0.29/0.58      ! [P: rat > $o,L: rat] :
% 0.29/0.58        ( ( ? [X4: rat] : ( P @ ( times_times_rat @ L @ X4 ) ) )
% 0.29/0.58        = ( ? [X4: rat] :
% 0.29/0.58              ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X4 @ zero_zero_rat ) )
% 0.29/0.58              & ( P @ X4 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unity_coeff_ex
% 0.29/0.58  thf(fact_6611_unity__coeff__ex,axiom,
% 0.29/0.58      ! [P: nat > $o,L: nat] :
% 0.29/0.58        ( ( ? [X4: nat] : ( P @ ( times_times_nat @ L @ X4 ) ) )
% 0.29/0.58        = ( ? [X4: nat] :
% 0.29/0.58              ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
% 0.29/0.58              & ( P @ X4 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unity_coeff_ex
% 0.29/0.58  thf(fact_6612_unity__coeff__ex,axiom,
% 0.29/0.58      ! [P: int > $o,L: int] :
% 0.29/0.58        ( ( ? [X4: int] : ( P @ ( times_times_int @ L @ X4 ) ) )
% 0.29/0.58        = ( ? [X4: int] :
% 0.29/0.58              ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
% 0.29/0.58              & ( P @ X4 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unity_coeff_ex
% 0.29/0.58  thf(fact_6613_dvd__div__eq__mult,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.29/0.58        ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.29/0.58         => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.29/0.58              = C )
% 0.29/0.58            = ( B
% 0.29/0.58              = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_mult
% 0.29/0.58  thf(fact_6614_dvd__div__eq__mult,axiom,
% 0.29/0.58      ! [A: nat,B: nat,C: nat] :
% 0.29/0.58        ( ( A != zero_zero_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ A @ B )
% 0.29/0.58         => ( ( ( divide_divide_nat @ B @ A )
% 0.29/0.58              = C )
% 0.29/0.58            = ( B
% 0.29/0.58              = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_mult
% 0.29/0.58  thf(fact_6615_dvd__div__eq__mult,axiom,
% 0.29/0.58      ! [A: int,B: int,C: int] :
% 0.29/0.58        ( ( A != zero_zero_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ A @ B )
% 0.29/0.58         => ( ( ( divide_divide_int @ B @ A )
% 0.29/0.58              = C )
% 0.29/0.58            = ( B
% 0.29/0.58              = ( times_times_int @ C @ A ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_eq_mult
% 0.29/0.58  thf(fact_6616_div__dvd__iff__mult,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.58        ( ( B != zero_z3403309356797280102nteger )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.29/0.58         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.29/0.58            = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_dvd_iff_mult
% 0.29/0.58  thf(fact_6617_div__dvd__iff__mult,axiom,
% 0.29/0.58      ! [B: nat,A: nat,C: nat] :
% 0.29/0.58        ( ( B != zero_zero_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ B @ A )
% 0.29/0.58         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.29/0.58            = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_dvd_iff_mult
% 0.29/0.58  thf(fact_6618_div__dvd__iff__mult,axiom,
% 0.29/0.58      ! [B: int,A: int,C: int] :
% 0.29/0.58        ( ( B != zero_zero_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ B @ A )
% 0.29/0.58         => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.29/0.58            = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_dvd_iff_mult
% 0.29/0.58  thf(fact_6619_dvd__div__iff__mult,axiom,
% 0.29/0.58      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( C != zero_z3403309356797280102nteger )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.29/0.58         => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.29/0.58            = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_iff_mult
% 0.29/0.58  thf(fact_6620_dvd__div__iff__mult,axiom,
% 0.29/0.58      ! [C: nat,B: nat,A: nat] :
% 0.29/0.58        ( ( C != zero_zero_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ C @ B )
% 0.29/0.58         => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.29/0.58            = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_iff_mult
% 0.29/0.58  thf(fact_6621_dvd__div__iff__mult,axiom,
% 0.29/0.58      ! [C: int,B: int,A: int] :
% 0.29/0.58        ( ( C != zero_zero_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ C @ B )
% 0.29/0.58         => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.29/0.58            = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_iff_mult
% 0.29/0.58  thf(fact_6622_dvd__div__div__eq__mult,axiom,
% 0.29/0.58      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.29/0.58        ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.58       => ( ( C != zero_z3403309356797280102nteger )
% 0.29/0.58         => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.29/0.58           => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.29/0.58             => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.29/0.58                  = ( divide6298287555418463151nteger @ D @ C ) )
% 0.29/0.58                = ( ( times_3573771949741848930nteger @ B @ C )
% 0.29/0.58                  = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_div_eq_mult
% 0.29/0.58  thf(fact_6623_dvd__div__div__eq__mult,axiom,
% 0.29/0.58      ! [A: nat,C: nat,B: nat,D: nat] :
% 0.29/0.58        ( ( A != zero_zero_nat )
% 0.29/0.58       => ( ( C != zero_zero_nat )
% 0.29/0.58         => ( ( dvd_dvd_nat @ A @ B )
% 0.29/0.58           => ( ( dvd_dvd_nat @ C @ D )
% 0.29/0.58             => ( ( ( divide_divide_nat @ B @ A )
% 0.29/0.58                  = ( divide_divide_nat @ D @ C ) )
% 0.29/0.58                = ( ( times_times_nat @ B @ C )
% 0.29/0.58                  = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_div_eq_mult
% 0.29/0.58  thf(fact_6624_dvd__div__div__eq__mult,axiom,
% 0.29/0.58      ! [A: int,C: int,B: int,D: int] :
% 0.29/0.58        ( ( A != zero_zero_int )
% 0.29/0.58       => ( ( C != zero_zero_int )
% 0.29/0.58         => ( ( dvd_dvd_int @ A @ B )
% 0.29/0.58           => ( ( dvd_dvd_int @ C @ D )
% 0.29/0.58             => ( ( ( divide_divide_int @ B @ A )
% 0.29/0.58                  = ( divide_divide_int @ D @ C ) )
% 0.29/0.58                = ( ( times_times_int @ B @ C )
% 0.29/0.58                  = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_div_div_eq_mult
% 0.29/0.58  thf(fact_6625_unit__div__eq__0__iff,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.29/0.58            = zero_z3403309356797280102nteger )
% 0.29/0.58          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_eq_0_iff
% 0.29/0.58  thf(fact_6626_unit__div__eq__0__iff,axiom,
% 0.29/0.58      ! [B: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( ( divide_divide_nat @ A @ B )
% 0.29/0.58            = zero_zero_nat )
% 0.29/0.58          = ( A = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_eq_0_iff
% 0.29/0.58  thf(fact_6627_unit__div__eq__0__iff,axiom,
% 0.29/0.58      ! [B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( ( divide_divide_int @ A @ B )
% 0.29/0.58            = zero_zero_int )
% 0.29/0.58          = ( A = zero_zero_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_eq_0_iff
% 0.29/0.58  thf(fact_6628_even__numeral,axiom,
% 0.29/0.58      ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_numeral
% 0.29/0.58  thf(fact_6629_even__numeral,axiom,
% 0.29/0.58      ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_numeral
% 0.29/0.58  thf(fact_6630_even__numeral,axiom,
% 0.29/0.58      ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_numeral
% 0.29/0.58  thf(fact_6631_inf__period_I4_J,axiom,
% 0.29/0.58      ! [D: code_integer,D2: code_integer,T: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ D @ D2 )
% 0.29/0.58       => ! [X2: code_integer,K6: code_integer] :
% 0.29/0.58            ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T ) ) )
% 0.29/0.58            = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K6 @ D2 ) ) @ T ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % inf_period(4)
% 0.29/0.58  thf(fact_6632_inf__period_I4_J,axiom,
% 0.29/0.58      ! [D: complex,D2: complex,T: complex] :
% 0.29/0.58        ( ( dvd_dvd_complex @ D @ D2 )
% 0.29/0.58       => ! [X2: complex,K6: complex] :
% 0.29/0.58            ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T ) ) )
% 0.29/0.58            = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K6 @ D2 ) ) @ T ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % inf_period(4)
% 0.29/0.58  thf(fact_6633_inf__period_I4_J,axiom,
% 0.29/0.58      ! [D: real,D2: real,T: real] :
% 0.29/0.58        ( ( dvd_dvd_real @ D @ D2 )
% 0.29/0.58       => ! [X2: real,K6: real] :
% 0.29/0.58            ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) ) )
% 0.29/0.58            = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K6 @ D2 ) ) @ T ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % inf_period(4)
% 0.29/0.58  thf(fact_6634_inf__period_I4_J,axiom,
% 0.29/0.58      ! [D: rat,D2: rat,T: rat] :
% 0.29/0.58        ( ( dvd_dvd_rat @ D @ D2 )
% 0.29/0.58       => ! [X2: rat,K6: rat] :
% 0.29/0.58            ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) ) )
% 0.29/0.58            = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K6 @ D2 ) ) @ T ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % inf_period(4)
% 0.29/0.58  thf(fact_6635_inf__period_I4_J,axiom,
% 0.29/0.58      ! [D: int,D2: int,T: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ D @ D2 )
% 0.29/0.58       => ! [X2: int,K6: int] :
% 0.29/0.58            ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) ) )
% 0.29/0.58            = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K6 @ D2 ) ) @ T ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % inf_period(4)
% 0.29/0.58  thf(fact_6636_inf__period_I3_J,axiom,
% 0.29/0.58      ! [D: code_integer,D2: code_integer,T: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ D @ D2 )
% 0.29/0.58       => ! [X2: code_integer,K6: code_integer] :
% 0.29/0.58            ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T ) )
% 0.29/0.58            = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K6 @ D2 ) ) @ T ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % inf_period(3)
% 0.29/0.58  thf(fact_6637_inf__period_I3_J,axiom,
% 0.29/0.58      ! [D: complex,D2: complex,T: complex] :
% 0.29/0.58        ( ( dvd_dvd_complex @ D @ D2 )
% 0.29/0.58       => ! [X2: complex,K6: complex] :
% 0.29/0.58            ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T ) )
% 0.29/0.58            = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K6 @ D2 ) ) @ T ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % inf_period(3)
% 0.29/0.58  thf(fact_6638_inf__period_I3_J,axiom,
% 0.29/0.58      ! [D: real,D2: real,T: real] :
% 0.29/0.58        ( ( dvd_dvd_real @ D @ D2 )
% 0.29/0.58       => ! [X2: real,K6: real] :
% 0.29/0.58            ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) )
% 0.29/0.58            = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K6 @ D2 ) ) @ T ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % inf_period(3)
% 0.29/0.58  thf(fact_6639_inf__period_I3_J,axiom,
% 0.29/0.58      ! [D: rat,D2: rat,T: rat] :
% 0.29/0.58        ( ( dvd_dvd_rat @ D @ D2 )
% 0.29/0.58       => ! [X2: rat,K6: rat] :
% 0.29/0.58            ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) )
% 0.29/0.58            = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K6 @ D2 ) ) @ T ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % inf_period(3)
% 0.29/0.58  thf(fact_6640_inf__period_I3_J,axiom,
% 0.29/0.58      ! [D: int,D2: int,T: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ D @ D2 )
% 0.29/0.58       => ! [X2: int,K6: int] :
% 0.29/0.58            ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 0.29/0.58            = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K6 @ D2 ) ) @ T ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % inf_period(3)
% 0.29/0.58  thf(fact_6641_is__unit__div__mult2__eq,axiom,
% 0.29/0.58      ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.29/0.58         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.29/0.58            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_div_mult2_eq
% 0.29/0.58  thf(fact_6642_is__unit__div__mult2__eq,axiom,
% 0.29/0.58      ! [B: nat,C: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.29/0.58         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.29/0.58            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_div_mult2_eq
% 0.29/0.58  thf(fact_6643_is__unit__div__mult2__eq,axiom,
% 0.29/0.58      ! [B: int,C: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ C @ one_one_int )
% 0.29/0.58         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.29/0.58            = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_div_mult2_eq
% 0.29/0.58  thf(fact_6644_unit__div__mult__swap,axiom,
% 0.29/0.58      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.29/0.58       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.29/0.58          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_mult_swap
% 0.29/0.58  thf(fact_6645_unit__div__mult__swap,axiom,
% 0.29/0.58      ! [C: nat,A: nat,B: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.29/0.58       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.29/0.58          = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_mult_swap
% 0.29/0.58  thf(fact_6646_unit__div__mult__swap,axiom,
% 0.29/0.58      ! [C: int,A: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ C @ one_one_int )
% 0.29/0.58       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.29/0.58          = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_mult_swap
% 0.29/0.58  thf(fact_6647_unit__div__commute,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.29/0.58          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_commute
% 0.29/0.58  thf(fact_6648_unit__div__commute,axiom,
% 0.29/0.58      ! [B: nat,A: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.29/0.58          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_commute
% 0.29/0.58  thf(fact_6649_unit__div__commute,axiom,
% 0.29/0.58      ! [B: int,A: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.29/0.58          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_div_commute
% 0.29/0.58  thf(fact_6650_div__mult__unit2,axiom,
% 0.29/0.58      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.29/0.58         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.29/0.58            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_mult_unit2
% 0.29/0.58  thf(fact_6651_div__mult__unit2,axiom,
% 0.29/0.58      ! [C: nat,B: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ B @ A )
% 0.29/0.58         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.29/0.58            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_mult_unit2
% 0.29/0.58  thf(fact_6652_div__mult__unit2,axiom,
% 0.29/0.58      ! [C: int,B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ C @ one_one_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ B @ A )
% 0.29/0.58         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.29/0.58            = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % div_mult_unit2
% 0.29/0.58  thf(fact_6653_unit__eq__div2,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( A
% 0.29/0.58            = ( divide6298287555418463151nteger @ C @ B ) )
% 0.29/0.58          = ( ( times_3573771949741848930nteger @ A @ B )
% 0.29/0.58            = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_eq_div2
% 0.29/0.58  thf(fact_6654_unit__eq__div2,axiom,
% 0.29/0.58      ! [B: nat,A: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( A
% 0.29/0.58            = ( divide_divide_nat @ C @ B ) )
% 0.29/0.58          = ( ( times_times_nat @ A @ B )
% 0.29/0.58            = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_eq_div2
% 0.29/0.58  thf(fact_6655_unit__eq__div2,axiom,
% 0.29/0.58      ! [B: int,A: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( A
% 0.29/0.58            = ( divide_divide_int @ C @ B ) )
% 0.29/0.58          = ( ( times_times_int @ A @ B )
% 0.29/0.58            = C ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_eq_div2
% 0.29/0.58  thf(fact_6656_unit__eq__div1,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.29/0.58            = C )
% 0.29/0.58          = ( A
% 0.29/0.58            = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_eq_div1
% 0.29/0.58  thf(fact_6657_unit__eq__div1,axiom,
% 0.29/0.58      ! [B: nat,A: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( ( divide_divide_nat @ A @ B )
% 0.29/0.58            = C )
% 0.29/0.58          = ( A
% 0.29/0.58            = ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_eq_div1
% 0.29/0.58  thf(fact_6658_unit__eq__div1,axiom,
% 0.29/0.58      ! [B: int,A: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( ( divide_divide_int @ A @ B )
% 0.29/0.58            = C )
% 0.29/0.58          = ( A
% 0.29/0.58            = ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_eq_div1
% 0.29/0.58  thf(fact_6659_exp__minus__inverse,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
% 0.29/0.58        = one_one_real ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_minus_inverse
% 0.29/0.58  thf(fact_6660_exp__minus__inverse,axiom,
% 0.29/0.58      ! [X: complex] :
% 0.29/0.58        ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
% 0.29/0.58        = one_one_complex ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_minus_inverse
% 0.29/0.58  thf(fact_6661_unit__imp__mod__eq__0,axiom,
% 0.29/0.58      ! [B: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58       => ( ( modulo_modulo_nat @ A @ B )
% 0.29/0.58          = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_imp_mod_eq_0
% 0.29/0.58  thf(fact_6662_unit__imp__mod__eq__0,axiom,
% 0.29/0.58      ! [B: int,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58       => ( ( modulo_modulo_int @ A @ B )
% 0.29/0.58          = zero_zero_int ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_imp_mod_eq_0
% 0.29/0.58  thf(fact_6663_unit__imp__mod__eq__0,axiom,
% 0.29/0.58      ! [B: code_integer,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58       => ( ( modulo364778990260209775nteger @ A @ B )
% 0.29/0.58          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.58  
% 0.29/0.58  % unit_imp_mod_eq_0
% 0.29/0.58  thf(fact_6664_is__unit__power__iff,axiom,
% 0.29/0.58      ! [A: code_integer,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 0.29/0.58        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.58          | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_power_iff
% 0.29/0.58  thf(fact_6665_is__unit__power__iff,axiom,
% 0.29/0.58      ! [A: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 0.29/0.58        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.58          | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_power_iff
% 0.29/0.58  thf(fact_6666_is__unit__power__iff,axiom,
% 0.29/0.58      ! [A: int,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 0.29/0.58        = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.58          | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_power_iff
% 0.29/0.58  thf(fact_6667_exp__of__nat2__mult,axiom,
% 0.29/0.58      ! [X: complex,N: nat] :
% 0.29/0.58        ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.29/0.58        = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_of_nat2_mult
% 0.29/0.58  thf(fact_6668_exp__of__nat2__mult,axiom,
% 0.29/0.58      ! [X: real,N: nat] :
% 0.29/0.58        ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.29/0.58        = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_of_nat2_mult
% 0.29/0.58  thf(fact_6669_exp__of__nat__mult,axiom,
% 0.29/0.58      ! [N: nat,X: complex] :
% 0.29/0.58        ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
% 0.29/0.58        = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_of_nat_mult
% 0.29/0.58  thf(fact_6670_exp__of__nat__mult,axiom,
% 0.29/0.58      ! [N: nat,X: real] :
% 0.29/0.58        ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
% 0.29/0.58        = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_of_nat_mult
% 0.29/0.58  thf(fact_6671_log__ln,axiom,
% 0.29/0.58      ( ln_ln_real
% 0.29/0.58      = ( log @ ( exp_real @ one_one_real ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % log_ln
% 0.29/0.58  thf(fact_6672_dvd__imp__le,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ K @ N )
% 0.29/0.58       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58         => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_imp_le
% 0.29/0.58  thf(fact_6673_nat__mult__dvd__cancel1,axiom,
% 0.29/0.58      ! [K: nat,M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.58       => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.58          = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % nat_mult_dvd_cancel1
% 0.29/0.58  thf(fact_6674_dvd__mult__cancel,axiom,
% 0.29/0.58      ! [K: nat,M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.58       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.58         => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_cancel
% 0.29/0.58  thf(fact_6675_bezout__add__strong__nat,axiom,
% 0.29/0.58      ! [A: nat,B: nat] :
% 0.29/0.58        ( ( A != zero_zero_nat )
% 0.29/0.58       => ? [D5: nat,X3: nat,Y4: nat] :
% 0.29/0.58            ( ( dvd_dvd_nat @ D5 @ A )
% 0.29/0.58            & ( dvd_dvd_nat @ D5 @ B )
% 0.29/0.58            & ( ( times_times_nat @ A @ X3 )
% 0.29/0.58              = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D5 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bezout_add_strong_nat
% 0.29/0.58  thf(fact_6676_zdvd__imp__le,axiom,
% 0.29/0.58      ! [Z: int,N: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ Z @ N )
% 0.29/0.58       => ( ( ord_less_int @ zero_zero_int @ N )
% 0.29/0.58         => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zdvd_imp_le
% 0.29/0.58  thf(fact_6677_mod__greater__zero__iff__not__dvd,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.29/0.58        = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod_greater_zero_iff_not_dvd
% 0.29/0.58  thf(fact_6678_dvd__imp__le__int,axiom,
% 0.29/0.58      ! [I: int,D: int] :
% 0.29/0.58        ( ( I != zero_zero_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ D @ I )
% 0.29/0.58         => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_imp_le_int
% 0.29/0.58  thf(fact_6679_mod__eq__dvd__iff__nat,axiom,
% 0.29/0.58      ! [N: nat,M: nat,Q3: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58       => ( ( ( modulo_modulo_nat @ M @ Q3 )
% 0.29/0.58            = ( modulo_modulo_nat @ N @ Q3 ) )
% 0.29/0.58          = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod_eq_dvd_iff_nat
% 0.29/0.58  thf(fact_6680_real__of__nat__div,axiom,
% 0.29/0.58      ! [D: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ D @ N )
% 0.29/0.58       => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 0.29/0.58          = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % real_of_nat_div
% 0.29/0.58  thf(fact_6681_real__of__int__div,axiom,
% 0.29/0.58      ! [D: int,N: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ D @ N )
% 0.29/0.58       => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 0.29/0.58          = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % real_of_int_div
% 0.29/0.58  thf(fact_6682_exp__ge__add__one__self__aux,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.58       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_ge_add_one_self_aux
% 0.29/0.58  thf(fact_6683_lemma__exp__total,axiom,
% 0.29/0.58      ! [Y: real] :
% 0.29/0.58        ( ( ord_less_eq_real @ one_one_real @ Y )
% 0.29/0.58       => ? [X3: real] :
% 0.29/0.58            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.58            & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y @ one_one_real ) )
% 0.29/0.58            & ( ( exp_real @ X3 )
% 0.29/0.58              = Y ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % lemma_exp_total
% 0.29/0.58  thf(fact_6684_ln__ge__iff,axiom,
% 0.29/0.58      ! [X: real,Y: real] :
% 0.29/0.58        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.58       => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
% 0.29/0.58          = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % ln_ge_iff
% 0.29/0.58  thf(fact_6685_even__zero,axiom,
% 0.29/0.58      dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 0.29/0.58  
% 0.29/0.58  % even_zero
% 0.29/0.58  thf(fact_6686_even__zero,axiom,
% 0.29/0.58      dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 0.29/0.58  
% 0.29/0.58  % even_zero
% 0.29/0.58  thf(fact_6687_even__zero,axiom,
% 0.29/0.58      dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 0.29/0.58  
% 0.29/0.58  % even_zero
% 0.29/0.58  thf(fact_6688_ln__x__over__x__mono,axiom,
% 0.29/0.58      ! [X: real,Y: real] :
% 0.29/0.58        ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
% 0.29/0.58       => ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.58         => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % ln_x_over_x_mono
% 0.29/0.58  thf(fact_6689_is__unit__div__mult__cancel__right,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 0.29/0.58            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_div_mult_cancel_right
% 0.29/0.58  thf(fact_6690_is__unit__div__mult__cancel__right,axiom,
% 0.29/0.58      ! [A: nat,B: nat] :
% 0.29/0.58        ( ( A != zero_zero_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 0.29/0.58            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_div_mult_cancel_right
% 0.29/0.58  thf(fact_6691_is__unit__div__mult__cancel__right,axiom,
% 0.29/0.58      ! [A: int,B: int] :
% 0.29/0.58        ( ( A != zero_zero_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 0.29/0.58            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_div_mult_cancel_right
% 0.29/0.58  thf(fact_6692_is__unit__div__mult__cancel__left,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.29/0.58         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.29/0.58            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_div_mult_cancel_left
% 0.29/0.58  thf(fact_6693_is__unit__div__mult__cancel__left,axiom,
% 0.29/0.58      ! [A: nat,B: nat] :
% 0.29/0.58        ( ( A != zero_zero_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.29/0.58         => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 0.29/0.58            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_div_mult_cancel_left
% 0.29/0.58  thf(fact_6694_is__unit__div__mult__cancel__left,axiom,
% 0.29/0.58      ! [A: int,B: int] :
% 0.29/0.58        ( ( A != zero_zero_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.29/0.58         => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 0.29/0.58            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unit_div_mult_cancel_left
% 0.29/0.58  thf(fact_6695_is__unitE,axiom,
% 0.29/0.58      ! [A: code_integer,C: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.29/0.58       => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.58           => ! [B6: code_integer] :
% 0.29/0.58                ( ( B6 != zero_z3403309356797280102nteger )
% 0.29/0.58               => ( ( dvd_dvd_Code_integer @ B6 @ one_one_Code_integer )
% 0.29/0.58                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 0.29/0.58                      = B6 )
% 0.29/0.58                   => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B6 )
% 0.29/0.58                        = A )
% 0.29/0.58                     => ( ( ( times_3573771949741848930nteger @ A @ B6 )
% 0.29/0.58                          = one_one_Code_integer )
% 0.29/0.58                       => ( ( divide6298287555418463151nteger @ C @ A )
% 0.29/0.58                         != ( times_3573771949741848930nteger @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unitE
% 0.29/0.58  thf(fact_6696_is__unitE,axiom,
% 0.29/0.58      ! [A: nat,C: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.29/0.58       => ~ ( ( A != zero_zero_nat )
% 0.29/0.58           => ! [B6: nat] :
% 0.29/0.58                ( ( B6 != zero_zero_nat )
% 0.29/0.58               => ( ( dvd_dvd_nat @ B6 @ one_one_nat )
% 0.29/0.58                 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 0.29/0.58                      = B6 )
% 0.29/0.58                   => ( ( ( divide_divide_nat @ one_one_nat @ B6 )
% 0.29/0.58                        = A )
% 0.29/0.58                     => ( ( ( times_times_nat @ A @ B6 )
% 0.29/0.58                          = one_one_nat )
% 0.29/0.58                       => ( ( divide_divide_nat @ C @ A )
% 0.29/0.58                         != ( times_times_nat @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unitE
% 0.29/0.58  thf(fact_6697_is__unitE,axiom,
% 0.29/0.58      ! [A: int,C: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.29/0.58       => ~ ( ( A != zero_zero_int )
% 0.29/0.58           => ! [B6: int] :
% 0.29/0.58                ( ( B6 != zero_zero_int )
% 0.29/0.58               => ( ( dvd_dvd_int @ B6 @ one_one_int )
% 0.29/0.58                 => ( ( ( divide_divide_int @ one_one_int @ A )
% 0.29/0.58                      = B6 )
% 0.29/0.58                   => ( ( ( divide_divide_int @ one_one_int @ B6 )
% 0.29/0.58                        = A )
% 0.29/0.58                     => ( ( ( times_times_int @ A @ B6 )
% 0.29/0.58                          = one_one_int )
% 0.29/0.58                       => ( ( divide_divide_int @ C @ A )
% 0.29/0.58                         != ( times_times_int @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % is_unitE
% 0.29/0.58  thf(fact_6698_evenE,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ~ ! [B6: code_integer] :
% 0.29/0.58              ( A
% 0.29/0.58             != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % evenE
% 0.29/0.58  thf(fact_6699_evenE,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ~ ! [B6: nat] :
% 0.29/0.58              ( A
% 0.29/0.58             != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % evenE
% 0.29/0.58  thf(fact_6700_evenE,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ~ ! [B6: int] :
% 0.29/0.58              ( A
% 0.29/0.58             != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % evenE
% 0.29/0.58  thf(fact_6701_odd__one,axiom,
% 0.29/0.58      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_one
% 0.29/0.58  thf(fact_6702_odd__one,axiom,
% 0.29/0.58      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_one
% 0.29/0.58  thf(fact_6703_odd__one,axiom,
% 0.29/0.58      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_one
% 0.29/0.58  thf(fact_6704_odd__even__add,axiom,
% 0.29/0.58      ! [A: code_integer,B: code_integer] :
% 0.29/0.58        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 0.29/0.58         => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_even_add
% 0.29/0.58  thf(fact_6705_odd__even__add,axiom,
% 0.29/0.58      ! [A: nat,B: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.29/0.58         => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_even_add
% 0.29/0.58  thf(fact_6706_odd__even__add,axiom,
% 0.29/0.58      ! [A: int,B: int] :
% 0.29/0.58        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 0.29/0.58         => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_even_add
% 0.29/0.58  thf(fact_6707_bit__eq__rec,axiom,
% 0.29/0.58      ( ( ^ [Y3: code_integer,Z2: code_integer] : ( Y3 = Z2 ) )
% 0.29/0.58      = ( ^ [A3: code_integer,B4: code_integer] :
% 0.29/0.58            ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
% 0.29/0.58              = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) )
% 0.29/0.58            & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58              = ( divide6298287555418463151nteger @ B4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bit_eq_rec
% 0.29/0.58  thf(fact_6708_bit__eq__rec,axiom,
% 0.29/0.58      ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 0.29/0.58      = ( ^ [A3: nat,B4: nat] :
% 0.29/0.58            ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
% 0.29/0.58              = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
% 0.29/0.58            & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58              = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bit_eq_rec
% 0.29/0.58  thf(fact_6709_bit__eq__rec,axiom,
% 0.29/0.58      ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 0.29/0.58      = ( ^ [A3: int,B4: int] :
% 0.29/0.58            ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
% 0.29/0.58              = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
% 0.29/0.58            & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58              = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bit_eq_rec
% 0.29/0.58  thf(fact_6710_even__minus,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 0.29/0.58        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_minus
% 0.29/0.58  thf(fact_6711_even__minus,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.29/0.58        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_minus
% 0.29/0.58  thf(fact_6712_dvd__power__iff,axiom,
% 0.29/0.58      ! [X: code_integer,M: nat,N: nat] :
% 0.29/0.58        ( ( X != zero_z3403309356797280102nteger )
% 0.29/0.58       => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
% 0.29/0.58          = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 0.29/0.58            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_iff
% 0.29/0.58  thf(fact_6713_dvd__power__iff,axiom,
% 0.29/0.58      ! [X: nat,M: nat,N: nat] :
% 0.29/0.58        ( ( X != zero_zero_nat )
% 0.29/0.58       => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
% 0.29/0.58          = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 0.29/0.58            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_iff
% 0.29/0.58  thf(fact_6714_dvd__power__iff,axiom,
% 0.29/0.58      ! [X: int,M: nat,N: nat] :
% 0.29/0.58        ( ( X != zero_zero_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
% 0.29/0.58          = ( ( dvd_dvd_int @ X @ one_one_int )
% 0.29/0.58            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_iff
% 0.29/0.58  thf(fact_6715_odd__numeral,axiom,
% 0.29/0.58      ! [N: num] :
% 0.29/0.58        ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_numeral
% 0.29/0.58  thf(fact_6716_odd__numeral,axiom,
% 0.29/0.58      ! [N: num] :
% 0.29/0.58        ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_numeral
% 0.29/0.58  thf(fact_6717_odd__numeral,axiom,
% 0.29/0.58      ! [N: num] :
% 0.29/0.58        ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_numeral
% 0.29/0.58  thf(fact_6718_dvd__power,axiom,
% 0.29/0.58      ! [N: nat,X: code_integer] :
% 0.29/0.58        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58          | ( X = one_one_Code_integer ) )
% 0.29/0.58       => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power
% 0.29/0.58  thf(fact_6719_dvd__power,axiom,
% 0.29/0.58      ! [N: nat,X: rat] :
% 0.29/0.58        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58          | ( X = one_one_rat ) )
% 0.29/0.58       => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power
% 0.29/0.58  thf(fact_6720_dvd__power,axiom,
% 0.29/0.58      ! [N: nat,X: nat] :
% 0.29/0.58        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58          | ( X = one_one_nat ) )
% 0.29/0.58       => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power
% 0.29/0.58  thf(fact_6721_dvd__power,axiom,
% 0.29/0.58      ! [N: nat,X: real] :
% 0.29/0.58        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58          | ( X = one_one_real ) )
% 0.29/0.58       => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power
% 0.29/0.58  thf(fact_6722_dvd__power,axiom,
% 0.29/0.58      ! [N: nat,X: int] :
% 0.29/0.58        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58          | ( X = one_one_int ) )
% 0.29/0.58       => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power
% 0.29/0.58  thf(fact_6723_dvd__power,axiom,
% 0.29/0.58      ! [N: nat,X: complex] :
% 0.29/0.58        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58          | ( X = one_one_complex ) )
% 0.29/0.58       => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power
% 0.29/0.58  thf(fact_6724_even__signed__take__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 0.29/0.58        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_signed_take_bit_iff
% 0.29/0.58  thf(fact_6725_even__signed__take__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 0.29/0.58        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_signed_take_bit_iff
% 0.29/0.58  thf(fact_6726_even__even__mod__4__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_even_mod_4_iff
% 0.29/0.58  thf(fact_6727_dvd__mult__cancel2,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.58       => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 0.29/0.58          = ( N = one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_cancel2
% 0.29/0.58  thf(fact_6728_dvd__mult__cancel1,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.58       => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 0.29/0.58          = ( N = one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_mult_cancel1
% 0.29/0.58  thf(fact_6729_powr__def,axiom,
% 0.29/0.58      ( powr_real
% 0.29/0.58      = ( ^ [X4: real,A3: real] : ( if_real @ ( X4 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X4 ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % powr_def
% 0.29/0.58  thf(fact_6730_dvd__minus__add,axiom,
% 0.29/0.58      ! [Q3: nat,N: nat,R2: nat,M: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ Q3 @ N )
% 0.29/0.58       => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M ) )
% 0.29/0.58         => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q3 ) )
% 0.29/0.58            = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q3 ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_minus_add
% 0.29/0.58  thf(fact_6731_power__dvd__imp__le,axiom,
% 0.29/0.58      ! [I: nat,M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.29/0.58       => ( ( ord_less_nat @ one_one_nat @ I )
% 0.29/0.58         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_dvd_imp_le
% 0.29/0.58  thf(fact_6732_mod__nat__eqI,axiom,
% 0.29/0.58      ! [R2: nat,N: nat,M: nat] :
% 0.29/0.58        ( ( ord_less_nat @ R2 @ N )
% 0.29/0.58       => ( ( ord_less_eq_nat @ R2 @ M )
% 0.29/0.58         => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 0.29/0.58           => ( ( modulo_modulo_nat @ M @ N )
% 0.29/0.58              = R2 ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod_nat_eqI
% 0.29/0.58  thf(fact_6733_zdvd__mult__cancel1,axiom,
% 0.29/0.58      ! [M: int,N: int] :
% 0.29/0.58        ( ( M != zero_zero_int )
% 0.29/0.58       => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 0.29/0.58          = ( ( abs_abs_int @ N )
% 0.29/0.58            = one_one_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zdvd_mult_cancel1
% 0.29/0.58  thf(fact_6734_mod__int__pos__iff,axiom,
% 0.29/0.58      ! [K: int,L: int] :
% 0.29/0.58        ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 0.29/0.58        = ( ( dvd_dvd_int @ L @ K )
% 0.29/0.58          | ( ( L = zero_zero_int )
% 0.29/0.58            & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.29/0.58          | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod_int_pos_iff
% 0.29/0.58  thf(fact_6735_exp__le,axiom,
% 0.29/0.58      ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_le
% 0.29/0.58  thf(fact_6736_even__two__times__div__two,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.29/0.58          = A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_two_times_div_two
% 0.29/0.58  thf(fact_6737_even__two__times__div__two,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.58          = A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_two_times_div_two
% 0.29/0.58  thf(fact_6738_even__two__times__div__two,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.29/0.58          = A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_two_times_div_two
% 0.29/0.58  thf(fact_6739_even__iff__mod__2__eq__zero,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58          = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_iff_mod_2_eq_zero
% 0.29/0.58  thf(fact_6740_even__iff__mod__2__eq__zero,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58          = zero_zero_int ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_iff_mod_2_eq_zero
% 0.29/0.58  thf(fact_6741_even__iff__mod__2__eq__zero,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_iff_mod_2_eq_zero
% 0.29/0.58  thf(fact_6742_odd__iff__mod__2__eq__one,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.58        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58          = one_one_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_iff_mod_2_eq_one
% 0.29/0.58  thf(fact_6743_odd__iff__mod__2__eq__one,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.58        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58          = one_one_int ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_iff_mod_2_eq_one
% 0.29/0.58  thf(fact_6744_odd__iff__mod__2__eq__one,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.58        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58          = one_one_Code_integer ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_iff_mod_2_eq_one
% 0.29/0.58  thf(fact_6745_power__mono__odd,axiom,
% 0.29/0.58      ! [N: nat,A: real,B: real] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( ord_less_eq_real @ A @ B )
% 0.29/0.58         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_mono_odd
% 0.29/0.58  thf(fact_6746_power__mono__odd,axiom,
% 0.29/0.58      ! [N: nat,A: rat,B: rat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.58         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_mono_odd
% 0.29/0.58  thf(fact_6747_power__mono__odd,axiom,
% 0.29/0.58      ! [N: nat,A: int,B: int] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( ord_less_eq_int @ A @ B )
% 0.29/0.58         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_mono_odd
% 0.29/0.58  thf(fact_6748_exp__divide__power__eq,axiom,
% 0.29/0.58      ! [N: nat,X: complex] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58       => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 0.29/0.58          = ( exp_complex @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_divide_power_eq
% 0.29/0.58  thf(fact_6749_exp__divide__power__eq,axiom,
% 0.29/0.58      ! [N: nat,X: real] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58       => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 0.29/0.58          = ( exp_real @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_divide_power_eq
% 0.29/0.58  thf(fact_6750_uminus__power__if,axiom,
% 0.29/0.58      ! [N: nat,A: real] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.29/0.58            = ( power_power_real @ A @ N ) ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.29/0.58            = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % uminus_power_if
% 0.29/0.58  thf(fact_6751_uminus__power__if,axiom,
% 0.29/0.58      ! [N: nat,A: int] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.29/0.58            = ( power_power_int @ A @ N ) ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.29/0.58            = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % uminus_power_if
% 0.29/0.58  thf(fact_6752_uminus__power__if,axiom,
% 0.29/0.58      ! [N: nat,A: complex] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.29/0.58            = ( power_power_complex @ A @ N ) ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.29/0.58            = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % uminus_power_if
% 0.29/0.58  thf(fact_6753_uminus__power__if,axiom,
% 0.29/0.58      ! [N: nat,A: rat] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.29/0.58            = ( power_power_rat @ A @ N ) ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.29/0.58            = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % uminus_power_if
% 0.29/0.58  thf(fact_6754_uminus__power__if,axiom,
% 0.29/0.58      ! [N: nat,A: code_integer] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.29/0.58            = ( power_8256067586552552935nteger @ A @ N ) ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.29/0.58            = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % uminus_power_if
% 0.29/0.58  thf(fact_6755_odd__pos,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_pos
% 0.29/0.58  thf(fact_6756_power__even__abs,axiom,
% 0.29/0.58      ! [N: nat,A: rat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 0.29/0.58          = ( power_power_rat @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_even_abs
% 0.29/0.58  thf(fact_6757_power__even__abs,axiom,
% 0.29/0.58      ! [N: nat,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 0.29/0.58          = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_even_abs
% 0.29/0.58  thf(fact_6758_power__even__abs,axiom,
% 0.29/0.58      ! [N: nat,A: real] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 0.29/0.58          = ( power_power_real @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_even_abs
% 0.29/0.58  thf(fact_6759_power__even__abs,axiom,
% 0.29/0.58      ! [N: nat,A: int] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 0.29/0.58          = ( power_power_int @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_even_abs
% 0.29/0.58  thf(fact_6760_dvd__power__iff__le,axiom,
% 0.29/0.58      ! [K: nat,M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.29/0.58       => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 0.29/0.58          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_power_iff_le
% 0.29/0.58  thf(fact_6761_even__unset__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.29/0.58        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58          | ( M = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_unset_bit_iff
% 0.29/0.58  thf(fact_6762_even__unset__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.29/0.58        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58          | ( M = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_unset_bit_iff
% 0.29/0.58  thf(fact_6763_even__unset__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.29/0.58        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58          | ( M = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_unset_bit_iff
% 0.29/0.58  thf(fact_6764_even__set__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.29/0.58        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58          & ( M != zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_set_bit_iff
% 0.29/0.58  thf(fact_6765_even__set__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.29/0.58        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58          & ( M != zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_set_bit_iff
% 0.29/0.58  thf(fact_6766_even__set__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.29/0.58        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58          & ( M != zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_set_bit_iff
% 0.29/0.58  thf(fact_6767_even__flip__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.29/0.58        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         != ( M = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_flip_bit_iff
% 0.29/0.58  thf(fact_6768_even__flip__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.29/0.58        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         != ( M = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_flip_bit_iff
% 0.29/0.58  thf(fact_6769_even__flip__bit__iff,axiom,
% 0.29/0.58      ! [M: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.29/0.58        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         != ( M = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_flip_bit_iff
% 0.29/0.58  thf(fact_6770_tanh__altdef,axiom,
% 0.29/0.58      ( tanh_real
% 0.29/0.58      = ( ^ [X4: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % tanh_altdef
% 0.29/0.58  thf(fact_6771_tanh__altdef,axiom,
% 0.29/0.58      ( tanh_complex
% 0.29/0.58      = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % tanh_altdef
% 0.29/0.58  thf(fact_6772_even__diff__iff,axiom,
% 0.29/0.58      ! [K: int,L: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
% 0.29/0.58        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_diff_iff
% 0.29/0.58  thf(fact_6773_even__add__abs__iff,axiom,
% 0.29/0.58      ! [K: int,L: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
% 0.29/0.58        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_add_abs_iff
% 0.29/0.58  thf(fact_6774_even__abs__add__iff,axiom,
% 0.29/0.58      ! [K: int,L: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
% 0.29/0.58        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_abs_add_iff
% 0.29/0.58  thf(fact_6775_exp__half__le2,axiom,
% 0.29/0.58      ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_half_le2
% 0.29/0.58  thf(fact_6776_oddE,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ~ ! [B6: code_integer] :
% 0.29/0.58              ( A
% 0.29/0.58             != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B6 ) @ one_one_Code_integer ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % oddE
% 0.29/0.58  thf(fact_6777_oddE,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ~ ! [B6: nat] :
% 0.29/0.58              ( A
% 0.29/0.58             != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) @ one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % oddE
% 0.29/0.58  thf(fact_6778_oddE,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58       => ~ ! [B6: int] :
% 0.29/0.58              ( A
% 0.29/0.58             != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) @ one_one_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % oddE
% 0.29/0.58  thf(fact_6779_parity__cases,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58           != zero_zero_nat ) )
% 0.29/0.58       => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58           => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58             != one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % parity_cases
% 0.29/0.58  thf(fact_6780_parity__cases,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58           != zero_zero_int ) )
% 0.29/0.58       => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58           => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58             != one_one_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % parity_cases
% 0.29/0.58  thf(fact_6781_parity__cases,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58           != zero_z3403309356797280102nteger ) )
% 0.29/0.58       => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58           => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58             != one_one_Code_integer ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % parity_cases
% 0.29/0.58  thf(fact_6782_mod2__eq__if,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58            = zero_zero_nat ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58            = one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod2_eq_if
% 0.29/0.58  thf(fact_6783_mod2__eq__if,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58            = zero_zero_int ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58            = one_one_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod2_eq_if
% 0.29/0.58  thf(fact_6784_mod2__eq__if,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58            = zero_z3403309356797280102nteger ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.58         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58            = one_one_Code_integer ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mod2_eq_if
% 0.29/0.58  thf(fact_6785_exp__double,axiom,
% 0.29/0.58      ! [Z: complex] :
% 0.29/0.58        ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 0.29/0.58        = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_double
% 0.29/0.58  thf(fact_6786_exp__double,axiom,
% 0.29/0.58      ! [Z: real] :
% 0.29/0.58        ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 0.29/0.58        = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_double
% 0.29/0.58  thf(fact_6787_zero__le__even__power,axiom,
% 0.29/0.58      ! [N: nat,A: real] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_le_even_power
% 0.29/0.58  thf(fact_6788_zero__le__even__power,axiom,
% 0.29/0.58      ! [N: nat,A: rat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_le_even_power
% 0.29/0.58  thf(fact_6789_zero__le__even__power,axiom,
% 0.29/0.58      ! [N: nat,A: int] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_le_even_power
% 0.29/0.58  thf(fact_6790_zero__le__odd__power,axiom,
% 0.29/0.58      ! [N: nat,A: real] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.29/0.58          = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_le_odd_power
% 0.29/0.58  thf(fact_6791_zero__le__odd__power,axiom,
% 0.29/0.58      ! [N: nat,A: rat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.29/0.58          = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_le_odd_power
% 0.29/0.58  thf(fact_6792_zero__le__odd__power,axiom,
% 0.29/0.58      ! [N: nat,A: int] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.29/0.58          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_le_odd_power
% 0.29/0.58  thf(fact_6793_zero__le__power__eq,axiom,
% 0.29/0.58      ! [A: real,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.29/0.58        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_le_power_eq
% 0.29/0.58  thf(fact_6794_zero__le__power__eq,axiom,
% 0.29/0.58      ! [A: rat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.29/0.58        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_le_power_eq
% 0.29/0.58  thf(fact_6795_zero__le__power__eq,axiom,
% 0.29/0.58      ! [A: int,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.29/0.58        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_le_power_eq
% 0.29/0.58  thf(fact_6796_minus__one__power__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.29/0.58            = one_one_real ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.29/0.58            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minus_one_power_iff
% 0.29/0.58  thf(fact_6797_minus__one__power__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.29/0.58            = one_one_int ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.29/0.58            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minus_one_power_iff
% 0.29/0.58  thf(fact_6798_minus__one__power__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.29/0.58            = one_one_complex ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.29/0.58            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minus_one_power_iff
% 0.29/0.58  thf(fact_6799_minus__one__power__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.29/0.58            = one_one_rat ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.29/0.58            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minus_one_power_iff
% 0.29/0.58  thf(fact_6800_minus__one__power__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.29/0.58            = one_one_Code_integer ) )
% 0.29/0.58        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.29/0.58            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % minus_one_power_iff
% 0.29/0.58  thf(fact_6801_power__mono__even,axiom,
% 0.29/0.58      ! [N: nat,A: code_integer,B: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 0.29/0.58         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_mono_even
% 0.29/0.58  thf(fact_6802_power__mono__even,axiom,
% 0.29/0.58      ! [N: nat,A: real,B: real] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 0.29/0.58         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_mono_even
% 0.29/0.58  thf(fact_6803_power__mono__even,axiom,
% 0.29/0.58      ! [N: nat,A: rat,B: rat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 0.29/0.58         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_mono_even
% 0.29/0.58  thf(fact_6804_power__mono__even,axiom,
% 0.29/0.58      ! [N: nat,A: int,B: int] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 0.29/0.58         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_mono_even
% 0.29/0.58  thf(fact_6805_central__binomial__odd,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.58          = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % central_binomial_odd
% 0.29/0.58  thf(fact_6806_add__0__iff,axiom,
% 0.29/0.58      ! [B: complex,A: complex] :
% 0.29/0.58        ( ( B
% 0.29/0.58          = ( plus_plus_complex @ B @ A ) )
% 0.29/0.58        = ( A = zero_zero_complex ) ) ).
% 0.29/0.58  
% 0.29/0.58  % add_0_iff
% 0.29/0.58  thf(fact_6807_add__0__iff,axiom,
% 0.29/0.58      ! [B: real,A: real] :
% 0.29/0.58        ( ( B
% 0.29/0.58          = ( plus_plus_real @ B @ A ) )
% 0.29/0.58        = ( A = zero_zero_real ) ) ).
% 0.29/0.58  
% 0.29/0.58  % add_0_iff
% 0.29/0.58  thf(fact_6808_add__0__iff,axiom,
% 0.29/0.58      ! [B: rat,A: rat] :
% 0.29/0.58        ( ( B
% 0.29/0.58          = ( plus_plus_rat @ B @ A ) )
% 0.29/0.58        = ( A = zero_zero_rat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % add_0_iff
% 0.29/0.58  thf(fact_6809_add__0__iff,axiom,
% 0.29/0.58      ! [B: nat,A: nat] :
% 0.29/0.58        ( ( B
% 0.29/0.58          = ( plus_plus_nat @ B @ A ) )
% 0.29/0.58        = ( A = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % add_0_iff
% 0.29/0.58  thf(fact_6810_add__0__iff,axiom,
% 0.29/0.58      ! [B: int,A: int] :
% 0.29/0.58        ( ( B
% 0.29/0.58          = ( plus_plus_int @ B @ A ) )
% 0.29/0.58        = ( A = zero_zero_int ) ) ).
% 0.29/0.58  
% 0.29/0.58  % add_0_iff
% 0.29/0.58  thf(fact_6811_crossproduct__noteq,axiom,
% 0.29/0.58      ! [A: complex,B: complex,C: complex,D: complex] :
% 0.29/0.58        ( ( ( A != B )
% 0.29/0.58          & ( C != D ) )
% 0.29/0.58        = ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) )
% 0.29/0.58         != ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % crossproduct_noteq
% 0.29/0.58  thf(fact_6812_crossproduct__noteq,axiom,
% 0.29/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.58        ( ( ( A != B )
% 0.29/0.58          & ( C != D ) )
% 0.29/0.58        = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 0.29/0.58         != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % crossproduct_noteq
% 0.29/0.58  thf(fact_6813_crossproduct__noteq,axiom,
% 0.29/0.58      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.58        ( ( ( A != B )
% 0.29/0.58          & ( C != D ) )
% 0.29/0.58        = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 0.29/0.58         != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % crossproduct_noteq
% 0.29/0.58  thf(fact_6814_crossproduct__noteq,axiom,
% 0.29/0.58      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.58        ( ( ( A != B )
% 0.29/0.58          & ( C != D ) )
% 0.29/0.58        = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 0.29/0.58         != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % crossproduct_noteq
% 0.29/0.58  thf(fact_6815_crossproduct__noteq,axiom,
% 0.29/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.58        ( ( ( A != B )
% 0.29/0.58          & ( C != D ) )
% 0.29/0.58        = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 0.29/0.58         != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % crossproduct_noteq
% 0.29/0.58  thf(fact_6816_crossproduct__eq,axiom,
% 0.29/0.58      ! [W: complex,Y: complex,X: complex,Z: complex] :
% 0.29/0.58        ( ( ( plus_plus_complex @ ( times_times_complex @ W @ Y ) @ ( times_times_complex @ X @ Z ) )
% 0.29/0.58          = ( plus_plus_complex @ ( times_times_complex @ W @ Z ) @ ( times_times_complex @ X @ Y ) ) )
% 0.29/0.58        = ( ( W = X )
% 0.29/0.58          | ( Y = Z ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % crossproduct_eq
% 0.29/0.58  thf(fact_6817_crossproduct__eq,axiom,
% 0.29/0.58      ! [W: real,Y: real,X: real,Z: real] :
% 0.29/0.58        ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
% 0.29/0.58          = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
% 0.29/0.58        = ( ( W = X )
% 0.29/0.58          | ( Y = Z ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % crossproduct_eq
% 0.29/0.58  thf(fact_6818_crossproduct__eq,axiom,
% 0.29/0.58      ! [W: rat,Y: rat,X: rat,Z: rat] :
% 0.29/0.58        ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
% 0.29/0.58          = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
% 0.29/0.58        = ( ( W = X )
% 0.29/0.58          | ( Y = Z ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % crossproduct_eq
% 0.29/0.58  thf(fact_6819_crossproduct__eq,axiom,
% 0.29/0.58      ! [W: nat,Y: nat,X: nat,Z: nat] :
% 0.29/0.58        ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
% 0.29/0.58          = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
% 0.29/0.58        = ( ( W = X )
% 0.29/0.58          | ( Y = Z ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % crossproduct_eq
% 0.29/0.58  thf(fact_6820_crossproduct__eq,axiom,
% 0.29/0.58      ! [W: int,Y: int,X: int,Z: int] :
% 0.29/0.58        ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
% 0.29/0.58          = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
% 0.29/0.58        = ( ( W = X )
% 0.29/0.58          | ( Y = Z ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % crossproduct_eq
% 0.29/0.58  thf(fact_6821_list__decode_Ocases,axiom,
% 0.29/0.58      ! [X: nat] :
% 0.29/0.58        ( ( X != zero_zero_nat )
% 0.29/0.58       => ~ ! [N2: nat] :
% 0.29/0.58              ( X
% 0.29/0.58             != ( suc @ N2 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % list_decode.cases
% 0.29/0.58  thf(fact_6822_zero__less__power__eq,axiom,
% 0.29/0.58      ! [A: real,N: nat] :
% 0.29/0.58        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.29/0.58        = ( ( N = zero_zero_nat )
% 0.29/0.58          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            & ( A != zero_zero_real ) )
% 0.29/0.58          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_power_eq
% 0.29/0.58  thf(fact_6823_zero__less__power__eq,axiom,
% 0.29/0.58      ! [A: rat,N: nat] :
% 0.29/0.58        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.29/0.58        = ( ( N = zero_zero_nat )
% 0.29/0.58          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            & ( A != zero_zero_rat ) )
% 0.29/0.58          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_power_eq
% 0.29/0.58  thf(fact_6824_zero__less__power__eq,axiom,
% 0.29/0.58      ! [A: int,N: nat] :
% 0.29/0.58        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.29/0.58        = ( ( N = zero_zero_nat )
% 0.29/0.58          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            & ( A != zero_zero_int ) )
% 0.29/0.58          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_power_eq
% 0.29/0.58  thf(fact_6825_exp__bound__half,axiom,
% 0.29/0.58      ! [Z: real] :
% 0.29/0.58        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.58       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_bound_half
% 0.29/0.58  thf(fact_6826_exp__bound__half,axiom,
% 0.29/0.58      ! [Z: complex] :
% 0.29/0.58        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.58       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_bound_half
% 0.29/0.58  thf(fact_6827_exp__bound,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.58       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.58         => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_bound
% 0.29/0.58  thf(fact_6828_even__mask__div__iff_H,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( 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 ) ) )
% 0.29/0.58        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_mask_div_iff'
% 0.29/0.58  thf(fact_6829_even__mask__div__iff_H,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( 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 ) ) )
% 0.29/0.58        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_mask_div_iff'
% 0.29/0.58  thf(fact_6830_even__mask__div__iff_H,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( 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 ) ) )
% 0.29/0.58        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_mask_div_iff'
% 0.29/0.58  thf(fact_6831_power__le__zero__eq,axiom,
% 0.29/0.58      ! [A: real,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.29/0.58        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.29/0.58            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58              & ( A = zero_zero_real ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_zero_eq
% 0.29/0.58  thf(fact_6832_power__le__zero__eq,axiom,
% 0.29/0.58      ! [A: rat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.29/0.58        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.29/0.58            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58              & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_zero_eq
% 0.29/0.58  thf(fact_6833_power__le__zero__eq,axiom,
% 0.29/0.58      ! [A: int,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.29/0.58        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.29/0.58            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58              & ( A = zero_zero_int ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % power_le_zero_eq
% 0.29/0.58  thf(fact_6834_even__mod__4__div__2,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.58          = ( suc @ zero_zero_nat ) )
% 0.29/0.58       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_mod_4_div_2
% 0.29/0.58  thf(fact_6835_real__exp__bound__lemma,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.58       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.58         => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % real_exp_bound_lemma
% 0.29/0.58  thf(fact_6836_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.29/0.58      ! [N: nat,X: real] :
% 0.29/0.58        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
% 0.29/0.58       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58         => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_ge_one_plus_x_over_n_power_n
% 0.29/0.58  thf(fact_6837_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.29/0.58      ! [X: real,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 0.29/0.58       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58         => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_ge_one_minus_x_over_n_power_n
% 0.29/0.58  thf(fact_6838_even__mask__div__iff,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( 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 ) ) )
% 0.29/0.58        = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            = zero_z3403309356797280102nteger )
% 0.29/0.58          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_mask_div_iff
% 0.29/0.58  thf(fact_6839_even__mask__div__iff,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( 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 ) ) )
% 0.29/0.58        = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            = zero_zero_nat )
% 0.29/0.58          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_mask_div_iff
% 0.29/0.58  thf(fact_6840_even__mask__div__iff,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( 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 ) ) )
% 0.29/0.58        = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            = zero_zero_int )
% 0.29/0.58          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_mask_div_iff
% 0.29/0.58  thf(fact_6841_exp__bound__lemma,axiom,
% 0.29/0.58      ! [Z: real] :
% 0.29/0.58        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.58       => ( 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 ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_bound_lemma
% 0.29/0.58  thf(fact_6842_exp__bound__lemma,axiom,
% 0.29/0.58      ! [Z: complex] :
% 0.29/0.58        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.58       => ( 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 ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_bound_lemma
% 0.29/0.58  thf(fact_6843_odd__mod__4__div__2,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.58          = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.29/0.58       => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_mod_4_div_2
% 0.29/0.58  thf(fact_6844_Bernoulli__inequality__even,axiom,
% 0.29/0.58      ! [N: nat,X: real] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % Bernoulli_inequality_even
% 0.29/0.58  thf(fact_6845_exp__lower__Taylor__quadratic,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.58       => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_lower_Taylor_quadratic
% 0.29/0.58  thf(fact_6846_log__base__10__eq2,axiom,
% 0.29/0.58      ! [X: real] :
% 0.29/0.58        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.58       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.58          = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % log_base_10_eq2
% 0.29/0.58  thf(fact_6847_tanh__real__altdef,axiom,
% 0.29/0.58      ( tanh_real
% 0.29/0.58      = ( ^ [X4: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % tanh_real_altdef
% 0.29/0.58  thf(fact_6848_even__mult__exp__div__exp__iff,axiom,
% 0.29/0.58      ! [A: code_integer,M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.58        = ( ( ord_less_nat @ N @ M )
% 0.29/0.58          | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            = zero_z3403309356797280102nteger )
% 0.29/0.58          | ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58            & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_mult_exp_div_exp_iff
% 0.29/0.58  thf(fact_6849_even__mult__exp__div__exp__iff,axiom,
% 0.29/0.58      ! [A: nat,M: nat,N: nat] :
% 0.29/0.58        ( ( 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 ) ) )
% 0.29/0.58        = ( ( ord_less_nat @ N @ M )
% 0.29/0.58          | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            = zero_zero_nat )
% 0.29/0.58          | ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58            & ( 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 ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_mult_exp_div_exp_iff
% 0.29/0.58  thf(fact_6850_even__mult__exp__div__exp__iff,axiom,
% 0.29/0.58      ! [A: int,M: nat,N: nat] :
% 0.29/0.58        ( ( 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 ) ) )
% 0.29/0.58        = ( ( ord_less_nat @ N @ M )
% 0.29/0.58          | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.29/0.58            = zero_zero_int )
% 0.29/0.58          | ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58            & ( 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 ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_mult_exp_div_exp_iff
% 0.29/0.58  thf(fact_6851_triangle__def,axiom,
% 0.29/0.58      ( nat_triangle
% 0.29/0.58      = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % triangle_def
% 0.29/0.58  thf(fact_6852_flip__bit__0,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 0.29/0.58        = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % flip_bit_0
% 0.29/0.58  thf(fact_6853_flip__bit__0,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 0.29/0.58        = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % flip_bit_0
% 0.29/0.58  thf(fact_6854_flip__bit__0,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 0.29/0.58        = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % flip_bit_0
% 0.29/0.58  thf(fact_6855_set__decode__Suc,axiom,
% 0.29/0.58      ! [N: nat,X: nat] :
% 0.29/0.58        ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
% 0.29/0.58        = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % set_decode_Suc
% 0.29/0.58  thf(fact_6856_VEBT__internal_Oheight_Opelims,axiom,
% 0.29/0.58      ! [X: vEBT_VEBT,Y: nat] :
% 0.29/0.58        ( ( ( vEBT_VEBT_height @ X )
% 0.29/0.58          = Y )
% 0.29/0.58       => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X )
% 0.29/0.58         => ( ! [A5: $o,B6: $o] :
% 0.29/0.58                ( ( X
% 0.29/0.58                  = ( vEBT_Leaf @ A5 @ B6 ) )
% 0.29/0.58               => ( ( Y = zero_zero_nat )
% 0.29/0.58                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A5 @ B6 ) ) ) )
% 0.29/0.58           => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.29/0.58                  ( ( X
% 0.29/0.58                    = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.58                 => ( ( Y
% 0.29/0.58                      = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList3 ) ) ) ) ) )
% 0.29/0.58                   => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % VEBT_internal.height.pelims
% 0.29/0.58  thf(fact_6857_fact__double,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( 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 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_double
% 0.29/0.58  thf(fact_6858_fact__double,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( 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 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_double
% 0.29/0.58  thf(fact_6859_fact__double,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( 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 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_double
% 0.29/0.58  thf(fact_6860_mask__numeral,axiom,
% 0.29/0.58      ! [N: num] :
% 0.29/0.58        ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
% 0.29/0.58        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_numeral
% 0.29/0.58  thf(fact_6861_mask__numeral,axiom,
% 0.29/0.58      ! [N: num] :
% 0.29/0.58        ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
% 0.29/0.58        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_numeral
% 0.29/0.58  thf(fact_6862_mask__nat__positive__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.29/0.58        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_nat_positive_iff
% 0.29/0.58  thf(fact_6863_of__bool__less__eq__iff,axiom,
% 0.29/0.58      ! [P: $o,Q: $o] :
% 0.29/0.58        ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.29/0.58        = ( P
% 0.29/0.58         => Q ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_eq_iff
% 0.29/0.58  thf(fact_6864_of__bool__less__eq__iff,axiom,
% 0.29/0.58      ! [P: $o,Q: $o] :
% 0.29/0.58        ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.29/0.58        = ( P
% 0.29/0.58         => Q ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_eq_iff
% 0.29/0.58  thf(fact_6865_of__bool__less__eq__iff,axiom,
% 0.29/0.58      ! [P: $o,Q: $o] :
% 0.29/0.58        ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.29/0.58        = ( P
% 0.29/0.58         => Q ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_eq_iff
% 0.29/0.58  thf(fact_6866_of__bool__less__eq__iff,axiom,
% 0.29/0.58      ! [P: $o,Q: $o] :
% 0.29/0.58        ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.29/0.58        = ( P
% 0.29/0.58         => Q ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_eq_iff
% 0.29/0.58  thf(fact_6867_of__bool__eq__0__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n1201886186963655149omplex @ P )
% 0.29/0.58          = zero_zero_complex )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_0_iff
% 0.29/0.58  thf(fact_6868_of__bool__eq__0__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n3304061248610475627l_real @ P )
% 0.29/0.58          = zero_zero_real )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_0_iff
% 0.29/0.58  thf(fact_6869_of__bool__eq__0__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.29/0.58          = zero_zero_rat )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_0_iff
% 0.29/0.58  thf(fact_6870_of__bool__eq__0__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.29/0.58          = zero_zero_nat )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_0_iff
% 0.29/0.58  thf(fact_6871_of__bool__eq__0__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.29/0.58          = zero_zero_int )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_0_iff
% 0.29/0.58  thf(fact_6872_of__bool__eq__0__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n356916108424825756nteger @ P )
% 0.29/0.58          = zero_z3403309356797280102nteger )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_0_iff
% 0.29/0.58  thf(fact_6873_of__bool__eq_I1_J,axiom,
% 0.29/0.58      ( ( zero_n1201886186963655149omplex @ $false )
% 0.29/0.58      = zero_zero_complex ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(1)
% 0.29/0.58  thf(fact_6874_of__bool__eq_I1_J,axiom,
% 0.29/0.58      ( ( zero_n3304061248610475627l_real @ $false )
% 0.29/0.58      = zero_zero_real ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(1)
% 0.29/0.58  thf(fact_6875_of__bool__eq_I1_J,axiom,
% 0.29/0.58      ( ( zero_n2052037380579107095ol_rat @ $false )
% 0.29/0.58      = zero_zero_rat ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(1)
% 0.29/0.58  thf(fact_6876_of__bool__eq_I1_J,axiom,
% 0.29/0.58      ( ( zero_n2687167440665602831ol_nat @ $false )
% 0.29/0.58      = zero_zero_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(1)
% 0.29/0.58  thf(fact_6877_of__bool__eq_I1_J,axiom,
% 0.29/0.58      ( ( zero_n2684676970156552555ol_int @ $false )
% 0.29/0.58      = zero_zero_int ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(1)
% 0.29/0.58  thf(fact_6878_of__bool__eq_I1_J,axiom,
% 0.29/0.58      ( ( zero_n356916108424825756nteger @ $false )
% 0.29/0.58      = zero_z3403309356797280102nteger ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(1)
% 0.29/0.58  thf(fact_6879_of__bool__less__iff,axiom,
% 0.29/0.58      ! [P: $o,Q: $o] :
% 0.29/0.58        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.29/0.58        = ( ~ P
% 0.29/0.58          & Q ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_iff
% 0.29/0.58  thf(fact_6880_of__bool__less__iff,axiom,
% 0.29/0.58      ! [P: $o,Q: $o] :
% 0.29/0.58        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.29/0.58        = ( ~ P
% 0.29/0.58          & Q ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_iff
% 0.29/0.58  thf(fact_6881_of__bool__less__iff,axiom,
% 0.29/0.58      ! [P: $o,Q: $o] :
% 0.29/0.58        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.29/0.58        = ( ~ P
% 0.29/0.58          & Q ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_iff
% 0.29/0.58  thf(fact_6882_of__bool__less__iff,axiom,
% 0.29/0.58      ! [P: $o,Q: $o] :
% 0.29/0.58        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.29/0.58        = ( ~ P
% 0.29/0.58          & Q ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_iff
% 0.29/0.58  thf(fact_6883_of__bool__less__iff,axiom,
% 0.29/0.58      ! [P: $o,Q: $o] :
% 0.29/0.58        ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.29/0.58        = ( ~ P
% 0.29/0.58          & Q ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_iff
% 0.29/0.58  thf(fact_6884_of__bool__eq_I2_J,axiom,
% 0.29/0.58      ( ( zero_n1201886186963655149omplex @ $true )
% 0.29/0.58      = one_one_complex ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(2)
% 0.29/0.58  thf(fact_6885_of__bool__eq_I2_J,axiom,
% 0.29/0.58      ( ( zero_n3304061248610475627l_real @ $true )
% 0.29/0.58      = one_one_real ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(2)
% 0.29/0.58  thf(fact_6886_of__bool__eq_I2_J,axiom,
% 0.29/0.58      ( ( zero_n2052037380579107095ol_rat @ $true )
% 0.29/0.58      = one_one_rat ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(2)
% 0.29/0.58  thf(fact_6887_of__bool__eq_I2_J,axiom,
% 0.29/0.58      ( ( zero_n2687167440665602831ol_nat @ $true )
% 0.29/0.58      = one_one_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(2)
% 0.29/0.58  thf(fact_6888_of__bool__eq_I2_J,axiom,
% 0.29/0.58      ( ( zero_n2684676970156552555ol_int @ $true )
% 0.29/0.58      = one_one_int ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(2)
% 0.29/0.58  thf(fact_6889_of__bool__eq_I2_J,axiom,
% 0.29/0.58      ( ( zero_n356916108424825756nteger @ $true )
% 0.29/0.58      = one_one_Code_integer ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq(2)
% 0.29/0.58  thf(fact_6890_of__bool__eq__1__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n1201886186963655149omplex @ P )
% 0.29/0.58          = one_one_complex )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_1_iff
% 0.29/0.58  thf(fact_6891_of__bool__eq__1__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n3304061248610475627l_real @ P )
% 0.29/0.58          = one_one_real )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_1_iff
% 0.29/0.58  thf(fact_6892_of__bool__eq__1__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.29/0.58          = one_one_rat )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_1_iff
% 0.29/0.58  thf(fact_6893_of__bool__eq__1__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.29/0.58          = one_one_nat )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_1_iff
% 0.29/0.58  thf(fact_6894_of__bool__eq__1__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.29/0.58          = one_one_int )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_1_iff
% 0.29/0.58  thf(fact_6895_of__bool__eq__1__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ( zero_n356916108424825756nteger @ P )
% 0.29/0.58          = one_one_Code_integer )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_eq_1_iff
% 0.29/0.58  thf(fact_6896_of__nat__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri681578069525770553at_rat @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.29/0.58        = ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_fact
% 0.29/0.58  thf(fact_6897_of__nat__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri1314217659103216013at_int @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.29/0.58        = ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_fact
% 0.29/0.58  thf(fact_6898_of__nat__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri1316708129612266289at_nat @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.29/0.58        = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_fact
% 0.29/0.58  thf(fact_6899_of__nat__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri5074537144036343181t_real @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.29/0.58        = ( semiri2265585572941072030t_real @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_fact
% 0.29/0.58  thf(fact_6900_of__nat__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri8010041392384452111omplex @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.29/0.58        = ( semiri5044797733671781792omplex @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_fact
% 0.29/0.58  thf(fact_6901_of__nat__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.29/0.58        = ( zero_n1201886186963655149omplex @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_of_bool
% 0.29/0.58  thf(fact_6902_of__nat__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.29/0.58        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_of_bool
% 0.29/0.58  thf(fact_6903_of__nat__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.29/0.58        = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_of_bool
% 0.29/0.58  thf(fact_6904_of__nat__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.29/0.58        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_of_bool
% 0.29/0.58  thf(fact_6905_of__nat__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.29/0.58        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_of_bool
% 0.29/0.58  thf(fact_6906_of__nat__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.29/0.58        = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_of_bool
% 0.29/0.58  thf(fact_6907_norm__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( real_V7735802525324610683m_real @ ( semiri2265585572941072030t_real @ N ) )
% 0.29/0.58        = ( semiri2265585572941072030t_real @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % norm_fact
% 0.29/0.58  thf(fact_6908_norm__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( real_V1022390504157884413omplex @ ( semiri5044797733671781792omplex @ N ) )
% 0.29/0.58        = ( semiri2265585572941072030t_real @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % norm_fact
% 0.29/0.58  thf(fact_6909_of__int__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ring_1_of_int_rat @ ( semiri1406184849735516958ct_int @ N ) )
% 0.29/0.58        = ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_fact
% 0.29/0.58  thf(fact_6910_of__int__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ring_18347121197199848620nteger @ ( semiri1406184849735516958ct_int @ N ) )
% 0.29/0.58        = ( semiri3624122377584611663nteger @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_fact
% 0.29/0.58  thf(fact_6911_of__int__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ring_1_of_int_real @ ( semiri1406184849735516958ct_int @ N ) )
% 0.29/0.58        = ( semiri2265585572941072030t_real @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_fact
% 0.29/0.58  thf(fact_6912_of__int__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ring_17405671764205052669omplex @ ( semiri1406184849735516958ct_int @ N ) )
% 0.29/0.58        = ( semiri5044797733671781792omplex @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_fact
% 0.29/0.58  thf(fact_6913_of__int__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.29/0.58        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_of_bool
% 0.29/0.58  thf(fact_6914_of__int__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.29/0.58        = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_of_bool
% 0.29/0.58  thf(fact_6915_of__int__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ring_17405671764205052669omplex @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.29/0.58        = ( zero_n1201886186963655149omplex @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_of_bool
% 0.29/0.58  thf(fact_6916_of__int__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.29/0.58        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_of_bool
% 0.29/0.58  thf(fact_6917_of__int__of__bool,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.29/0.58        = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_of_bool
% 0.29/0.58  thf(fact_6918_of__real__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( real_V1803761363581548252l_real @ ( semiri2265585572941072030t_real @ N ) )
% 0.29/0.58        = ( semiri2265585572941072030t_real @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_real_fact
% 0.29/0.58  thf(fact_6919_of__real__fact,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( real_V4546457046886955230omplex @ ( semiri2265585572941072030t_real @ N ) )
% 0.29/0.58        = ( semiri5044797733671781792omplex @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_real_fact
% 0.29/0.58  thf(fact_6920_triangle__0,axiom,
% 0.29/0.58      ( ( nat_triangle @ zero_zero_nat )
% 0.29/0.58      = zero_zero_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % triangle_0
% 0.29/0.58  thf(fact_6921_zero__less__of__bool__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_of_bool_iff
% 0.29/0.58  thf(fact_6922_zero__less__of__bool__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_of_bool_iff
% 0.29/0.58  thf(fact_6923_zero__less__of__bool__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_of_bool_iff
% 0.29/0.58  thf(fact_6924_zero__less__of__bool__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_of_bool_iff
% 0.29/0.58  thf(fact_6925_zero__less__of__bool__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 0.29/0.58        = P ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_of_bool_iff
% 0.29/0.58  thf(fact_6926_of__bool__less__one__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_one_iff
% 0.29/0.58  thf(fact_6927_of__bool__less__one__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_one_iff
% 0.29/0.58  thf(fact_6928_of__bool__less__one__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_one_iff
% 0.29/0.58  thf(fact_6929_of__bool__less__one__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_one_iff
% 0.29/0.58  thf(fact_6930_of__bool__less__one__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 0.29/0.58        = ( ~ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_one_iff
% 0.29/0.58  thf(fact_6931_fact__0,axiom,
% 0.29/0.58      ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 0.29/0.58      = one_one_rat ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_0
% 0.29/0.58  thf(fact_6932_fact__0,axiom,
% 0.29/0.58      ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 0.29/0.58      = one_one_int ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_0
% 0.29/0.58  thf(fact_6933_fact__0,axiom,
% 0.29/0.58      ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 0.29/0.58      = one_one_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_0
% 0.29/0.58  thf(fact_6934_fact__0,axiom,
% 0.29/0.58      ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 0.29/0.58      = one_one_real ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_0
% 0.29/0.58  thf(fact_6935_fact__0,axiom,
% 0.29/0.58      ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 0.29/0.58      = one_one_complex ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_0
% 0.29/0.58  thf(fact_6936_of__bool__not__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( zero_n1201886186963655149omplex @ ~ P )
% 0.29/0.58        = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_not_iff
% 0.29/0.58  thf(fact_6937_of__bool__not__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( zero_n3304061248610475627l_real @ ~ P )
% 0.29/0.58        = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_not_iff
% 0.29/0.58  thf(fact_6938_of__bool__not__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 0.29/0.58        = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_not_iff
% 0.29/0.58  thf(fact_6939_of__bool__not__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( zero_n2684676970156552555ol_int @ ~ P )
% 0.29/0.58        = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_not_iff
% 0.29/0.58  thf(fact_6940_of__bool__not__iff,axiom,
% 0.29/0.58      ! [P: $o] :
% 0.29/0.58        ( ( zero_n356916108424825756nteger @ ~ P )
% 0.29/0.58        = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_not_iff
% 0.29/0.58  thf(fact_6941_Suc__0__mod__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.58        = ( zero_n2687167440665602831ol_nat
% 0.29/0.58          @ ( N
% 0.29/0.58           != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % Suc_0_mod_eq
% 0.29/0.58  thf(fact_6942_mask__eq__0__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( bit_se2002935070580805687sk_nat @ N )
% 0.29/0.58          = zero_zero_nat )
% 0.29/0.58        = ( N = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_eq_0_iff
% 0.29/0.58  thf(fact_6943_mask__eq__0__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( bit_se2000444600071755411sk_int @ N )
% 0.29/0.58          = zero_zero_int )
% 0.29/0.58        = ( N = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_eq_0_iff
% 0.29/0.58  thf(fact_6944_mask__0,axiom,
% 0.29/0.58      ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
% 0.29/0.58      = zero_zero_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_0
% 0.29/0.58  thf(fact_6945_mask__0,axiom,
% 0.29/0.58      ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
% 0.29/0.58      = zero_zero_int ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_0
% 0.29/0.58  thf(fact_6946_fact__1,axiom,
% 0.29/0.58      ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 0.29/0.58      = one_one_rat ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_1
% 0.29/0.58  thf(fact_6947_fact__1,axiom,
% 0.29/0.58      ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 0.29/0.58      = one_one_int ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_1
% 0.29/0.58  thf(fact_6948_fact__1,axiom,
% 0.29/0.58      ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 0.29/0.58      = one_one_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_1
% 0.29/0.58  thf(fact_6949_fact__1,axiom,
% 0.29/0.58      ( ( semiri2265585572941072030t_real @ one_one_nat )
% 0.29/0.58      = one_one_real ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_1
% 0.29/0.58  thf(fact_6950_fact__1,axiom,
% 0.29/0.58      ( ( semiri5044797733671781792omplex @ one_one_nat )
% 0.29/0.58      = one_one_complex ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_1
% 0.29/0.58  thf(fact_6951_triangle__Suc,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( nat_triangle @ ( suc @ N ) )
% 0.29/0.58        = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % triangle_Suc
% 0.29/0.58  thf(fact_6952_fact__Suc__0,axiom,
% 0.29/0.58      ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 0.29/0.58      = one_one_rat ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_Suc_0
% 0.29/0.58  thf(fact_6953_fact__Suc__0,axiom,
% 0.29/0.58      ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 0.29/0.58      = one_one_int ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_Suc_0
% 0.29/0.58  thf(fact_6954_fact__Suc__0,axiom,
% 0.29/0.58      ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 0.29/0.58      = one_one_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_Suc_0
% 0.29/0.58  thf(fact_6955_fact__Suc__0,axiom,
% 0.29/0.58      ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 0.29/0.58      = one_one_real ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_Suc_0
% 0.29/0.58  thf(fact_6956_fact__Suc__0,axiom,
% 0.29/0.58      ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 0.29/0.58      = one_one_complex ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_Suc_0
% 0.29/0.58  thf(fact_6957_fact__Suc,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 0.29/0.58        = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_Suc
% 0.29/0.58  thf(fact_6958_fact__Suc,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 0.29/0.58        = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_Suc
% 0.29/0.58  thf(fact_6959_fact__Suc,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 0.29/0.58        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_Suc
% 0.29/0.58  thf(fact_6960_fact__Suc,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 0.29/0.58        = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_Suc
% 0.29/0.58  thf(fact_6961_fact__Suc,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
% 0.29/0.58        = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_Suc
% 0.29/0.58  thf(fact_6962_mask__Suc__0,axiom,
% 0.29/0.58      ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
% 0.29/0.58      = one_one_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_Suc_0
% 0.29/0.58  thf(fact_6963_mask__Suc__0,axiom,
% 0.29/0.58      ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
% 0.29/0.58      = one_one_int ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_Suc_0
% 0.29/0.58  thf(fact_6964_fact__2,axiom,
% 0.29/0.58      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_2
% 0.29/0.58  thf(fact_6965_fact__2,axiom,
% 0.29/0.58      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_2
% 0.29/0.58  thf(fact_6966_fact__2,axiom,
% 0.29/0.58      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_2
% 0.29/0.58  thf(fact_6967_fact__2,axiom,
% 0.29/0.58      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_2
% 0.29/0.58  thf(fact_6968_fact__2,axiom,
% 0.29/0.58      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_2
% 0.29/0.58  thf(fact_6969_odd__of__bool__self,axiom,
% 0.29/0.58      ! [P5: $o] :
% 0.29/0.58        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P5 ) ) )
% 0.29/0.58        = P5 ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_of_bool_self
% 0.29/0.58  thf(fact_6970_odd__of__bool__self,axiom,
% 0.29/0.58      ! [P5: $o] :
% 0.29/0.58        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P5 ) ) )
% 0.29/0.58        = P5 ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_of_bool_self
% 0.29/0.58  thf(fact_6971_odd__of__bool__self,axiom,
% 0.29/0.58      ! [P5: $o] :
% 0.29/0.58        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P5 ) ) )
% 0.29/0.58        = P5 ) ).
% 0.29/0.58  
% 0.29/0.58  % odd_of_bool_self
% 0.29/0.58  thf(fact_6972_of__bool__half__eq__0,axiom,
% 0.29/0.58      ! [B: $o] :
% 0.29/0.58        ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58        = zero_zero_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_half_eq_0
% 0.29/0.58  thf(fact_6973_of__bool__half__eq__0,axiom,
% 0.29/0.58      ! [B: $o] :
% 0.29/0.58        ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58        = zero_zero_int ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_half_eq_0
% 0.29/0.58  thf(fact_6974_of__bool__half__eq__0,axiom,
% 0.29/0.58      ! [B: $o] :
% 0.29/0.58        ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58        = zero_z3403309356797280102nteger ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_half_eq_0
% 0.29/0.58  thf(fact_6975_set__decode__0,axiom,
% 0.29/0.58      ! [X: nat] :
% 0.29/0.58        ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 0.29/0.58        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % set_decode_0
% 0.29/0.58  thf(fact_6976_bits__1__div__exp,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bits_1_div_exp
% 0.29/0.58  thf(fact_6977_bits__1__div__exp,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bits_1_div_exp
% 0.29/0.58  thf(fact_6978_bits__1__div__exp,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bits_1_div_exp
% 0.29/0.58  thf(fact_6979_one__div__2__pow__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % one_div_2_pow_eq
% 0.29/0.58  thf(fact_6980_one__div__2__pow__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % one_div_2_pow_eq
% 0.29/0.58  thf(fact_6981_one__div__2__pow__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % one_div_2_pow_eq
% 0.29/0.58  thf(fact_6982_one__mod__2__pow__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % one_mod_2_pow_eq
% 0.29/0.58  thf(fact_6983_one__mod__2__pow__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % one_mod_2_pow_eq
% 0.29/0.58  thf(fact_6984_one__mod__2__pow__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % one_mod_2_pow_eq
% 0.29/0.58  thf(fact_6985_dvd__antisym,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ M @ N )
% 0.29/0.58       => ( ( dvd_dvd_nat @ N @ M )
% 0.29/0.58         => ( M = N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_antisym
% 0.29/0.58  thf(fact_6986_of__nat__mask__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.29/0.58        = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_mask_eq
% 0.29/0.58  thf(fact_6987_of__nat__mask__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.29/0.58        = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_nat_mask_eq
% 0.29/0.58  thf(fact_6988_of__int__mask__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ring_18347121197199848620nteger @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.29/0.58        = ( bit_se2119862282449309892nteger @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_mask_eq
% 0.29/0.58  thf(fact_6989_of__int__mask__eq,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.29/0.58        = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_int_mask_eq
% 0.29/0.58  thf(fact_6990_fact__ge__self,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_ge_self
% 0.29/0.58  thf(fact_6991_fact__mono__nat,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_mono_nat
% 0.29/0.58  thf(fact_6992_fact__nonzero,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri773545260158071498ct_rat @ N )
% 0.29/0.58       != zero_zero_rat ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_nonzero
% 0.29/0.58  thf(fact_6993_fact__nonzero,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri1406184849735516958ct_int @ N )
% 0.29/0.58       != zero_zero_int ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_nonzero
% 0.29/0.58  thf(fact_6994_fact__nonzero,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri1408675320244567234ct_nat @ N )
% 0.29/0.58       != zero_zero_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_nonzero
% 0.29/0.58  thf(fact_6995_fact__nonzero,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri2265585572941072030t_real @ N )
% 0.29/0.58       != zero_zero_real ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_nonzero
% 0.29/0.58  thf(fact_6996_fact__nonzero,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( semiri5044797733671781792omplex @ N )
% 0.29/0.58       != zero_zero_complex ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_nonzero
% 0.29/0.58  thf(fact_6997_less__eq__mask,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % less_eq_mask
% 0.29/0.58  thf(fact_6998_subset__decode__imp__le,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 0.29/0.58       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % subset_decode_imp_le
% 0.29/0.58  thf(fact_6999_fact__less__mono__nat,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.58       => ( ( ord_less_nat @ M @ N )
% 0.29/0.58         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_less_mono_nat
% 0.29/0.58  thf(fact_7000_zero__less__eq__of__bool,axiom,
% 0.29/0.58      ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_eq_of_bool
% 0.29/0.58  thf(fact_7001_zero__less__eq__of__bool,axiom,
% 0.29/0.58      ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_eq_of_bool
% 0.29/0.58  thf(fact_7002_zero__less__eq__of__bool,axiom,
% 0.29/0.58      ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_eq_of_bool
% 0.29/0.58  thf(fact_7003_zero__less__eq__of__bool,axiom,
% 0.29/0.58      ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_eq_of_bool
% 0.29/0.58  thf(fact_7004_zero__less__eq__of__bool,axiom,
% 0.29/0.58      ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 0.29/0.58  
% 0.29/0.58  % zero_less_eq_of_bool
% 0.29/0.58  thf(fact_7005_of__bool__less__eq__one,axiom,
% 0.29/0.58      ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_eq_one
% 0.29/0.58  thf(fact_7006_of__bool__less__eq__one,axiom,
% 0.29/0.58      ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_eq_one
% 0.29/0.58  thf(fact_7007_of__bool__less__eq__one,axiom,
% 0.29/0.58      ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_eq_one
% 0.29/0.58  thf(fact_7008_of__bool__less__eq__one,axiom,
% 0.29/0.58      ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_eq_one
% 0.29/0.58  thf(fact_7009_of__bool__less__eq__one,axiom,
% 0.29/0.58      ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_less_eq_one
% 0.29/0.58  thf(fact_7010_of__bool__def,axiom,
% 0.29/0.58      ( zero_n1201886186963655149omplex
% 0.29/0.58      = ( ^ [P4: $o] : ( if_complex @ P4 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_def
% 0.29/0.58  thf(fact_7011_of__bool__def,axiom,
% 0.29/0.58      ( zero_n3304061248610475627l_real
% 0.29/0.58      = ( ^ [P4: $o] : ( if_real @ P4 @ one_one_real @ zero_zero_real ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_def
% 0.29/0.58  thf(fact_7012_of__bool__def,axiom,
% 0.29/0.58      ( zero_n2052037380579107095ol_rat
% 0.29/0.58      = ( ^ [P4: $o] : ( if_rat @ P4 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_def
% 0.29/0.58  thf(fact_7013_of__bool__def,axiom,
% 0.29/0.58      ( zero_n2687167440665602831ol_nat
% 0.29/0.58      = ( ^ [P4: $o] : ( if_nat @ P4 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_def
% 0.29/0.58  thf(fact_7014_of__bool__def,axiom,
% 0.29/0.58      ( zero_n2684676970156552555ol_int
% 0.29/0.58      = ( ^ [P4: $o] : ( if_int @ P4 @ one_one_int @ zero_zero_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_def
% 0.29/0.58  thf(fact_7015_of__bool__def,axiom,
% 0.29/0.58      ( zero_n356916108424825756nteger
% 0.29/0.58      = ( ^ [P4: $o] : ( if_Code_integer @ P4 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_def
% 0.29/0.58  thf(fact_7016_split__of__bool,axiom,
% 0.29/0.58      ! [P: complex > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
% 0.29/0.58        = ( ( P5
% 0.29/0.58           => ( P @ one_one_complex ) )
% 0.29/0.58          & ( ~ P5
% 0.29/0.58           => ( P @ zero_zero_complex ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool
% 0.29/0.58  thf(fact_7017_split__of__bool,axiom,
% 0.29/0.58      ! [P: real > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
% 0.29/0.58        = ( ( P5
% 0.29/0.58           => ( P @ one_one_real ) )
% 0.29/0.58          & ( ~ P5
% 0.29/0.58           => ( P @ zero_zero_real ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool
% 0.29/0.58  thf(fact_7018_split__of__bool,axiom,
% 0.29/0.58      ! [P: rat > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
% 0.29/0.58        = ( ( P5
% 0.29/0.58           => ( P @ one_one_rat ) )
% 0.29/0.58          & ( ~ P5
% 0.29/0.58           => ( P @ zero_zero_rat ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool
% 0.29/0.58  thf(fact_7019_split__of__bool,axiom,
% 0.29/0.58      ! [P: nat > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
% 0.29/0.58        = ( ( P5
% 0.29/0.58           => ( P @ one_one_nat ) )
% 0.29/0.58          & ( ~ P5
% 0.29/0.58           => ( P @ zero_zero_nat ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool
% 0.29/0.58  thf(fact_7020_split__of__bool,axiom,
% 0.29/0.58      ! [P: int > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
% 0.29/0.58        = ( ( P5
% 0.29/0.58           => ( P @ one_one_int ) )
% 0.29/0.58          & ( ~ P5
% 0.29/0.58           => ( P @ zero_zero_int ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool
% 0.29/0.58  thf(fact_7021_split__of__bool,axiom,
% 0.29/0.58      ! [P: code_integer > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
% 0.29/0.58        = ( ( P5
% 0.29/0.58           => ( P @ one_one_Code_integer ) )
% 0.29/0.58          & ( ~ P5
% 0.29/0.58           => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool
% 0.29/0.58  thf(fact_7022_split__of__bool__asm,axiom,
% 0.29/0.58      ! [P: complex > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
% 0.29/0.58        = ( ~ ( ( P5
% 0.29/0.58                & ~ ( P @ one_one_complex ) )
% 0.29/0.58              | ( ~ P5
% 0.29/0.58                & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool_asm
% 0.29/0.58  thf(fact_7023_split__of__bool__asm,axiom,
% 0.29/0.58      ! [P: real > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
% 0.29/0.58        = ( ~ ( ( P5
% 0.29/0.58                & ~ ( P @ one_one_real ) )
% 0.29/0.58              | ( ~ P5
% 0.29/0.58                & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool_asm
% 0.29/0.58  thf(fact_7024_split__of__bool__asm,axiom,
% 0.29/0.58      ! [P: rat > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
% 0.29/0.58        = ( ~ ( ( P5
% 0.29/0.58                & ~ ( P @ one_one_rat ) )
% 0.29/0.58              | ( ~ P5
% 0.29/0.58                & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool_asm
% 0.29/0.58  thf(fact_7025_split__of__bool__asm,axiom,
% 0.29/0.58      ! [P: nat > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
% 0.29/0.58        = ( ~ ( ( P5
% 0.29/0.58                & ~ ( P @ one_one_nat ) )
% 0.29/0.58              | ( ~ P5
% 0.29/0.58                & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool_asm
% 0.29/0.58  thf(fact_7026_split__of__bool__asm,axiom,
% 0.29/0.58      ! [P: int > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
% 0.29/0.58        = ( ~ ( ( P5
% 0.29/0.58                & ~ ( P @ one_one_int ) )
% 0.29/0.58              | ( ~ P5
% 0.29/0.58                & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool_asm
% 0.29/0.58  thf(fact_7027_split__of__bool__asm,axiom,
% 0.29/0.58      ! [P: code_integer > $o,P5: $o] :
% 0.29/0.58        ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
% 0.29/0.58        = ( ~ ( ( P5
% 0.29/0.58                & ~ ( P @ one_one_Code_integer ) )
% 0.29/0.58              | ( ~ P5
% 0.29/0.58                & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % split_of_bool_asm
% 0.29/0.58  thf(fact_7028_dvd__fact,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.29/0.58       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58         => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % dvd_fact
% 0.29/0.58  thf(fact_7029_fact__ge__zero,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_ge_zero
% 0.29/0.58  thf(fact_7030_fact__ge__zero,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_ge_zero
% 0.29/0.58  thf(fact_7031_fact__ge__zero,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_ge_zero
% 0.29/0.58  thf(fact_7032_fact__ge__zero,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_ge_zero
% 0.29/0.58  thf(fact_7033_fact__not__neg,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_not_neg
% 0.29/0.58  thf(fact_7034_fact__not__neg,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_not_neg
% 0.29/0.58  thf(fact_7035_fact__not__neg,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_not_neg
% 0.29/0.58  thf(fact_7036_fact__not__neg,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_not_neg
% 0.29/0.58  thf(fact_7037_fact__gt__zero,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_gt_zero
% 0.29/0.58  thf(fact_7038_fact__gt__zero,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_gt_zero
% 0.29/0.58  thf(fact_7039_fact__gt__zero,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_gt_zero
% 0.29/0.58  thf(fact_7040_fact__gt__zero,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_gt_zero
% 0.29/0.58  thf(fact_7041_fact__ge__1,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_ge_1
% 0.29/0.58  thf(fact_7042_fact__ge__1,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_ge_1
% 0.29/0.58  thf(fact_7043_fact__ge__1,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_ge_1
% 0.29/0.58  thf(fact_7044_fact__ge__1,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_ge_1
% 0.29/0.58  thf(fact_7045_fact__mono,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_mono
% 0.29/0.58  thf(fact_7046_fact__mono,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_mono
% 0.29/0.58  thf(fact_7047_fact__mono,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_mono
% 0.29/0.58  thf(fact_7048_fact__mono,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_mono
% 0.29/0.58  thf(fact_7049_fact__dvd,axiom,
% 0.29/0.58      ! [N: nat,M: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58       => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_dvd
% 0.29/0.58  thf(fact_7050_fact__dvd,axiom,
% 0.29/0.58      ! [N: nat,M: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58       => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_dvd
% 0.29/0.58  thf(fact_7051_fact__dvd,axiom,
% 0.29/0.58      ! [N: nat,M: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58       => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_dvd
% 0.29/0.58  thf(fact_7052_fact__dvd,axiom,
% 0.29/0.58      ! [N: nat,M: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58       => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_dvd
% 0.29/0.58  thf(fact_7053_pochhammer__fact,axiom,
% 0.29/0.58      ( semiri773545260158071498ct_rat
% 0.29/0.58      = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pochhammer_fact
% 0.29/0.58  thf(fact_7054_pochhammer__fact,axiom,
% 0.29/0.58      ( semiri1406184849735516958ct_int
% 0.29/0.58      = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pochhammer_fact
% 0.29/0.58  thf(fact_7055_pochhammer__fact,axiom,
% 0.29/0.58      ( semiri1408675320244567234ct_nat
% 0.29/0.58      = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pochhammer_fact
% 0.29/0.58  thf(fact_7056_pochhammer__fact,axiom,
% 0.29/0.58      ( semiri2265585572941072030t_real
% 0.29/0.58      = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pochhammer_fact
% 0.29/0.58  thf(fact_7057_pochhammer__fact,axiom,
% 0.29/0.58      ( semiri5044797733671781792omplex
% 0.29/0.58      = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pochhammer_fact
% 0.29/0.58  thf(fact_7058_mask__nonnegative__int,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_nonnegative_int
% 0.29/0.58  thf(fact_7059_not__mask__negative__int,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 0.29/0.58  
% 0.29/0.58  % not_mask_negative_int
% 0.29/0.58  thf(fact_7060_fact__ge__Suc__0__nat,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_ge_Suc_0_nat
% 0.29/0.58  thf(fact_7061_fact__less__mono,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.58       => ( ( ord_less_nat @ M @ N )
% 0.29/0.58         => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_less_mono
% 0.29/0.58  thf(fact_7062_fact__less__mono,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.58       => ( ( ord_less_nat @ M @ N )
% 0.29/0.58         => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_less_mono
% 0.29/0.58  thf(fact_7063_fact__less__mono,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.58       => ( ( ord_less_nat @ M @ N )
% 0.29/0.58         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_less_mono
% 0.29/0.58  thf(fact_7064_fact__less__mono,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.58       => ( ( ord_less_nat @ M @ N )
% 0.29/0.58         => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_less_mono
% 0.29/0.58  thf(fact_7065_fact__fact__dvd__fact,axiom,
% 0.29/0.58      ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_fact_dvd_fact
% 0.29/0.58  thf(fact_7066_fact__fact__dvd__fact,axiom,
% 0.29/0.58      ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_fact_dvd_fact
% 0.29/0.58  thf(fact_7067_fact__fact__dvd__fact,axiom,
% 0.29/0.58      ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_fact_dvd_fact
% 0.29/0.58  thf(fact_7068_fact__fact__dvd__fact,axiom,
% 0.29/0.58      ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_fact_dvd_fact
% 0.29/0.58  thf(fact_7069_fact__fact__dvd__fact,axiom,
% 0.29/0.58      ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_fact_dvd_fact
% 0.29/0.58  thf(fact_7070_fact__mod,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 0.29/0.58          = zero_zero_int ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_mod
% 0.29/0.58  thf(fact_7071_fact__mod,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 0.29/0.58          = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_mod
% 0.29/0.58  thf(fact_7072_fact__mod,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 0.29/0.58          = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_mod
% 0.29/0.58  thf(fact_7073_fact__le__power,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_le_power
% 0.29/0.58  thf(fact_7074_fact__le__power,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_le_power
% 0.29/0.58  thf(fact_7075_fact__le__power,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_le_power
% 0.29/0.58  thf(fact_7076_fact__le__power,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_le_power
% 0.29/0.58  thf(fact_7077_less__mask,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.58       => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % less_mask
% 0.29/0.58  thf(fact_7078_fact__diff__Suc,axiom,
% 0.29/0.58      ! [N: nat,M: nat] :
% 0.29/0.58        ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.29/0.58       => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 0.29/0.58          = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_diff_Suc
% 0.29/0.58  thf(fact_7079_fact__div__fact__le__pow,axiom,
% 0.29/0.58      ! [R2: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ R2 @ N )
% 0.29/0.58       => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_div_fact_le_pow
% 0.29/0.58  thf(fact_7080_binomial__fact__lemma,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 0.29/0.58          = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % binomial_fact_lemma
% 0.29/0.58  thf(fact_7081_choose__dvd,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % choose_dvd
% 0.29/0.58  thf(fact_7082_choose__dvd,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % choose_dvd
% 0.29/0.58  thf(fact_7083_choose__dvd,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % choose_dvd
% 0.29/0.58  thf(fact_7084_choose__dvd,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % choose_dvd
% 0.29/0.58  thf(fact_7085_choose__dvd,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % choose_dvd
% 0.29/0.58  thf(fact_7086_fact__numeral,axiom,
% 0.29/0.58      ! [K: num] :
% 0.29/0.58        ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 0.29/0.58        = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_numeral
% 0.29/0.58  thf(fact_7087_fact__numeral,axiom,
% 0.29/0.58      ! [K: num] :
% 0.29/0.58        ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 0.29/0.58        = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_numeral
% 0.29/0.58  thf(fact_7088_fact__numeral,axiom,
% 0.29/0.58      ! [K: num] :
% 0.29/0.58        ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 0.29/0.58        = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_numeral
% 0.29/0.58  thf(fact_7089_fact__numeral,axiom,
% 0.29/0.58      ! [K: num] :
% 0.29/0.58        ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 0.29/0.58        = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_numeral
% 0.29/0.58  thf(fact_7090_fact__numeral,axiom,
% 0.29/0.58      ! [K: num] :
% 0.29/0.58        ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 0.29/0.58        = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_numeral
% 0.29/0.58  thf(fact_7091_of__bool__odd__eq__mod__2,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( zero_n2687167440665602831ol_nat
% 0.29/0.58          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.58        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_odd_eq_mod_2
% 0.29/0.58  thf(fact_7092_of__bool__odd__eq__mod__2,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( zero_n2684676970156552555ol_int
% 0.29/0.58          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.58        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_odd_eq_mod_2
% 0.29/0.58  thf(fact_7093_of__bool__odd__eq__mod__2,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( zero_n356916108424825756nteger
% 0.29/0.58          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.58        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % of_bool_odd_eq_mod_2
% 0.29/0.58  thf(fact_7094_binomial__altdef__nat,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( ( binomial @ N @ K )
% 0.29/0.58          = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % binomial_altdef_nat
% 0.29/0.58  thf(fact_7095_square__fact__le__2__fact,axiom,
% 0.29/0.58      ! [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 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % square_fact_le_2_fact
% 0.29/0.58  thf(fact_7096_Suc__mask__eq__exp,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.29/0.58        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % Suc_mask_eq_exp
% 0.29/0.58  thf(fact_7097_mask__nat__less__exp,axiom,
% 0.29/0.58      ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_nat_less_exp
% 0.29/0.58  thf(fact_7098_bits__induct,axiom,
% 0.29/0.58      ! [P: nat > $o,A: nat] :
% 0.29/0.58        ( ! [A5: nat] :
% 0.29/0.58            ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58              = A5 )
% 0.29/0.58           => ( P @ A5 ) )
% 0.29/0.58       => ( ! [A5: nat,B6: $o] :
% 0.29/0.58              ( ( P @ A5 )
% 0.29/0.58             => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B6 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58                  = A5 )
% 0.29/0.58               => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B6 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 0.29/0.58         => ( P @ A ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bits_induct
% 0.29/0.58  thf(fact_7099_bits__induct,axiom,
% 0.29/0.58      ! [P: int > $o,A: int] :
% 0.29/0.58        ( ! [A5: int] :
% 0.29/0.58            ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58              = A5 )
% 0.29/0.58           => ( P @ A5 ) )
% 0.29/0.58       => ( ! [A5: int,B6: $o] :
% 0.29/0.58              ( ( P @ A5 )
% 0.29/0.58             => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B6 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58                  = A5 )
% 0.29/0.58               => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B6 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 0.29/0.58         => ( P @ A ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bits_induct
% 0.29/0.58  thf(fact_7100_bits__induct,axiom,
% 0.29/0.58      ! [P: code_integer > $o,A: code_integer] :
% 0.29/0.58        ( ! [A5: code_integer] :
% 0.29/0.58            ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58              = A5 )
% 0.29/0.58           => ( P @ A5 ) )
% 0.29/0.58       => ( ! [A5: code_integer,B6: $o] :
% 0.29/0.58              ( ( P @ A5 )
% 0.29/0.58             => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B6 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58                  = A5 )
% 0.29/0.58               => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B6 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 0.29/0.58         => ( P @ A ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % bits_induct
% 0.29/0.58  thf(fact_7101_fact__num__eq__if,axiom,
% 0.29/0.58      ( semiri773545260158071498ct_rat
% 0.29/0.58      = ( ^ [M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M6 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_num_eq_if
% 0.29/0.58  thf(fact_7102_fact__num__eq__if,axiom,
% 0.29/0.58      ( semiri1406184849735516958ct_int
% 0.29/0.58      = ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_num_eq_if
% 0.29/0.58  thf(fact_7103_fact__num__eq__if,axiom,
% 0.29/0.58      ( semiri1408675320244567234ct_nat
% 0.29/0.58      = ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_num_eq_if
% 0.29/0.58  thf(fact_7104_fact__num__eq__if,axiom,
% 0.29/0.58      ( semiri2265585572941072030t_real
% 0.29/0.58      = ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_num_eq_if
% 0.29/0.58  thf(fact_7105_fact__num__eq__if,axiom,
% 0.29/0.58      ( semiri5044797733671781792omplex
% 0.29/0.58      = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_num_eq_if
% 0.29/0.58  thf(fact_7106_fact__reduce,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58       => ( ( semiri773545260158071498ct_rat @ N )
% 0.29/0.58          = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_reduce
% 0.29/0.58  thf(fact_7107_fact__reduce,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58       => ( ( semiri1406184849735516958ct_int @ N )
% 0.29/0.58          = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_reduce
% 0.29/0.58  thf(fact_7108_fact__reduce,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58       => ( ( semiri1408675320244567234ct_nat @ N )
% 0.29/0.58          = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_reduce
% 0.29/0.58  thf(fact_7109_fact__reduce,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58       => ( ( semiri2265585572941072030t_real @ N )
% 0.29/0.58          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_reduce
% 0.29/0.58  thf(fact_7110_fact__reduce,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.58       => ( ( semiri5044797733671781792omplex @ N )
% 0.29/0.58          = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_reduce
% 0.29/0.58  thf(fact_7111_pochhammer__same,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
% 0.29/0.58        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pochhammer_same
% 0.29/0.58  thf(fact_7112_pochhammer__same,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
% 0.29/0.58        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pochhammer_same
% 0.29/0.58  thf(fact_7113_pochhammer__same,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
% 0.29/0.58        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pochhammer_same
% 0.29/0.58  thf(fact_7114_pochhammer__same,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
% 0.29/0.58        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pochhammer_same
% 0.29/0.58  thf(fact_7115_pochhammer__same,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
% 0.29/0.58        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % pochhammer_same
% 0.29/0.58  thf(fact_7116_binomial__fact,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 0.29/0.58          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % binomial_fact
% 0.29/0.58  thf(fact_7117_binomial__fact,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 0.29/0.58          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % binomial_fact
% 0.29/0.58  thf(fact_7118_binomial__fact,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 0.29/0.58          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % binomial_fact
% 0.29/0.58  thf(fact_7119_fact__binomial,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 0.29/0.58          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_binomial
% 0.29/0.58  thf(fact_7120_fact__binomial,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 0.29/0.58          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_binomial
% 0.29/0.58  thf(fact_7121_fact__binomial,axiom,
% 0.29/0.58      ! [K: nat,N: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.58       => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 0.29/0.58          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_binomial
% 0.29/0.58  thf(fact_7122_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
% 0.29/0.58        = ( N = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % semiring_bit_operations_class.even_mask_iff
% 0.29/0.58  thf(fact_7123_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.29/0.58        = ( N = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % semiring_bit_operations_class.even_mask_iff
% 0.29/0.58  thf(fact_7124_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.29/0.58        = ( N = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % semiring_bit_operations_class.even_mask_iff
% 0.29/0.58  thf(fact_7125_exp__mod__exp,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_mod_exp
% 0.29/0.58  thf(fact_7126_exp__mod__exp,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_mod_exp
% 0.29/0.58  thf(fact_7127_exp__mod__exp,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_mod_exp
% 0.29/0.58  thf(fact_7128_mask__nat__def,axiom,
% 0.29/0.58      ( bit_se2002935070580805687sk_nat
% 0.29/0.58      = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_nat_def
% 0.29/0.58  thf(fact_7129_mask__half__int,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58        = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_half_int
% 0.29/0.58  thf(fact_7130_mask__int__def,axiom,
% 0.29/0.58      ( bit_se2000444600071755411sk_int
% 0.29/0.58      = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_int_def
% 0.29/0.58  thf(fact_7131_mask__eq__exp__minus__1,axiom,
% 0.29/0.58      ( bit_se2002935070580805687sk_nat
% 0.29/0.58      = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_eq_exp_minus_1
% 0.29/0.58  thf(fact_7132_mask__eq__exp__minus__1,axiom,
% 0.29/0.58      ( bit_se2000444600071755411sk_int
% 0.29/0.58      = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % mask_eq_exp_minus_1
% 0.29/0.58  thf(fact_7133_set__decode__plus__power__2,axiom,
% 0.29/0.58      ! [N: nat,Z: nat] :
% 0.29/0.58        ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 0.29/0.58       => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 0.29/0.58          = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % set_decode_plus_power_2
% 0.29/0.58  thf(fact_7134_exp__div__exp__eq,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( times_times_nat
% 0.29/0.58          @ ( zero_n2687167440665602831ol_nat
% 0.29/0.58            @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.29/0.58               != zero_zero_nat )
% 0.29/0.58              & ( ord_less_eq_nat @ N @ M ) ) )
% 0.29/0.58          @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_div_exp_eq
% 0.29/0.58  thf(fact_7135_exp__div__exp__eq,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( times_times_int
% 0.29/0.58          @ ( zero_n2684676970156552555ol_int
% 0.29/0.58            @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.29/0.58               != zero_zero_int )
% 0.29/0.58              & ( ord_less_eq_nat @ N @ M ) ) )
% 0.29/0.58          @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_div_exp_eq
% 0.29/0.58  thf(fact_7136_exp__div__exp__eq,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( times_3573771949741848930nteger
% 0.29/0.58          @ ( zero_n356916108424825756nteger
% 0.29/0.58            @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.29/0.58               != zero_z3403309356797280102nteger )
% 0.29/0.58              & ( ord_less_eq_nat @ N @ M ) ) )
% 0.29/0.58          @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % exp_div_exp_eq
% 0.29/0.58  thf(fact_7137_sin__coeff__def,axiom,
% 0.29/0.58      ( sin_coeff
% 0.29/0.58      = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % sin_coeff_def
% 0.29/0.58  thf(fact_7138_binomial__code,axiom,
% 0.29/0.58      ( binomial
% 0.29/0.58      = ( ^ [N3: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K2 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % binomial_code
% 0.29/0.58  thf(fact_7139_cos__coeff__def,axiom,
% 0.29/0.58      ( cos_coeff
% 0.29/0.58      = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % cos_coeff_def
% 0.29/0.58  thf(fact_7140_fact__code,axiom,
% 0.29/0.58      ( semiri773545260158071498ct_rat
% 0.29/0.58      = ( ^ [N3: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_code
% 0.29/0.58  thf(fact_7141_fact__code,axiom,
% 0.29/0.58      ( semiri1406184849735516958ct_int
% 0.29/0.58      = ( ^ [N3: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_code
% 0.29/0.58  thf(fact_7142_fact__code,axiom,
% 0.29/0.58      ( semiri1408675320244567234ct_nat
% 0.29/0.58      = ( ^ [N3: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_code
% 0.29/0.58  thf(fact_7143_fact__code,axiom,
% 0.29/0.58      ( semiri2265585572941072030t_real
% 0.29/0.58      = ( ^ [N3: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_code
% 0.29/0.58  thf(fact_7144_fact__code,axiom,
% 0.29/0.58      ( semiri5044797733671781792omplex
% 0.29/0.58      = ( ^ [N3: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % fact_code
% 0.29/0.58  thf(fact_7145_num_Osize__gen_I3_J,axiom,
% 0.29/0.58      ! [X32: num] :
% 0.29/0.58        ( ( size_num @ ( bit1 @ X32 ) )
% 0.29/0.58        = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % num.size_gen(3)
% 0.29/0.58  thf(fact_7146_take__bit__rec,axiom,
% 0.29/0.58      ( bit_se1745604003318907178nteger
% 0.29/0.58      = ( ^ [N3: nat,A3: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_rec
% 0.29/0.58  thf(fact_7147_take__bit__rec,axiom,
% 0.29/0.58      ( bit_se2923211474154528505it_int
% 0.29/0.58      = ( ^ [N3: nat,A3: int] : ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_rec
% 0.29/0.58  thf(fact_7148_take__bit__rec,axiom,
% 0.29/0.58      ( bit_se2925701944663578781it_nat
% 0.29/0.58      = ( ^ [N3: nat,A3: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_rec
% 0.29/0.58  thf(fact_7149_take__bit__of__0,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 0.29/0.58        = zero_zero_int ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_0
% 0.29/0.58  thf(fact_7150_take__bit__of__0,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 0.29/0.58        = zero_zero_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_0
% 0.29/0.58  thf(fact_7151_concat__bit__of__zero__2,axiom,
% 0.29/0.58      ! [N: nat,K: int] :
% 0.29/0.58        ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 0.29/0.58        = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.29/0.58  
% 0.29/0.58  % concat_bit_of_zero_2
% 0.29/0.58  thf(fact_7152_take__bit__0,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 0.29/0.58        = zero_zero_int ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_0
% 0.29/0.58  thf(fact_7153_take__bit__0,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 0.29/0.58        = zero_zero_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_0
% 0.29/0.58  thf(fact_7154_take__bit__Suc__1,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
% 0.29/0.58        = one_one_int ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_Suc_1
% 0.29/0.58  thf(fact_7155_take__bit__Suc__1,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
% 0.29/0.58        = one_one_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_Suc_1
% 0.29/0.58  thf(fact_7156_take__bit__numeral__1,axiom,
% 0.29/0.58      ! [L: num] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
% 0.29/0.58        = one_one_int ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_numeral_1
% 0.29/0.58  thf(fact_7157_take__bit__numeral__1,axiom,
% 0.29/0.58      ! [L: num] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
% 0.29/0.58        = one_one_nat ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_numeral_1
% 0.29/0.58  thf(fact_7158_sin__coeff__0,axiom,
% 0.29/0.58      ( ( sin_coeff @ zero_zero_nat )
% 0.29/0.58      = zero_zero_real ) ).
% 0.29/0.58  
% 0.29/0.58  % sin_coeff_0
% 0.29/0.58  thf(fact_7159_cos__coeff__0,axiom,
% 0.29/0.58      ( ( cos_coeff @ zero_zero_nat )
% 0.29/0.58      = one_one_real ) ).
% 0.29/0.58  
% 0.29/0.58  % cos_coeff_0
% 0.29/0.58  thf(fact_7160_take__bit__of__1__eq__0__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.29/0.58          = zero_zero_int )
% 0.29/0.58        = ( N = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_1_eq_0_iff
% 0.29/0.58  thf(fact_7161_take__bit__of__1__eq__0__iff,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.29/0.58          = zero_zero_nat )
% 0.29/0.58        = ( N = zero_zero_nat ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_1_eq_0_iff
% 0.29/0.58  thf(fact_7162_take__bit__minus__one__eq__mask,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.58        = ( bit_se2119862282449309892nteger @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_minus_one_eq_mask
% 0.29/0.58  thf(fact_7163_take__bit__minus__one__eq__mask,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.58        = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_minus_one_eq_mask
% 0.29/0.58  thf(fact_7164_take__bit__of__Suc__0,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.58        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_Suc_0
% 0.29/0.58  thf(fact_7165_take__bit__of__1,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
% 0.29/0.58        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_1
% 0.29/0.58  thf(fact_7166_take__bit__of__1,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.29/0.58        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_1
% 0.29/0.58  thf(fact_7167_take__bit__of__1,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.29/0.58        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_1
% 0.29/0.58  thf(fact_7168_even__take__bit__eq,axiom,
% 0.29/0.58      ! [N: nat,A: code_integer] :
% 0.29/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 0.29/0.58        = ( ( N = zero_zero_nat )
% 0.29/0.58          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_take_bit_eq
% 0.29/0.58  thf(fact_7169_even__take__bit__eq,axiom,
% 0.29/0.58      ! [N: nat,A: int] :
% 0.29/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.29/0.58        = ( ( N = zero_zero_nat )
% 0.29/0.58          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_take_bit_eq
% 0.29/0.58  thf(fact_7170_even__take__bit__eq,axiom,
% 0.29/0.58      ! [N: nat,A: nat] :
% 0.29/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.29/0.58        = ( ( N = zero_zero_nat )
% 0.29/0.58          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % even_take_bit_eq
% 0.29/0.58  thf(fact_7171_take__bit__Suc__0,axiom,
% 0.29/0.58      ! [A: code_integer] :
% 0.29/0.58        ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 0.29/0.58        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_Suc_0
% 0.29/0.58  thf(fact_7172_take__bit__Suc__0,axiom,
% 0.29/0.58      ! [A: int] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 0.29/0.58        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_Suc_0
% 0.29/0.58  thf(fact_7173_take__bit__Suc__0,axiom,
% 0.29/0.58      ! [A: nat] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 0.29/0.58        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_Suc_0
% 0.29/0.58  thf(fact_7174_take__bit__of__exp,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_exp
% 0.29/0.58  thf(fact_7175_take__bit__of__exp,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_exp
% 0.29/0.58  thf(fact_7176_take__bit__of__exp,axiom,
% 0.29/0.58      ! [M: nat,N: nat] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.58        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_exp
% 0.29/0.58  thf(fact_7177_take__bit__of__2,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.58        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_2
% 0.29/0.58  thf(fact_7178_take__bit__of__2,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.58        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_2
% 0.29/0.58  thf(fact_7179_take__bit__of__2,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.58        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_2
% 0.29/0.58  thf(fact_7180_take__bit__add,axiom,
% 0.29/0.58      ! [N: nat,A: int,B: int] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
% 0.29/0.58        = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_add
% 0.29/0.58  thf(fact_7181_take__bit__add,axiom,
% 0.29/0.58      ! [N: nat,A: nat,B: nat] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
% 0.29/0.58        = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_add
% 0.29/0.58  thf(fact_7182_take__bit__of__int,axiom,
% 0.29/0.58      ! [N: nat,K: int] :
% 0.29/0.58        ( ( bit_se1745604003318907178nteger @ N @ ( ring_18347121197199848620nteger @ K ) )
% 0.29/0.58        = ( ring_18347121197199848620nteger @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_int
% 0.29/0.58  thf(fact_7183_take__bit__of__int,axiom,
% 0.29/0.58      ! [N: nat,K: int] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
% 0.29/0.58        = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_int
% 0.29/0.58  thf(fact_7184_take__bit__of__nat,axiom,
% 0.29/0.58      ! [N: nat,M: nat] :
% 0.29/0.58        ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.29/0.58        = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_nat
% 0.29/0.58  thf(fact_7185_take__bit__of__nat,axiom,
% 0.29/0.58      ! [N: nat,M: nat] :
% 0.29/0.58        ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.29/0.58        = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_of_nat
% 0.29/0.58  thf(fact_7186_take__bit__tightened,axiom,
% 0.29/0.58      ! [N: nat,A: int,B: int,M: nat] :
% 0.29/0.58        ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.29/0.58          = ( bit_se2923211474154528505it_int @ N @ B ) )
% 0.29/0.58       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58         => ( ( bit_se2923211474154528505it_int @ M @ A )
% 0.29/0.58            = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_tightened
% 0.29/0.58  thf(fact_7187_take__bit__tightened,axiom,
% 0.29/0.58      ! [N: nat,A: nat,B: nat,M: nat] :
% 0.29/0.58        ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.29/0.58          = ( bit_se2925701944663578781it_nat @ N @ B ) )
% 0.29/0.58       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58         => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 0.29/0.58            = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_tightened
% 0.29/0.58  thf(fact_7188_take__bit__nat__less__eq__self,axiom,
% 0.29/0.58      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_nat_less_eq_self
% 0.29/0.58  thf(fact_7189_take__bit__tightened__less__eq__nat,axiom,
% 0.29/0.58      ! [M: nat,N: nat,Q3: nat] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_tightened_less_eq_nat
% 0.29/0.58  thf(fact_7190_take__bit__tightened__less__eq__int,axiom,
% 0.29/0.58      ! [M: nat,N: nat,K: int] :
% 0.29/0.58        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.58       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_tightened_less_eq_int
% 0.29/0.58  thf(fact_7191_take__bit__int__less__eq__self__iff,axiom,
% 0.29/0.58      ! [N: nat,K: int] :
% 0.29/0.58        ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.29/0.58        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_int_less_eq_self_iff
% 0.29/0.58  thf(fact_7192_take__bit__nonnegative,axiom,
% 0.29/0.58      ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_nonnegative
% 0.29/0.58  thf(fact_7193_take__bit__int__greater__self__iff,axiom,
% 0.29/0.58      ! [K: int,N: nat] :
% 0.29/0.58        ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.29/0.58        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_int_greater_self_iff
% 0.29/0.58  thf(fact_7194_not__take__bit__negative,axiom,
% 0.29/0.58      ! [N: nat,K: int] :
% 0.29/0.58        ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 0.29/0.58  
% 0.29/0.58  % not_take_bit_negative
% 0.29/0.58  thf(fact_7195_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 0.29/0.58      ! [N: nat,A: int,B: int] :
% 0.29/0.58        ( ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.29/0.58          = ( bit_ri631733984087533419it_int @ N @ B ) )
% 0.29/0.58        = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 0.29/0.58          = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % signed_take_bit_eq_iff_take_bit_eq
% 0.29/0.58  thf(fact_7196_signed__take__bit__take__bit,axiom,
% 0.29/0.58      ! [M: nat,N: nat,A: int] :
% 0.29/0.58        ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.29/0.58        = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 0.29/0.58  
% 0.29/0.58  % signed_take_bit_take_bit
% 0.29/0.58  thf(fact_7197_take__bit__unset__bit__eq,axiom,
% 0.29/0.58      ! [N: nat,M: nat,A: int] :
% 0.29/0.58        ( ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.29/0.58            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.29/0.58        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.29/0.58            = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_unset_bit_eq
% 0.29/0.58  thf(fact_7198_take__bit__unset__bit__eq,axiom,
% 0.29/0.58      ! [N: nat,M: nat,A: nat] :
% 0.29/0.58        ( ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.29/0.58            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.29/0.58        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.29/0.58            = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_unset_bit_eq
% 0.29/0.58  thf(fact_7199_take__bit__set__bit__eq,axiom,
% 0.29/0.58      ! [N: nat,M: nat,A: int] :
% 0.29/0.58        ( ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.29/0.58            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.29/0.58        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.29/0.58            = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_set_bit_eq
% 0.29/0.58  thf(fact_7200_take__bit__set__bit__eq,axiom,
% 0.29/0.58      ! [N: nat,M: nat,A: nat] :
% 0.29/0.58        ( ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.29/0.58            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.29/0.58        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.29/0.58            = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_set_bit_eq
% 0.29/0.58  thf(fact_7201_take__bit__flip__bit__eq,axiom,
% 0.29/0.58      ! [N: nat,M: nat,A: int] :
% 0.29/0.58        ( ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.29/0.58            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.29/0.58        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.29/0.58            = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_flip_bit_eq
% 0.29/0.58  thf(fact_7202_take__bit__flip__bit__eq,axiom,
% 0.29/0.58      ! [N: nat,M: nat,A: nat] :
% 0.29/0.58        ( ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.29/0.58            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.29/0.58        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.29/0.58         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.29/0.58            = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.29/0.58  
% 0.29/0.58  % take_bit_flip_bit_eq
% 0.29/0.58  thf(fact_7203_sin__coeff__Suc,axiom,
% 0.29/0.58      ! [N: nat] :
% 0.29/0.58        ( ( sin_coeff @ ( suc @ N ) )
% 0.29/0.58        = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_coeff_Suc
% 0.29/0.59  thf(fact_7204_take__bit__signed__take__bit,axiom,
% 0.29/0.59      ! [M: nat,N: nat,A: int] :
% 0.29/0.59        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.29/0.59       => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 0.29/0.59          = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_signed_take_bit
% 0.29/0.59  thf(fact_7205_take__bit__eq__mask__iff,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.29/0.59          = ( bit_se2000444600071755411sk_int @ N ) )
% 0.29/0.59        = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.29/0.59          = zero_zero_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_eq_mask_iff
% 0.29/0.59  thf(fact_7206_take__bit__decr__eq,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.29/0.59         != zero_zero_int )
% 0.29/0.59       => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 0.29/0.59          = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_decr_eq
% 0.29/0.59  thf(fact_7207_cos__coeff__Suc,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( cos_coeff @ ( suc @ N ) )
% 0.29/0.59        = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_coeff_Suc
% 0.29/0.59  thf(fact_7208_take__bit__Suc__bit0,axiom,
% 0.29/0.59      ! [N: nat,K: num] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.29/0.59        = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc_bit0
% 0.29/0.59  thf(fact_7209_take__bit__Suc__bit0,axiom,
% 0.29/0.59      ! [N: nat,K: num] :
% 0.29/0.59        ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.29/0.59        = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc_bit0
% 0.29/0.59  thf(fact_7210_take__bit__eq__mod,axiom,
% 0.29/0.59      ( bit_se1745604003318907178nteger
% 0.29/0.59      = ( ^ [N3: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_eq_mod
% 0.29/0.59  thf(fact_7211_take__bit__eq__mod,axiom,
% 0.29/0.59      ( bit_se2923211474154528505it_int
% 0.29/0.59      = ( ^ [N3: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_eq_mod
% 0.29/0.59  thf(fact_7212_take__bit__eq__mod,axiom,
% 0.29/0.59      ( bit_se2925701944663578781it_nat
% 0.29/0.59      = ( ^ [N3: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_eq_mod
% 0.29/0.59  thf(fact_7213_take__bit__nat__eq__self,axiom,
% 0.29/0.59      ! [M: nat,N: nat] :
% 0.29/0.59        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.59       => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.29/0.59          = M ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_nat_eq_self
% 0.29/0.59  thf(fact_7214_take__bit__nat__less__exp,axiom,
% 0.29/0.59      ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_nat_less_exp
% 0.29/0.59  thf(fact_7215_take__bit__nat__eq__self__iff,axiom,
% 0.29/0.59      ! [N: nat,M: nat] :
% 0.29/0.59        ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.29/0.59          = M )
% 0.29/0.59        = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_nat_eq_self_iff
% 0.29/0.59  thf(fact_7216_take__bit__nat__def,axiom,
% 0.29/0.59      ( bit_se2925701944663578781it_nat
% 0.29/0.59      = ( ^ [N3: nat,M6: nat] : ( modulo_modulo_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_nat_def
% 0.29/0.59  thf(fact_7217_take__bit__int__less__exp,axiom,
% 0.29/0.59      ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_int_less_exp
% 0.29/0.59  thf(fact_7218_fold__atLeastAtMost__nat_Oelims,axiom,
% 0.29/0.59      ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
% 0.29/0.59        ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
% 0.29/0.59          = Y )
% 0.29/0.59       => ( ( ( ord_less_nat @ Xb @ Xa )
% 0.29/0.59           => ( Y = Xc ) )
% 0.29/0.59          & ( ~ ( ord_less_nat @ Xb @ Xa )
% 0.29/0.59           => ( Y
% 0.29/0.59              = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % fold_atLeastAtMost_nat.elims
% 0.29/0.59  thf(fact_7219_fold__atLeastAtMost__nat_Osimps,axiom,
% 0.29/0.59      ( set_fo2584398358068434914at_nat
% 0.29/0.59      = ( ^ [F2: nat > nat > nat,A3: nat,B4: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A3 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B4 @ ( F2 @ A3 @ Acc ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % fold_atLeastAtMost_nat.simps
% 0.29/0.59  thf(fact_7220_take__bit__int__def,axiom,
% 0.29/0.59      ( bit_se2923211474154528505it_int
% 0.29/0.59      = ( ^ [N3: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_int_def
% 0.29/0.59  thf(fact_7221_num_Osize__gen_I1_J,axiom,
% 0.29/0.59      ( ( size_num @ one )
% 0.29/0.59      = zero_zero_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % num.size_gen(1)
% 0.29/0.59  thf(fact_7222_take__bit__eq__0__iff,axiom,
% 0.29/0.59      ! [N: nat,A: code_integer] :
% 0.29/0.59        ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.29/0.59          = zero_z3403309356797280102nteger )
% 0.29/0.59        = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_eq_0_iff
% 0.29/0.59  thf(fact_7223_take__bit__eq__0__iff,axiom,
% 0.29/0.59      ! [N: nat,A: int] :
% 0.29/0.59        ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.29/0.59          = zero_zero_int )
% 0.29/0.59        = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_eq_0_iff
% 0.29/0.59  thf(fact_7224_take__bit__eq__0__iff,axiom,
% 0.29/0.59      ! [N: nat,A: nat] :
% 0.29/0.59        ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.29/0.59          = zero_zero_nat )
% 0.29/0.59        = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_eq_0_iff
% 0.29/0.59  thf(fact_7225_take__bit__numeral__bit0,axiom,
% 0.29/0.59      ! [L: num,K: num] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.29/0.59        = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_numeral_bit0
% 0.29/0.59  thf(fact_7226_take__bit__numeral__bit0,axiom,
% 0.29/0.59      ! [L: num,K: num] :
% 0.29/0.59        ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.29/0.59        = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_numeral_bit0
% 0.29/0.59  thf(fact_7227_take__bit__nat__less__self__iff,axiom,
% 0.29/0.59      ! [N: nat,M: nat] :
% 0.29/0.59        ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 0.29/0.59        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_nat_less_self_iff
% 0.29/0.59  thf(fact_7228_take__bit__Suc__minus__bit0,axiom,
% 0.29/0.59      ! [N: nat,K: num] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.29/0.59        = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc_minus_bit0
% 0.29/0.59  thf(fact_7229_take__bit__int__less__self__iff,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.29/0.59        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_int_less_self_iff
% 0.29/0.59  thf(fact_7230_take__bit__int__greater__eq__self__iff,axiom,
% 0.29/0.59      ! [K: int,N: nat] :
% 0.29/0.59        ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.29/0.59        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_int_greater_eq_self_iff
% 0.29/0.59  thf(fact_7231_take__bit__int__eq__self,axiom,
% 0.29/0.59      ! [K: int,N: nat] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.59       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.59         => ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.29/0.59            = K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_int_eq_self
% 0.29/0.59  thf(fact_7232_take__bit__int__eq__self__iff,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.29/0.59          = K )
% 0.29/0.59        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.59          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_int_eq_self_iff
% 0.29/0.59  thf(fact_7233_take__bit__numeral__minus__bit0,axiom,
% 0.29/0.59      ! [L: num,K: num] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.29/0.59        = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_numeral_minus_bit0
% 0.29/0.59  thf(fact_7234_take__bit__incr__eq,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.29/0.59         != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.29/0.59       => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.29/0.59          = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_incr_eq
% 0.29/0.59  thf(fact_7235_take__bit__eq__mask__iff__exp__dvd,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.29/0.59          = ( bit_se2000444600071755411sk_int @ N ) )
% 0.29/0.59        = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_eq_mask_iff_exp_dvd
% 0.29/0.59  thf(fact_7236_take__bit__Suc__minus__1__eq,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.59        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc_minus_1_eq
% 0.29/0.59  thf(fact_7237_take__bit__Suc__minus__1__eq,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59        = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc_minus_1_eq
% 0.29/0.59  thf(fact_7238_take__bit__Suc__bit1,axiom,
% 0.29/0.59      ! [N: nat,K: num] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.29/0.59        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc_bit1
% 0.29/0.59  thf(fact_7239_take__bit__Suc__bit1,axiom,
% 0.29/0.59      ! [N: nat,K: num] :
% 0.29/0.59        ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.29/0.59        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc_bit1
% 0.29/0.59  thf(fact_7240_take__bit__numeral__minus__1__eq,axiom,
% 0.29/0.59      ! [K: num] :
% 0.29/0.59        ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.59        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_numeral_minus_1_eq
% 0.29/0.59  thf(fact_7241_take__bit__numeral__minus__1__eq,axiom,
% 0.29/0.59      ! [K: num] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59        = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_numeral_minus_1_eq
% 0.29/0.59  thf(fact_7242_take__bit__Suc,axiom,
% 0.29/0.59      ! [N: nat,A: code_integer] :
% 0.29/0.59        ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
% 0.29/0.59        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc
% 0.29/0.59  thf(fact_7243_take__bit__Suc,axiom,
% 0.29/0.59      ! [N: nat,A: int] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 0.29/0.59        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc
% 0.29/0.59  thf(fact_7244_take__bit__Suc,axiom,
% 0.29/0.59      ! [N: nat,A: nat] :
% 0.29/0.59        ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
% 0.29/0.59        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc
% 0.29/0.59  thf(fact_7245_take__bit__int__less__eq,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.29/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.59         => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_int_less_eq
% 0.29/0.59  thf(fact_7246_take__bit__int__greater__eq,axiom,
% 0.29/0.59      ! [K: int,N: nat] :
% 0.29/0.59        ( ( ord_less_int @ K @ zero_zero_int )
% 0.29/0.59       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_int_greater_eq
% 0.29/0.59  thf(fact_7247_signed__take__bit__eq__take__bit__shift,axiom,
% 0.29/0.59      ( bit_ri631733984087533419it_int
% 0.29/0.59      = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % signed_take_bit_eq_take_bit_shift
% 0.29/0.59  thf(fact_7248_stable__imp__take__bit__eq,axiom,
% 0.29/0.59      ! [A: code_integer,N: nat] :
% 0.29/0.59        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.59          = A )
% 0.29/0.59       => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.59           => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.29/0.59              = zero_z3403309356797280102nteger ) )
% 0.29/0.59          & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.59           => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.29/0.59              = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % stable_imp_take_bit_eq
% 0.29/0.59  thf(fact_7249_stable__imp__take__bit__eq,axiom,
% 0.29/0.59      ! [A: int,N: nat] :
% 0.29/0.59        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.59          = A )
% 0.29/0.59       => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.59           => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.29/0.59              = zero_zero_int ) )
% 0.29/0.59          & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.59           => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.29/0.59              = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % stable_imp_take_bit_eq
% 0.29/0.59  thf(fact_7250_stable__imp__take__bit__eq,axiom,
% 0.29/0.59      ! [A: nat,N: nat] :
% 0.29/0.59        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.59          = A )
% 0.29/0.59       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.59           => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.29/0.59              = zero_zero_nat ) )
% 0.29/0.59          & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.59           => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.29/0.59              = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % stable_imp_take_bit_eq
% 0.29/0.59  thf(fact_7251_take__bit__numeral__bit1,axiom,
% 0.29/0.59      ! [L: num,K: num] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.29/0.59        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_numeral_bit1
% 0.29/0.59  thf(fact_7252_take__bit__numeral__bit1,axiom,
% 0.29/0.59      ! [L: num,K: num] :
% 0.29/0.59        ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.29/0.59        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_numeral_bit1
% 0.29/0.59  thf(fact_7253_take__bit__minus__small__eq,axiom,
% 0.29/0.59      ! [K: int,N: nat] :
% 0.29/0.59        ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.59       => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.59         => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 0.29/0.59            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_minus_small_eq
% 0.29/0.59  thf(fact_7254_num_Osize__gen_I2_J,axiom,
% 0.29/0.59      ! [X22: num] :
% 0.29/0.59        ( ( size_num @ ( bit0 @ X22 ) )
% 0.29/0.59        = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % num.size_gen(2)
% 0.29/0.59  thf(fact_7255_take__bit__numeral__minus__bit1,axiom,
% 0.29/0.59      ! [L: num,K: num] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.29/0.59        = ( 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 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_numeral_minus_bit1
% 0.29/0.59  thf(fact_7256_take__bit__Suc__minus__bit1,axiom,
% 0.29/0.59      ! [N: nat,K: num] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.29/0.59        = ( 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 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc_minus_bit1
% 0.29/0.59  thf(fact_7257_modulo__int__unfold,axiom,
% 0.29/0.59      ! [L: int,K: int,N: nat,M: nat] :
% 0.29/0.59        ( ( ( ( ( sgn_sgn_int @ L )
% 0.29/0.59              = zero_zero_int )
% 0.29/0.59            | ( ( sgn_sgn_int @ K )
% 0.29/0.59              = zero_zero_int )
% 0.29/0.59            | ( N = zero_zero_nat ) )
% 0.29/0.59         => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.29/0.59            = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.29/0.59        & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.29/0.59                = zero_zero_int )
% 0.29/0.59              | ( ( sgn_sgn_int @ K )
% 0.29/0.59                = zero_zero_int )
% 0.29/0.59              | ( N = zero_zero_nat ) )
% 0.29/0.59         => ( ( ( ( sgn_sgn_int @ K )
% 0.29/0.59                = ( sgn_sgn_int @ L ) )
% 0.29/0.59             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.29/0.59                = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 0.29/0.59            & ( ( ( sgn_sgn_int @ K )
% 0.29/0.59               != ( sgn_sgn_int @ L ) )
% 0.29/0.59             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.29/0.59                = ( times_times_int @ ( sgn_sgn_int @ L )
% 0.29/0.59                  @ ( minus_minus_int
% 0.29/0.59                    @ ( semiri1314217659103216013at_int
% 0.29/0.59                      @ ( times_times_nat @ N
% 0.29/0.59                        @ ( zero_n2687167440665602831ol_nat
% 0.29/0.59                          @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 0.29/0.59                    @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % modulo_int_unfold
% 0.29/0.59  thf(fact_7258_powr__int,axiom,
% 0.29/0.59      ! [X: real,I: int] :
% 0.29/0.59        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.29/0.59           => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 0.29/0.59              = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
% 0.29/0.59          & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 0.29/0.59           => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 0.29/0.59              = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % powr_int
% 0.29/0.59  thf(fact_7259_divide__int__unfold,axiom,
% 0.29/0.59      ! [L: int,K: int,N: nat,M: nat] :
% 0.29/0.59        ( ( ( ( ( sgn_sgn_int @ L )
% 0.29/0.59              = zero_zero_int )
% 0.29/0.59            | ( ( sgn_sgn_int @ K )
% 0.29/0.59              = zero_zero_int )
% 0.29/0.59            | ( N = zero_zero_nat ) )
% 0.29/0.59         => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.29/0.59            = zero_zero_int ) )
% 0.29/0.59        & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.29/0.59                = zero_zero_int )
% 0.29/0.59              | ( ( sgn_sgn_int @ K )
% 0.29/0.59                = zero_zero_int )
% 0.29/0.59              | ( N = zero_zero_nat ) )
% 0.29/0.59         => ( ( ( ( sgn_sgn_int @ K )
% 0.29/0.59                = ( sgn_sgn_int @ L ) )
% 0.29/0.59             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.29/0.59                = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 0.29/0.59            & ( ( ( sgn_sgn_int @ K )
% 0.29/0.59               != ( sgn_sgn_int @ L ) )
% 0.29/0.59             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.29/0.59                = ( uminus_uminus_int
% 0.29/0.59                  @ ( semiri1314217659103216013at_int
% 0.29/0.59                    @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 0.29/0.59                      @ ( zero_n2687167440665602831ol_nat
% 0.29/0.59                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divide_int_unfold
% 0.29/0.59  thf(fact_7260_invar__vebt_Ocases,axiom,
% 0.29/0.59      ! [A1: vEBT_VEBT,A22: nat] :
% 0.29/0.59        ( ( vEBT_invar_vebt @ A1 @ A22 )
% 0.29/0.59       => ( ( ? [A5: $o,B6: $o] :
% 0.29/0.59                ( A1
% 0.29/0.59                = ( vEBT_Leaf @ A5 @ B6 ) )
% 0.29/0.59           => ( A22
% 0.29/0.59             != ( suc @ zero_zero_nat ) ) )
% 0.29/0.59         => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
% 0.29/0.59                ( ( A1
% 0.29/0.59                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.59               => ( ( A22 = Deg2 )
% 0.29/0.59                 => ( ! [X2: vEBT_VEBT] :
% 0.29/0.59                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.59                       => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 0.29/0.59                   => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
% 0.29/0.59                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.59                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.29/0.59                       => ( ( M5 = N2 )
% 0.29/0.59                         => ( ( Deg2
% 0.29/0.59                              = ( plus_plus_nat @ N2 @ M5 ) )
% 0.29/0.59                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
% 0.29/0.59                             => ~ ! [X2: vEBT_VEBT] :
% 0.29/0.59                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.59                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
% 0.29/0.59           => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
% 0.29/0.59                  ( ( A1
% 0.29/0.59                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.59                 => ( ( A22 = Deg2 )
% 0.29/0.59                   => ( ! [X2: vEBT_VEBT] :
% 0.29/0.59                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.59                         => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 0.29/0.59                     => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
% 0.29/0.59                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.59                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.29/0.59                         => ( ( M5
% 0.29/0.59                              = ( suc @ N2 ) )
% 0.29/0.59                           => ( ( Deg2
% 0.29/0.59                                = ( plus_plus_nat @ N2 @ M5 ) )
% 0.29/0.59                             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
% 0.29/0.59                               => ~ ! [X2: vEBT_VEBT] :
% 0.29/0.59                                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.59                                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
% 0.29/0.59             => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi: nat,Ma2: nat] :
% 0.29/0.59                    ( ( A1
% 0.29/0.59                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.59                   => ( ( A22 = Deg2 )
% 0.29/0.59                     => ( ! [X2: vEBT_VEBT] :
% 0.29/0.59                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.59                           => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 0.29/0.59                       => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
% 0.29/0.59                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.59                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.29/0.59                           => ( ( M5 = N2 )
% 0.29/0.59                             => ( ( Deg2
% 0.29/0.59                                  = ( plus_plus_nat @ N2 @ M5 ) )
% 0.29/0.59                               => ( ! [I4: nat] :
% 0.29/0.59                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.29/0.59                                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.29/0.59                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.29/0.59                                 => ( ( ( Mi = Ma2 )
% 0.29/0.59                                     => ! [X2: vEBT_VEBT] :
% 0.29/0.59                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.59                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
% 0.29/0.59                                   => ( ( ord_less_eq_nat @ Mi @ Ma2 )
% 0.29/0.59                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.59                                       => ~ ( ( Mi != Ma2 )
% 0.29/0.59                                           => ! [I4: nat] :
% 0.29/0.59                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.29/0.59                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 0.29/0.59                                                      = I4 )
% 0.29/0.59                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 0.29/0.59                                                  & ! [X2: nat] :
% 0.29/0.59                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
% 0.29/0.59                                                          = I4 )
% 0.29/0.59                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
% 0.29/0.59                                                     => ( ( ord_less_nat @ Mi @ X2 )
% 0.29/0.59                                                        & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.29/0.59               => ~ ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi: nat,Ma2: nat] :
% 0.29/0.59                      ( ( A1
% 0.29/0.59                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.59                     => ( ( A22 = Deg2 )
% 0.29/0.59                       => ( ! [X2: vEBT_VEBT] :
% 0.29/0.59                              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.59                             => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 0.29/0.59                         => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
% 0.29/0.59                           => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.59                                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.29/0.59                             => ( ( M5
% 0.29/0.59                                  = ( suc @ N2 ) )
% 0.29/0.59                               => ( ( Deg2
% 0.29/0.59                                    = ( plus_plus_nat @ N2 @ M5 ) )
% 0.29/0.59                                 => ( ! [I4: nat] :
% 0.29/0.59                                        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.29/0.59                                       => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.29/0.59                                          = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.29/0.59                                   => ( ( ( Mi = Ma2 )
% 0.29/0.59                                       => ! [X2: vEBT_VEBT] :
% 0.29/0.59                                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.59                                           => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
% 0.29/0.59                                     => ( ( ord_less_eq_nat @ Mi @ Ma2 )
% 0.29/0.59                                       => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.59                                         => ~ ( ( Mi != Ma2 )
% 0.29/0.59                                             => ! [I4: nat] :
% 0.29/0.59                                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.29/0.59                                                 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 0.29/0.59                                                        = I4 )
% 0.29/0.59                                                     => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 0.29/0.59                                                    & ! [X2: nat] :
% 0.29/0.59                                                        ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
% 0.29/0.59                                                            = I4 )
% 0.29/0.59                                                          & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
% 0.29/0.59                                                       => ( ( ord_less_nat @ Mi @ X2 )
% 0.29/0.59                                                          & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % invar_vebt.cases
% 0.29/0.59  thf(fact_7261_sgn__zero,axiom,
% 0.29/0.59      ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.29/0.59      = zero_zero_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_zero
% 0.29/0.59  thf(fact_7262_sgn__zero,axiom,
% 0.29/0.59      ( ( sgn_sgn_real @ zero_zero_real )
% 0.29/0.59      = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_zero
% 0.29/0.59  thf(fact_7263_sgn__0,axiom,
% 0.29/0.59      ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 0.29/0.59      = zero_z3403309356797280102nteger ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_0
% 0.29/0.59  thf(fact_7264_sgn__0,axiom,
% 0.29/0.59      ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.29/0.59      = zero_zero_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_0
% 0.29/0.59  thf(fact_7265_sgn__0,axiom,
% 0.29/0.59      ( ( sgn_sgn_real @ zero_zero_real )
% 0.29/0.59      = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_0
% 0.29/0.59  thf(fact_7266_sgn__0,axiom,
% 0.29/0.59      ( ( sgn_sgn_rat @ zero_zero_rat )
% 0.29/0.59      = zero_zero_rat ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_0
% 0.29/0.59  thf(fact_7267_sgn__0,axiom,
% 0.29/0.59      ( ( sgn_sgn_int @ zero_zero_int )
% 0.29/0.59      = zero_zero_int ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_0
% 0.29/0.59  thf(fact_7268_sgn__one,axiom,
% 0.29/0.59      ( ( sgn_sgn_real @ one_one_real )
% 0.29/0.59      = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_one
% 0.29/0.59  thf(fact_7269_sgn__one,axiom,
% 0.29/0.59      ( ( sgn_sgn_complex @ one_one_complex )
% 0.29/0.59      = one_one_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_one
% 0.29/0.59  thf(fact_7270_sgn__1,axiom,
% 0.29/0.59      ( ( sgn_sgn_int @ one_one_int )
% 0.29/0.59      = one_one_int ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1
% 0.29/0.59  thf(fact_7271_sgn__1,axiom,
% 0.29/0.59      ( ( sgn_sgn_real @ one_one_real )
% 0.29/0.59      = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1
% 0.29/0.59  thf(fact_7272_sgn__1,axiom,
% 0.29/0.59      ( ( sgn_sgn_complex @ one_one_complex )
% 0.29/0.59      = one_one_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1
% 0.29/0.59  thf(fact_7273_sgn__1,axiom,
% 0.29/0.59      ( ( sgn_sgn_rat @ one_one_rat )
% 0.29/0.59      = one_one_rat ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1
% 0.29/0.59  thf(fact_7274_sgn__1,axiom,
% 0.29/0.59      ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
% 0.29/0.59      = one_one_Code_integer ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1
% 0.29/0.59  thf(fact_7275_sgn__divide,axiom,
% 0.29/0.59      ! [A: complex,B: complex] :
% 0.29/0.59        ( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.59        = ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_divide
% 0.29/0.59  thf(fact_7276_sgn__divide,axiom,
% 0.29/0.59      ! [A: real,B: real] :
% 0.29/0.59        ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.59        = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_divide
% 0.29/0.59  thf(fact_7277_sgn__divide,axiom,
% 0.29/0.59      ! [A: rat,B: rat] :
% 0.29/0.59        ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B ) )
% 0.29/0.59        = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_divide
% 0.29/0.59  thf(fact_7278_power__sgn,axiom,
% 0.29/0.59      ! [A: rat,N: nat] :
% 0.29/0.59        ( ( sgn_sgn_rat @ ( power_power_rat @ A @ N ) )
% 0.29/0.59        = ( power_power_rat @ ( sgn_sgn_rat @ A ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % power_sgn
% 0.29/0.59  thf(fact_7279_power__sgn,axiom,
% 0.29/0.59      ! [A: code_integer,N: nat] :
% 0.29/0.59        ( ( sgn_sgn_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.29/0.59        = ( power_8256067586552552935nteger @ ( sgn_sgn_Code_integer @ A ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % power_sgn
% 0.29/0.59  thf(fact_7280_power__sgn,axiom,
% 0.29/0.59      ! [A: real,N: nat] :
% 0.29/0.59        ( ( sgn_sgn_real @ ( power_power_real @ A @ N ) )
% 0.29/0.59        = ( power_power_real @ ( sgn_sgn_real @ A ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % power_sgn
% 0.29/0.59  thf(fact_7281_power__sgn,axiom,
% 0.29/0.59      ! [A: int,N: nat] :
% 0.29/0.59        ( ( sgn_sgn_int @ ( power_power_int @ A @ N ) )
% 0.29/0.59        = ( power_power_int @ ( sgn_sgn_int @ A ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % power_sgn
% 0.29/0.59  thf(fact_7282_not__Some__eq,axiom,
% 0.29/0.59      ! [X: option4927543243414619207at_nat] :
% 0.29/0.59        ( ( ! [Y6: product_prod_nat_nat] :
% 0.29/0.59              ( X
% 0.29/0.59             != ( some_P7363390416028606310at_nat @ Y6 ) ) )
% 0.29/0.59        = ( X = none_P5556105721700978146at_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % not_Some_eq
% 0.29/0.59  thf(fact_7283_not__Some__eq,axiom,
% 0.29/0.59      ! [X: option_num] :
% 0.29/0.59        ( ( ! [Y6: num] :
% 0.29/0.59              ( X
% 0.29/0.59             != ( some_num @ Y6 ) ) )
% 0.29/0.59        = ( X = none_num ) ) ).
% 0.29/0.59  
% 0.29/0.59  % not_Some_eq
% 0.29/0.59  thf(fact_7284_not__None__eq,axiom,
% 0.29/0.59      ! [X: option4927543243414619207at_nat] :
% 0.29/0.59        ( ( X != none_P5556105721700978146at_nat )
% 0.29/0.59        = ( ? [Y6: product_prod_nat_nat] :
% 0.29/0.59              ( X
% 0.29/0.59              = ( some_P7363390416028606310at_nat @ Y6 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % not_None_eq
% 0.29/0.59  thf(fact_7285_not__None__eq,axiom,
% 0.29/0.59      ! [X: option_num] :
% 0.29/0.59        ( ( X != none_num )
% 0.29/0.59        = ( ? [Y6: num] :
% 0.29/0.59              ( X
% 0.29/0.59              = ( some_num @ Y6 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % not_None_eq
% 0.29/0.59  thf(fact_7286_nat__int,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
% 0.29/0.59        = N ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_int
% 0.29/0.59  thf(fact_7287_mi__ma__2__deg,axiom,
% 0.29/0.59      ! [Mi2: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.29/0.59        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 0.29/0.59       => ( ( ord_less_eq_nat @ Mi2 @ Ma )
% 0.29/0.59          & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % mi_ma_2_deg
% 0.29/0.59  thf(fact_7288_sgn__greater,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
% 0.29/0.59        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_greater
% 0.29/0.59  thf(fact_7289_sgn__greater,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 0.29/0.59        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_greater
% 0.29/0.59  thf(fact_7290_sgn__greater,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
% 0.29/0.59        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_greater
% 0.29/0.59  thf(fact_7291_sgn__greater,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
% 0.29/0.59        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_greater
% 0.29/0.59  thf(fact_7292_sgn__less,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.29/0.59        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_less
% 0.29/0.59  thf(fact_7293_sgn__less,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 0.29/0.59        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_less
% 0.29/0.59  thf(fact_7294_sgn__less,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
% 0.29/0.59        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_less
% 0.29/0.59  thf(fact_7295_sgn__less,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
% 0.29/0.59        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_less
% 0.29/0.59  thf(fact_7296_divide__sgn,axiom,
% 0.29/0.59      ! [A: real,B: real] :
% 0.29/0.59        ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B ) )
% 0.29/0.59        = ( times_times_real @ A @ ( sgn_sgn_real @ B ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divide_sgn
% 0.29/0.59  thf(fact_7297_divide__sgn,axiom,
% 0.29/0.59      ! [A: rat,B: rat] :
% 0.29/0.59        ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
% 0.29/0.59        = ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divide_sgn
% 0.29/0.59  thf(fact_7298_nat__numeral,axiom,
% 0.29/0.59      ! [K: num] :
% 0.29/0.59        ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.29/0.59        = ( numeral_numeral_nat @ K ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_numeral
% 0.29/0.59  thf(fact_7299_pred__numeral__inc,axiom,
% 0.29/0.59      ! [K: num] :
% 0.29/0.59        ( ( pred_numeral @ ( inc @ K ) )
% 0.29/0.59        = ( numeral_numeral_nat @ K ) ) ).
% 0.29/0.59  
% 0.29/0.59  % pred_numeral_inc
% 0.29/0.59  thf(fact_7300_both__member__options__from__complete__tree__to__child,axiom,
% 0.29/0.59      ! [Deg: nat,Mi2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.29/0.59        ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.29/0.59       => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.29/0.59         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.59            | ( X = Mi2 )
% 0.29/0.59            | ( X = Ma ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % both_member_options_from_complete_tree_to_child
% 0.29/0.59  thf(fact_7301_nat__of__bool,axiom,
% 0.29/0.59      ! [P: $o] :
% 0.29/0.59        ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.29/0.59        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_of_bool
% 0.29/0.59  thf(fact_7302_sgn__pos,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.59       => ( ( sgn_sgn_Code_integer @ A )
% 0.29/0.59          = one_one_Code_integer ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_pos
% 0.29/0.59  thf(fact_7303_sgn__pos,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.59       => ( ( sgn_sgn_real @ A )
% 0.29/0.59          = one_one_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_pos
% 0.29/0.59  thf(fact_7304_sgn__pos,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.59       => ( ( sgn_sgn_rat @ A )
% 0.29/0.59          = one_one_rat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_pos
% 0.29/0.59  thf(fact_7305_sgn__pos,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.59       => ( ( sgn_sgn_int @ A )
% 0.29/0.59          = one_one_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_pos
% 0.29/0.59  thf(fact_7306_abs__sgn__eq__1,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.59       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.29/0.59          = one_one_Code_integer ) ) ).
% 0.29/0.59  
% 0.29/0.59  % abs_sgn_eq_1
% 0.29/0.59  thf(fact_7307_abs__sgn__eq__1,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( A != zero_zero_real )
% 0.29/0.59       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.29/0.59          = one_one_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % abs_sgn_eq_1
% 0.29/0.59  thf(fact_7308_abs__sgn__eq__1,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( A != zero_zero_rat )
% 0.29/0.59       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.29/0.59          = one_one_rat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % abs_sgn_eq_1
% 0.29/0.59  thf(fact_7309_abs__sgn__eq__1,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( A != zero_zero_int )
% 0.29/0.59       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.29/0.59          = one_one_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % abs_sgn_eq_1
% 0.29/0.59  thf(fact_7310_nat__1,axiom,
% 0.29/0.59      ( ( nat2 @ one_one_int )
% 0.29/0.59      = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_1
% 0.29/0.59  thf(fact_7311_sgn__mult__self__eq,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
% 0.29/0.59        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_mult_self_eq
% 0.29/0.59  thf(fact_7312_sgn__mult__self__eq,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 0.29/0.59        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_mult_self_eq
% 0.29/0.59  thf(fact_7313_sgn__mult__self__eq,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
% 0.29/0.59        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_mult_self_eq
% 0.29/0.59  thf(fact_7314_sgn__mult__self__eq,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 0.29/0.59        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_mult_self_eq
% 0.29/0.59  thf(fact_7315_nat__le__0,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.29/0.59       => ( ( nat2 @ Z )
% 0.29/0.59          = zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_le_0
% 0.29/0.59  thf(fact_7316_nat__0__iff,axiom,
% 0.29/0.59      ! [I: int] :
% 0.29/0.59        ( ( ( nat2 @ I )
% 0.29/0.59          = zero_zero_nat )
% 0.29/0.59        = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_0_iff
% 0.29/0.59  thf(fact_7317_zless__nat__conj,axiom,
% 0.29/0.59      ! [W: int,Z: int] :
% 0.29/0.59        ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.29/0.59        = ( ( ord_less_int @ zero_zero_int @ Z )
% 0.29/0.59          & ( ord_less_int @ W @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % zless_nat_conj
% 0.29/0.59  thf(fact_7318_nat__neg__numeral,axiom,
% 0.29/0.59      ! [K: num] :
% 0.29/0.59        ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.59        = zero_zero_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_neg_numeral
% 0.29/0.59  thf(fact_7319_both__member__options__from__chilf__to__complete__tree,axiom,
% 0.29/0.59      ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi2: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.29/0.59        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.29/0.59       => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.29/0.59         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.59           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % both_member_options_from_chilf_to_complete_tree
% 0.29/0.59  thf(fact_7320_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.29/0.59      ! [A: complex] :
% 0.29/0.59        ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
% 0.29/0.59        = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % idom_abs_sgn_class.abs_sgn
% 0.29/0.59  thf(fact_7321_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
% 0.29/0.59        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % idom_abs_sgn_class.abs_sgn
% 0.29/0.59  thf(fact_7322_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
% 0.29/0.59        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % idom_abs_sgn_class.abs_sgn
% 0.29/0.59  thf(fact_7323_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
% 0.29/0.59        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % idom_abs_sgn_class.abs_sgn
% 0.29/0.59  thf(fact_7324_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.29/0.59        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % idom_abs_sgn_class.abs_sgn
% 0.29/0.59  thf(fact_7325_sgn__abs,axiom,
% 0.29/0.59      ! [A: complex] :
% 0.29/0.59        ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
% 0.29/0.59        = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_abs
% 0.29/0.59  thf(fact_7326_sgn__abs,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.29/0.59        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_abs
% 0.29/0.59  thf(fact_7327_sgn__abs,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.29/0.59        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_abs
% 0.29/0.59  thf(fact_7328_sgn__abs,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.29/0.59        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_abs
% 0.29/0.59  thf(fact_7329_sgn__abs,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.29/0.59        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_abs
% 0.29/0.59  thf(fact_7330_nat__zminus__int,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.29/0.59        = zero_zero_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_zminus_int
% 0.29/0.59  thf(fact_7331_int__nat__eq,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.29/0.59            = Z ) )
% 0.29/0.59        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.29/0.59            = zero_zero_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % int_nat_eq
% 0.29/0.59  thf(fact_7332_of__nat__nat__take__bit__eq,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( semiri4939895301339042750nteger @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.29/0.59        = ( ring_18347121197199848620nteger @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_nat_take_bit_eq
% 0.29/0.59  thf(fact_7333_of__nat__nat__take__bit__eq,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.29/0.59        = ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_nat_take_bit_eq
% 0.29/0.59  thf(fact_7334_of__nat__nat__take__bit__eq,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.29/0.59        = ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_nat_take_bit_eq
% 0.29/0.59  thf(fact_7335_of__nat__nat__take__bit__eq,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.29/0.59        = ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_nat_take_bit_eq
% 0.29/0.59  thf(fact_7336_of__nat__nat__take__bit__eq,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.29/0.59        = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_nat_take_bit_eq
% 0.29/0.59  thf(fact_7337_dvd__mult__sgn__iff,axiom,
% 0.29/0.59      ! [L: int,K: int,R2: int] :
% 0.29/0.59        ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
% 0.29/0.59        = ( ( dvd_dvd_int @ L @ K )
% 0.29/0.59          | ( R2 = zero_zero_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % dvd_mult_sgn_iff
% 0.29/0.59  thf(fact_7338_dvd__sgn__mult__iff,axiom,
% 0.29/0.59      ! [L: int,R2: int,K: int] :
% 0.29/0.59        ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
% 0.29/0.59        = ( ( dvd_dvd_int @ L @ K )
% 0.29/0.59          | ( R2 = zero_zero_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % dvd_sgn_mult_iff
% 0.29/0.59  thf(fact_7339_mult__sgn__dvd__iff,axiom,
% 0.29/0.59      ! [L: int,R2: int,K: int] :
% 0.29/0.59        ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
% 0.29/0.59        = ( ( dvd_dvd_int @ L @ K )
% 0.29/0.59          & ( ( R2 = zero_zero_int )
% 0.29/0.59           => ( K = zero_zero_int ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % mult_sgn_dvd_iff
% 0.29/0.59  thf(fact_7340_sgn__mult__dvd__iff,axiom,
% 0.29/0.59      ! [R2: int,L: int,K: int] :
% 0.29/0.59        ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
% 0.29/0.59        = ( ( dvd_dvd_int @ L @ K )
% 0.29/0.59          & ( ( R2 = zero_zero_int )
% 0.29/0.59           => ( K = zero_zero_int ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_mult_dvd_iff
% 0.29/0.59  thf(fact_7341_sgn__neg,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.59       => ( ( sgn_sgn_real @ A )
% 0.29/0.59          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_neg
% 0.29/0.59  thf(fact_7342_sgn__neg,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.59       => ( ( sgn_sgn_int @ A )
% 0.29/0.59          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_neg
% 0.29/0.59  thf(fact_7343_sgn__neg,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.59       => ( ( sgn_sgn_rat @ A )
% 0.29/0.59          = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_neg
% 0.29/0.59  thf(fact_7344_sgn__neg,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.29/0.59       => ( ( sgn_sgn_Code_integer @ A )
% 0.29/0.59          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_neg
% 0.29/0.59  thf(fact_7345_zero__less__nat__eq,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 0.29/0.59        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.29/0.59  
% 0.29/0.59  % zero_less_nat_eq
% 0.29/0.59  thf(fact_7346_add__neg__numeral__special_I6_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_neg_numeral_special(6)
% 0.29/0.59  thf(fact_7347_add__neg__numeral__special_I6_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_neg_numeral_special(6)
% 0.29/0.59  thf(fact_7348_add__neg__numeral__special_I6_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.59        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_neg_numeral_special(6)
% 0.29/0.59  thf(fact_7349_add__neg__numeral__special_I6_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.59        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_neg_numeral_special(6)
% 0.29/0.59  thf(fact_7350_add__neg__numeral__special_I6_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.59        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_neg_numeral_special(6)
% 0.29/0.59  thf(fact_7351_add__neg__numeral__special_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_neg_numeral_special(5)
% 0.29/0.59  thf(fact_7352_add__neg__numeral__special_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.59        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_neg_numeral_special(5)
% 0.29/0.59  thf(fact_7353_add__neg__numeral__special_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.29/0.59        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_neg_numeral_special(5)
% 0.29/0.59  thf(fact_7354_add__neg__numeral__special_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.29/0.59        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_neg_numeral_special(5)
% 0.29/0.59  thf(fact_7355_add__neg__numeral__special_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.59        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_neg_numeral_special(5)
% 0.29/0.59  thf(fact_7356_diff__numeral__special_I6_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.59        = ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_numeral_special(6)
% 0.29/0.59  thf(fact_7357_diff__numeral__special_I6_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59        = ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_numeral_special(6)
% 0.29/0.59  thf(fact_7358_diff__numeral__special_I6_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.59        = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_numeral_special(6)
% 0.29/0.59  thf(fact_7359_diff__numeral__special_I6_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.59        = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_numeral_special(6)
% 0.29/0.59  thf(fact_7360_diff__numeral__special_I6_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.59        = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_numeral_special(6)
% 0.29/0.59  thf(fact_7361_diff__numeral__special_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_numeral_special(5)
% 0.29/0.59  thf(fact_7362_diff__numeral__special_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.59        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_numeral_special(5)
% 0.29/0.59  thf(fact_7363_diff__numeral__special_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 0.29/0.59        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_numeral_special(5)
% 0.29/0.59  thf(fact_7364_diff__numeral__special_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 0.29/0.59        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_numeral_special(5)
% 0.29/0.59  thf(fact_7365_diff__numeral__special_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 0.29/0.59        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_numeral_special(5)
% 0.29/0.59  thf(fact_7366_of__nat__nat,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( semiri4939895301339042750nteger @ ( nat2 @ Z ) )
% 0.29/0.59          = ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_nat
% 0.29/0.59  thf(fact_7367_of__nat__nat,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( semiri8010041392384452111omplex @ ( nat2 @ Z ) )
% 0.29/0.59          = ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_nat
% 0.29/0.59  thf(fact_7368_of__nat__nat,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
% 0.29/0.59          = ( ring_1_of_int_real @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_nat
% 0.29/0.59  thf(fact_7369_of__nat__nat,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
% 0.29/0.59          = ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_nat
% 0.29/0.59  thf(fact_7370_of__nat__nat,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.29/0.59          = ( ring_1_of_int_int @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_nat
% 0.29/0.59  thf(fact_7371_sgn__of__nat,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.29/0.59        = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_of_nat
% 0.29/0.59  thf(fact_7372_sgn__of__nat,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.29/0.59        = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_of_nat
% 0.29/0.59  thf(fact_7373_sgn__of__nat,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.29/0.59        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_of_nat
% 0.29/0.59  thf(fact_7374_sgn__of__nat,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.29/0.59        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_of_nat
% 0.29/0.59  thf(fact_7375_diff__nat__numeral,axiom,
% 0.29/0.59      ! [V: num,V2: num] :
% 0.29/0.59        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
% 0.29/0.59        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % diff_nat_numeral
% 0.29/0.59  thf(fact_7376_nat__eq__numeral__power__cancel__iff,axiom,
% 0.29/0.59      ! [Y: int,X: num,N: nat] :
% 0.29/0.59        ( ( ( nat2 @ Y )
% 0.29/0.59          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.29/0.59        = ( Y
% 0.29/0.59          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_eq_numeral_power_cancel_iff
% 0.29/0.59  thf(fact_7377_numeral__power__eq__nat__cancel__iff,axiom,
% 0.29/0.59      ! [X: num,N: nat,Y: int] :
% 0.29/0.59        ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.29/0.59          = ( nat2 @ Y ) )
% 0.29/0.59        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.29/0.59          = Y ) ) ).
% 0.29/0.59  
% 0.29/0.59  % numeral_power_eq_nat_cancel_iff
% 0.29/0.59  thf(fact_7378_nat__abs__dvd__iff,axiom,
% 0.29/0.59      ! [K: int,N: nat] :
% 0.29/0.59        ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.29/0.59        = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_abs_dvd_iff
% 0.29/0.59  thf(fact_7379_dvd__nat__abs__iff,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.29/0.59        = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 0.29/0.59  
% 0.29/0.59  % dvd_nat_abs_iff
% 0.29/0.59  thf(fact_7380_nat__ceiling__le__eq,axiom,
% 0.29/0.59      ! [X: real,A: nat] :
% 0.29/0.59        ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 0.29/0.59        = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_ceiling_le_eq
% 0.29/0.59  thf(fact_7381_one__less__nat__eq,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 0.29/0.59        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.29/0.59  
% 0.29/0.59  % one_less_nat_eq
% 0.29/0.59  thf(fact_7382_nat__numeral__diff__1,axiom,
% 0.29/0.59      ! [V: num] :
% 0.29/0.59        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 0.29/0.59        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_numeral_diff_1
% 0.29/0.59  thf(fact_7383_numeral__power__less__nat__cancel__iff,axiom,
% 0.29/0.59      ! [X: num,N: nat,A: int] :
% 0.29/0.59        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.29/0.59        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % numeral_power_less_nat_cancel_iff
% 0.29/0.59  thf(fact_7384_nat__less__numeral__power__cancel__iff,axiom,
% 0.29/0.59      ! [A: int,X: num,N: nat] :
% 0.29/0.59        ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.29/0.59        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_less_numeral_power_cancel_iff
% 0.29/0.59  thf(fact_7385_numeral__power__le__nat__cancel__iff,axiom,
% 0.29/0.59      ! [X: num,N: nat,A: int] :
% 0.29/0.59        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.29/0.59        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % numeral_power_le_nat_cancel_iff
% 0.29/0.59  thf(fact_7386_nat__le__numeral__power__cancel__iff,axiom,
% 0.29/0.59      ! [A: int,X: num,N: nat] :
% 0.29/0.59        ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.29/0.59        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_le_numeral_power_cancel_iff
% 0.29/0.59  thf(fact_7387_sgn__zero__iff,axiom,
% 0.29/0.59      ! [X: complex] :
% 0.29/0.59        ( ( ( sgn_sgn_complex @ X )
% 0.29/0.59          = zero_zero_complex )
% 0.29/0.59        = ( X = zero_zero_complex ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_zero_iff
% 0.29/0.59  thf(fact_7388_sgn__zero__iff,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ( sgn_sgn_real @ X )
% 0.29/0.59          = zero_zero_real )
% 0.29/0.59        = ( X = zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_zero_iff
% 0.29/0.59  thf(fact_7389_sgn__eq__0__iff,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( ( sgn_sgn_Code_integer @ A )
% 0.29/0.59          = zero_z3403309356797280102nteger )
% 0.29/0.59        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_eq_0_iff
% 0.29/0.59  thf(fact_7390_sgn__eq__0__iff,axiom,
% 0.29/0.59      ! [A: complex] :
% 0.29/0.59        ( ( ( sgn_sgn_complex @ A )
% 0.29/0.59          = zero_zero_complex )
% 0.29/0.59        = ( A = zero_zero_complex ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_eq_0_iff
% 0.29/0.59  thf(fact_7391_sgn__eq__0__iff,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( ( sgn_sgn_real @ A )
% 0.29/0.59          = zero_zero_real )
% 0.29/0.59        = ( A = zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_eq_0_iff
% 0.29/0.59  thf(fact_7392_sgn__eq__0__iff,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( ( sgn_sgn_rat @ A )
% 0.29/0.59          = zero_zero_rat )
% 0.29/0.59        = ( A = zero_zero_rat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_eq_0_iff
% 0.29/0.59  thf(fact_7393_sgn__eq__0__iff,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( ( sgn_sgn_int @ A )
% 0.29/0.59          = zero_zero_int )
% 0.29/0.59        = ( A = zero_zero_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_eq_0_iff
% 0.29/0.59  thf(fact_7394_sgn__0__0,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( ( sgn_sgn_Code_integer @ A )
% 0.29/0.59          = zero_z3403309356797280102nteger )
% 0.29/0.59        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_0_0
% 0.29/0.59  thf(fact_7395_sgn__0__0,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( ( sgn_sgn_real @ A )
% 0.29/0.59          = zero_zero_real )
% 0.29/0.59        = ( A = zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_0_0
% 0.29/0.59  thf(fact_7396_sgn__0__0,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( ( sgn_sgn_rat @ A )
% 0.29/0.59          = zero_zero_rat )
% 0.29/0.59        = ( A = zero_zero_rat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_0_0
% 0.29/0.59  thf(fact_7397_sgn__0__0,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( ( sgn_sgn_int @ A )
% 0.29/0.59          = zero_zero_int )
% 0.29/0.59        = ( A = zero_zero_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_0_0
% 0.29/0.59  thf(fact_7398_same__sgn__sgn__add,axiom,
% 0.29/0.59      ! [B: code_integer,A: code_integer] :
% 0.29/0.59        ( ( ( sgn_sgn_Code_integer @ B )
% 0.29/0.59          = ( sgn_sgn_Code_integer @ A ) )
% 0.29/0.59       => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.29/0.59          = ( sgn_sgn_Code_integer @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % same_sgn_sgn_add
% 0.29/0.59  thf(fact_7399_same__sgn__sgn__add,axiom,
% 0.29/0.59      ! [B: real,A: real] :
% 0.29/0.59        ( ( ( sgn_sgn_real @ B )
% 0.29/0.59          = ( sgn_sgn_real @ A ) )
% 0.29/0.59       => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
% 0.29/0.59          = ( sgn_sgn_real @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % same_sgn_sgn_add
% 0.29/0.59  thf(fact_7400_same__sgn__sgn__add,axiom,
% 0.29/0.59      ! [B: rat,A: rat] :
% 0.29/0.59        ( ( ( sgn_sgn_rat @ B )
% 0.29/0.59          = ( sgn_sgn_rat @ A ) )
% 0.29/0.59       => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.59          = ( sgn_sgn_rat @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % same_sgn_sgn_add
% 0.29/0.59  thf(fact_7401_same__sgn__sgn__add,axiom,
% 0.29/0.59      ! [B: int,A: int] :
% 0.29/0.59        ( ( ( sgn_sgn_int @ B )
% 0.29/0.59          = ( sgn_sgn_int @ A ) )
% 0.29/0.59       => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
% 0.29/0.59          = ( sgn_sgn_int @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % same_sgn_sgn_add
% 0.29/0.59  thf(fact_7402_combine__options__cases,axiom,
% 0.29/0.59      ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 0.29/0.59        ( ( ( X = none_P5556105721700978146at_nat )
% 0.29/0.59         => ( P @ X @ Y ) )
% 0.29/0.59       => ( ( ( Y = none_P5556105721700978146at_nat )
% 0.29/0.59           => ( P @ X @ Y ) )
% 0.29/0.59         => ( ! [A5: product_prod_nat_nat,B6: product_prod_nat_nat] :
% 0.29/0.59                ( ( X
% 0.29/0.59                  = ( some_P7363390416028606310at_nat @ A5 ) )
% 0.29/0.59               => ( ( Y
% 0.29/0.59                    = ( some_P7363390416028606310at_nat @ B6 ) )
% 0.29/0.59                 => ( P @ X @ Y ) ) )
% 0.29/0.59           => ( P @ X @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % combine_options_cases
% 0.29/0.59  thf(fact_7403_combine__options__cases,axiom,
% 0.29/0.59      ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
% 0.29/0.59        ( ( ( X = none_P5556105721700978146at_nat )
% 0.29/0.59         => ( P @ X @ Y ) )
% 0.29/0.59       => ( ( ( Y = none_num )
% 0.29/0.59           => ( P @ X @ Y ) )
% 0.29/0.59         => ( ! [A5: product_prod_nat_nat,B6: num] :
% 0.29/0.59                ( ( X
% 0.29/0.59                  = ( some_P7363390416028606310at_nat @ A5 ) )
% 0.29/0.59               => ( ( Y
% 0.29/0.59                    = ( some_num @ B6 ) )
% 0.29/0.59                 => ( P @ X @ Y ) ) )
% 0.29/0.59           => ( P @ X @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % combine_options_cases
% 0.29/0.59  thf(fact_7404_combine__options__cases,axiom,
% 0.29/0.59      ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 0.29/0.59        ( ( ( X = none_num )
% 0.29/0.59         => ( P @ X @ Y ) )
% 0.29/0.59       => ( ( ( Y = none_P5556105721700978146at_nat )
% 0.29/0.59           => ( P @ X @ Y ) )
% 0.29/0.59         => ( ! [A5: num,B6: product_prod_nat_nat] :
% 0.29/0.59                ( ( X
% 0.29/0.59                  = ( some_num @ A5 ) )
% 0.29/0.59               => ( ( Y
% 0.29/0.59                    = ( some_P7363390416028606310at_nat @ B6 ) )
% 0.29/0.59                 => ( P @ X @ Y ) ) )
% 0.29/0.59           => ( P @ X @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % combine_options_cases
% 0.29/0.59  thf(fact_7405_combine__options__cases,axiom,
% 0.29/0.59      ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
% 0.29/0.59        ( ( ( X = none_num )
% 0.29/0.59         => ( P @ X @ Y ) )
% 0.29/0.59       => ( ( ( Y = none_num )
% 0.29/0.59           => ( P @ X @ Y ) )
% 0.29/0.59         => ( ! [A5: num,B6: num] :
% 0.29/0.59                ( ( X
% 0.29/0.59                  = ( some_num @ A5 ) )
% 0.29/0.59               => ( ( Y
% 0.29/0.59                    = ( some_num @ B6 ) )
% 0.29/0.59                 => ( P @ X @ Y ) ) )
% 0.29/0.59           => ( P @ X @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % combine_options_cases
% 0.29/0.59  thf(fact_7406_split__option__all,axiom,
% 0.29/0.59      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.29/0.59          ! [X5: option4927543243414619207at_nat] : ( P2 @ X5 ) )
% 0.29/0.59      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.29/0.59            ( ( P3 @ none_P5556105721700978146at_nat )
% 0.29/0.59            & ! [X4: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X4 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % split_option_all
% 0.29/0.59  thf(fact_7407_split__option__all,axiom,
% 0.29/0.59      ( ( ^ [P2: option_num > $o] :
% 0.29/0.59          ! [X5: option_num] : ( P2 @ X5 ) )
% 0.29/0.59      = ( ^ [P3: option_num > $o] :
% 0.29/0.59            ( ( P3 @ none_num )
% 0.29/0.59            & ! [X4: num] : ( P3 @ ( some_num @ X4 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % split_option_all
% 0.29/0.59  thf(fact_7408_split__option__ex,axiom,
% 0.29/0.59      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.29/0.59          ? [X5: option4927543243414619207at_nat] : ( P2 @ X5 ) )
% 0.29/0.59      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.29/0.59            ( ( P3 @ none_P5556105721700978146at_nat )
% 0.29/0.59            | ? [X4: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X4 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % split_option_ex
% 0.29/0.59  thf(fact_7409_split__option__ex,axiom,
% 0.29/0.59      ( ( ^ [P2: option_num > $o] :
% 0.29/0.59          ? [X5: option_num] : ( P2 @ X5 ) )
% 0.29/0.59      = ( ^ [P3: option_num > $o] :
% 0.29/0.59            ( ( P3 @ none_num )
% 0.29/0.59            | ? [X4: num] : ( P3 @ ( some_num @ X4 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % split_option_ex
% 0.29/0.59  thf(fact_7410_option_Oexhaust,axiom,
% 0.29/0.59      ! [Y: option4927543243414619207at_nat] :
% 0.29/0.59        ( ( Y != none_P5556105721700978146at_nat )
% 0.29/0.59       => ~ ! [X23: product_prod_nat_nat] :
% 0.29/0.59              ( Y
% 0.29/0.59             != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.exhaust
% 0.29/0.59  thf(fact_7411_option_Oexhaust,axiom,
% 0.29/0.59      ! [Y: option_num] :
% 0.29/0.59        ( ( Y != none_num )
% 0.29/0.59       => ~ ! [X23: num] :
% 0.29/0.59              ( Y
% 0.29/0.59             != ( some_num @ X23 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.exhaust
% 0.29/0.59  thf(fact_7412_option_OdiscI,axiom,
% 0.29/0.59      ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 0.29/0.59        ( ( Option
% 0.29/0.59          = ( some_P7363390416028606310at_nat @ X22 ) )
% 0.29/0.59       => ( Option != none_P5556105721700978146at_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.discI
% 0.29/0.59  thf(fact_7413_option_OdiscI,axiom,
% 0.29/0.59      ! [Option: option_num,X22: num] :
% 0.29/0.59        ( ( Option
% 0.29/0.59          = ( some_num @ X22 ) )
% 0.29/0.59       => ( Option != none_num ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.discI
% 0.29/0.59  thf(fact_7414_option_Odistinct_I1_J,axiom,
% 0.29/0.59      ! [X22: product_prod_nat_nat] :
% 0.29/0.59        ( none_P5556105721700978146at_nat
% 0.29/0.59       != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.distinct(1)
% 0.29/0.59  thf(fact_7415_option_Odistinct_I1_J,axiom,
% 0.29/0.59      ! [X22: num] :
% 0.29/0.59        ( none_num
% 0.29/0.59       != ( some_num @ X22 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.distinct(1)
% 0.29/0.59  thf(fact_7416_None__notin__image__Some,axiom,
% 0.29/0.59      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.29/0.59        ~ ( member3954567711264315760at_nat @ none_P5556105721700978146at_nat @ ( image_4198897800814241419at_nat @ some_P7363390416028606310at_nat @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % None_notin_image_Some
% 0.29/0.59  thf(fact_7417_None__notin__image__Some,axiom,
% 0.29/0.59      ! [A2: set_num] :
% 0.29/0.59        ~ ( member_option_num @ none_num @ ( image_num_option_num @ some_num @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % None_notin_image_Some
% 0.29/0.59  thf(fact_7418_num__induct,axiom,
% 0.29/0.59      ! [P: num > $o,X: num] :
% 0.29/0.59        ( ( P @ one )
% 0.29/0.59       => ( ! [X3: num] :
% 0.29/0.59              ( ( P @ X3 )
% 0.29/0.59             => ( P @ ( inc @ X3 ) ) )
% 0.29/0.59         => ( P @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % num_induct
% 0.29/0.59  thf(fact_7419_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 0.29/0.59      ! [Mi2: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
% 0.29/0.59        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X )
% 0.29/0.59        = ( ( X = Mi2 )
% 0.29/0.59          | ( X = Ma ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % VEBT_internal.membermima.simps(3)
% 0.29/0.59  thf(fact_7420_add__inc,axiom,
% 0.29/0.59      ! [X: num,Y: num] :
% 0.29/0.59        ( ( plus_plus_num @ X @ ( inc @ Y ) )
% 0.29/0.59        = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_inc
% 0.29/0.59  thf(fact_7421_sgn__not__eq__imp,axiom,
% 0.29/0.59      ! [B: real,A: real] :
% 0.29/0.59        ( ( ( sgn_sgn_real @ B )
% 0.29/0.59         != ( sgn_sgn_real @ A ) )
% 0.29/0.59       => ( ( ( sgn_sgn_real @ A )
% 0.29/0.59           != zero_zero_real )
% 0.29/0.59         => ( ( ( sgn_sgn_real @ B )
% 0.29/0.59             != zero_zero_real )
% 0.29/0.59           => ( ( sgn_sgn_real @ A )
% 0.29/0.59              = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_not_eq_imp
% 0.29/0.59  thf(fact_7422_sgn__not__eq__imp,axiom,
% 0.29/0.59      ! [B: int,A: int] :
% 0.29/0.59        ( ( ( sgn_sgn_int @ B )
% 0.29/0.59         != ( sgn_sgn_int @ A ) )
% 0.29/0.59       => ( ( ( sgn_sgn_int @ A )
% 0.29/0.59           != zero_zero_int )
% 0.29/0.59         => ( ( ( sgn_sgn_int @ B )
% 0.29/0.59             != zero_zero_int )
% 0.29/0.59           => ( ( sgn_sgn_int @ A )
% 0.29/0.59              = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_not_eq_imp
% 0.29/0.59  thf(fact_7423_sgn__not__eq__imp,axiom,
% 0.29/0.59      ! [B: rat,A: rat] :
% 0.29/0.59        ( ( ( sgn_sgn_rat @ B )
% 0.29/0.59         != ( sgn_sgn_rat @ A ) )
% 0.29/0.59       => ( ( ( sgn_sgn_rat @ A )
% 0.29/0.59           != zero_zero_rat )
% 0.29/0.59         => ( ( ( sgn_sgn_rat @ B )
% 0.29/0.59             != zero_zero_rat )
% 0.29/0.59           => ( ( sgn_sgn_rat @ A )
% 0.29/0.59              = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_not_eq_imp
% 0.29/0.59  thf(fact_7424_sgn__not__eq__imp,axiom,
% 0.29/0.59      ! [B: code_integer,A: code_integer] :
% 0.29/0.59        ( ( ( sgn_sgn_Code_integer @ B )
% 0.29/0.59         != ( sgn_sgn_Code_integer @ A ) )
% 0.29/0.59       => ( ( ( sgn_sgn_Code_integer @ A )
% 0.29/0.59           != zero_z3403309356797280102nteger )
% 0.29/0.59         => ( ( ( sgn_sgn_Code_integer @ B )
% 0.29/0.59             != zero_z3403309356797280102nteger )
% 0.29/0.59           => ( ( sgn_sgn_Code_integer @ A )
% 0.29/0.59              = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_not_eq_imp
% 0.29/0.59  thf(fact_7425_sgn__minus__1,axiom,
% 0.29/0.59      ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.59      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_minus_1
% 0.29/0.59  thf(fact_7426_sgn__minus__1,axiom,
% 0.29/0.59      ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_minus_1
% 0.29/0.59  thf(fact_7427_sgn__minus__1,axiom,
% 0.29/0.59      ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.59      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_minus_1
% 0.29/0.59  thf(fact_7428_sgn__minus__1,axiom,
% 0.29/0.59      ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.59      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_minus_1
% 0.29/0.59  thf(fact_7429_sgn__minus__1,axiom,
% 0.29/0.59      ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.59      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_minus_1
% 0.29/0.59  thf(fact_7430_nat__zero__as__int,axiom,
% 0.29/0.59      ( zero_zero_nat
% 0.29/0.59      = ( nat2 @ zero_zero_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_zero_as_int
% 0.29/0.59  thf(fact_7431_nat__numeral__as__int,axiom,
% 0.29/0.59      ( numeral_numeral_nat
% 0.29/0.59      = ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_numeral_as_int
% 0.29/0.59  thf(fact_7432_int__sgnE,axiom,
% 0.29/0.59      ! [K: int] :
% 0.29/0.59        ~ ! [N2: nat,L2: int] :
% 0.29/0.59            ( K
% 0.29/0.59           != ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % int_sgnE
% 0.29/0.59  thf(fact_7433_nat__mono,axiom,
% 0.29/0.59      ! [X: int,Y: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ X @ Y )
% 0.29/0.59       => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_mono
% 0.29/0.59  thf(fact_7434_same__sgn__abs__add,axiom,
% 0.29/0.59      ! [B: code_integer,A: code_integer] :
% 0.29/0.59        ( ( ( sgn_sgn_Code_integer @ B )
% 0.29/0.59          = ( sgn_sgn_Code_integer @ A ) )
% 0.29/0.59       => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.29/0.59          = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % same_sgn_abs_add
% 0.29/0.59  thf(fact_7435_same__sgn__abs__add,axiom,
% 0.29/0.59      ! [B: real,A: real] :
% 0.29/0.59        ( ( ( sgn_sgn_real @ B )
% 0.29/0.59          = ( sgn_sgn_real @ A ) )
% 0.29/0.59       => ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
% 0.29/0.59          = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % same_sgn_abs_add
% 0.29/0.59  thf(fact_7436_same__sgn__abs__add,axiom,
% 0.29/0.59      ! [B: rat,A: rat] :
% 0.29/0.59        ( ( ( sgn_sgn_rat @ B )
% 0.29/0.59          = ( sgn_sgn_rat @ A ) )
% 0.29/0.59       => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.59          = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % same_sgn_abs_add
% 0.29/0.59  thf(fact_7437_same__sgn__abs__add,axiom,
% 0.29/0.59      ! [B: int,A: int] :
% 0.29/0.59        ( ( ( sgn_sgn_int @ B )
% 0.29/0.59          = ( sgn_sgn_int @ A ) )
% 0.29/0.59       => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
% 0.29/0.59          = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % same_sgn_abs_add
% 0.29/0.59  thf(fact_7438_ex__nat,axiom,
% 0.29/0.59      ( ( ^ [P2: nat > $o] :
% 0.29/0.59          ? [X5: nat] : ( P2 @ X5 ) )
% 0.29/0.59      = ( ^ [P3: nat > $o] :
% 0.29/0.59          ? [X4: int] :
% 0.29/0.59            ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.29/0.59            & ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % ex_nat
% 0.29/0.59  thf(fact_7439_all__nat,axiom,
% 0.29/0.59      ( ( ^ [P2: nat > $o] :
% 0.29/0.59          ! [X5: nat] : ( P2 @ X5 ) )
% 0.29/0.59      = ( ^ [P3: nat > $o] :
% 0.29/0.59          ! [X4: int] :
% 0.29/0.59            ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.29/0.59           => ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % all_nat
% 0.29/0.59  thf(fact_7440_eq__nat__nat__iff,axiom,
% 0.29/0.59      ! [Z: int,Z6: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.29/0.59         => ( ( ( nat2 @ Z )
% 0.29/0.59              = ( nat2 @ Z6 ) )
% 0.29/0.59            = ( Z = Z6 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % eq_nat_nat_iff
% 0.29/0.59  thf(fact_7441_take__bit__nat__eq,axiom,
% 0.29/0.59      ! [K: int,N: nat] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.59       => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 0.29/0.59          = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_nat_eq
% 0.29/0.59  thf(fact_7442_nat__take__bit__eq,axiom,
% 0.29/0.59      ! [K: int,N: nat] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.59       => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.29/0.59          = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_take_bit_eq
% 0.29/0.59  thf(fact_7443_nat__one__as__int,axiom,
% 0.29/0.59      ( one_one_nat
% 0.29/0.59      = ( nat2 @ one_one_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_one_as_int
% 0.29/0.59  thf(fact_7444_div__eq__sgn__abs,axiom,
% 0.29/0.59      ! [K: int,L: int] :
% 0.29/0.59        ( ( ( sgn_sgn_int @ K )
% 0.29/0.59          = ( sgn_sgn_int @ L ) )
% 0.29/0.59       => ( ( divide_divide_int @ K @ L )
% 0.29/0.59          = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % div_eq_sgn_abs
% 0.29/0.59  thf(fact_7445_unset__bit__nat__def,axiom,
% 0.29/0.59      ( bit_se4205575877204974255it_nat
% 0.29/0.59      = ( ^ [M6: nat,N3: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M6 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % unset_bit_nat_def
% 0.29/0.59  thf(fact_7446_inc_Osimps_I1_J,axiom,
% 0.29/0.59      ( ( inc @ one )
% 0.29/0.59      = ( bit0 @ one ) ) ).
% 0.29/0.59  
% 0.29/0.59  % inc.simps(1)
% 0.29/0.59  thf(fact_7447_inc_Osimps_I3_J,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( inc @ ( bit1 @ X ) )
% 0.29/0.59        = ( bit0 @ ( inc @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % inc.simps(3)
% 0.29/0.59  thf(fact_7448_inc_Osimps_I2_J,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( inc @ ( bit0 @ X ) )
% 0.29/0.59        = ( bit1 @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % inc.simps(2)
% 0.29/0.59  thf(fact_7449_sgn__1__pos,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( ( sgn_sgn_Code_integer @ A )
% 0.29/0.59          = one_one_Code_integer )
% 0.29/0.59        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1_pos
% 0.29/0.59  thf(fact_7450_sgn__1__pos,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( ( sgn_sgn_real @ A )
% 0.29/0.59          = one_one_real )
% 0.29/0.59        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1_pos
% 0.29/0.59  thf(fact_7451_sgn__1__pos,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( ( sgn_sgn_rat @ A )
% 0.29/0.59          = one_one_rat )
% 0.29/0.59        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1_pos
% 0.29/0.59  thf(fact_7452_sgn__1__pos,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( ( sgn_sgn_int @ A )
% 0.29/0.59          = one_one_int )
% 0.29/0.59        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1_pos
% 0.29/0.59  thf(fact_7453_add__One,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( plus_plus_num @ X @ one )
% 0.29/0.59        = ( inc @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % add_One
% 0.29/0.59  thf(fact_7454_abs__sgn__eq,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( ( A = zero_z3403309356797280102nteger )
% 0.29/0.59         => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.29/0.59            = zero_z3403309356797280102nteger ) )
% 0.29/0.59        & ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.59         => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.29/0.59            = one_one_Code_integer ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % abs_sgn_eq
% 0.29/0.59  thf(fact_7455_abs__sgn__eq,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( ( A = zero_zero_real )
% 0.29/0.59         => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.29/0.59            = zero_zero_real ) )
% 0.29/0.59        & ( ( A != zero_zero_real )
% 0.29/0.59         => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.29/0.59            = one_one_real ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % abs_sgn_eq
% 0.29/0.59  thf(fact_7456_abs__sgn__eq,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( ( A = zero_zero_rat )
% 0.29/0.59         => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.29/0.59            = zero_zero_rat ) )
% 0.29/0.59        & ( ( A != zero_zero_rat )
% 0.29/0.59         => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.29/0.59            = one_one_rat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % abs_sgn_eq
% 0.29/0.59  thf(fact_7457_abs__sgn__eq,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( ( A = zero_zero_int )
% 0.29/0.59         => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.29/0.59            = zero_zero_int ) )
% 0.29/0.59        & ( ( A != zero_zero_int )
% 0.29/0.59         => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.29/0.59            = one_one_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % abs_sgn_eq
% 0.29/0.59  thf(fact_7458_nat__mono__iff,axiom,
% 0.29/0.59      ! [Z: int,W: int] :
% 0.29/0.59        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.29/0.59          = ( ord_less_int @ W @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_mono_iff
% 0.29/0.59  thf(fact_7459_of__nat__ceiling,axiom,
% 0.29/0.59      ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_ceiling
% 0.29/0.59  thf(fact_7460_of__nat__ceiling,axiom,
% 0.29/0.59      ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_ceiling
% 0.29/0.59  thf(fact_7461_zless__nat__eq__int__zless,axiom,
% 0.29/0.59      ! [M: nat,Z: int] :
% 0.29/0.59        ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 0.29/0.59        = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 0.29/0.59  
% 0.29/0.59  % zless_nat_eq_int_zless
% 0.29/0.59  thf(fact_7462_nat__le__iff,axiom,
% 0.29/0.59      ! [X: int,N: nat] :
% 0.29/0.59        ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 0.29/0.59        = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_le_iff
% 0.29/0.59  thf(fact_7463_int__eq__iff,axiom,
% 0.29/0.59      ! [M: nat,Z: int] :
% 0.29/0.59        ( ( ( semiri1314217659103216013at_int @ M )
% 0.29/0.59          = Z )
% 0.29/0.59        = ( ( M
% 0.29/0.59            = ( nat2 @ Z ) )
% 0.29/0.59          & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % int_eq_iff
% 0.29/0.59  thf(fact_7464_nat__0__le,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.29/0.59          = Z ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_0_le
% 0.29/0.59  thf(fact_7465_nat__int__add,axiom,
% 0.29/0.59      ! [A: nat,B: nat] :
% 0.29/0.59        ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 0.29/0.59        = ( plus_plus_nat @ A @ B ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_int_add
% 0.29/0.59  thf(fact_7466_sgn__mod,axiom,
% 0.29/0.59      ! [L: int,K: int] :
% 0.29/0.59        ( ( L != zero_zero_int )
% 0.29/0.59       => ( ~ ( dvd_dvd_int @ L @ K )
% 0.29/0.59         => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
% 0.29/0.59            = ( sgn_sgn_int @ L ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_mod
% 0.29/0.59  thf(fact_7467_int__minus,axiom,
% 0.29/0.59      ! [N: nat,M: nat] :
% 0.29/0.59        ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 0.29/0.59        = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % int_minus
% 0.29/0.59  thf(fact_7468_nat__abs__mult__distrib,axiom,
% 0.29/0.59      ! [W: int,Z: int] :
% 0.29/0.59        ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 0.29/0.59        = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_abs_mult_distrib
% 0.29/0.59  thf(fact_7469_mult__inc,axiom,
% 0.29/0.59      ! [X: num,Y: num] :
% 0.29/0.59        ( ( times_times_num @ X @ ( inc @ Y ) )
% 0.29/0.59        = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % mult_inc
% 0.29/0.59  thf(fact_7470_real__nat__ceiling__ge,axiom,
% 0.29/0.59      ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_nat_ceiling_ge
% 0.29/0.59  thf(fact_7471_option_Osize_I4_J,axiom,
% 0.29/0.59      ! [X22: product_prod_nat_nat] :
% 0.29/0.59        ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.29/0.59        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.size(4)
% 0.29/0.59  thf(fact_7472_option_Osize_I4_J,axiom,
% 0.29/0.59      ! [X22: num] :
% 0.29/0.59        ( ( size_size_option_num @ ( some_num @ X22 ) )
% 0.29/0.59        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.size(4)
% 0.29/0.59  thf(fact_7473_sgn__if,axiom,
% 0.29/0.59      ( sgn_sgn_real
% 0.29/0.59      = ( ^ [X4: real] : ( if_real @ ( X4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_if
% 0.29/0.59  thf(fact_7474_sgn__if,axiom,
% 0.29/0.59      ( sgn_sgn_int
% 0.29/0.59      = ( ^ [X4: int] : ( if_int @ ( X4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_if
% 0.29/0.59  thf(fact_7475_sgn__if,axiom,
% 0.29/0.59      ( sgn_sgn_rat
% 0.29/0.59      = ( ^ [X4: rat] : ( if_rat @ ( X4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_if
% 0.29/0.59  thf(fact_7476_sgn__if,axiom,
% 0.29/0.59      ( sgn_sgn_Code_integer
% 0.29/0.59      = ( ^ [X4: code_integer] : ( if_Code_integer @ ( X4 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X4 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_if
% 0.29/0.59  thf(fact_7477_sgn__1__neg,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ( ( ( sgn_sgn_real @ A )
% 0.29/0.59          = ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.59        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1_neg
% 0.29/0.59  thf(fact_7478_sgn__1__neg,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( ( sgn_sgn_int @ A )
% 0.29/0.59          = ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1_neg
% 0.29/0.59  thf(fact_7479_sgn__1__neg,axiom,
% 0.29/0.59      ! [A: rat] :
% 0.29/0.59        ( ( ( sgn_sgn_rat @ A )
% 0.29/0.59          = ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.59        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1_neg
% 0.29/0.59  thf(fact_7480_sgn__1__neg,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( ( sgn_sgn_Code_integer @ A )
% 0.29/0.59          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.59        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_1_neg
% 0.29/0.59  thf(fact_7481_of__nat__floor,axiom,
% 0.29/0.59      ! [R2: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.29/0.59       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_floor
% 0.29/0.59  thf(fact_7482_of__nat__floor,axiom,
% 0.29/0.59      ! [R2: rat] :
% 0.29/0.59        ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.29/0.59       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_floor
% 0.29/0.59  thf(fact_7483_zsgn__def,axiom,
% 0.29/0.59      ( sgn_sgn_int
% 0.29/0.59      = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % zsgn_def
% 0.29/0.59  thf(fact_7484_numeral__inc,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( numera6690914467698888265omplex @ ( inc @ X ) )
% 0.29/0.59        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 0.29/0.59  
% 0.29/0.59  % numeral_inc
% 0.29/0.59  thf(fact_7485_numeral__inc,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( numeral_numeral_real @ ( inc @ X ) )
% 0.29/0.59        = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % numeral_inc
% 0.29/0.59  thf(fact_7486_numeral__inc,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( numeral_numeral_rat @ ( inc @ X ) )
% 0.29/0.59        = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % numeral_inc
% 0.29/0.59  thf(fact_7487_numeral__inc,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( numeral_numeral_nat @ ( inc @ X ) )
% 0.29/0.59        = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % numeral_inc
% 0.29/0.59  thf(fact_7488_numeral__inc,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( numeral_numeral_int @ ( inc @ X ) )
% 0.29/0.59        = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % numeral_inc
% 0.29/0.59  thf(fact_7489_nat__less__eq__zless,axiom,
% 0.29/0.59      ! [W: int,Z: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.29/0.59       => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.29/0.59          = ( ord_less_int @ W @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_less_eq_zless
% 0.29/0.59  thf(fact_7490_norm__sgn,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ( X = zero_zero_real )
% 0.29/0.59         => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 0.29/0.59            = zero_zero_real ) )
% 0.29/0.59        & ( ( X != zero_zero_real )
% 0.29/0.59         => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 0.29/0.59            = one_one_real ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % norm_sgn
% 0.29/0.59  thf(fact_7491_norm__sgn,axiom,
% 0.29/0.59      ! [X: complex] :
% 0.29/0.59        ( ( ( X = zero_zero_complex )
% 0.29/0.59         => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 0.29/0.59            = zero_zero_real ) )
% 0.29/0.59        & ( ( X != zero_zero_complex )
% 0.29/0.59         => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 0.29/0.59            = one_one_real ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % norm_sgn
% 0.29/0.59  thf(fact_7492_nat__le__eq__zle,axiom,
% 0.29/0.59      ! [W: int,Z: int] :
% 0.29/0.59        ( ( ( ord_less_int @ zero_zero_int @ W )
% 0.29/0.59          | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 0.29/0.59       => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.29/0.59          = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_le_eq_zle
% 0.29/0.59  thf(fact_7493_nat__eq__iff2,axiom,
% 0.29/0.59      ! [M: nat,W: int] :
% 0.29/0.59        ( ( M
% 0.29/0.59          = ( nat2 @ W ) )
% 0.29/0.59        = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.29/0.59           => ( W
% 0.29/0.59              = ( semiri1314217659103216013at_int @ M ) ) )
% 0.29/0.59          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.29/0.59           => ( M = zero_zero_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_eq_iff2
% 0.29/0.59  thf(fact_7494_nat__eq__iff,axiom,
% 0.29/0.59      ! [W: int,M: nat] :
% 0.29/0.59        ( ( ( nat2 @ W )
% 0.29/0.59          = M )
% 0.29/0.59        = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.29/0.59           => ( W
% 0.29/0.59              = ( semiri1314217659103216013at_int @ M ) ) )
% 0.29/0.59          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.29/0.59           => ( M = zero_zero_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_eq_iff
% 0.29/0.59  thf(fact_7495_split__nat,axiom,
% 0.29/0.59      ! [P: nat > $o,I: int] :
% 0.29/0.59        ( ( P @ ( nat2 @ I ) )
% 0.29/0.59        = ( ! [N3: nat] :
% 0.29/0.59              ( ( I
% 0.29/0.59                = ( semiri1314217659103216013at_int @ N3 ) )
% 0.29/0.59             => ( P @ N3 ) )
% 0.29/0.59          & ( ( ord_less_int @ I @ zero_zero_int )
% 0.29/0.59           => ( P @ zero_zero_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % split_nat
% 0.29/0.59  thf(fact_7496_le__mult__nat__floor,axiom,
% 0.29/0.59      ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % le_mult_nat_floor
% 0.29/0.59  thf(fact_7497_le__mult__nat__floor,axiom,
% 0.29/0.59      ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % le_mult_nat_floor
% 0.29/0.59  thf(fact_7498_le__nat__iff,axiom,
% 0.29/0.59      ! [K: int,N: nat] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.59       => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.29/0.59          = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % le_nat_iff
% 0.29/0.59  thf(fact_7499_nat__add__distrib,axiom,
% 0.29/0.59      ! [Z: int,Z6: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.29/0.59         => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
% 0.29/0.59            = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_add_distrib
% 0.29/0.59  thf(fact_7500_div__sgn__abs__cancel,axiom,
% 0.29/0.59      ! [V: int,K: int,L: int] :
% 0.29/0.59        ( ( V != zero_zero_int )
% 0.29/0.59       => ( ( 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 ) ) )
% 0.29/0.59          = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % div_sgn_abs_cancel
% 0.29/0.59  thf(fact_7501_nat__mult__distrib,axiom,
% 0.29/0.59      ! [Z: int,Z6: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 0.29/0.59          = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_mult_distrib
% 0.29/0.59  thf(fact_7502_Suc__as__int,axiom,
% 0.29/0.59      ( suc
% 0.29/0.59      = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Suc_as_int
% 0.29/0.59  thf(fact_7503_nat__diff__distrib_H,axiom,
% 0.29/0.59      ! [X: int,Y: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.59       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.59         => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
% 0.29/0.59            = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_diff_distrib'
% 0.29/0.59  thf(fact_7504_nat__diff__distrib,axiom,
% 0.29/0.59      ! [Z6: int,Z: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.29/0.59       => ( ( ord_less_eq_int @ Z6 @ Z )
% 0.29/0.59         => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
% 0.29/0.59            = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_diff_distrib
% 0.29/0.59  thf(fact_7505_nat__abs__triangle__ineq,axiom,
% 0.29/0.59      ! [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 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_abs_triangle_ineq
% 0.29/0.59  thf(fact_7506_nat__div__distrib_H,axiom,
% 0.29/0.59      ! [Y: int,X: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.59       => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 0.29/0.59          = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_div_distrib'
% 0.29/0.59  thf(fact_7507_nat__div__distrib,axiom,
% 0.29/0.59      ! [X: int,Y: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.59       => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 0.29/0.59          = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_div_distrib
% 0.29/0.59  thf(fact_7508_div__dvd__sgn__abs,axiom,
% 0.29/0.59      ! [L: int,K: int] :
% 0.29/0.59        ( ( dvd_dvd_int @ L @ K )
% 0.29/0.59       => ( ( divide_divide_int @ K @ L )
% 0.29/0.59          = ( 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 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % div_dvd_sgn_abs
% 0.29/0.59  thf(fact_7509_nat__floor__neg,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.59       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.29/0.59          = zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_floor_neg
% 0.29/0.59  thf(fact_7510_nat__power__eq,axiom,
% 0.29/0.59      ! [Z: int,N: nat] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 0.29/0.59          = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_power_eq
% 0.29/0.59  thf(fact_7511_nat__mod__distrib,axiom,
% 0.29/0.59      ! [X: int,Y: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.59       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.59         => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
% 0.29/0.59            = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_mod_distrib
% 0.29/0.59  thf(fact_7512_div__abs__eq__div__nat,axiom,
% 0.29/0.59      ! [K: int,L: int] :
% 0.29/0.59        ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 0.29/0.59        = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % div_abs_eq_div_nat
% 0.29/0.59  thf(fact_7513_floor__eq3,axiom,
% 0.29/0.59      ! [N: nat,X: real] :
% 0.29/0.59        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 0.29/0.59       => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.29/0.59         => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.29/0.59            = N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % floor_eq3
% 0.29/0.59  thf(fact_7514_le__nat__floor,axiom,
% 0.29/0.59      ! [X: nat,A: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 0.29/0.59       => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % le_nat_floor
% 0.29/0.59  thf(fact_7515_mod__abs__eq__div__nat,axiom,
% 0.29/0.59      ! [K: int,L: int] :
% 0.29/0.59        ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 0.29/0.59        = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % mod_abs_eq_div_nat
% 0.29/0.59  thf(fact_7516_divide__int__def,axiom,
% 0.29/0.59      ( divide_divide_int
% 0.29/0.59      = ( ^ [K2: int,L3: int] :
% 0.29/0.59            ( if_int @ ( L3 = zero_zero_int ) @ zero_zero_int
% 0.29/0.59            @ ( if_int
% 0.29/0.59              @ ( ( sgn_sgn_int @ K2 )
% 0.29/0.59                = ( sgn_sgn_int @ L3 ) )
% 0.29/0.59              @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) )
% 0.29/0.59              @ ( uminus_uminus_int
% 0.29/0.59                @ ( semiri1314217659103216013at_int
% 0.29/0.59                  @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) )
% 0.29/0.59                    @ ( zero_n2687167440665602831ol_nat
% 0.29/0.59                      @ ~ ( dvd_dvd_int @ L3 @ K2 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divide_int_def
% 0.29/0.59  thf(fact_7517_modulo__int__def,axiom,
% 0.29/0.59      ( modulo_modulo_int
% 0.29/0.59      = ( ^ [K2: int,L3: int] :
% 0.29/0.59            ( if_int @ ( L3 = zero_zero_int ) @ K2
% 0.29/0.59            @ ( if_int
% 0.29/0.59              @ ( ( sgn_sgn_int @ K2 )
% 0.29/0.59                = ( sgn_sgn_int @ L3 ) )
% 0.29/0.59              @ ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) )
% 0.29/0.59              @ ( times_times_int @ ( sgn_sgn_int @ L3 )
% 0.29/0.59                @ ( minus_minus_int
% 0.29/0.59                  @ ( times_times_int @ ( abs_abs_int @ L3 )
% 0.29/0.59                    @ ( zero_n2684676970156552555ol_int
% 0.29/0.59                      @ ~ ( dvd_dvd_int @ L3 @ K2 ) ) )
% 0.29/0.59                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % modulo_int_def
% 0.29/0.59  thf(fact_7518_nat__2,axiom,
% 0.29/0.59      ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.59      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_2
% 0.29/0.59  thf(fact_7519_Suc__nat__eq__nat__zadd1,axiom,
% 0.29/0.59      ! [Z: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59       => ( ( suc @ ( nat2 @ Z ) )
% 0.29/0.59          = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Suc_nat_eq_nat_zadd1
% 0.29/0.59  thf(fact_7520_nat__less__iff,axiom,
% 0.29/0.59      ! [W: int,M: nat] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.29/0.59       => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 0.29/0.59          = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_less_iff
% 0.29/0.59  thf(fact_7521_nat__mult__distrib__neg,axiom,
% 0.29/0.59      ! [Z: int,Z6: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.29/0.59       => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 0.29/0.59          = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_mult_distrib_neg
% 0.29/0.59  thf(fact_7522_nat__abs__int__diff,axiom,
% 0.29/0.59      ! [A: nat,B: nat] :
% 0.29/0.59        ( ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.59         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.29/0.59            = ( minus_minus_nat @ B @ A ) ) )
% 0.29/0.59        & ( ~ ( ord_less_eq_nat @ A @ B )
% 0.29/0.59         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.29/0.59            = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_abs_int_diff
% 0.29/0.59  thf(fact_7523_floor__eq4,axiom,
% 0.29/0.59      ! [N: nat,X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 0.29/0.59       => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.29/0.59         => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.29/0.59            = N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % floor_eq4
% 0.29/0.59  thf(fact_7524_of__int__of__nat,axiom,
% 0.29/0.59      ( ring_18347121197199848620nteger
% 0.29/0.59      = ( ^ [K2: int] : ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_int_of_nat
% 0.29/0.59  thf(fact_7525_of__int__of__nat,axiom,
% 0.29/0.59      ( ring_17405671764205052669omplex
% 0.29/0.59      = ( ^ [K2: int] : ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_int_of_nat
% 0.29/0.59  thf(fact_7526_of__int__of__nat,axiom,
% 0.29/0.59      ( ring_1_of_int_real
% 0.29/0.59      = ( ^ [K2: int] : ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_int_of_nat
% 0.29/0.59  thf(fact_7527_of__int__of__nat,axiom,
% 0.29/0.59      ( ring_1_of_int_rat
% 0.29/0.59      = ( ^ [K2: int] : ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_int_of_nat
% 0.29/0.59  thf(fact_7528_of__int__of__nat,axiom,
% 0.29/0.59      ( ring_1_of_int_int
% 0.29/0.59      = ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_int_of_nat
% 0.29/0.59  thf(fact_7529_nat__dvd__iff,axiom,
% 0.29/0.59      ! [Z: int,M: nat] :
% 0.29/0.59        ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 0.29/0.59        = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59           => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.29/0.59          & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.59           => ( M = zero_zero_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % nat_dvd_iff
% 0.29/0.59  thf(fact_7530_div__noneq__sgn__abs,axiom,
% 0.29/0.59      ! [L: int,K: int] :
% 0.29/0.59        ( ( L != zero_zero_int )
% 0.29/0.59       => ( ( ( sgn_sgn_int @ K )
% 0.29/0.59           != ( sgn_sgn_int @ L ) )
% 0.29/0.59         => ( ( divide_divide_int @ K @ L )
% 0.29/0.59            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
% 0.29/0.59              @ ( zero_n2684676970156552555ol_int
% 0.29/0.59                @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % div_noneq_sgn_abs
% 0.29/0.59  thf(fact_7531_even__nat__iff,axiom,
% 0.29/0.59      ! [K: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.59       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 0.29/0.59          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % even_nat_iff
% 0.29/0.59  thf(fact_7532_invar__vebt_Ointros_I4_J,axiom,
% 0.29/0.59      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma: nat] :
% 0.29/0.59        ( ! [X3: vEBT_VEBT] :
% 0.29/0.59            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.59           => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.29/0.59       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.29/0.59         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.29/0.59              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.59           => ( ( M = N )
% 0.29/0.59             => ( ( Deg
% 0.29/0.59                  = ( plus_plus_nat @ N @ M ) )
% 0.29/0.59               => ( ! [I2: nat] :
% 0.29/0.59                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.59                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 0.29/0.59                        = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.29/0.59                 => ( ( ( Mi2 = Ma )
% 0.29/0.59                     => ! [X3: vEBT_VEBT] :
% 0.29/0.59                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.59                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
% 0.29/0.59                   => ( ( ord_less_eq_nat @ Mi2 @ Ma )
% 0.29/0.59                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.29/0.59                       => ( ( ( Mi2 != Ma )
% 0.29/0.59                           => ! [I2: nat] :
% 0.29/0.59                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.59                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.29/0.59                                      = I2 )
% 0.29/0.59                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.29/0.59                                  & ! [X3: nat] :
% 0.29/0.59                                      ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 0.29/0.59                                          = I2 )
% 0.29/0.59                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 0.29/0.59                                     => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.29/0.59                                        & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 0.29/0.59                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % invar_vebt.intros(4)
% 0.29/0.59  thf(fact_7533_invar__vebt_Ointros_I5_J,axiom,
% 0.29/0.59      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma: nat] :
% 0.29/0.59        ( ! [X3: vEBT_VEBT] :
% 0.29/0.59            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.59           => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.29/0.59       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.29/0.59         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.29/0.59              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.59           => ( ( M
% 0.29/0.59                = ( suc @ N ) )
% 0.29/0.59             => ( ( Deg
% 0.29/0.59                  = ( plus_plus_nat @ N @ M ) )
% 0.29/0.59               => ( ! [I2: nat] :
% 0.29/0.59                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.59                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 0.29/0.59                        = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.29/0.59                 => ( ( ( Mi2 = Ma )
% 0.29/0.59                     => ! [X3: vEBT_VEBT] :
% 0.29/0.59                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.59                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
% 0.29/0.59                   => ( ( ord_less_eq_nat @ Mi2 @ Ma )
% 0.29/0.59                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.29/0.59                       => ( ( ( Mi2 != Ma )
% 0.29/0.59                           => ! [I2: nat] :
% 0.29/0.59                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.59                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.29/0.59                                      = I2 )
% 0.29/0.59                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.29/0.59                                  & ! [X3: nat] :
% 0.29/0.59                                      ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 0.29/0.59                                          = I2 )
% 0.29/0.59                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 0.29/0.59                                     => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.29/0.59                                        & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 0.29/0.59                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % invar_vebt.intros(5)
% 0.29/0.59  thf(fact_7534_invar__vebt_Osimps,axiom,
% 0.29/0.59      ( vEBT_invar_vebt
% 0.29/0.59      = ( ^ [A12: vEBT_VEBT,A23: nat] :
% 0.29/0.59            ( ( ? [A3: $o,B4: $o] :
% 0.29/0.59                  ( A12
% 0.29/0.59                  = ( vEBT_Leaf @ A3 @ B4 ) )
% 0.29/0.59              & ( A23
% 0.29/0.59                = ( suc @ zero_zero_nat ) ) )
% 0.29/0.59            | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
% 0.29/0.59                ( ( A12
% 0.29/0.59                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList2 @ Summary3 ) )
% 0.29/0.59                & ! [X4: vEBT_VEBT] :
% 0.29/0.59                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.59                   => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 0.29/0.59                & ( vEBT_invar_vebt @ Summary3 @ N3 )
% 0.29/0.59                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.29/0.59                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.29/0.59                & ( A23
% 0.29/0.59                  = ( plus_plus_nat @ N3 @ N3 ) )
% 0.29/0.59                & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.29/0.59                & ! [X4: vEBT_VEBT] :
% 0.29/0.59                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.59                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.59            | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
% 0.29/0.59                ( ( A12
% 0.29/0.59                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList2 @ Summary3 ) )
% 0.29/0.59                & ! [X4: vEBT_VEBT] :
% 0.29/0.59                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.59                   => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 0.29/0.59                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
% 0.29/0.59                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.29/0.59                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.29/0.59                & ( A23
% 0.29/0.59                  = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 0.29/0.59                & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.29/0.59                & ! [X4: vEBT_VEBT] :
% 0.29/0.59                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.59                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.59            | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.29/0.59                ( ( A12
% 0.29/0.59                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList2 @ Summary3 ) )
% 0.29/0.59                & ! [X4: vEBT_VEBT] :
% 0.29/0.59                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.59                   => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 0.29/0.59                & ( vEBT_invar_vebt @ Summary3 @ N3 )
% 0.29/0.59                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.29/0.59                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.29/0.59                & ( A23
% 0.29/0.59                  = ( plus_plus_nat @ N3 @ N3 ) )
% 0.29/0.59                & ! [I3: nat] :
% 0.29/0.59                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.29/0.59                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
% 0.29/0.59                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 0.29/0.59                & ( ( Mi3 = Ma3 )
% 0.29/0.59                 => ! [X4: vEBT_VEBT] :
% 0.29/0.59                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.59                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.59                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.59                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 0.29/0.59                & ( ( Mi3 != Ma3 )
% 0.29/0.59                 => ! [I3: nat] :
% 0.29/0.59                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.29/0.59                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 0.29/0.59                            = I3 )
% 0.29/0.59                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 0.29/0.59                        & ! [X4: nat] :
% 0.29/0.59                            ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
% 0.29/0.59                                = I3 )
% 0.29/0.59                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
% 0.29/0.59                           => ( ( ord_less_nat @ Mi3 @ X4 )
% 0.29/0.59                              & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) )
% 0.29/0.59            | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.29/0.59                ( ( A12
% 0.29/0.59                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList2 @ Summary3 ) )
% 0.29/0.59                & ! [X4: vEBT_VEBT] :
% 0.29/0.59                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.59                   => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 0.29/0.59                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
% 0.29/0.59                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.29/0.59                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.29/0.59                & ( A23
% 0.29/0.59                  = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 0.29/0.59                & ! [I3: nat] :
% 0.29/0.59                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.29/0.59                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
% 0.29/0.59                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 0.29/0.59                & ( ( Mi3 = Ma3 )
% 0.29/0.59                 => ! [X4: vEBT_VEBT] :
% 0.29/0.59                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.59                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.59                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.59                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 0.29/0.59                & ( ( Mi3 != Ma3 )
% 0.29/0.59                 => ! [I3: nat] :
% 0.29/0.59                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.29/0.59                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 0.29/0.59                            = I3 )
% 0.29/0.59                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 0.29/0.59                        & ! [X4: nat] :
% 0.29/0.59                            ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
% 0.29/0.59                                = I3 )
% 0.29/0.59                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
% 0.29/0.59                           => ( ( ord_less_nat @ Mi3 @ X4 )
% 0.29/0.59                              & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % invar_vebt.simps
% 0.29/0.59  thf(fact_7535_divmod__step__eq,axiom,
% 0.29/0.59      ! [L: num,R2: nat,Q3: nat] :
% 0.29/0.59        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 0.29/0.59         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 0.29/0.59            = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
% 0.29/0.59        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 0.29/0.59         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 0.29/0.59            = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_step_eq
% 0.29/0.59  thf(fact_7536_divmod__step__eq,axiom,
% 0.29/0.59      ! [L: num,R2: int,Q3: int] :
% 0.29/0.59        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 0.29/0.59         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.29/0.59            = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
% 0.29/0.59        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 0.29/0.59         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.29/0.59            = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_step_eq
% 0.29/0.59  thf(fact_7537_divmod__step__eq,axiom,
% 0.29/0.59      ! [L: num,R2: code_integer,Q3: code_integer] :
% 0.29/0.59        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 0.29/0.59         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 0.29/0.59            = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 0.29/0.59        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 0.29/0.59         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 0.29/0.59            = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_step_eq
% 0.29/0.59  thf(fact_7538_divides__aux__eq,axiom,
% 0.29/0.59      ! [Q3: nat,R2: nat] :
% 0.29/0.59        ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 0.29/0.59        = ( R2 = zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divides_aux_eq
% 0.29/0.59  thf(fact_7539_divides__aux__eq,axiom,
% 0.29/0.59      ! [Q3: int,R2: int] :
% 0.29/0.59        ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.29/0.59        = ( R2 = zero_zero_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divides_aux_eq
% 0.29/0.59  thf(fact_7540_product__nth,axiom,
% 0.29/0.59      ! [N: nat,Xs: list_num,Ys2: list_num] :
% 0.29/0.59        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs ) @ ( size_size_list_num @ Ys2 ) ) )
% 0.29/0.59       => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs @ Ys2 ) @ N )
% 0.29/0.59          = ( product_Pair_num_num @ ( nth_num @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % product_nth
% 0.29/0.59  thf(fact_7541_product__nth,axiom,
% 0.29/0.59      ! [N: nat,Xs: list_nat,Ys2: list_num] :
% 0.29/0.59        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_num @ Ys2 ) ) )
% 0.29/0.59       => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs @ Ys2 ) @ N )
% 0.29/0.59          = ( product_Pair_nat_num @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % product_nth
% 0.29/0.59  thf(fact_7542_product__nth,axiom,
% 0.29/0.59      ! [N: nat,Xs: list_nat,Ys2: list_nat] :
% 0.29/0.59        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
% 0.29/0.59       => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys2 ) @ N )
% 0.29/0.59          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % product_nth
% 0.29/0.59  thf(fact_7543_product__nth,axiom,
% 0.29/0.59      ! [N: nat,Xs: list_nat,Ys2: list_VEBT_VEBT] :
% 0.29/0.59        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 0.29/0.59       => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys2 ) @ N )
% 0.29/0.59          = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % product_nth
% 0.29/0.59  thf(fact_7544_product__nth,axiom,
% 0.29/0.59      ! [N: nat,Xs: list_nat,Ys2: list_o] :
% 0.29/0.59        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
% 0.29/0.59       => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys2 ) @ N )
% 0.29/0.59          = ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % product_nth
% 0.29/0.59  thf(fact_7545_product__nth,axiom,
% 0.29/0.59      ! [N: nat,Xs: list_Code_integer,Ys2: list_o] :
% 0.29/0.59        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
% 0.29/0.59       => ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs @ Ys2 ) @ N )
% 0.29/0.59          = ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % product_nth
% 0.29/0.59  thf(fact_7546_product__nth,axiom,
% 0.29/0.59      ! [N: nat,Xs: list_nat,Ys2: list_int] :
% 0.29/0.59        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
% 0.29/0.59       => ( ( nth_Pr3440142176431000676at_int @ ( product_nat_int @ Xs @ Ys2 ) @ N )
% 0.29/0.59          = ( product_Pair_nat_int @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % product_nth
% 0.29/0.59  thf(fact_7547_product__nth,axiom,
% 0.29/0.59      ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
% 0.29/0.59        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
% 0.29/0.59       => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys2 ) @ N )
% 0.29/0.59          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % product_nth
% 0.29/0.59  thf(fact_7548_product__nth,axiom,
% 0.29/0.59      ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.29/0.59        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 0.29/0.59       => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) @ N )
% 0.29/0.59          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % product_nth
% 0.29/0.59  thf(fact_7549_product__nth,axiom,
% 0.29/0.59      ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
% 0.29/0.59        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
% 0.29/0.59       => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys2 ) @ N )
% 0.29/0.59          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % product_nth
% 0.29/0.59  thf(fact_7550_option_Osize__gen_I2_J,axiom,
% 0.29/0.59      ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 0.29/0.59        ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.29/0.59        = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.size_gen(2)
% 0.29/0.59  thf(fact_7551_option_Osize__gen_I2_J,axiom,
% 0.29/0.59      ! [X: num > nat,X22: num] :
% 0.29/0.59        ( ( size_option_num @ X @ ( some_num @ X22 ) )
% 0.29/0.59        = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.size_gen(2)
% 0.29/0.59  thf(fact_7552_signed__take__bit__eq__take__bit__minus,axiom,
% 0.29/0.59      ( bit_ri631733984087533419it_int
% 0.29/0.59      = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % signed_take_bit_eq_take_bit_minus
% 0.29/0.59  thf(fact_7553_zero__le__sgn__iff,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
% 0.29/0.59        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % zero_le_sgn_iff
% 0.29/0.59  thf(fact_7554_sgn__le__0__iff,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
% 0.29/0.59        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_le_0_iff
% 0.29/0.59  thf(fact_7555_length__product,axiom,
% 0.29/0.59      ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.29/0.59        ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) )
% 0.29/0.59        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % length_product
% 0.29/0.59  thf(fact_7556_length__product,axiom,
% 0.29/0.59      ! [Xs: list_VEBT_VEBT,Ys2: list_o] :
% 0.29/0.59        ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys2 ) )
% 0.29/0.59        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % length_product
% 0.29/0.59  thf(fact_7557_length__product,axiom,
% 0.29/0.59      ! [Xs: list_VEBT_VEBT,Ys2: list_int] :
% 0.29/0.59        ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys2 ) )
% 0.29/0.59        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % length_product
% 0.29/0.59  thf(fact_7558_length__product,axiom,
% 0.29/0.59      ! [Xs: list_o,Ys2: list_VEBT_VEBT] :
% 0.29/0.59        ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys2 ) )
% 0.29/0.59        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % length_product
% 0.29/0.59  thf(fact_7559_length__product,axiom,
% 0.29/0.59      ! [Xs: list_o,Ys2: list_o] :
% 0.29/0.59        ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys2 ) )
% 0.29/0.59        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % length_product
% 0.29/0.59  thf(fact_7560_length__product,axiom,
% 0.29/0.59      ! [Xs: list_o,Ys2: list_int] :
% 0.29/0.59        ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys2 ) )
% 0.29/0.59        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % length_product
% 0.29/0.59  thf(fact_7561_length__product,axiom,
% 0.29/0.59      ! [Xs: list_int,Ys2: list_VEBT_VEBT] :
% 0.29/0.59        ( ( size_s6639371672096860321T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys2 ) )
% 0.29/0.59        = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % length_product
% 0.29/0.59  thf(fact_7562_length__product,axiom,
% 0.29/0.59      ! [Xs: list_int,Ys2: list_o] :
% 0.29/0.59        ( ( size_s4246224855604898693_int_o @ ( product_int_o @ Xs @ Ys2 ) )
% 0.29/0.59        = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % length_product
% 0.29/0.59  thf(fact_7563_length__product,axiom,
% 0.29/0.59      ! [Xs: list_int,Ys2: list_int] :
% 0.29/0.59        ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys2 ) )
% 0.29/0.59        = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % length_product
% 0.29/0.59  thf(fact_7564_bit__numeral__Bit0__Suc__iff,axiom,
% 0.29/0.59      ! [M: num,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.29/0.59        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_Bit0_Suc_iff
% 0.29/0.59  thf(fact_7565_bit__numeral__Bit0__Suc__iff,axiom,
% 0.29/0.59      ! [M: num,N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.29/0.59        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_Bit0_Suc_iff
% 0.29/0.59  thf(fact_7566_bit__numeral__Bit1__Suc__iff,axiom,
% 0.29/0.59      ! [M: num,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.29/0.59        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_Bit1_Suc_iff
% 0.29/0.59  thf(fact_7567_bit__numeral__Bit1__Suc__iff,axiom,
% 0.29/0.59      ! [M: num,N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.29/0.59        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_Bit1_Suc_iff
% 0.29/0.59  thf(fact_7568_signed__take__bit__nonnegative__iff,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.29/0.59        = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % signed_take_bit_nonnegative_iff
% 0.29/0.59  thf(fact_7569_signed__take__bit__negative__iff,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 0.29/0.59        = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % signed_take_bit_negative_iff
% 0.29/0.59  thf(fact_7570_bit__numeral__simps_I2_J,axiom,
% 0.29/0.59      ! [W: num,N: num] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.59        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_simps(2)
% 0.29/0.59  thf(fact_7571_bit__numeral__simps_I2_J,axiom,
% 0.29/0.59      ! [W: num,N: num] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.59        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_simps(2)
% 0.29/0.59  thf(fact_7572_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 0.29/0.59      ! [W: num,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 0.29/0.59        = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_minus_numeral_Bit0_Suc_iff
% 0.29/0.59  thf(fact_7573_bit__numeral__simps_I3_J,axiom,
% 0.29/0.59      ! [W: num,N: num] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.59        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_simps(3)
% 0.29/0.59  thf(fact_7574_bit__numeral__simps_I3_J,axiom,
% 0.29/0.59      ! [W: num,N: num] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.59        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_simps(3)
% 0.29/0.59  thf(fact_7575_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 0.29/0.59      ! [W: num,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 0.29/0.59        = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_minus_numeral_Bit1_Suc_iff
% 0.29/0.59  thf(fact_7576_bit__0,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 0.29/0.59        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_0
% 0.29/0.59  thf(fact_7577_bit__0,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 0.29/0.59        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_0
% 0.29/0.59  thf(fact_7578_bit__0,axiom,
% 0.29/0.59      ! [A: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 0.29/0.59        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_0
% 0.29/0.59  thf(fact_7579_bit__minus__numeral__int_I1_J,axiom,
% 0.29/0.59      ! [W: num,N: num] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.59        = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_minus_numeral_int(1)
% 0.29/0.59  thf(fact_7580_bit__minus__numeral__int_I2_J,axiom,
% 0.29/0.59      ! [W: num,N: num] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.59        = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_minus_numeral_int(2)
% 0.29/0.59  thf(fact_7581_bit__mod__2__iff,axiom,
% 0.29/0.59      ! [A: code_integer,N: nat] :
% 0.29/0.59        ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
% 0.29/0.59        = ( ( N = zero_zero_nat )
% 0.29/0.59          & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_mod_2_iff
% 0.29/0.59  thf(fact_7582_bit__mod__2__iff,axiom,
% 0.29/0.59      ! [A: int,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
% 0.29/0.59        = ( ( N = zero_zero_nat )
% 0.29/0.59          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_mod_2_iff
% 0.29/0.59  thf(fact_7583_bit__mod__2__iff,axiom,
% 0.29/0.59      ! [A: nat,N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.29/0.59        = ( ( N = zero_zero_nat )
% 0.29/0.59          & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_mod_2_iff
% 0.29/0.59  thf(fact_7584_bit__disjunctive__add__iff,axiom,
% 0.29/0.59      ! [A: int,B: int,N: nat] :
% 0.29/0.59        ( ! [N2: nat] :
% 0.29/0.59            ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.29/0.59            | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
% 0.29/0.59       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N )
% 0.29/0.59          = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.29/0.59            | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_disjunctive_add_iff
% 0.29/0.59  thf(fact_7585_bit__disjunctive__add__iff,axiom,
% 0.29/0.59      ! [A: nat,B: nat,N: nat] :
% 0.29/0.59        ( ! [N2: nat] :
% 0.29/0.59            ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.29/0.59            | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
% 0.29/0.59       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 0.29/0.59          = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.29/0.59            | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_disjunctive_add_iff
% 0.29/0.59  thf(fact_7586_bit__of__nat__iff__bit,axiom,
% 0.29/0.59      ! [M: nat,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
% 0.29/0.59        = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_of_nat_iff_bit
% 0.29/0.59  thf(fact_7587_bit__of__nat__iff__bit,axiom,
% 0.29/0.59      ! [M: nat,N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
% 0.29/0.59        = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_of_nat_iff_bit
% 0.29/0.59  thf(fact_7588_bit__numeral__iff,axiom,
% 0.29/0.59      ! [M: num,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
% 0.29/0.59        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_iff
% 0.29/0.59  thf(fact_7589_bit__numeral__iff,axiom,
% 0.29/0.59      ! [M: num,N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
% 0.29/0.59        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_iff
% 0.29/0.59  thf(fact_7590_sgn__eq,axiom,
% 0.29/0.59      ( sgn_sgn_complex
% 0.29/0.59      = ( ^ [Z4: complex] : ( divide1717551699836669952omplex @ Z4 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z4 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_eq
% 0.29/0.59  thf(fact_7591_not__bit__1__Suc,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % not_bit_1_Suc
% 0.29/0.59  thf(fact_7592_not__bit__1__Suc,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % not_bit_1_Suc
% 0.29/0.59  thf(fact_7593_bit__1__iff,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
% 0.29/0.59        = ( N = zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_1_iff
% 0.29/0.59  thf(fact_7594_bit__1__iff,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
% 0.29/0.59        = ( N = zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_1_iff
% 0.29/0.59  thf(fact_7595_bit__numeral__simps_I1_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_simps(1)
% 0.29/0.59  thf(fact_7596_bit__numeral__simps_I1_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_numeral_simps(1)
% 0.29/0.59  thf(fact_7597_bit__take__bit__iff,axiom,
% 0.29/0.59      ! [M: nat,A: int,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
% 0.29/0.59        = ( ( ord_less_nat @ N @ M )
% 0.29/0.59          & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_take_bit_iff
% 0.29/0.59  thf(fact_7598_bit__take__bit__iff,axiom,
% 0.29/0.59      ! [M: nat,A: nat,N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
% 0.29/0.59        = ( ( ord_less_nat @ N @ M )
% 0.29/0.59          & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_take_bit_iff
% 0.29/0.59  thf(fact_7599_bit__of__bool__iff,axiom,
% 0.29/0.59      ! [B: $o,N: nat] :
% 0.29/0.59        ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N )
% 0.29/0.59        = ( B
% 0.29/0.59          & ( N = zero_zero_nat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_of_bool_iff
% 0.29/0.59  thf(fact_7600_bit__of__bool__iff,axiom,
% 0.29/0.59      ! [B: $o,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N )
% 0.29/0.59        = ( B
% 0.29/0.59          & ( N = zero_zero_nat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_of_bool_iff
% 0.29/0.59  thf(fact_7601_bit__of__bool__iff,axiom,
% 0.29/0.59      ! [B: $o,N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N )
% 0.29/0.59        = ( B
% 0.29/0.59          & ( N = zero_zero_nat ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_of_bool_iff
% 0.29/0.59  thf(fact_7602_real__sgn__eq,axiom,
% 0.29/0.59      ( sgn_sgn_real
% 0.29/0.59      = ( ^ [X4: real] : ( divide_divide_real @ X4 @ ( abs_abs_real @ X4 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sgn_eq
% 0.29/0.59  thf(fact_7603_VEBT__internal_Ovalid_H_Ocases,axiom,
% 0.29/0.59      ! [X: produc9072475918466114483BT_nat] :
% 0.29/0.59        ( ! [Uu2: $o,Uv2: $o,D5: nat] :
% 0.29/0.59            ( X
% 0.29/0.59           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D5 ) )
% 0.29/0.59       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 0.29/0.59              ( X
% 0.29/0.59             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % VEBT_internal.valid'.cases
% 0.29/0.59  thf(fact_7604_bit__not__int__iff_H,axiom,
% 0.29/0.59      ! [K: int,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 0.29/0.59        = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_not_int_iff'
% 0.29/0.59  thf(fact_7605_xor__num_Ocases,axiom,
% 0.29/0.59      ! [X: product_prod_num_num] :
% 0.29/0.59        ( ( X
% 0.29/0.59         != ( product_Pair_num_num @ one @ one ) )
% 0.29/0.59       => ( ! [N2: num] :
% 0.29/0.59              ( X
% 0.29/0.59             != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
% 0.29/0.59         => ( ! [N2: num] :
% 0.29/0.59                ( X
% 0.29/0.59               != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
% 0.29/0.59           => ( ! [M5: num] :
% 0.29/0.59                  ( X
% 0.29/0.59                 != ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
% 0.29/0.59             => ( ! [M5: num,N2: num] :
% 0.29/0.59                    ( X
% 0.29/0.59                   != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N2 ) ) )
% 0.29/0.59               => ( ! [M5: num,N2: num] :
% 0.29/0.59                      ( X
% 0.29/0.59                     != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N2 ) ) )
% 0.29/0.59                 => ( ! [M5: num] :
% 0.29/0.59                        ( X
% 0.29/0.59                       != ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
% 0.29/0.59                   => ( ! [M5: num,N2: num] :
% 0.29/0.59                          ( X
% 0.29/0.59                         != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N2 ) ) )
% 0.29/0.59                     => ~ ! [M5: num,N2: num] :
% 0.29/0.59                            ( X
% 0.29/0.59                           != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % xor_num.cases
% 0.29/0.59  thf(fact_7606_sgn__real__def,axiom,
% 0.29/0.59      ( sgn_sgn_real
% 0.29/0.59      = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_real_def
% 0.29/0.59  thf(fact_7607_bit__imp__take__bit__positive,axiom,
% 0.29/0.59      ! [N: nat,M: nat,K: int] :
% 0.29/0.59        ( ( ord_less_nat @ N @ M )
% 0.29/0.59       => ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.29/0.59         => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_imp_take_bit_positive
% 0.29/0.59  thf(fact_7608_bit__concat__bit__iff,axiom,
% 0.29/0.59      ! [M: nat,K: int,L: int,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 0.29/0.59        = ( ( ( ord_less_nat @ N @ M )
% 0.29/0.59            & ( bit_se1146084159140164899it_int @ K @ N ) )
% 0.29/0.59          | ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.59            & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_concat_bit_iff
% 0.29/0.59  thf(fact_7609_VEBT__internal_Onaive__member_Ocases,axiom,
% 0.29/0.59      ! [X: produc9072475918466114483BT_nat] :
% 0.29/0.59        ( ! [A5: $o,B6: $o,X3: nat] :
% 0.29/0.59            ( X
% 0.29/0.59           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ X3 ) )
% 0.29/0.59       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 0.29/0.59              ( X
% 0.29/0.59             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 0.29/0.59         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
% 0.29/0.59                ( X
% 0.29/0.59               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ X3 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % VEBT_internal.naive_member.cases
% 0.29/0.59  thf(fact_7610_exp__eq__0__imp__not__bit,axiom,
% 0.29/0.59      ! [N: nat,A: int] :
% 0.29/0.59        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.29/0.59          = zero_zero_int )
% 0.29/0.59       => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % exp_eq_0_imp_not_bit
% 0.29/0.59  thf(fact_7611_exp__eq__0__imp__not__bit,axiom,
% 0.29/0.59      ! [N: nat,A: nat] :
% 0.29/0.59        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.59          = zero_zero_nat )
% 0.29/0.59       => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % exp_eq_0_imp_not_bit
% 0.29/0.59  thf(fact_7612_bit__Suc,axiom,
% 0.29/0.59      ! [A: int,N: nat] :
% 0.29/0.59        ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
% 0.29/0.59        = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_Suc
% 0.29/0.59  thf(fact_7613_bit__Suc,axiom,
% 0.29/0.59      ! [A: nat,N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
% 0.29/0.59        = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_Suc
% 0.29/0.59  thf(fact_7614_stable__imp__bit__iff__odd,axiom,
% 0.29/0.59      ! [A: code_integer,N: nat] :
% 0.29/0.59        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.59          = A )
% 0.29/0.59       => ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.29/0.59          = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % stable_imp_bit_iff_odd
% 0.29/0.59  thf(fact_7615_stable__imp__bit__iff__odd,axiom,
% 0.29/0.59      ! [A: int,N: nat] :
% 0.29/0.59        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.59          = A )
% 0.29/0.59       => ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.29/0.59          = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % stable_imp_bit_iff_odd
% 0.29/0.59  thf(fact_7616_stable__imp__bit__iff__odd,axiom,
% 0.29/0.59      ! [A: nat,N: nat] :
% 0.29/0.59        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.59          = A )
% 0.29/0.59       => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.29/0.59          = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % stable_imp_bit_iff_odd
% 0.29/0.59  thf(fact_7617_bit__iff__idd__imp__stable,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ! [N2: nat] :
% 0.29/0.59            ( ( bit_se9216721137139052372nteger @ A @ N2 )
% 0.29/0.59            = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.29/0.59       => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.59          = A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_iff_idd_imp_stable
% 0.29/0.59  thf(fact_7618_bit__iff__idd__imp__stable,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ! [N2: nat] :
% 0.29/0.59            ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.29/0.59            = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.29/0.59       => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.59          = A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_iff_idd_imp_stable
% 0.29/0.59  thf(fact_7619_bit__iff__idd__imp__stable,axiom,
% 0.29/0.59      ! [A: nat] :
% 0.29/0.59        ( ! [N2: nat] :
% 0.29/0.59            ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.29/0.59            = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.29/0.59       => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.59          = A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_iff_idd_imp_stable
% 0.29/0.59  thf(fact_7620_sgn__power__injE,axiom,
% 0.29/0.59      ! [A: real,N: nat,X: real,B: real] :
% 0.29/0.59        ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.29/0.59          = X )
% 0.29/0.59       => ( ( X
% 0.29/0.59            = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 0.29/0.59         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.59           => ( A = B ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sgn_power_injE
% 0.29/0.59  thf(fact_7621_int__bit__bound,axiom,
% 0.29/0.59      ! [K: int] :
% 0.29/0.59        ~ ! [N2: nat] :
% 0.29/0.59            ( ! [M3: nat] :
% 0.29/0.59                ( ( ord_less_eq_nat @ N2 @ M3 )
% 0.29/0.59               => ( ( bit_se1146084159140164899it_int @ K @ M3 )
% 0.29/0.59                  = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
% 0.29/0.59           => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.59               => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 0.29/0.59                  = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % int_bit_bound
% 0.29/0.59  thf(fact_7622_bit__iff__odd,axiom,
% 0.29/0.59      ( bit_se9216721137139052372nteger
% 0.29/0.59      = ( ^ [A3: code_integer,N3: nat] :
% 0.29/0.59            ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_iff_odd
% 0.29/0.59  thf(fact_7623_bit__iff__odd,axiom,
% 0.29/0.59      ( bit_se1146084159140164899it_int
% 0.29/0.59      = ( ^ [A3: int,N3: nat] :
% 0.29/0.59            ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_iff_odd
% 0.29/0.59  thf(fact_7624_bit__iff__odd,axiom,
% 0.29/0.59      ( bit_se1148574629649215175it_nat
% 0.29/0.59      = ( ^ [A3: nat,N3: nat] :
% 0.29/0.59            ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_iff_odd
% 0.29/0.59  thf(fact_7625_bit__int__def,axiom,
% 0.29/0.59      ( bit_se1146084159140164899it_int
% 0.29/0.59      = ( ^ [K2: int,N3: nat] :
% 0.29/0.59            ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_int_def
% 0.29/0.59  thf(fact_7626_even__bit__succ__iff,axiom,
% 0.29/0.59      ! [A: code_integer,N: nat] :
% 0.29/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.59       => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
% 0.29/0.59          = ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.29/0.59            | ( N = zero_zero_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % even_bit_succ_iff
% 0.29/0.59  thf(fact_7627_even__bit__succ__iff,axiom,
% 0.29/0.59      ! [A: int,N: nat] :
% 0.29/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.59       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
% 0.29/0.59          = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.29/0.59            | ( N = zero_zero_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % even_bit_succ_iff
% 0.29/0.59  thf(fact_7628_even__bit__succ__iff,axiom,
% 0.29/0.59      ! [A: nat,N: nat] :
% 0.29/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.59       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
% 0.29/0.59          = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.29/0.59            | ( N = zero_zero_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % even_bit_succ_iff
% 0.29/0.59  thf(fact_7629_odd__bit__iff__bit__pred,axiom,
% 0.29/0.59      ! [A: code_integer,N: nat] :
% 0.29/0.59        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.59       => ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.29/0.59          = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
% 0.29/0.59            | ( N = zero_zero_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % odd_bit_iff_bit_pred
% 0.29/0.59  thf(fact_7630_odd__bit__iff__bit__pred,axiom,
% 0.29/0.59      ! [A: int,N: nat] :
% 0.29/0.59        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.59       => ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.29/0.59          = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
% 0.29/0.59            | ( N = zero_zero_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % odd_bit_iff_bit_pred
% 0.29/0.59  thf(fact_7631_odd__bit__iff__bit__pred,axiom,
% 0.29/0.59      ! [A: nat,N: nat] :
% 0.29/0.59        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.59       => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.29/0.59          = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
% 0.29/0.59            | ( N = zero_zero_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % odd_bit_iff_bit_pred
% 0.29/0.59  thf(fact_7632_bit__sum__mult__2__cases,axiom,
% 0.29/0.59      ! [A: code_integer,B: code_integer,N: nat] :
% 0.29/0.59        ( ! [J3: nat] :
% 0.29/0.59            ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J3 ) )
% 0.29/0.59       => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.29/0.59          = ( ( ( N = zero_zero_nat )
% 0.29/0.59             => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.59            & ( ( N != zero_zero_nat )
% 0.29/0.59             => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_sum_mult_2_cases
% 0.29/0.59  thf(fact_7633_bit__sum__mult__2__cases,axiom,
% 0.29/0.59      ! [A: int,B: int,N: nat] :
% 0.29/0.59        ( ! [J3: nat] :
% 0.29/0.59            ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J3 ) )
% 0.29/0.59       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.29/0.59          = ( ( ( N = zero_zero_nat )
% 0.29/0.59             => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.59            & ( ( N != zero_zero_nat )
% 0.29/0.59             => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_sum_mult_2_cases
% 0.29/0.59  thf(fact_7634_bit__sum__mult__2__cases,axiom,
% 0.29/0.59      ! [A: nat,B: nat,N: nat] :
% 0.29/0.59        ( ! [J3: nat] :
% 0.29/0.59            ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J3 ) )
% 0.29/0.59       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.29/0.59          = ( ( ( N = zero_zero_nat )
% 0.29/0.59             => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.29/0.59            & ( ( N != zero_zero_nat )
% 0.29/0.59             => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_sum_mult_2_cases
% 0.29/0.59  thf(fact_7635_bit__rec,axiom,
% 0.29/0.59      ( bit_se9216721137139052372nteger
% 0.29/0.59      = ( ^ [A3: code_integer,N3: nat] :
% 0.29/0.59            ( ( ( N3 = zero_zero_nat )
% 0.29/0.59             => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
% 0.29/0.59            & ( ( N3 != zero_zero_nat )
% 0.29/0.59             => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_rec
% 0.29/0.59  thf(fact_7636_bit__rec,axiom,
% 0.29/0.59      ( bit_se1146084159140164899it_int
% 0.29/0.59      = ( ^ [A3: int,N3: nat] :
% 0.29/0.59            ( ( ( N3 = zero_zero_nat )
% 0.29/0.59             => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
% 0.29/0.59            & ( ( N3 != zero_zero_nat )
% 0.29/0.59             => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_rec
% 0.29/0.59  thf(fact_7637_bit__rec,axiom,
% 0.29/0.59      ( bit_se1148574629649215175it_nat
% 0.29/0.59      = ( ^ [A3: nat,N3: nat] :
% 0.29/0.59            ( ( ( N3 = zero_zero_nat )
% 0.29/0.59             => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
% 0.29/0.59            & ( ( N3 != zero_zero_nat )
% 0.29/0.59             => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_rec
% 0.29/0.59  thf(fact_7638_VEBT__internal_Omembermima_Ocases,axiom,
% 0.29/0.59      ! [X: produc9072475918466114483BT_nat] :
% 0.29/0.59        ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 0.29/0.59            ( X
% 0.29/0.59           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 0.29/0.59       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 0.29/0.59              ( X
% 0.29/0.59             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 0.29/0.59         => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
% 0.29/0.59                ( X
% 0.29/0.59               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X3 ) )
% 0.29/0.59           => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
% 0.29/0.59                  ( X
% 0.29/0.59                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) @ X3 ) )
% 0.29/0.59             => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
% 0.29/0.59                    ( X
% 0.29/0.59                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ X3 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % VEBT_internal.membermima.cases
% 0.29/0.59  thf(fact_7639_option_Osize__gen_I1_J,axiom,
% 0.29/0.59      ! [X: product_prod_nat_nat > nat] :
% 0.29/0.59        ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
% 0.29/0.59        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.size_gen(1)
% 0.29/0.59  thf(fact_7640_option_Osize__gen_I1_J,axiom,
% 0.29/0.59      ! [X: num > nat] :
% 0.29/0.59        ( ( size_option_num @ X @ none_num )
% 0.29/0.59        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % option.size_gen(1)
% 0.29/0.59  thf(fact_7641_set__bit__eq,axiom,
% 0.29/0.59      ( bit_se7879613467334960850it_int
% 0.29/0.59      = ( ^ [N3: nat,K2: int] :
% 0.29/0.59            ( plus_plus_int @ K2
% 0.29/0.59            @ ( times_times_int
% 0.29/0.59              @ ( zero_n2684676970156552555ol_int
% 0.29/0.59                @ ~ ( bit_se1146084159140164899it_int @ K2 @ N3 ) )
% 0.29/0.59              @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % set_bit_eq
% 0.29/0.59  thf(fact_7642_unset__bit__eq,axiom,
% 0.29/0.59      ( bit_se4203085406695923979it_int
% 0.29/0.59      = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % unset_bit_eq
% 0.29/0.59  thf(fact_7643_take__bit__Suc__from__most,axiom,
% 0.29/0.59      ! [N: nat,K: int] :
% 0.29/0.59        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 0.29/0.59        = ( 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 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % take_bit_Suc_from_most
% 0.29/0.59  thf(fact_7644_divmod__algorithm__code_I7_J,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( ( ord_less_eq_num @ M @ N )
% 0.29/0.59         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 0.29/0.59        & ( ~ ( ord_less_eq_num @ M @ N )
% 0.29/0.59         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(7)
% 0.29/0.59  thf(fact_7645_divmod__algorithm__code_I7_J,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( ( ord_less_eq_num @ M @ N )
% 0.29/0.59         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 0.29/0.59        & ( ~ ( ord_less_eq_num @ M @ N )
% 0.29/0.59         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(7)
% 0.29/0.59  thf(fact_7646_divmod__algorithm__code_I7_J,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( ( ord_less_eq_num @ M @ N )
% 0.29/0.59         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 0.29/0.59        & ( ~ ( ord_less_eq_num @ M @ N )
% 0.29/0.59         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(7)
% 0.29/0.59  thf(fact_7647_divmod__algorithm__code_I8_J,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( ( ord_less_num @ M @ N )
% 0.29/0.59         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 0.29/0.59        & ( ~ ( ord_less_num @ M @ N )
% 0.29/0.59         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(8)
% 0.29/0.59  thf(fact_7648_divmod__algorithm__code_I8_J,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( ( ord_less_num @ M @ N )
% 0.29/0.59         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 0.29/0.59        & ( ~ ( ord_less_num @ M @ N )
% 0.29/0.59         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(8)
% 0.29/0.59  thf(fact_7649_divmod__algorithm__code_I8_J,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( ( ord_less_num @ M @ N )
% 0.29/0.59         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 0.29/0.59        & ( ~ ( ord_less_num @ M @ N )
% 0.29/0.59         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.59            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(8)
% 0.29/0.59  thf(fact_7650_neg__eucl__rel__int__mult__2,axiom,
% 0.29/0.59      ! [B: int,A: int,Q3: int,R2: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.29/0.59       => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.29/0.59         => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % neg_eucl_rel_int_mult_2
% 0.29/0.59  thf(fact_7651_Divides_Oadjust__div__eq,axiom,
% 0.29/0.59      ! [Q3: int,R2: int] :
% 0.29/0.59        ( ( adjust_div @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.29/0.59        = ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Divides.adjust_div_eq
% 0.29/0.59  thf(fact_7652_pos__eucl__rel__int__mult__2,axiom,
% 0.29/0.59      ! [B: int,A: int,Q3: int,R2: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.59       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.29/0.59         => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % pos_eucl_rel_int_mult_2
% 0.29/0.59  thf(fact_7653_numeral__div__minus__numeral,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.59        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % numeral_div_minus_numeral
% 0.29/0.59  thf(fact_7654_minus__numeral__div__numeral,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.59        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % minus_numeral_div_numeral
% 0.29/0.59  thf(fact_7655_dvd__numeral__simp,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.59        = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % dvd_numeral_simp
% 0.29/0.59  thf(fact_7656_dvd__numeral__simp,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.29/0.59        = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % dvd_numeral_simp
% 0.29/0.59  thf(fact_7657_dvd__numeral__simp,axiom,
% 0.29/0.59      ! [M: num,N: num] :
% 0.29/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 0.29/0.59        = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % dvd_numeral_simp
% 0.29/0.59  thf(fact_7658_divmod__algorithm__code_I2_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( unique5052692396658037445od_int @ M @ one )
% 0.29/0.59        = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(2)
% 0.29/0.59  thf(fact_7659_divmod__algorithm__code_I2_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( unique5055182867167087721od_nat @ M @ one )
% 0.29/0.59        = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(2)
% 0.29/0.59  thf(fact_7660_divmod__algorithm__code_I2_J,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( unique3479559517661332726nteger @ M @ one )
% 0.29/0.59        = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(2)
% 0.29/0.59  thf(fact_7661_divmod__algorithm__code_I3_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 0.29/0.59        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(3)
% 0.29/0.59  thf(fact_7662_divmod__algorithm__code_I3_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 0.29/0.59        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(3)
% 0.29/0.59  thf(fact_7663_divmod__algorithm__code_I3_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 0.29/0.59        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(3)
% 0.29/0.59  thf(fact_7664_divmod__algorithm__code_I4_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 0.29/0.59        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(4)
% 0.29/0.59  thf(fact_7665_divmod__algorithm__code_I4_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 0.29/0.59        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(4)
% 0.29/0.59  thf(fact_7666_divmod__algorithm__code_I4_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 0.29/0.59        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_algorithm_code(4)
% 0.29/0.59  thf(fact_7667_one__div__minus__numeral,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.59        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % one_div_minus_numeral
% 0.29/0.59  thf(fact_7668_minus__one__div__numeral,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.59        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % minus_one_div_numeral
% 0.29/0.59  thf(fact_7669_eucl__rel__int__by0,axiom,
% 0.29/0.59      ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 0.29/0.59  
% 0.29/0.59  % eucl_rel_int_by0
% 0.29/0.59  thf(fact_7670_div__int__unique,axiom,
% 0.29/0.59      ! [K: int,L: int,Q3: int,R2: int] :
% 0.29/0.59        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.29/0.59       => ( ( divide_divide_int @ K @ L )
% 0.29/0.59          = Q3 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % div_int_unique
% 0.29/0.59  thf(fact_7671_bit__Suc__0__iff,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.59        = ( N = zero_zero_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_Suc_0_iff
% 0.29/0.59  thf(fact_7672_not__bit__Suc__0__Suc,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % not_bit_Suc_0_Suc
% 0.29/0.59  thf(fact_7673_not__bit__Suc__0__numeral,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % not_bit_Suc_0_numeral
% 0.29/0.59  thf(fact_7674_eucl__rel__int__dividesI,axiom,
% 0.29/0.59      ! [L: int,K: int,Q3: int] :
% 0.29/0.59        ( ( L != zero_zero_int )
% 0.29/0.59       => ( ( K
% 0.29/0.59            = ( times_times_int @ Q3 @ L ) )
% 0.29/0.59         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % eucl_rel_int_dividesI
% 0.29/0.59  thf(fact_7675_eucl__rel__int,axiom,
% 0.29/0.59      ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % eucl_rel_int
% 0.29/0.59  thf(fact_7676_bit__nat__iff,axiom,
% 0.29/0.59      ! [K: int,N: nat] :
% 0.29/0.59        ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 0.29/0.59        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.59          & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_nat_iff
% 0.29/0.59  thf(fact_7677_divmod__int__def,axiom,
% 0.29/0.59      ( unique5052692396658037445od_int
% 0.29/0.59      = ( ^ [M6: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_int_def
% 0.29/0.59  thf(fact_7678_divmod__def,axiom,
% 0.29/0.59      ( unique5052692396658037445od_int
% 0.29/0.59      = ( ^ [M6: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_def
% 0.29/0.59  thf(fact_7679_divmod__def,axiom,
% 0.29/0.59      ( unique5055182867167087721od_nat
% 0.29/0.59      = ( ^ [M6: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_def
% 0.29/0.59  thf(fact_7680_divmod__def,axiom,
% 0.29/0.59      ( unique3479559517661332726nteger
% 0.29/0.59      = ( ^ [M6: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_def
% 0.29/0.59  thf(fact_7681_bit__nat__def,axiom,
% 0.29/0.59      ( bit_se1148574629649215175it_nat
% 0.29/0.59      = ( ^ [M6: nat,N3: nat] :
% 0.29/0.59            ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bit_nat_def
% 0.29/0.59  thf(fact_7682_divmod_H__nat__def,axiom,
% 0.29/0.59      ( unique5055182867167087721od_nat
% 0.29/0.59      = ( ^ [M6: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod'_nat_def
% 0.29/0.59  thf(fact_7683_zminus1__lemma,axiom,
% 0.29/0.59      ! [A: int,B: int,Q3: int,R2: int] :
% 0.29/0.59        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.29/0.59       => ( ( B != zero_zero_int )
% 0.29/0.59         => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % zminus1_lemma
% 0.29/0.59  thf(fact_7684_eucl__rel__int__iff,axiom,
% 0.29/0.59      ! [K: int,L: int,Q3: int,R2: int] :
% 0.29/0.59        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.29/0.59        = ( ( K
% 0.29/0.59            = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
% 0.29/0.59          & ( ( ord_less_int @ zero_zero_int @ L )
% 0.29/0.59           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.29/0.59              & ( ord_less_int @ R2 @ L ) ) )
% 0.29/0.59          & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 0.29/0.59           => ( ( ( ord_less_int @ L @ zero_zero_int )
% 0.29/0.59               => ( ( ord_less_int @ L @ R2 )
% 0.29/0.59                  & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 0.29/0.59              & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 0.29/0.59               => ( Q3 = zero_zero_int ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % eucl_rel_int_iff
% 0.29/0.59  thf(fact_7685_eucl__rel__int__remainderI,axiom,
% 0.29/0.59      ! [R2: int,L: int,K: int,Q3: int] :
% 0.29/0.59        ( ( ( sgn_sgn_int @ R2 )
% 0.29/0.59          = ( sgn_sgn_int @ L ) )
% 0.29/0.59       => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 0.29/0.59         => ( ( K
% 0.29/0.59              = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
% 0.29/0.59           => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % eucl_rel_int_remainderI
% 0.29/0.59  thf(fact_7686_eucl__rel__int_Ocases,axiom,
% 0.29/0.59      ! [A1: int,A22: int,A32: product_prod_int_int] :
% 0.29/0.59        ( ( eucl_rel_int @ A1 @ A22 @ A32 )
% 0.29/0.59       => ( ( ( A22 = zero_zero_int )
% 0.29/0.59           => ( A32
% 0.29/0.59             != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
% 0.29/0.59         => ( ! [Q2: int] :
% 0.29/0.59                ( ( A32
% 0.29/0.59                  = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
% 0.29/0.59               => ( ( A22 != zero_zero_int )
% 0.29/0.59                 => ( A1
% 0.29/0.59                   != ( times_times_int @ Q2 @ A22 ) ) ) )
% 0.29/0.59           => ~ ! [R3: int,Q2: int] :
% 0.29/0.59                  ( ( A32
% 0.29/0.59                    = ( product_Pair_int_int @ Q2 @ R3 ) )
% 0.29/0.59                 => ( ( ( sgn_sgn_int @ R3 )
% 0.29/0.59                      = ( sgn_sgn_int @ A22 ) )
% 0.29/0.59                   => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A22 ) )
% 0.29/0.59                     => ( A1
% 0.29/0.59                       != ( plus_plus_int @ ( times_times_int @ Q2 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % eucl_rel_int.cases
% 0.29/0.59  thf(fact_7687_eucl__rel__int_Osimps,axiom,
% 0.29/0.59      ( eucl_rel_int
% 0.29/0.59      = ( ^ [A12: int,A23: int,A33: product_prod_int_int] :
% 0.29/0.59            ( ? [K2: int] :
% 0.29/0.59                ( ( A12 = K2 )
% 0.29/0.59                & ( A23 = zero_zero_int )
% 0.29/0.59                & ( A33
% 0.29/0.59                  = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
% 0.29/0.59            | ? [L3: int,K2: int,Q4: int] :
% 0.29/0.59                ( ( A12 = K2 )
% 0.29/0.59                & ( A23 = L3 )
% 0.29/0.59                & ( A33
% 0.29/0.59                  = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 0.29/0.59                & ( L3 != zero_zero_int )
% 0.29/0.59                & ( K2
% 0.29/0.59                  = ( times_times_int @ Q4 @ L3 ) ) )
% 0.29/0.59            | ? [R5: int,L3: int,K2: int,Q4: int] :
% 0.29/0.59                ( ( A12 = K2 )
% 0.29/0.59                & ( A23 = L3 )
% 0.29/0.59                & ( A33
% 0.29/0.59                  = ( product_Pair_int_int @ Q4 @ R5 ) )
% 0.29/0.59                & ( ( sgn_sgn_int @ R5 )
% 0.29/0.59                  = ( sgn_sgn_int @ L3 ) )
% 0.29/0.59                & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L3 ) )
% 0.29/0.59                & ( K2
% 0.29/0.59                  = ( plus_plus_int @ ( times_times_int @ Q4 @ L3 ) @ R5 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % eucl_rel_int.simps
% 0.29/0.59  thf(fact_7688_divmod__divmod__step,axiom,
% 0.29/0.59      ( unique5055182867167087721od_nat
% 0.29/0.59      = ( ^ [M6: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_divmod_step
% 0.29/0.59  thf(fact_7689_divmod__divmod__step,axiom,
% 0.29/0.59      ( unique5052692396658037445od_int
% 0.29/0.59      = ( ^ [M6: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_divmod_step
% 0.29/0.59  thf(fact_7690_divmod__divmod__step,axiom,
% 0.29/0.59      ( unique3479559517661332726nteger
% 0.29/0.59      = ( ^ [M6: num,N3: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N3 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N3 @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_divmod_step
% 0.29/0.59  thf(fact_7691_arctan__inverse,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( X != zero_zero_real )
% 0.29/0.59       => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
% 0.29/0.59          = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % arctan_inverse
% 0.29/0.59  thf(fact_7692_and__int__unfold,axiom,
% 0.29/0.59      ( bit_se725231765392027082nd_int
% 0.29/0.59      = ( ^ [K2: int,L3: int] :
% 0.29/0.59            ( if_int
% 0.29/0.59            @ ( ( K2 = zero_zero_int )
% 0.29/0.59              | ( L3 = zero_zero_int ) )
% 0.29/0.59            @ zero_zero_int
% 0.29/0.59            @ ( if_int
% 0.29/0.59              @ ( K2
% 0.29/0.59                = ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59              @ L3
% 0.29/0.59              @ ( if_int
% 0.29/0.59                @ ( L3
% 0.29/0.59                  = ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59                @ K2
% 0.29/0.59                @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_int_unfold
% 0.29/0.59  thf(fact_7693_arctan__half,axiom,
% 0.29/0.59      ( arctan
% 0.29/0.59      = ( ^ [X4: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X4 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % arctan_half
% 0.29/0.59  thf(fact_7694_machin,axiom,
% 0.29/0.59      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.59      = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % machin
% 0.29/0.59  thf(fact_7695_machin__Euler,axiom,
% 0.29/0.59      ( ( 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 ) ) ) ) ) ) ) ) ) ) )
% 0.29/0.59      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % machin_Euler
% 0.29/0.59  thf(fact_7696_and__zero__eq,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 0.29/0.59        = zero_zero_int ) ).
% 0.29/0.59  
% 0.29/0.59  % and_zero_eq
% 0.29/0.59  thf(fact_7697_and__zero__eq,axiom,
% 0.29/0.59      ! [A: nat] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 0.29/0.59        = zero_zero_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % and_zero_eq
% 0.29/0.59  thf(fact_7698_zero__and__eq,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 0.29/0.59        = zero_zero_int ) ).
% 0.29/0.59  
% 0.29/0.59  % zero_and_eq
% 0.29/0.59  thf(fact_7699_zero__and__eq,axiom,
% 0.29/0.59      ! [A: nat] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 0.29/0.59        = zero_zero_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % zero_and_eq
% 0.29/0.59  thf(fact_7700_bit_Oconj__zero__left,axiom,
% 0.29/0.59      ! [X: int] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
% 0.29/0.59        = zero_zero_int ) ).
% 0.29/0.59  
% 0.29/0.59  % bit.conj_zero_left
% 0.29/0.59  thf(fact_7701_bit_Oconj__zero__right,axiom,
% 0.29/0.59      ! [X: int] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
% 0.29/0.59        = zero_zero_int ) ).
% 0.29/0.59  
% 0.29/0.59  % bit.conj_zero_right
% 0.29/0.59  thf(fact_7702_real__sqrt__zero,axiom,
% 0.29/0.59      ( ( sqrt @ zero_zero_real )
% 0.29/0.59      = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_zero
% 0.29/0.59  thf(fact_7703_real__sqrt__eq__zero__cancel__iff,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ( sqrt @ X )
% 0.29/0.59          = zero_zero_real )
% 0.29/0.59        = ( X = zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_eq_zero_cancel_iff
% 0.29/0.59  thf(fact_7704_real__sqrt__less__iff,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 0.29/0.59        = ( ord_less_real @ X @ Y ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_less_iff
% 0.29/0.59  thf(fact_7705_real__sqrt__le__iff,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 0.29/0.59        = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_le_iff
% 0.29/0.59  thf(fact_7706_real__sqrt__eq__1__iff,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ( sqrt @ X )
% 0.29/0.59          = one_one_real )
% 0.29/0.59        = ( X = one_one_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_eq_1_iff
% 0.29/0.59  thf(fact_7707_real__sqrt__one,axiom,
% 0.29/0.59      ( ( sqrt @ one_one_real )
% 0.29/0.59      = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_one
% 0.29/0.59  thf(fact_7708_and_Oleft__neutral,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
% 0.29/0.59        = A ) ).
% 0.29/0.59  
% 0.29/0.59  % and.left_neutral
% 0.29/0.59  thf(fact_7709_and_Oleft__neutral,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
% 0.29/0.59        = A ) ).
% 0.29/0.59  
% 0.29/0.59  % and.left_neutral
% 0.29/0.59  thf(fact_7710_and_Oright__neutral,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.59        = A ) ).
% 0.29/0.59  
% 0.29/0.59  % and.right_neutral
% 0.29/0.59  thf(fact_7711_and_Oright__neutral,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59        = A ) ).
% 0.29/0.59  
% 0.29/0.59  % and.right_neutral
% 0.29/0.59  thf(fact_7712_bit_Oconj__one__right,axiom,
% 0.29/0.59      ! [X: code_integer] :
% 0.29/0.59        ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.59        = X ) ).
% 0.29/0.59  
% 0.29/0.59  % bit.conj_one_right
% 0.29/0.59  thf(fact_7713_bit_Oconj__one__right,axiom,
% 0.29/0.59      ! [X: int] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59        = X ) ).
% 0.29/0.59  
% 0.29/0.59  % bit.conj_one_right
% 0.29/0.59  thf(fact_7714_real__sqrt__gt__0__iff,axiom,
% 0.29/0.59      ! [Y: real] :
% 0.29/0.59        ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
% 0.29/0.59        = ( ord_less_real @ zero_zero_real @ Y ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_gt_0_iff
% 0.29/0.59  thf(fact_7715_real__sqrt__lt__0__iff,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.29/0.59        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_lt_0_iff
% 0.29/0.59  thf(fact_7716_real__sqrt__le__0__iff,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.29/0.59        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_le_0_iff
% 0.29/0.59  thf(fact_7717_real__sqrt__ge__0__iff,axiom,
% 0.29/0.59      ! [Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
% 0.29/0.59        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_ge_0_iff
% 0.29/0.59  thf(fact_7718_real__sqrt__gt__1__iff,axiom,
% 0.29/0.59      ! [Y: real] :
% 0.29/0.59        ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
% 0.29/0.59        = ( ord_less_real @ one_one_real @ Y ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_gt_1_iff
% 0.29/0.59  thf(fact_7719_real__sqrt__lt__1__iff,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 0.29/0.59        = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_lt_1_iff
% 0.29/0.59  thf(fact_7720_real__sqrt__le__1__iff,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
% 0.29/0.59        = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_le_1_iff
% 0.29/0.59  thf(fact_7721_real__sqrt__ge__1__iff,axiom,
% 0.29/0.59      ! [Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
% 0.29/0.59        = ( ord_less_eq_real @ one_one_real @ Y ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_ge_1_iff
% 0.29/0.59  thf(fact_7722_and__nonnegative__int__iff,axiom,
% 0.29/0.59      ! [K: int,L: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.29/0.59        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.59          | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_nonnegative_int_iff
% 0.29/0.59  thf(fact_7723_and__negative__int__iff,axiom,
% 0.29/0.59      ! [K: int,L: int] :
% 0.29/0.59        ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 0.29/0.59        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.29/0.59          & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_negative_int_iff
% 0.29/0.59  thf(fact_7724_and__numerals_I2_J,axiom,
% 0.29/0.59      ! [Y: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.29/0.59        = one_one_int ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(2)
% 0.29/0.59  thf(fact_7725_and__numerals_I2_J,axiom,
% 0.29/0.59      ! [Y: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.29/0.59        = one_one_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(2)
% 0.29/0.59  thf(fact_7726_and__numerals_I8_J,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 0.29/0.59        = one_one_int ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(8)
% 0.29/0.59  thf(fact_7727_and__numerals_I8_J,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 0.29/0.59        = one_one_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(8)
% 0.29/0.59  thf(fact_7728_real__sqrt__four,axiom,
% 0.29/0.59      ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.59      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_four
% 0.29/0.59  thf(fact_7729_and__numerals_I1_J,axiom,
% 0.29/0.59      ! [Y: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.29/0.59        = zero_zero_int ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(1)
% 0.29/0.59  thf(fact_7730_and__numerals_I1_J,axiom,
% 0.29/0.59      ! [Y: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.29/0.59        = zero_zero_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(1)
% 0.29/0.59  thf(fact_7731_and__numerals_I5_J,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 0.29/0.59        = zero_zero_int ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(5)
% 0.29/0.59  thf(fact_7732_and__numerals_I5_J,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 0.29/0.59        = zero_zero_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(5)
% 0.29/0.59  thf(fact_7733_and__numerals_I3_J,axiom,
% 0.29/0.59      ! [X: num,Y: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.29/0.59        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(3)
% 0.29/0.59  thf(fact_7734_and__numerals_I3_J,axiom,
% 0.29/0.59      ! [X: num,Y: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.29/0.59        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(3)
% 0.29/0.59  thf(fact_7735_and__minus__numerals_I6_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.29/0.59        = one_one_int ) ).
% 0.29/0.59  
% 0.29/0.59  % and_minus_numerals(6)
% 0.29/0.59  thf(fact_7736_and__minus__numerals_I2_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.29/0.59        = one_one_int ) ).
% 0.29/0.59  
% 0.29/0.59  % and_minus_numerals(2)
% 0.29/0.59  thf(fact_7737_and__numerals_I4_J,axiom,
% 0.29/0.59      ! [X: num,Y: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.29/0.59        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(4)
% 0.29/0.59  thf(fact_7738_and__numerals_I4_J,axiom,
% 0.29/0.59      ! [X: num,Y: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.29/0.59        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(4)
% 0.29/0.59  thf(fact_7739_and__numerals_I6_J,axiom,
% 0.29/0.59      ! [X: num,Y: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.29/0.59        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(6)
% 0.29/0.59  thf(fact_7740_and__numerals_I6_J,axiom,
% 0.29/0.59      ! [X: num,Y: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.29/0.59        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(6)
% 0.29/0.59  thf(fact_7741_real__sqrt__abs,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.59        = ( abs_abs_real @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_abs
% 0.29/0.59  thf(fact_7742_and__minus__numerals_I5_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.29/0.59        = zero_zero_int ) ).
% 0.29/0.59  
% 0.29/0.59  % and_minus_numerals(5)
% 0.29/0.59  thf(fact_7743_and__minus__numerals_I1_J,axiom,
% 0.29/0.59      ! [N: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.29/0.59        = zero_zero_int ) ).
% 0.29/0.59  
% 0.29/0.59  % and_minus_numerals(1)
% 0.29/0.59  thf(fact_7744_real__sqrt__pow2__iff,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.59          = X )
% 0.29/0.59        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_pow2_iff
% 0.29/0.59  thf(fact_7745_real__sqrt__pow2,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.59          = X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_pow2
% 0.29/0.59  thf(fact_7746_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 0.29/0.59      ! [X: real,Y: real,Xa: real,Ya: real] :
% 0.29/0.59        ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.59        = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_sum_squares_mult_squared_eq
% 0.29/0.59  thf(fact_7747_and__numerals_I7_J,axiom,
% 0.29/0.59      ! [X: num,Y: num] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.29/0.59        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(7)
% 0.29/0.59  thf(fact_7748_and__numerals_I7_J,axiom,
% 0.29/0.59      ! [X: num,Y: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.29/0.59        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_numerals(7)
% 0.29/0.59  thf(fact_7749_real__sqrt__divide,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
% 0.29/0.59        = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_divide
% 0.29/0.59  thf(fact_7750_real__sqrt__less__mono,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_real @ X @ Y )
% 0.29/0.59       => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_less_mono
% 0.29/0.59  thf(fact_7751_real__sqrt__le__mono,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.59       => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_le_mono
% 0.29/0.59  thf(fact_7752_of__int__and__eq,axiom,
% 0.29/0.59      ! [K: int,L: int] :
% 0.29/0.59        ( ( ring_18347121197199848620nteger @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.29/0.59        = ( bit_se3949692690581998587nteger @ ( ring_18347121197199848620nteger @ K ) @ ( ring_18347121197199848620nteger @ L ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_int_and_eq
% 0.29/0.59  thf(fact_7753_of__int__and__eq,axiom,
% 0.29/0.59      ! [K: int,L: int] :
% 0.29/0.59        ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.29/0.59        = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_int_and_eq
% 0.29/0.59  thf(fact_7754_of__nat__and__eq,axiom,
% 0.29/0.59      ! [M: nat,N: nat] :
% 0.29/0.59        ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 0.29/0.59        = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_and_eq
% 0.29/0.59  thf(fact_7755_of__nat__and__eq,axiom,
% 0.29/0.59      ! [M: nat,N: nat] :
% 0.29/0.59        ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 0.29/0.59        = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % of_nat_and_eq
% 0.29/0.59  thf(fact_7756_real__sqrt__power,axiom,
% 0.29/0.59      ! [X: real,K: nat] :
% 0.29/0.59        ( ( sqrt @ ( power_power_real @ X @ K ) )
% 0.29/0.59        = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_power
% 0.29/0.59  thf(fact_7757_pi__neq__zero,axiom,
% 0.29/0.59      pi != zero_zero_real ).
% 0.29/0.59  
% 0.29/0.59  % pi_neq_zero
% 0.29/0.59  thf(fact_7758_and__eq__minus__1__iff,axiom,
% 0.29/0.59      ! [A: code_integer,B: code_integer] :
% 0.29/0.59        ( ( ( bit_se3949692690581998587nteger @ A @ B )
% 0.29/0.59          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.59        = ( ( A
% 0.29/0.59            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.59          & ( B
% 0.29/0.59            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_eq_minus_1_iff
% 0.29/0.59  thf(fact_7759_and__eq__minus__1__iff,axiom,
% 0.29/0.59      ! [A: int,B: int] :
% 0.29/0.59        ( ( ( bit_se725231765392027082nd_int @ A @ B )
% 0.29/0.59          = ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59        = ( ( A
% 0.29/0.59            = ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.59          & ( B
% 0.29/0.59            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_eq_minus_1_iff
% 0.29/0.59  thf(fact_7760_real__sqrt__gt__zero,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_gt_zero
% 0.29/0.59  thf(fact_7761_real__sqrt__eq__zero__cancel,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( ( sqrt @ X )
% 0.29/0.59            = zero_zero_real )
% 0.29/0.59         => ( X = zero_zero_real ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_eq_zero_cancel
% 0.29/0.59  thf(fact_7762_real__sqrt__ge__zero,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_ge_zero
% 0.29/0.59  thf(fact_7763_real__sqrt__ge__one,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ one_one_real @ X )
% 0.29/0.59       => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_ge_one
% 0.29/0.59  thf(fact_7764_AND__upper2_H,axiom,
% 0.29/0.59      ! [Y: int,Z: int,X: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.59       => ( ( ord_less_eq_int @ Y @ Z )
% 0.29/0.59         => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % AND_upper2'
% 0.29/0.59  thf(fact_7765_AND__upper1_H,axiom,
% 0.29/0.59      ! [Y: int,Z: int,Ya: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.59       => ( ( ord_less_eq_int @ Y @ Z )
% 0.29/0.59         => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % AND_upper1'
% 0.29/0.59  thf(fact_7766_AND__upper2,axiom,
% 0.29/0.59      ! [Y: int,X: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.59       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).
% 0.29/0.59  
% 0.29/0.59  % AND_upper2
% 0.29/0.59  thf(fact_7767_AND__upper1,axiom,
% 0.29/0.59      ! [X: int,Y: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.59       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % AND_upper1
% 0.29/0.59  thf(fact_7768_AND__lower,axiom,
% 0.29/0.59      ! [X: int,Y: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.59       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % AND_lower
% 0.29/0.59  thf(fact_7769_pi__not__less__zero,axiom,
% 0.29/0.59      ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % pi_not_less_zero
% 0.29/0.59  thf(fact_7770_pi__gt__zero,axiom,
% 0.29/0.59      ord_less_real @ zero_zero_real @ pi ).
% 0.29/0.59  
% 0.29/0.59  % pi_gt_zero
% 0.29/0.59  thf(fact_7771_pi__ge__zero,axiom,
% 0.29/0.59      ord_less_eq_real @ zero_zero_real @ pi ).
% 0.29/0.59  
% 0.29/0.59  % pi_ge_zero
% 0.29/0.59  thf(fact_7772_real__div__sqrt,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
% 0.29/0.59          = ( sqrt @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_div_sqrt
% 0.29/0.59  thf(fact_7773_sqrt__add__le__add__sqrt,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.59         => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sqrt_add_le_add_sqrt
% 0.29/0.59  thf(fact_7774_le__real__sqrt__sumsq,axiom,
% 0.29/0.59      ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % le_real_sqrt_sumsq
% 0.29/0.59  thf(fact_7775_AND__upper2_H_H,axiom,
% 0.29/0.59      ! [Y: int,Z: int,X: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.59       => ( ( ord_less_int @ Y @ Z )
% 0.29/0.59         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % AND_upper2''
% 0.29/0.59  thf(fact_7776_AND__upper1_H_H,axiom,
% 0.29/0.59      ! [Y: int,Z: int,Ya: int] :
% 0.29/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.59       => ( ( ord_less_int @ Y @ Z )
% 0.29/0.59         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % AND_upper1''
% 0.29/0.59  thf(fact_7777_and__less__eq,axiom,
% 0.29/0.59      ! [L: int,K: int] :
% 0.29/0.59        ( ( ord_less_int @ L @ zero_zero_int )
% 0.29/0.59       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_less_eq
% 0.29/0.59  thf(fact_7778_even__and__iff,axiom,
% 0.29/0.59      ! [A: code_integer,B: code_integer] :
% 0.29/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
% 0.29/0.59        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.59          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % even_and_iff
% 0.29/0.59  thf(fact_7779_even__and__iff,axiom,
% 0.29/0.59      ! [A: int,B: int] :
% 0.29/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.29/0.59        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.59          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % even_and_iff
% 0.29/0.59  thf(fact_7780_even__and__iff,axiom,
% 0.29/0.59      ! [A: nat,B: nat] :
% 0.29/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.29/0.59        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.59          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % even_and_iff
% 0.29/0.59  thf(fact_7781_sqrt2__less__2,axiom,
% 0.29/0.59      ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sqrt2_less_2
% 0.29/0.59  thf(fact_7782_even__and__iff__int,axiom,
% 0.29/0.59      ! [K: int,L: int] :
% 0.29/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.29/0.59        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.29/0.59          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % even_and_iff_int
% 0.29/0.59  thf(fact_7783_pi__less__4,axiom,
% 0.29/0.59      ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % pi_less_4
% 0.29/0.59  thf(fact_7784_pi__ge__two,axiom,
% 0.29/0.59      ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 0.29/0.59  
% 0.29/0.59  % pi_ge_two
% 0.29/0.59  thf(fact_7785_pi__half__neq__two,axiom,
% 0.29/0.59      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.59     != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % pi_half_neq_two
% 0.29/0.59  thf(fact_7786_one__and__eq,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
% 0.29/0.59        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % one_and_eq
% 0.29/0.59  thf(fact_7787_one__and__eq,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
% 0.29/0.59        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % one_and_eq
% 0.29/0.59  thf(fact_7788_one__and__eq,axiom,
% 0.29/0.59      ! [A: nat] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
% 0.29/0.59        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % one_and_eq
% 0.29/0.59  thf(fact_7789_and__one__eq,axiom,
% 0.29/0.59      ! [A: code_integer] :
% 0.29/0.59        ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
% 0.29/0.59        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_one_eq
% 0.29/0.59  thf(fact_7790_and__one__eq,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
% 0.29/0.59        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_one_eq
% 0.29/0.59  thf(fact_7791_and__one__eq,axiom,
% 0.29/0.59      ! [A: nat] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
% 0.29/0.59        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_one_eq
% 0.29/0.59  thf(fact_7792_real__less__rsqrt,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 0.29/0.59       => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_less_rsqrt
% 0.29/0.59  thf(fact_7793_sqrt__le__D,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
% 0.29/0.59       => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sqrt_le_D
% 0.29/0.59  thf(fact_7794_real__le__rsqrt,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 0.29/0.59       => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_le_rsqrt
% 0.29/0.59  thf(fact_7795_pi__half__neq__zero,axiom,
% 0.29/0.59      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.59     != zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % pi_half_neq_zero
% 0.29/0.59  thf(fact_7796_pi__half__less__two,axiom,
% 0.29/0.59      ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.29/0.59  
% 0.29/0.59  % pi_half_less_two
% 0.29/0.59  thf(fact_7797_pi__half__le__two,axiom,
% 0.29/0.59      ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.29/0.59  
% 0.29/0.59  % pi_half_le_two
% 0.29/0.59  thf(fact_7798_and__exp__eq__0__iff__not__bit,axiom,
% 0.29/0.59      ! [A: int,N: nat] :
% 0.29/0.59        ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.59          = zero_zero_int )
% 0.29/0.59        = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_exp_eq_0_iff_not_bit
% 0.29/0.59  thf(fact_7799_and__exp__eq__0__iff__not__bit,axiom,
% 0.29/0.59      ! [A: nat,N: nat] :
% 0.29/0.59        ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.59          = zero_zero_nat )
% 0.29/0.59        = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_exp_eq_0_iff_not_bit
% 0.29/0.59  thf(fact_7800_real__sqrt__unique,axiom,
% 0.29/0.59      ! [Y: real,X: real] :
% 0.29/0.59        ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.59          = X )
% 0.29/0.59       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.59         => ( ( sqrt @ X )
% 0.29/0.59            = Y ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_unique
% 0.29/0.59  thf(fact_7801_real__le__lsqrt,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.59         => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.59           => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_le_lsqrt
% 0.29/0.59  thf(fact_7802_lemma__real__divide__sqrt__less,axiom,
% 0.29/0.59      ! [U: real] :
% 0.29/0.59        ( ( ord_less_real @ zero_zero_real @ U )
% 0.29/0.59       => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.29/0.59  
% 0.29/0.59  % lemma_real_divide_sqrt_less
% 0.29/0.59  thf(fact_7803_real__sqrt__sum__squares__eq__cancel,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.59          = X )
% 0.29/0.59       => ( Y = zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_sum_squares_eq_cancel
% 0.29/0.59  thf(fact_7804_real__sqrt__sum__squares__eq__cancel2,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.59          = Y )
% 0.29/0.59       => ( X = zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_sum_squares_eq_cancel2
% 0.29/0.59  thf(fact_7805_real__sqrt__sum__squares__ge1,axiom,
% 0.29/0.59      ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_sum_squares_ge1
% 0.29/0.59  thf(fact_7806_real__sqrt__sum__squares__ge2,axiom,
% 0.29/0.59      ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_sum_squares_ge2
% 0.29/0.59  thf(fact_7807_real__sqrt__sum__squares__triangle__ineq,axiom,
% 0.29/0.59      ! [A: real,C: real,B: 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 @ B @ 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 @ B @ ( 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 ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_sum_squares_triangle_ineq
% 0.29/0.59  thf(fact_7808_sqrt__ge__absD,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
% 0.29/0.59       => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sqrt_ge_absD
% 0.29/0.59  thf(fact_7809_pi__half__gt__zero,axiom,
% 0.29/0.59      ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % pi_half_gt_zero
% 0.29/0.59  thf(fact_7810_pi__half__ge__zero,axiom,
% 0.29/0.59      ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % pi_half_ge_zero
% 0.29/0.59  thf(fact_7811_m2pi__less__pi,axiom,
% 0.29/0.59      ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 0.29/0.59  
% 0.29/0.59  % m2pi_less_pi
% 0.29/0.59  thf(fact_7812_arctan__ubound,axiom,
% 0.29/0.59      ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % arctan_ubound
% 0.29/0.59  thf(fact_7813_arctan__one,axiom,
% 0.29/0.59      ( ( arctan @ one_one_real )
% 0.29/0.59      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % arctan_one
% 0.29/0.59  thf(fact_7814_real__less__lsqrt,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.59         => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.59           => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_less_lsqrt
% 0.29/0.59  thf(fact_7815_sqrt__sum__squares__le__sum,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.59         => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sqrt_sum_squares_le_sum
% 0.29/0.59  thf(fact_7816_ln__sqrt,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( ln_ln_real @ ( sqrt @ X ) )
% 0.29/0.59          = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % ln_sqrt
% 0.29/0.59  thf(fact_7817_sqrt__sum__squares__le__sum__abs,axiom,
% 0.29/0.59      ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sqrt_sum_squares_le_sum_abs
% 0.29/0.59  thf(fact_7818_real__sqrt__ge__abs2,axiom,
% 0.29/0.59      ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_ge_abs2
% 0.29/0.59  thf(fact_7819_real__sqrt__ge__abs1,axiom,
% 0.29/0.59      ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_ge_abs1
% 0.29/0.59  thf(fact_7820_sqrt__even__pow2,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.59       => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.59          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sqrt_even_pow2
% 0.29/0.59  thf(fact_7821_minus__pi__half__less__zero,axiom,
% 0.29/0.59      ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 0.29/0.59  
% 0.29/0.59  % minus_pi_half_less_zero
% 0.29/0.59  thf(fact_7822_arctan__bounded,axiom,
% 0.29/0.59      ! [Y: real] :
% 0.29/0.59        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 0.29/0.59        & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % arctan_bounded
% 0.29/0.59  thf(fact_7823_arctan__lbound,axiom,
% 0.29/0.59      ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% 0.29/0.59  
% 0.29/0.59  % arctan_lbound
% 0.29/0.59  thf(fact_7824_arsinh__real__aux,axiom,
% 0.29/0.59      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % arsinh_real_aux
% 0.29/0.59  thf(fact_7825_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 0.29/0.59      ! [X: real,Y: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_sum_squares_mult_ge_zero
% 0.29/0.59  thf(fact_7826_real__sqrt__power__even,axiom,
% 0.29/0.59      ! [N: nat,X: real] :
% 0.29/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.59       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59         => ( ( power_power_real @ ( sqrt @ X ) @ N )
% 0.29/0.59            = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_power_even
% 0.29/0.59  thf(fact_7827_arith__geo__mean__sqrt,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.59         => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % arith_geo_mean_sqrt
% 0.29/0.59  thf(fact_7828_powr__half__sqrt,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59       => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.59          = ( sqrt @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % powr_half_sqrt
% 0.29/0.59  thf(fact_7829_arsinh__real__def,axiom,
% 0.29/0.59      ( arsinh_real
% 0.29/0.59      = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % arsinh_real_def
% 0.29/0.59  thf(fact_7830_and__int__rec,axiom,
% 0.29/0.59      ( bit_se725231765392027082nd_int
% 0.29/0.59      = ( ^ [K2: int,L3: int] :
% 0.29/0.59            ( plus_plus_int
% 0.29/0.59            @ ( zero_n2684676970156552555ol_int
% 0.29/0.59              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.29/0.59                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 0.29/0.59            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_int_rec
% 0.29/0.59  thf(fact_7831_cos__x__y__le__one,axiom,
% 0.29/0.59      ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_x_y_le_one
% 0.29/0.59  thf(fact_7832_real__sqrt__sum__squares__less,axiom,
% 0.29/0.59      ! [X: real,U: real,Y: real] :
% 0.29/0.59        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.29/0.59       => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.29/0.59         => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % real_sqrt_sum_squares_less
% 0.29/0.59  thf(fact_7833_arcosh__real__def,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ord_less_eq_real @ one_one_real @ X )
% 0.29/0.59       => ( ( arcosh_real @ X )
% 0.29/0.59          = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % arcosh_real_def
% 0.29/0.59  thf(fact_7834_sqrt__sum__squares__half__less,axiom,
% 0.29/0.59      ! [X: real,U: real,Y: real] :
% 0.29/0.59        ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.59       => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.59         => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.59           => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.59             => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sqrt_sum_squares_half_less
% 0.29/0.59  thf(fact_7835_sin__cos__npi,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( 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 ) ) ) )
% 0.29/0.59        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_cos_npi
% 0.29/0.59  thf(fact_7836_and__int_Osimps,axiom,
% 0.29/0.59      ( bit_se725231765392027082nd_int
% 0.29/0.59      = ( ^ [K2: int,L3: int] :
% 0.29/0.59            ( if_int
% 0.29/0.59            @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.29/0.59              & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.29/0.59            @ ( uminus_uminus_int
% 0.29/0.59              @ ( zero_n2684676970156552555ol_int
% 0.29/0.59                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.29/0.59                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
% 0.29/0.59            @ ( plus_plus_int
% 0.29/0.59              @ ( zero_n2684676970156552555ol_int
% 0.29/0.59                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.29/0.59                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 0.29/0.59              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_int.simps
% 0.29/0.59  thf(fact_7837_and__int_Oelims,axiom,
% 0.29/0.59      ! [X: int,Xa: int,Y: int] :
% 0.29/0.59        ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 0.29/0.59          = Y )
% 0.29/0.59       => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.29/0.59              & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.29/0.59           => ( Y
% 0.29/0.59              = ( uminus_uminus_int
% 0.29/0.59                @ ( zero_n2684676970156552555ol_int
% 0.29/0.59                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.29/0.59                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.29/0.59          & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.29/0.59                & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.29/0.59           => ( Y
% 0.29/0.59              = ( plus_plus_int
% 0.29/0.59                @ ( zero_n2684676970156552555ol_int
% 0.29/0.59                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.29/0.59                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.29/0.59                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_int.elims
% 0.29/0.59  thf(fact_7838_cos__pi__eq__zero,axiom,
% 0.29/0.59      ! [M: nat] :
% 0.29/0.59        ( ( 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 ) ) ) )
% 0.29/0.59        = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_pi_eq_zero
% 0.29/0.59  thf(fact_7839_divmod__BitM__2__eq,axiom,
% 0.29/0.59      ! [M: num] :
% 0.29/0.59        ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 0.29/0.59        = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % divmod_BitM_2_eq
% 0.29/0.59  thf(fact_7840_empty__iff,axiom,
% 0.29/0.59      ! [C: complex] :
% 0.29/0.59        ~ ( member_complex @ C @ bot_bot_set_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_iff
% 0.29/0.59  thf(fact_7841_empty__iff,axiom,
% 0.29/0.59      ! [C: set_nat] :
% 0.29/0.59        ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_iff
% 0.29/0.59  thf(fact_7842_empty__iff,axiom,
% 0.29/0.59      ! [C: real] :
% 0.29/0.59        ~ ( member_real @ C @ bot_bot_set_real ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_iff
% 0.29/0.59  thf(fact_7843_empty__iff,axiom,
% 0.29/0.59      ! [C: $o] :
% 0.29/0.59        ~ ( member_o @ C @ bot_bot_set_o ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_iff
% 0.29/0.59  thf(fact_7844_empty__iff,axiom,
% 0.29/0.59      ! [C: nat] :
% 0.29/0.59        ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_iff
% 0.29/0.59  thf(fact_7845_empty__iff,axiom,
% 0.29/0.59      ! [C: int] :
% 0.29/0.59        ~ ( member_int @ C @ bot_bot_set_int ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_iff
% 0.29/0.59  thf(fact_7846_all__not__in__conv,axiom,
% 0.29/0.59      ! [A2: set_complex] :
% 0.29/0.59        ( ( ! [X4: complex] :
% 0.29/0.59              ~ ( member_complex @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_complex ) ) ).
% 0.29/0.59  
% 0.29/0.59  % all_not_in_conv
% 0.29/0.59  thf(fact_7847_all__not__in__conv,axiom,
% 0.29/0.59      ! [A2: set_set_nat] :
% 0.29/0.59        ( ( ! [X4: set_nat] :
% 0.29/0.59              ~ ( member_set_nat @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % all_not_in_conv
% 0.29/0.59  thf(fact_7848_all__not__in__conv,axiom,
% 0.29/0.59      ! [A2: set_real] :
% 0.29/0.59        ( ( ! [X4: real] :
% 0.29/0.59              ~ ( member_real @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % all_not_in_conv
% 0.29/0.59  thf(fact_7849_all__not__in__conv,axiom,
% 0.29/0.59      ! [A2: set_o] :
% 0.29/0.59        ( ( ! [X4: $o] :
% 0.29/0.59              ~ ( member_o @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % all_not_in_conv
% 0.29/0.59  thf(fact_7850_all__not__in__conv,axiom,
% 0.29/0.59      ! [A2: set_nat] :
% 0.29/0.59        ( ( ! [X4: nat] :
% 0.29/0.59              ~ ( member_nat @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % all_not_in_conv
% 0.29/0.59  thf(fact_7851_all__not__in__conv,axiom,
% 0.29/0.59      ! [A2: set_int] :
% 0.29/0.59        ( ( ! [X4: int] :
% 0.29/0.59              ~ ( member_int @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % all_not_in_conv
% 0.29/0.59  thf(fact_7852_Collect__empty__eq,axiom,
% 0.29/0.59      ! [P: complex > $o] :
% 0.29/0.59        ( ( ( collect_complex @ P )
% 0.29/0.59          = bot_bot_set_complex )
% 0.29/0.59        = ( ! [X4: complex] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Collect_empty_eq
% 0.29/0.59  thf(fact_7853_Collect__empty__eq,axiom,
% 0.29/0.59      ! [P: set_nat > $o] :
% 0.29/0.59        ( ( ( collect_set_nat @ P )
% 0.29/0.59          = bot_bot_set_set_nat )
% 0.29/0.59        = ( ! [X4: set_nat] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Collect_empty_eq
% 0.29/0.59  thf(fact_7854_Collect__empty__eq,axiom,
% 0.29/0.59      ! [P: real > $o] :
% 0.29/0.59        ( ( ( collect_real @ P )
% 0.29/0.59          = bot_bot_set_real )
% 0.29/0.59        = ( ! [X4: real] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Collect_empty_eq
% 0.29/0.59  thf(fact_7855_Collect__empty__eq,axiom,
% 0.29/0.59      ! [P: $o > $o] :
% 0.29/0.59        ( ( ( collect_o @ P )
% 0.29/0.59          = bot_bot_set_o )
% 0.29/0.59        = ( ! [X4: $o] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Collect_empty_eq
% 0.29/0.59  thf(fact_7856_Collect__empty__eq,axiom,
% 0.29/0.59      ! [P: nat > $o] :
% 0.29/0.59        ( ( ( collect_nat @ P )
% 0.29/0.59          = bot_bot_set_nat )
% 0.29/0.59        = ( ! [X4: nat] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Collect_empty_eq
% 0.29/0.59  thf(fact_7857_Collect__empty__eq,axiom,
% 0.29/0.59      ! [P: int > $o] :
% 0.29/0.59        ( ( ( collect_int @ P )
% 0.29/0.59          = bot_bot_set_int )
% 0.29/0.59        = ( ! [X4: int] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Collect_empty_eq
% 0.29/0.59  thf(fact_7858_empty__Collect__eq,axiom,
% 0.29/0.59      ! [P: complex > $o] :
% 0.29/0.59        ( ( bot_bot_set_complex
% 0.29/0.59          = ( collect_complex @ P ) )
% 0.29/0.59        = ( ! [X4: complex] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_Collect_eq
% 0.29/0.59  thf(fact_7859_empty__Collect__eq,axiom,
% 0.29/0.59      ! [P: set_nat > $o] :
% 0.29/0.59        ( ( bot_bot_set_set_nat
% 0.29/0.59          = ( collect_set_nat @ P ) )
% 0.29/0.59        = ( ! [X4: set_nat] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_Collect_eq
% 0.29/0.59  thf(fact_7860_empty__Collect__eq,axiom,
% 0.29/0.59      ! [P: real > $o] :
% 0.29/0.59        ( ( bot_bot_set_real
% 0.29/0.59          = ( collect_real @ P ) )
% 0.29/0.59        = ( ! [X4: real] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_Collect_eq
% 0.29/0.59  thf(fact_7861_empty__Collect__eq,axiom,
% 0.29/0.59      ! [P: $o > $o] :
% 0.29/0.59        ( ( bot_bot_set_o
% 0.29/0.59          = ( collect_o @ P ) )
% 0.29/0.59        = ( ! [X4: $o] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_Collect_eq
% 0.29/0.59  thf(fact_7862_empty__Collect__eq,axiom,
% 0.29/0.59      ! [P: nat > $o] :
% 0.29/0.59        ( ( bot_bot_set_nat
% 0.29/0.59          = ( collect_nat @ P ) )
% 0.29/0.59        = ( ! [X4: nat] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_Collect_eq
% 0.29/0.59  thf(fact_7863_empty__Collect__eq,axiom,
% 0.29/0.59      ! [P: int > $o] :
% 0.29/0.59        ( ( bot_bot_set_int
% 0.29/0.59          = ( collect_int @ P ) )
% 0.29/0.59        = ( ! [X4: int] :
% 0.29/0.59              ~ ( P @ X4 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_Collect_eq
% 0.29/0.59  thf(fact_7864_image__is__empty,axiom,
% 0.29/0.59      ! [F: real > real,A2: set_real] :
% 0.29/0.59        ( ( ( image_real_real @ F @ A2 )
% 0.29/0.59          = bot_bot_set_real )
% 0.29/0.59        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % image_is_empty
% 0.29/0.59  thf(fact_7865_image__is__empty,axiom,
% 0.29/0.59      ! [F: $o > real,A2: set_o] :
% 0.29/0.59        ( ( ( image_o_real @ F @ A2 )
% 0.29/0.59          = bot_bot_set_real )
% 0.29/0.59        = ( A2 = bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % image_is_empty
% 0.29/0.59  thf(fact_7866_image__is__empty,axiom,
% 0.29/0.59      ! [F: nat > real,A2: set_nat] :
% 0.29/0.59        ( ( ( image_nat_real @ F @ A2 )
% 0.29/0.59          = bot_bot_set_real )
% 0.29/0.59        = ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % image_is_empty
% 0.29/0.59  thf(fact_7867_image__is__empty,axiom,
% 0.29/0.59      ! [F: int > real,A2: set_int] :
% 0.29/0.59        ( ( ( image_int_real @ F @ A2 )
% 0.29/0.59          = bot_bot_set_real )
% 0.29/0.59        = ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % image_is_empty
% 0.29/0.59  thf(fact_7868_image__is__empty,axiom,
% 0.29/0.59      ! [F: real > $o,A2: set_real] :
% 0.29/0.59        ( ( ( image_real_o @ F @ A2 )
% 0.29/0.59          = bot_bot_set_o )
% 0.29/0.59        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % image_is_empty
% 0.29/0.59  thf(fact_7869_image__is__empty,axiom,
% 0.29/0.59      ! [F: $o > $o,A2: set_o] :
% 0.29/0.59        ( ( ( image_o_o @ F @ A2 )
% 0.29/0.59          = bot_bot_set_o )
% 0.29/0.59        = ( A2 = bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % image_is_empty
% 0.29/0.59  thf(fact_7870_image__is__empty,axiom,
% 0.29/0.59      ! [F: nat > $o,A2: set_nat] :
% 0.29/0.59        ( ( ( image_nat_o @ F @ A2 )
% 0.29/0.59          = bot_bot_set_o )
% 0.29/0.59        = ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % image_is_empty
% 0.29/0.59  thf(fact_7871_image__is__empty,axiom,
% 0.29/0.59      ! [F: int > $o,A2: set_int] :
% 0.29/0.59        ( ( ( image_int_o @ F @ A2 )
% 0.29/0.59          = bot_bot_set_o )
% 0.29/0.59        = ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % image_is_empty
% 0.29/0.59  thf(fact_7872_image__is__empty,axiom,
% 0.29/0.59      ! [F: real > nat,A2: set_real] :
% 0.29/0.59        ( ( ( image_real_nat @ F @ A2 )
% 0.29/0.59          = bot_bot_set_nat )
% 0.29/0.59        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % image_is_empty
% 0.29/0.59  thf(fact_7873_image__is__empty,axiom,
% 0.29/0.59      ! [F: $o > nat,A2: set_o] :
% 0.29/0.59        ( ( ( image_o_nat @ F @ A2 )
% 0.29/0.59          = bot_bot_set_nat )
% 0.29/0.59        = ( A2 = bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % image_is_empty
% 0.29/0.59  thf(fact_7874_empty__is__image,axiom,
% 0.29/0.59      ! [F: real > real,A2: set_real] :
% 0.29/0.59        ( ( bot_bot_set_real
% 0.29/0.59          = ( image_real_real @ F @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_is_image
% 0.29/0.59  thf(fact_7875_empty__is__image,axiom,
% 0.29/0.59      ! [F: $o > real,A2: set_o] :
% 0.29/0.59        ( ( bot_bot_set_real
% 0.29/0.59          = ( image_o_real @ F @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_is_image
% 0.29/0.59  thf(fact_7876_empty__is__image,axiom,
% 0.29/0.59      ! [F: nat > real,A2: set_nat] :
% 0.29/0.59        ( ( bot_bot_set_real
% 0.29/0.59          = ( image_nat_real @ F @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_is_image
% 0.29/0.59  thf(fact_7877_empty__is__image,axiom,
% 0.29/0.59      ! [F: int > real,A2: set_int] :
% 0.29/0.59        ( ( bot_bot_set_real
% 0.29/0.59          = ( image_int_real @ F @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_is_image
% 0.29/0.59  thf(fact_7878_empty__is__image,axiom,
% 0.29/0.59      ! [F: real > $o,A2: set_real] :
% 0.29/0.59        ( ( bot_bot_set_o
% 0.29/0.59          = ( image_real_o @ F @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_is_image
% 0.29/0.59  thf(fact_7879_empty__is__image,axiom,
% 0.29/0.59      ! [F: $o > $o,A2: set_o] :
% 0.29/0.59        ( ( bot_bot_set_o
% 0.29/0.59          = ( image_o_o @ F @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_is_image
% 0.29/0.59  thf(fact_7880_empty__is__image,axiom,
% 0.29/0.59      ! [F: nat > $o,A2: set_nat] :
% 0.29/0.59        ( ( bot_bot_set_o
% 0.29/0.59          = ( image_nat_o @ F @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_is_image
% 0.29/0.59  thf(fact_7881_empty__is__image,axiom,
% 0.29/0.59      ! [F: int > $o,A2: set_int] :
% 0.29/0.59        ( ( bot_bot_set_o
% 0.29/0.59          = ( image_int_o @ F @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_is_image
% 0.29/0.59  thf(fact_7882_empty__is__image,axiom,
% 0.29/0.59      ! [F: real > nat,A2: set_real] :
% 0.29/0.59        ( ( bot_bot_set_nat
% 0.29/0.59          = ( image_real_nat @ F @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_is_image
% 0.29/0.59  thf(fact_7883_empty__is__image,axiom,
% 0.29/0.59      ! [F: $o > nat,A2: set_o] :
% 0.29/0.59        ( ( bot_bot_set_nat
% 0.29/0.59          = ( image_o_nat @ F @ A2 ) )
% 0.29/0.59        = ( A2 = bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_is_image
% 0.29/0.59  thf(fact_7884_image__empty,axiom,
% 0.29/0.59      ! [F: real > real] :
% 0.29/0.59        ( ( image_real_real @ F @ bot_bot_set_real )
% 0.29/0.59        = bot_bot_set_real ) ).
% 0.29/0.59  
% 0.29/0.59  % image_empty
% 0.29/0.59  thf(fact_7885_image__empty,axiom,
% 0.29/0.59      ! [F: real > $o] :
% 0.29/0.59        ( ( image_real_o @ F @ bot_bot_set_real )
% 0.29/0.59        = bot_bot_set_o ) ).
% 0.29/0.59  
% 0.29/0.59  % image_empty
% 0.29/0.59  thf(fact_7886_image__empty,axiom,
% 0.29/0.59      ! [F: real > nat] :
% 0.29/0.59        ( ( image_real_nat @ F @ bot_bot_set_real )
% 0.29/0.59        = bot_bot_set_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % image_empty
% 0.29/0.59  thf(fact_7887_image__empty,axiom,
% 0.29/0.59      ! [F: real > int] :
% 0.29/0.59        ( ( image_real_int @ F @ bot_bot_set_real )
% 0.29/0.59        = bot_bot_set_int ) ).
% 0.29/0.59  
% 0.29/0.59  % image_empty
% 0.29/0.59  thf(fact_7888_image__empty,axiom,
% 0.29/0.59      ! [F: $o > real] :
% 0.29/0.59        ( ( image_o_real @ F @ bot_bot_set_o )
% 0.29/0.59        = bot_bot_set_real ) ).
% 0.29/0.59  
% 0.29/0.59  % image_empty
% 0.29/0.59  thf(fact_7889_image__empty,axiom,
% 0.29/0.59      ! [F: $o > $o] :
% 0.29/0.59        ( ( image_o_o @ F @ bot_bot_set_o )
% 0.29/0.59        = bot_bot_set_o ) ).
% 0.29/0.59  
% 0.29/0.59  % image_empty
% 0.29/0.59  thf(fact_7890_image__empty,axiom,
% 0.29/0.59      ! [F: $o > nat] :
% 0.29/0.59        ( ( image_o_nat @ F @ bot_bot_set_o )
% 0.29/0.59        = bot_bot_set_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % image_empty
% 0.29/0.59  thf(fact_7891_image__empty,axiom,
% 0.29/0.59      ! [F: $o > int] :
% 0.29/0.59        ( ( image_o_int @ F @ bot_bot_set_o )
% 0.29/0.59        = bot_bot_set_int ) ).
% 0.29/0.59  
% 0.29/0.59  % image_empty
% 0.29/0.59  thf(fact_7892_image__empty,axiom,
% 0.29/0.59      ! [F: nat > real] :
% 0.29/0.59        ( ( image_nat_real @ F @ bot_bot_set_nat )
% 0.29/0.59        = bot_bot_set_real ) ).
% 0.29/0.59  
% 0.29/0.59  % image_empty
% 0.29/0.59  thf(fact_7893_image__empty,axiom,
% 0.29/0.59      ! [F: nat > $o] :
% 0.29/0.59        ( ( image_nat_o @ F @ bot_bot_set_nat )
% 0.29/0.59        = bot_bot_set_o ) ).
% 0.29/0.59  
% 0.29/0.59  % image_empty
% 0.29/0.59  thf(fact_7894_subset__empty,axiom,
% 0.29/0.59      ! [A2: set_real] :
% 0.29/0.59        ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 0.29/0.59        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_empty
% 0.29/0.59  thf(fact_7895_subset__empty,axiom,
% 0.29/0.59      ! [A2: set_o] :
% 0.29/0.59        ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
% 0.29/0.59        = ( A2 = bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_empty
% 0.29/0.59  thf(fact_7896_subset__empty,axiom,
% 0.29/0.59      ! [A2: set_nat] :
% 0.29/0.59        ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 0.29/0.59        = ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_empty
% 0.29/0.59  thf(fact_7897_subset__empty,axiom,
% 0.29/0.59      ! [A2: set_int] :
% 0.29/0.59        ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 0.29/0.59        = ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_empty
% 0.29/0.59  thf(fact_7898_empty__subsetI,axiom,
% 0.29/0.59      ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_subsetI
% 0.29/0.59  thf(fact_7899_empty__subsetI,axiom,
% 0.29/0.59      ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_subsetI
% 0.29/0.59  thf(fact_7900_empty__subsetI,axiom,
% 0.29/0.59      ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_subsetI
% 0.29/0.59  thf(fact_7901_empty__subsetI,axiom,
% 0.29/0.59      ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_subsetI
% 0.29/0.59  thf(fact_7902_singletonI,axiom,
% 0.29/0.59      ! [A: vEBT_VEBT] : ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singletonI
% 0.29/0.59  thf(fact_7903_singletonI,axiom,
% 0.29/0.59      ! [A: complex] : ( member_complex @ A @ ( insert_complex @ A @ bot_bot_set_complex ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singletonI
% 0.29/0.59  thf(fact_7904_singletonI,axiom,
% 0.29/0.59      ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singletonI
% 0.29/0.59  thf(fact_7905_singletonI,axiom,
% 0.29/0.59      ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singletonI
% 0.29/0.59  thf(fact_7906_singletonI,axiom,
% 0.29/0.59      ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singletonI
% 0.29/0.59  thf(fact_7907_singletonI,axiom,
% 0.29/0.59      ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singletonI
% 0.29/0.59  thf(fact_7908_singletonI,axiom,
% 0.29/0.59      ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singletonI
% 0.29/0.59  thf(fact_7909_Diff__empty,axiom,
% 0.29/0.59      ! [A2: set_real] :
% 0.29/0.59        ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 0.29/0.59        = A2 ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_empty
% 0.29/0.59  thf(fact_7910_Diff__empty,axiom,
% 0.29/0.59      ! [A2: set_o] :
% 0.29/0.59        ( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
% 0.29/0.59        = A2 ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_empty
% 0.29/0.59  thf(fact_7911_Diff__empty,axiom,
% 0.29/0.59      ! [A2: set_int] :
% 0.29/0.59        ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 0.29/0.59        = A2 ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_empty
% 0.29/0.59  thf(fact_7912_Diff__empty,axiom,
% 0.29/0.59      ! [A2: set_nat] :
% 0.29/0.59        ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 0.29/0.59        = A2 ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_empty
% 0.29/0.59  thf(fact_7913_empty__Diff,axiom,
% 0.29/0.59      ! [A2: set_real] :
% 0.29/0.59        ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 0.29/0.59        = bot_bot_set_real ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_Diff
% 0.29/0.59  thf(fact_7914_empty__Diff,axiom,
% 0.29/0.59      ! [A2: set_o] :
% 0.29/0.59        ( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
% 0.29/0.59        = bot_bot_set_o ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_Diff
% 0.29/0.59  thf(fact_7915_empty__Diff,axiom,
% 0.29/0.59      ! [A2: set_int] :
% 0.29/0.59        ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 0.29/0.59        = bot_bot_set_int ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_Diff
% 0.29/0.59  thf(fact_7916_empty__Diff,axiom,
% 0.29/0.59      ! [A2: set_nat] :
% 0.29/0.59        ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 0.29/0.59        = bot_bot_set_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % empty_Diff
% 0.29/0.59  thf(fact_7917_Diff__cancel,axiom,
% 0.29/0.59      ! [A2: set_real] :
% 0.29/0.59        ( ( minus_minus_set_real @ A2 @ A2 )
% 0.29/0.59        = bot_bot_set_real ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_cancel
% 0.29/0.59  thf(fact_7918_Diff__cancel,axiom,
% 0.29/0.59      ! [A2: set_o] :
% 0.29/0.59        ( ( minus_minus_set_o @ A2 @ A2 )
% 0.29/0.59        = bot_bot_set_o ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_cancel
% 0.29/0.59  thf(fact_7919_Diff__cancel,axiom,
% 0.29/0.59      ! [A2: set_int] :
% 0.29/0.59        ( ( minus_minus_set_int @ A2 @ A2 )
% 0.29/0.59        = bot_bot_set_int ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_cancel
% 0.29/0.59  thf(fact_7920_Diff__cancel,axiom,
% 0.29/0.59      ! [A2: set_nat] :
% 0.29/0.59        ( ( minus_minus_set_nat @ A2 @ A2 )
% 0.29/0.59        = bot_bot_set_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_cancel
% 0.29/0.59  thf(fact_7921_sin__zero,axiom,
% 0.29/0.59      ( ( sin_complex @ zero_zero_complex )
% 0.29/0.59      = zero_zero_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_zero
% 0.29/0.59  thf(fact_7922_sin__zero,axiom,
% 0.29/0.59      ( ( sin_real @ zero_zero_real )
% 0.29/0.59      = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_zero
% 0.29/0.59  thf(fact_7923_cos__minus,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( cos_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.59        = ( cos_real @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_minus
% 0.29/0.59  thf(fact_7924_cos__minus,axiom,
% 0.29/0.59      ! [X: complex] :
% 0.29/0.59        ( ( cos_complex @ ( uminus1482373934393186551omplex @ X ) )
% 0.29/0.59        = ( cos_complex @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_minus
% 0.29/0.59  thf(fact_7925_sin__minus,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( sin_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_minus
% 0.29/0.59  thf(fact_7926_sin__minus,axiom,
% 0.29/0.59      ! [X: complex] :
% 0.29/0.59        ( ( sin_complex @ ( uminus1482373934393186551omplex @ X ) )
% 0.29/0.59        = ( uminus1482373934393186551omplex @ ( sin_complex @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_minus
% 0.29/0.59  thf(fact_7927_singleton__insert__inj__eq,axiom,
% 0.29/0.59      ! [B: vEBT_VEBT,A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.29/0.59        ( ( ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT )
% 0.29/0.59          = ( insert_VEBT_VEBT @ A @ A2 ) )
% 0.29/0.59        = ( ( A = B )
% 0.29/0.59          & ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singleton_insert_inj_eq
% 0.29/0.59  thf(fact_7928_singleton__insert__inj__eq,axiom,
% 0.29/0.59      ! [B: real,A: real,A2: set_real] :
% 0.29/0.59        ( ( ( insert_real @ B @ bot_bot_set_real )
% 0.29/0.59          = ( insert_real @ A @ A2 ) )
% 0.29/0.59        = ( ( A = B )
% 0.29/0.59          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singleton_insert_inj_eq
% 0.29/0.59  thf(fact_7929_singleton__insert__inj__eq,axiom,
% 0.29/0.59      ! [B: $o,A: $o,A2: set_o] :
% 0.29/0.59        ( ( ( insert_o @ B @ bot_bot_set_o )
% 0.29/0.59          = ( insert_o @ A @ A2 ) )
% 0.29/0.59        = ( ( A = B )
% 0.29/0.59          & ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singleton_insert_inj_eq
% 0.29/0.59  thf(fact_7930_singleton__insert__inj__eq,axiom,
% 0.29/0.59      ! [B: nat,A: nat,A2: set_nat] :
% 0.29/0.59        ( ( ( insert_nat @ B @ bot_bot_set_nat )
% 0.29/0.59          = ( insert_nat @ A @ A2 ) )
% 0.29/0.59        = ( ( A = B )
% 0.29/0.59          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singleton_insert_inj_eq
% 0.29/0.59  thf(fact_7931_singleton__insert__inj__eq,axiom,
% 0.29/0.59      ! [B: int,A: int,A2: set_int] :
% 0.29/0.59        ( ( ( insert_int @ B @ bot_bot_set_int )
% 0.29/0.59          = ( insert_int @ A @ A2 ) )
% 0.29/0.59        = ( ( A = B )
% 0.29/0.59          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singleton_insert_inj_eq
% 0.29/0.59  thf(fact_7932_singleton__insert__inj__eq_H,axiom,
% 0.29/0.59      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT] :
% 0.29/0.59        ( ( ( insert_VEBT_VEBT @ A @ A2 )
% 0.29/0.59          = ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
% 0.29/0.59        = ( ( A = B )
% 0.29/0.59          & ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singleton_insert_inj_eq'
% 0.29/0.59  thf(fact_7933_singleton__insert__inj__eq_H,axiom,
% 0.29/0.59      ! [A: real,A2: set_real,B: real] :
% 0.29/0.59        ( ( ( insert_real @ A @ A2 )
% 0.29/0.59          = ( insert_real @ B @ bot_bot_set_real ) )
% 0.29/0.59        = ( ( A = B )
% 0.29/0.59          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singleton_insert_inj_eq'
% 0.29/0.59  thf(fact_7934_singleton__insert__inj__eq_H,axiom,
% 0.29/0.59      ! [A: $o,A2: set_o,B: $o] :
% 0.29/0.59        ( ( ( insert_o @ A @ A2 )
% 0.29/0.59          = ( insert_o @ B @ bot_bot_set_o ) )
% 0.29/0.59        = ( ( A = B )
% 0.29/0.59          & ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singleton_insert_inj_eq'
% 0.29/0.59  thf(fact_7935_singleton__insert__inj__eq_H,axiom,
% 0.29/0.59      ! [A: nat,A2: set_nat,B: nat] :
% 0.29/0.59        ( ( ( insert_nat @ A @ A2 )
% 0.29/0.59          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.29/0.59        = ( ( A = B )
% 0.29/0.59          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singleton_insert_inj_eq'
% 0.29/0.59  thf(fact_7936_singleton__insert__inj__eq_H,axiom,
% 0.29/0.59      ! [A: int,A2: set_int,B: int] :
% 0.29/0.59        ( ( ( insert_int @ A @ A2 )
% 0.29/0.59          = ( insert_int @ B @ bot_bot_set_int ) )
% 0.29/0.59        = ( ( A = B )
% 0.29/0.59          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % singleton_insert_inj_eq'
% 0.29/0.59  thf(fact_7937_cos__zero,axiom,
% 0.29/0.59      ( ( cos_complex @ zero_zero_complex )
% 0.29/0.59      = one_one_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_zero
% 0.29/0.59  thf(fact_7938_cos__zero,axiom,
% 0.29/0.59      ( ( cos_real @ zero_zero_real )
% 0.29/0.59      = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_zero
% 0.29/0.59  thf(fact_7939_Diff__eq__empty__iff,axiom,
% 0.29/0.59      ! [A2: set_real,B2: set_real] :
% 0.29/0.59        ( ( ( minus_minus_set_real @ A2 @ B2 )
% 0.29/0.59          = bot_bot_set_real )
% 0.29/0.59        = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_eq_empty_iff
% 0.29/0.59  thf(fact_7940_Diff__eq__empty__iff,axiom,
% 0.29/0.59      ! [A2: set_o,B2: set_o] :
% 0.29/0.59        ( ( ( minus_minus_set_o @ A2 @ B2 )
% 0.29/0.59          = bot_bot_set_o )
% 0.29/0.59        = ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_eq_empty_iff
% 0.29/0.59  thf(fact_7941_Diff__eq__empty__iff,axiom,
% 0.29/0.59      ! [A2: set_nat,B2: set_nat] :
% 0.29/0.59        ( ( ( minus_minus_set_nat @ A2 @ B2 )
% 0.29/0.59          = bot_bot_set_nat )
% 0.29/0.59        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_eq_empty_iff
% 0.29/0.59  thf(fact_7942_Diff__eq__empty__iff,axiom,
% 0.29/0.59      ! [A2: set_int,B2: set_int] :
% 0.29/0.59        ( ( ( minus_minus_set_int @ A2 @ B2 )
% 0.29/0.59          = bot_bot_set_int )
% 0.29/0.59        = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Diff_eq_empty_iff
% 0.29/0.59  thf(fact_7943_insert__Diff__single,axiom,
% 0.29/0.59      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.29/0.59        ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
% 0.29/0.59        = ( insert_VEBT_VEBT @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % insert_Diff_single
% 0.29/0.59  thf(fact_7944_insert__Diff__single,axiom,
% 0.29/0.59      ! [A: real,A2: set_real] :
% 0.29/0.59        ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.29/0.59        = ( insert_real @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % insert_Diff_single
% 0.29/0.59  thf(fact_7945_insert__Diff__single,axiom,
% 0.29/0.59      ! [A: $o,A2: set_o] :
% 0.29/0.59        ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
% 0.29/0.59        = ( insert_o @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % insert_Diff_single
% 0.29/0.59  thf(fact_7946_insert__Diff__single,axiom,
% 0.29/0.59      ! [A: int,A2: set_int] :
% 0.29/0.59        ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.29/0.59        = ( insert_int @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % insert_Diff_single
% 0.29/0.59  thf(fact_7947_insert__Diff__single,axiom,
% 0.29/0.59      ! [A: nat,A2: set_nat] :
% 0.29/0.59        ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.29/0.59        = ( insert_nat @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % insert_Diff_single
% 0.29/0.59  thf(fact_7948_Max__singleton,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( lattic4275903605611617917x_real @ ( insert_real @ X @ bot_bot_set_real ) )
% 0.29/0.59        = X ) ).
% 0.29/0.59  
% 0.29/0.59  % Max_singleton
% 0.29/0.59  thf(fact_7949_Max__singleton,axiom,
% 0.29/0.59      ! [X: $o] :
% 0.29/0.59        ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.29/0.59        = X ) ).
% 0.29/0.59  
% 0.29/0.59  % Max_singleton
% 0.29/0.59  thf(fact_7950_Max__singleton,axiom,
% 0.29/0.59      ! [X: nat] :
% 0.29/0.59        ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.29/0.59        = X ) ).
% 0.29/0.59  
% 0.29/0.59  % Max_singleton
% 0.29/0.59  thf(fact_7951_Max__singleton,axiom,
% 0.29/0.59      ! [X: int] :
% 0.29/0.59        ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.29/0.59        = X ) ).
% 0.29/0.59  
% 0.29/0.59  % Max_singleton
% 0.29/0.59  thf(fact_7952_sin__pi,axiom,
% 0.29/0.59      ( ( sin_real @ pi )
% 0.29/0.59      = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_pi
% 0.29/0.59  thf(fact_7953_sin__pi__minus,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
% 0.29/0.59        = ( sin_real @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_pi_minus
% 0.29/0.59  thf(fact_7954_dbl__dec__simps_I5_J,axiom,
% 0.29/0.59      ! [K: num] :
% 0.29/0.59        ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.29/0.59        = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % dbl_dec_simps(5)
% 0.29/0.59  thf(fact_7955_dbl__dec__simps_I5_J,axiom,
% 0.29/0.59      ! [K: num] :
% 0.29/0.59        ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
% 0.29/0.59        = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % dbl_dec_simps(5)
% 0.29/0.59  thf(fact_7956_dbl__dec__simps_I5_J,axiom,
% 0.29/0.59      ! [K: num] :
% 0.29/0.59        ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
% 0.29/0.59        = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % dbl_dec_simps(5)
% 0.29/0.59  thf(fact_7957_dbl__dec__simps_I5_J,axiom,
% 0.29/0.59      ! [K: num] :
% 0.29/0.59        ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
% 0.29/0.59        = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % dbl_dec_simps(5)
% 0.29/0.59  thf(fact_7958_subset__Compl__singleton,axiom,
% 0.29/0.59      ! [A2: set_VEBT_VEBT,B: vEBT_VEBT] :
% 0.29/0.59        ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) )
% 0.29/0.59        = ( ~ ( member_VEBT_VEBT @ B @ A2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_Compl_singleton
% 0.29/0.59  thf(fact_7959_subset__Compl__singleton,axiom,
% 0.29/0.59      ! [A2: set_complex,B: complex] :
% 0.29/0.59        ( ( ord_le211207098394363844omplex @ A2 @ ( uminus8566677241136511917omplex @ ( insert_complex @ B @ bot_bot_set_complex ) ) )
% 0.29/0.59        = ( ~ ( member_complex @ B @ A2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_Compl_singleton
% 0.29/0.59  thf(fact_7960_subset__Compl__singleton,axiom,
% 0.29/0.59      ! [A2: set_set_nat,B: set_nat] :
% 0.29/0.59        ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
% 0.29/0.59        = ( ~ ( member_set_nat @ B @ A2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_Compl_singleton
% 0.29/0.59  thf(fact_7961_subset__Compl__singleton,axiom,
% 0.29/0.59      ! [A2: set_real,B: real] :
% 0.29/0.59        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
% 0.29/0.59        = ( ~ ( member_real @ B @ A2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_Compl_singleton
% 0.29/0.59  thf(fact_7962_subset__Compl__singleton,axiom,
% 0.29/0.59      ! [A2: set_o,B: $o] :
% 0.29/0.59        ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.29/0.59        = ( ~ ( member_o @ B @ A2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_Compl_singleton
% 0.29/0.59  thf(fact_7963_subset__Compl__singleton,axiom,
% 0.29/0.59      ! [A2: set_nat,B: nat] :
% 0.29/0.59        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.29/0.59        = ( ~ ( member_nat @ B @ A2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_Compl_singleton
% 0.29/0.59  thf(fact_7964_subset__Compl__singleton,axiom,
% 0.29/0.59      ! [A2: set_int,B: int] :
% 0.29/0.59        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 0.29/0.59        = ( ~ ( member_int @ B @ A2 ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % subset_Compl_singleton
% 0.29/0.59  thf(fact_7965_sin__of__real__pi,axiom,
% 0.29/0.59      ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
% 0.29/0.59      = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_of_real_pi
% 0.29/0.59  thf(fact_7966_sin__of__real__pi,axiom,
% 0.29/0.59      ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 0.29/0.59      = zero_zero_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_of_real_pi
% 0.29/0.59  thf(fact_7967_cos__pi,axiom,
% 0.29/0.59      ( ( cos_real @ pi )
% 0.29/0.59      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_pi
% 0.29/0.59  thf(fact_7968_cos__periodic__pi,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_periodic_pi
% 0.29/0.59  thf(fact_7969_cos__periodic__pi2,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_periodic_pi2
% 0.29/0.59  thf(fact_7970_sin__periodic__pi,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_periodic_pi
% 0.29/0.59  thf(fact_7971_sin__periodic__pi2,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_periodic_pi2
% 0.29/0.59  thf(fact_7972_cos__minus__pi,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_minus_pi
% 0.29/0.59  thf(fact_7973_cos__pi__minus,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_pi_minus
% 0.29/0.59  thf(fact_7974_sin__minus__pi,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_minus_pi
% 0.29/0.59  thf(fact_7975_pred__numeral__simps_I2_J,axiom,
% 0.29/0.59      ! [K: num] :
% 0.29/0.59        ( ( pred_numeral @ ( bit0 @ K ) )
% 0.29/0.59        = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % pred_numeral_simps(2)
% 0.29/0.59  thf(fact_7976_sin__cos__squared__add3,axiom,
% 0.29/0.59      ! [X: complex] :
% 0.29/0.59        ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
% 0.29/0.59        = one_one_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_cos_squared_add3
% 0.29/0.59  thf(fact_7977_sin__cos__squared__add3,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
% 0.29/0.59        = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_cos_squared_add3
% 0.29/0.59  thf(fact_7978_cos__of__real__pi,axiom,
% 0.29/0.59      ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
% 0.29/0.59      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_of_real_pi
% 0.29/0.59  thf(fact_7979_cos__of__real__pi,axiom,
% 0.29/0.59      ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 0.29/0.59      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_of_real_pi
% 0.29/0.59  thf(fact_7980_sin__npi2,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.29/0.59        = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_npi2
% 0.29/0.59  thf(fact_7981_sin__npi,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.29/0.59        = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_npi
% 0.29/0.59  thf(fact_7982_and__nat__numerals_I3_J,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.29/0.59        = zero_zero_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % and_nat_numerals(3)
% 0.29/0.59  thf(fact_7983_and__nat__numerals_I1_J,axiom,
% 0.29/0.59      ! [Y: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.29/0.59        = zero_zero_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % and_nat_numerals(1)
% 0.29/0.59  thf(fact_7984_sin__npi__int,axiom,
% 0.29/0.59      ! [N: int] :
% 0.29/0.59        ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.29/0.59        = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_npi_int
% 0.29/0.59  thf(fact_7985_and__nat__numerals_I4_J,axiom,
% 0.29/0.59      ! [X: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.29/0.59        = one_one_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % and_nat_numerals(4)
% 0.29/0.59  thf(fact_7986_and__nat__numerals_I2_J,axiom,
% 0.29/0.59      ! [Y: num] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.29/0.59        = one_one_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % and_nat_numerals(2)
% 0.29/0.59  thf(fact_7987_cos__pi__half,axiom,
% 0.29/0.59      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.59      = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_pi_half
% 0.29/0.59  thf(fact_7988_sin__two__pi,axiom,
% 0.29/0.59      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.29/0.59      = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_two_pi
% 0.29/0.59  thf(fact_7989_sin__pi__half,axiom,
% 0.29/0.59      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.59      = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_pi_half
% 0.29/0.59  thf(fact_7990_cos__two__pi,axiom,
% 0.29/0.59      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.29/0.59      = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_two_pi
% 0.29/0.59  thf(fact_7991_cos__periodic,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.29/0.59        = ( cos_real @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_periodic
% 0.29/0.59  thf(fact_7992_sin__periodic,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.29/0.59        = ( sin_real @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_periodic
% 0.29/0.59  thf(fact_7993_cos__2pi__minus,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 0.29/0.59        = ( cos_real @ X ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_2pi_minus
% 0.29/0.59  thf(fact_7994_Suc__0__and__eq,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.59        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % Suc_0_and_eq
% 0.29/0.59  thf(fact_7995_and__Suc__0__eq,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.59        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % and_Suc_0_eq
% 0.29/0.59  thf(fact_7996_cos__npi,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.29/0.59        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_npi
% 0.29/0.59  thf(fact_7997_cos__npi2,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.29/0.59        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_npi2
% 0.29/0.59  thf(fact_7998_sin__cos__squared__add,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.59        = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_cos_squared_add
% 0.29/0.59  thf(fact_7999_sin__cos__squared__add,axiom,
% 0.29/0.59      ! [X: complex] :
% 0.29/0.59        ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.59        = one_one_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_cos_squared_add
% 0.29/0.59  thf(fact_8000_sin__cos__squared__add2,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.59        = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_cos_squared_add2
% 0.29/0.59  thf(fact_8001_sin__cos__squared__add2,axiom,
% 0.29/0.59      ! [X: complex] :
% 0.29/0.59        ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.59        = one_one_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_cos_squared_add2
% 0.29/0.59  thf(fact_8002_cos__of__real__pi__half,axiom,
% 0.29/0.59      ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.59      = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_of_real_pi_half
% 0.29/0.59  thf(fact_8003_cos__of__real__pi__half,axiom,
% 0.29/0.59      ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.29/0.59      = zero_zero_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_of_real_pi_half
% 0.29/0.59  thf(fact_8004_sin__of__real__pi__half,axiom,
% 0.29/0.59      ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.59      = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_of_real_pi_half
% 0.29/0.59  thf(fact_8005_sin__of__real__pi__half,axiom,
% 0.29/0.59      ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.29/0.59      = one_one_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_of_real_pi_half
% 0.29/0.59  thf(fact_8006_sin__2npi,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.29/0.59        = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_2npi
% 0.29/0.59  thf(fact_8007_cos__2npi,axiom,
% 0.29/0.59      ! [N: nat] :
% 0.29/0.59        ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.29/0.59        = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_2npi
% 0.29/0.59  thf(fact_8008_sin__2pi__minus,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 0.29/0.59        = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_2pi_minus
% 0.29/0.59  thf(fact_8009_sin__int__2pin,axiom,
% 0.29/0.59      ! [N: int] :
% 0.29/0.59        ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.29/0.59        = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_int_2pin
% 0.29/0.59  thf(fact_8010_cos__int__2pin,axiom,
% 0.29/0.59      ! [N: int] :
% 0.29/0.59        ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.29/0.59        = one_one_real ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_int_2pin
% 0.29/0.59  thf(fact_8011_cos__3over2__pi,axiom,
% 0.29/0.59      ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.29/0.59      = zero_zero_real ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_3over2_pi
% 0.29/0.59  thf(fact_8012_sin__3over2__pi,axiom,
% 0.29/0.59      ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.29/0.59      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_3over2_pi
% 0.29/0.59  thf(fact_8013_cos__npi__int,axiom,
% 0.29/0.59      ! [N: int] :
% 0.29/0.59        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.29/0.59         => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.29/0.59            = one_one_real ) )
% 0.29/0.59        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.29/0.59         => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.29/0.59            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_npi_int
% 0.29/0.59  thf(fact_8014_cos__one__sin__zero,axiom,
% 0.29/0.59      ! [X: complex] :
% 0.29/0.59        ( ( ( cos_complex @ X )
% 0.29/0.59          = one_one_complex )
% 0.29/0.59       => ( ( sin_complex @ X )
% 0.29/0.59          = zero_zero_complex ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_one_sin_zero
% 0.29/0.59  thf(fact_8015_cos__one__sin__zero,axiom,
% 0.29/0.59      ! [X: real] :
% 0.29/0.59        ( ( ( cos_real @ X )
% 0.29/0.59          = one_one_real )
% 0.29/0.59       => ( ( sin_real @ X )
% 0.29/0.59          = zero_zero_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % cos_one_sin_zero
% 0.29/0.59  thf(fact_8016_emptyE,axiom,
% 0.29/0.59      ! [A: complex] :
% 0.29/0.59        ~ ( member_complex @ A @ bot_bot_set_complex ) ).
% 0.29/0.59  
% 0.29/0.59  % emptyE
% 0.29/0.59  thf(fact_8017_emptyE,axiom,
% 0.29/0.59      ! [A: set_nat] :
% 0.29/0.59        ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % emptyE
% 0.29/0.59  thf(fact_8018_emptyE,axiom,
% 0.29/0.59      ! [A: real] :
% 0.29/0.59        ~ ( member_real @ A @ bot_bot_set_real ) ).
% 0.29/0.59  
% 0.29/0.59  % emptyE
% 0.29/0.59  thf(fact_8019_emptyE,axiom,
% 0.29/0.59      ! [A: $o] :
% 0.29/0.59        ~ ( member_o @ A @ bot_bot_set_o ) ).
% 0.29/0.59  
% 0.29/0.59  % emptyE
% 0.29/0.59  thf(fact_8020_emptyE,axiom,
% 0.29/0.59      ! [A: nat] :
% 0.29/0.59        ~ ( member_nat @ A @ bot_bot_set_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % emptyE
% 0.29/0.59  thf(fact_8021_emptyE,axiom,
% 0.29/0.59      ! [A: int] :
% 0.29/0.59        ~ ( member_int @ A @ bot_bot_set_int ) ).
% 0.29/0.59  
% 0.29/0.59  % emptyE
% 0.29/0.59  thf(fact_8022_equals0D,axiom,
% 0.29/0.59      ! [A2: set_complex,A: complex] :
% 0.29/0.59        ( ( A2 = bot_bot_set_complex )
% 0.29/0.59       => ~ ( member_complex @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0D
% 0.29/0.59  thf(fact_8023_equals0D,axiom,
% 0.29/0.59      ! [A2: set_set_nat,A: set_nat] :
% 0.29/0.59        ( ( A2 = bot_bot_set_set_nat )
% 0.29/0.59       => ~ ( member_set_nat @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0D
% 0.29/0.59  thf(fact_8024_equals0D,axiom,
% 0.29/0.59      ! [A2: set_real,A: real] :
% 0.29/0.59        ( ( A2 = bot_bot_set_real )
% 0.29/0.59       => ~ ( member_real @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0D
% 0.29/0.59  thf(fact_8025_equals0D,axiom,
% 0.29/0.59      ! [A2: set_o,A: $o] :
% 0.29/0.59        ( ( A2 = bot_bot_set_o )
% 0.29/0.59       => ~ ( member_o @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0D
% 0.29/0.59  thf(fact_8026_equals0D,axiom,
% 0.29/0.59      ! [A2: set_nat,A: nat] :
% 0.29/0.59        ( ( A2 = bot_bot_set_nat )
% 0.29/0.59       => ~ ( member_nat @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0D
% 0.29/0.59  thf(fact_8027_equals0D,axiom,
% 0.29/0.59      ! [A2: set_int,A: int] :
% 0.29/0.59        ( ( A2 = bot_bot_set_int )
% 0.29/0.59       => ~ ( member_int @ A @ A2 ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0D
% 0.29/0.59  thf(fact_8028_equals0I,axiom,
% 0.29/0.59      ! [A2: set_complex] :
% 0.29/0.59        ( ! [Y4: complex] :
% 0.29/0.59            ~ ( member_complex @ Y4 @ A2 )
% 0.29/0.59       => ( A2 = bot_bot_set_complex ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0I
% 0.29/0.59  thf(fact_8029_equals0I,axiom,
% 0.29/0.59      ! [A2: set_set_nat] :
% 0.29/0.59        ( ! [Y4: set_nat] :
% 0.29/0.59            ~ ( member_set_nat @ Y4 @ A2 )
% 0.29/0.59       => ( A2 = bot_bot_set_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0I
% 0.29/0.59  thf(fact_8030_equals0I,axiom,
% 0.29/0.59      ! [A2: set_real] :
% 0.29/0.59        ( ! [Y4: real] :
% 0.29/0.59            ~ ( member_real @ Y4 @ A2 )
% 0.29/0.59       => ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0I
% 0.29/0.59  thf(fact_8031_equals0I,axiom,
% 0.29/0.59      ! [A2: set_o] :
% 0.29/0.59        ( ! [Y4: $o] :
% 0.29/0.59            ~ ( member_o @ Y4 @ A2 )
% 0.29/0.59       => ( A2 = bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0I
% 0.29/0.59  thf(fact_8032_equals0I,axiom,
% 0.29/0.59      ! [A2: set_nat] :
% 0.29/0.59        ( ! [Y4: nat] :
% 0.29/0.59            ~ ( member_nat @ Y4 @ A2 )
% 0.29/0.59       => ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0I
% 0.29/0.59  thf(fact_8033_equals0I,axiom,
% 0.29/0.59      ! [A2: set_int] :
% 0.29/0.59        ( ! [Y4: int] :
% 0.29/0.59            ~ ( member_int @ Y4 @ A2 )
% 0.29/0.59       => ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % equals0I
% 0.29/0.59  thf(fact_8034_ex__in__conv,axiom,
% 0.29/0.59      ! [A2: set_complex] :
% 0.29/0.59        ( ( ? [X4: complex] : ( member_complex @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 != bot_bot_set_complex ) ) ).
% 0.29/0.59  
% 0.29/0.59  % ex_in_conv
% 0.29/0.59  thf(fact_8035_ex__in__conv,axiom,
% 0.29/0.59      ! [A2: set_set_nat] :
% 0.29/0.59        ( ( ? [X4: set_nat] : ( member_set_nat @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 != bot_bot_set_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % ex_in_conv
% 0.29/0.59  thf(fact_8036_ex__in__conv,axiom,
% 0.29/0.59      ! [A2: set_real] :
% 0.29/0.59        ( ( ? [X4: real] : ( member_real @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 != bot_bot_set_real ) ) ).
% 0.29/0.59  
% 0.29/0.59  % ex_in_conv
% 0.29/0.59  thf(fact_8037_ex__in__conv,axiom,
% 0.29/0.59      ! [A2: set_o] :
% 0.29/0.59        ( ( ? [X4: $o] : ( member_o @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 != bot_bot_set_o ) ) ).
% 0.29/0.59  
% 0.29/0.59  % ex_in_conv
% 0.29/0.59  thf(fact_8038_ex__in__conv,axiom,
% 0.29/0.59      ! [A2: set_nat] :
% 0.29/0.59        ( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 != bot_bot_set_nat ) ) ).
% 0.29/0.59  
% 0.29/0.59  % ex_in_conv
% 0.29/0.59  thf(fact_8039_ex__in__conv,axiom,
% 0.29/0.59      ! [A2: set_int] :
% 0.29/0.59        ( ( ? [X4: int] : ( member_int @ X4 @ A2 ) )
% 0.29/0.59        = ( A2 != bot_bot_set_int ) ) ).
% 0.29/0.59  
% 0.29/0.59  % ex_in_conv
% 0.29/0.59  thf(fact_8040_polar__Ex,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59      ? [R3: real,A5: real] :
% 0.29/0.59        ( ( X
% 0.29/0.59          = ( times_times_real @ R3 @ ( cos_real @ A5 ) ) )
% 0.29/0.59        & ( Y
% 0.29/0.59          = ( times_times_real @ R3 @ ( sin_real @ A5 ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % polar_Ex
% 0.29/0.59  thf(fact_8041_sin__diff,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( sin_real @ ( minus_minus_real @ X @ Y ) )
% 0.29/0.59        = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_diff
% 0.29/0.59  thf(fact_8042_sin__add,axiom,
% 0.29/0.59      ! [X: complex,Y: complex] :
% 0.29/0.59        ( ( sin_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.29/0.59        = ( plus_plus_complex @ ( times_times_complex @ ( sin_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_add
% 0.29/0.59  thf(fact_8043_sin__add,axiom,
% 0.29/0.59      ! [X: real,Y: real] :
% 0.29/0.59        ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.59        = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.29/0.59  
% 0.29/0.59  % sin_add
% 0.29/0.59  thf(fact_8044_bot_Oextremum__strict,axiom,
% 0.29/0.59      ! [A: set_real] :
% 0.29/0.59        ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 0.29/0.59  
% 0.29/0.59  % bot.extremum_strict
% 0.29/0.59  thf(fact_8045_bot_Oextremum__strict,axiom,
% 0.29/0.59      ! [A: set_o] :
% 0.29/0.59        ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% 0.29/0.59  
% 0.29/0.59  % bot.extremum_strict
% 0.29/0.59  thf(fact_8046_bot_Oextremum__strict,axiom,
% 0.29/0.59      ! [A: set_nat] :
% 0.29/0.59        ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % bot.extremum_strict
% 0.29/0.59  thf(fact_8047_bot_Oextremum__strict,axiom,
% 0.29/0.59      ! [A: set_int] :
% 0.29/0.59        ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 0.29/0.59  
% 0.29/0.59  % bot.extremum_strict
% 0.29/0.59  thf(fact_8048_bot_Oextremum__strict,axiom,
% 0.29/0.59      ! [A: nat] :
% 0.29/0.59        ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 0.29/0.59  
% 0.29/0.59  % bot.extremum_strict
% 0.29/0.59  thf(fact_8049_bot_Onot__eq__extremum,axiom,
% 0.29/0.59      ! [A: set_real] :
% 0.29/0.59        ( ( A != bot_bot_set_real )
% 0.29/0.59        = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bot.not_eq_extremum
% 0.29/0.59  thf(fact_8050_bot_Onot__eq__extremum,axiom,
% 0.29/0.59      ! [A: set_o] :
% 0.29/0.59        ( ( A != bot_bot_set_o )
% 0.29/0.59        = ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bot.not_eq_extremum
% 0.29/0.59  thf(fact_8051_bot_Onot__eq__extremum,axiom,
% 0.29/0.59      ! [A: set_nat] :
% 0.29/0.59        ( ( A != bot_bot_set_nat )
% 0.29/0.59        = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 0.29/0.59  
% 0.29/0.59  % bot.not_eq_extremum
% 0.29/0.59  thf(fact_8052_bot_Onot__eq__extremum,axiom,
% 0.29/0.59      ! [A: set_int] :
% 0.29/0.59        ( ( A != bot_bot_set_int )
% 0.29/0.60        = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.not_eq_extremum
% 0.29/0.60  thf(fact_8053_bot_Onot__eq__extremum,axiom,
% 0.29/0.60      ! [A: nat] :
% 0.29/0.60        ( ( A != bot_bot_nat )
% 0.29/0.60        = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.not_eq_extremum
% 0.29/0.60  thf(fact_8054_bot_Oextremum,axiom,
% 0.29/0.60      ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum
% 0.29/0.60  thf(fact_8055_bot_Oextremum,axiom,
% 0.29/0.60      ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum
% 0.29/0.60  thf(fact_8056_bot_Oextremum,axiom,
% 0.29/0.60      ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum
% 0.29/0.60  thf(fact_8057_bot_Oextremum,axiom,
% 0.29/0.60      ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum
% 0.29/0.60  thf(fact_8058_bot_Oextremum,axiom,
% 0.29/0.60      ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum
% 0.29/0.60  thf(fact_8059_bot_Oextremum__unique,axiom,
% 0.29/0.60      ! [A: set_real] :
% 0.29/0.60        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.29/0.60        = ( A = bot_bot_set_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum_unique
% 0.29/0.60  thf(fact_8060_bot_Oextremum__unique,axiom,
% 0.29/0.60      ! [A: set_o] :
% 0.29/0.60        ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
% 0.29/0.60        = ( A = bot_bot_set_o ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum_unique
% 0.29/0.60  thf(fact_8061_bot_Oextremum__unique,axiom,
% 0.29/0.60      ! [A: set_nat] :
% 0.29/0.60        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.29/0.60        = ( A = bot_bot_set_nat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum_unique
% 0.29/0.60  thf(fact_8062_bot_Oextremum__unique,axiom,
% 0.29/0.60      ! [A: set_int] :
% 0.29/0.60        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.29/0.60        = ( A = bot_bot_set_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum_unique
% 0.29/0.60  thf(fact_8063_bot_Oextremum__unique,axiom,
% 0.29/0.60      ! [A: nat] :
% 0.29/0.60        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.29/0.60        = ( A = bot_bot_nat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum_unique
% 0.29/0.60  thf(fact_8064_bot_Oextremum__uniqueI,axiom,
% 0.29/0.60      ! [A: set_real] :
% 0.29/0.60        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.29/0.60       => ( A = bot_bot_set_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum_uniqueI
% 0.29/0.60  thf(fact_8065_bot_Oextremum__uniqueI,axiom,
% 0.29/0.60      ! [A: set_o] :
% 0.29/0.60        ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
% 0.29/0.60       => ( A = bot_bot_set_o ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum_uniqueI
% 0.29/0.60  thf(fact_8066_bot_Oextremum__uniqueI,axiom,
% 0.29/0.60      ! [A: set_nat] :
% 0.29/0.60        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.29/0.60       => ( A = bot_bot_set_nat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum_uniqueI
% 0.29/0.60  thf(fact_8067_bot_Oextremum__uniqueI,axiom,
% 0.29/0.60      ! [A: set_int] :
% 0.29/0.60        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.29/0.60       => ( A = bot_bot_set_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum_uniqueI
% 0.29/0.60  thf(fact_8068_bot_Oextremum__uniqueI,axiom,
% 0.29/0.60      ! [A: nat] :
% 0.29/0.60        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.29/0.60       => ( A = bot_bot_nat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot.extremum_uniqueI
% 0.29/0.60  thf(fact_8069_cos__of__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cos_real @ ( real_V1803761363581548252l_real @ X ) )
% 0.29/0.60        = ( real_V1803761363581548252l_real @ ( cos_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_of_real
% 0.29/0.60  thf(fact_8070_cos__of__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cos_complex @ ( real_V4546457046886955230omplex @ X ) )
% 0.29/0.60        = ( real_V4546457046886955230omplex @ ( cos_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_of_real
% 0.29/0.60  thf(fact_8071_sin__of__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( sin_real @ ( real_V1803761363581548252l_real @ X ) )
% 0.29/0.60        = ( real_V1803761363581548252l_real @ ( sin_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_of_real
% 0.29/0.60  thf(fact_8072_sin__of__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( sin_complex @ ( real_V4546457046886955230omplex @ X ) )
% 0.29/0.60        = ( real_V4546457046886955230omplex @ ( sin_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_of_real
% 0.29/0.60  thf(fact_8073_singleton__inject,axiom,
% 0.29/0.60      ! [A: vEBT_VEBT,B: vEBT_VEBT] :
% 0.29/0.60        ( ( ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT )
% 0.29/0.60          = ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
% 0.29/0.60       => ( A = B ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_inject
% 0.29/0.60  thf(fact_8074_singleton__inject,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ( insert_real @ A @ bot_bot_set_real )
% 0.29/0.60          = ( insert_real @ B @ bot_bot_set_real ) )
% 0.29/0.60       => ( A = B ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_inject
% 0.29/0.60  thf(fact_8075_singleton__inject,axiom,
% 0.29/0.60      ! [A: $o,B: $o] :
% 0.29/0.60        ( ( ( insert_o @ A @ bot_bot_set_o )
% 0.29/0.60          = ( insert_o @ B @ bot_bot_set_o ) )
% 0.29/0.60       => ( A = B ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_inject
% 0.29/0.60  thf(fact_8076_singleton__inject,axiom,
% 0.29/0.60      ! [A: nat,B: nat] :
% 0.29/0.60        ( ( ( insert_nat @ A @ bot_bot_set_nat )
% 0.29/0.60          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.29/0.60       => ( A = B ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_inject
% 0.29/0.60  thf(fact_8077_singleton__inject,axiom,
% 0.29/0.60      ! [A: int,B: int] :
% 0.29/0.60        ( ( ( insert_int @ A @ bot_bot_set_int )
% 0.29/0.60          = ( insert_int @ B @ bot_bot_set_int ) )
% 0.29/0.60       => ( A = B ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_inject
% 0.29/0.60  thf(fact_8078_insert__not__empty,axiom,
% 0.29/0.60      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.29/0.60        ( ( insert_VEBT_VEBT @ A @ A2 )
% 0.29/0.60       != bot_bo8194388402131092736T_VEBT ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_not_empty
% 0.29/0.60  thf(fact_8079_insert__not__empty,axiom,
% 0.29/0.60      ! [A: real,A2: set_real] :
% 0.29/0.60        ( ( insert_real @ A @ A2 )
% 0.29/0.60       != bot_bot_set_real ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_not_empty
% 0.29/0.60  thf(fact_8080_insert__not__empty,axiom,
% 0.29/0.60      ! [A: $o,A2: set_o] :
% 0.29/0.60        ( ( insert_o @ A @ A2 )
% 0.29/0.60       != bot_bot_set_o ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_not_empty
% 0.29/0.60  thf(fact_8081_insert__not__empty,axiom,
% 0.29/0.60      ! [A: nat,A2: set_nat] :
% 0.29/0.60        ( ( insert_nat @ A @ A2 )
% 0.29/0.60       != bot_bot_set_nat ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_not_empty
% 0.29/0.60  thf(fact_8082_insert__not__empty,axiom,
% 0.29/0.60      ! [A: int,A2: set_int] :
% 0.29/0.60        ( ( insert_int @ A @ A2 )
% 0.29/0.60       != bot_bot_set_int ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_not_empty
% 0.29/0.60  thf(fact_8083_doubleton__eq__iff,axiom,
% 0.29/0.60      ! [A: vEBT_VEBT,B: vEBT_VEBT,C: vEBT_VEBT,D: vEBT_VEBT] :
% 0.29/0.60        ( ( ( insert_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
% 0.29/0.60          = ( insert_VEBT_VEBT @ C @ ( insert_VEBT_VEBT @ D @ bot_bo8194388402131092736T_VEBT ) ) )
% 0.29/0.60        = ( ( ( A = C )
% 0.29/0.60            & ( B = D ) )
% 0.29/0.60          | ( ( A = D )
% 0.29/0.60            & ( B = C ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % doubleton_eq_iff
% 0.29/0.60  thf(fact_8084_doubleton__eq__iff,axiom,
% 0.29/0.60      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.60        ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
% 0.29/0.60          = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
% 0.29/0.60        = ( ( ( A = C )
% 0.29/0.60            & ( B = D ) )
% 0.29/0.60          | ( ( A = D )
% 0.29/0.60            & ( B = C ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % doubleton_eq_iff
% 0.29/0.60  thf(fact_8085_doubleton__eq__iff,axiom,
% 0.29/0.60      ! [A: $o,B: $o,C: $o,D: $o] :
% 0.29/0.60        ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
% 0.29/0.60          = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
% 0.29/0.60        = ( ( ( A = C )
% 0.29/0.60            & ( B = D ) )
% 0.29/0.60          | ( ( A = D )
% 0.29/0.60            & ( B = C ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % doubleton_eq_iff
% 0.29/0.60  thf(fact_8086_doubleton__eq__iff,axiom,
% 0.29/0.60      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.60        ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.29/0.60          = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
% 0.29/0.60        = ( ( ( A = C )
% 0.29/0.60            & ( B = D ) )
% 0.29/0.60          | ( ( A = D )
% 0.29/0.60            & ( B = C ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % doubleton_eq_iff
% 0.29/0.60  thf(fact_8087_doubleton__eq__iff,axiom,
% 0.29/0.60      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.60        ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
% 0.29/0.60          = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
% 0.29/0.60        = ( ( ( A = C )
% 0.29/0.60            & ( B = D ) )
% 0.29/0.60          | ( ( A = D )
% 0.29/0.60            & ( B = C ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % doubleton_eq_iff
% 0.29/0.60  thf(fact_8088_singleton__iff,axiom,
% 0.29/0.60      ! [B: vEBT_VEBT,A: vEBT_VEBT] :
% 0.29/0.60        ( ( member_VEBT_VEBT @ B @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
% 0.29/0.60        = ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_iff
% 0.29/0.60  thf(fact_8089_singleton__iff,axiom,
% 0.29/0.60      ! [B: complex,A: complex] :
% 0.29/0.60        ( ( member_complex @ B @ ( insert_complex @ A @ bot_bot_set_complex ) )
% 0.29/0.60        = ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_iff
% 0.29/0.60  thf(fact_8090_singleton__iff,axiom,
% 0.29/0.60      ! [B: set_nat,A: set_nat] :
% 0.29/0.60        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 0.29/0.60        = ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_iff
% 0.29/0.60  thf(fact_8091_singleton__iff,axiom,
% 0.29/0.60      ! [B: real,A: real] :
% 0.29/0.60        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 0.29/0.60        = ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_iff
% 0.29/0.60  thf(fact_8092_singleton__iff,axiom,
% 0.29/0.60      ! [B: $o,A: $o] :
% 0.29/0.60        ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
% 0.29/0.60        = ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_iff
% 0.29/0.60  thf(fact_8093_singleton__iff,axiom,
% 0.29/0.60      ! [B: nat,A: nat] :
% 0.29/0.60        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 0.29/0.60        = ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_iff
% 0.29/0.60  thf(fact_8094_singleton__iff,axiom,
% 0.29/0.60      ! [B: int,A: int] :
% 0.29/0.60        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 0.29/0.60        = ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singleton_iff
% 0.29/0.60  thf(fact_8095_singletonD,axiom,
% 0.29/0.60      ! [B: vEBT_VEBT,A: vEBT_VEBT] :
% 0.29/0.60        ( ( member_VEBT_VEBT @ B @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
% 0.29/0.60       => ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singletonD
% 0.29/0.60  thf(fact_8096_singletonD,axiom,
% 0.29/0.60      ! [B: complex,A: complex] :
% 0.29/0.60        ( ( member_complex @ B @ ( insert_complex @ A @ bot_bot_set_complex ) )
% 0.29/0.60       => ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singletonD
% 0.29/0.60  thf(fact_8097_singletonD,axiom,
% 0.29/0.60      ! [B: set_nat,A: set_nat] :
% 0.29/0.60        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 0.29/0.60       => ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singletonD
% 0.29/0.60  thf(fact_8098_singletonD,axiom,
% 0.29/0.60      ! [B: real,A: real] :
% 0.29/0.60        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 0.29/0.60       => ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singletonD
% 0.29/0.60  thf(fact_8099_singletonD,axiom,
% 0.29/0.60      ! [B: $o,A: $o] :
% 0.29/0.60        ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
% 0.29/0.60       => ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singletonD
% 0.29/0.60  thf(fact_8100_singletonD,axiom,
% 0.29/0.60      ! [B: nat,A: nat] :
% 0.29/0.60        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 0.29/0.60       => ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singletonD
% 0.29/0.60  thf(fact_8101_singletonD,axiom,
% 0.29/0.60      ! [B: int,A: int] :
% 0.29/0.60        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 0.29/0.60       => ( B = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % singletonD
% 0.29/0.60  thf(fact_8102_cos__add,axiom,
% 0.29/0.60      ! [X: complex,Y: complex] :
% 0.29/0.60        ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.29/0.60        = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_add
% 0.29/0.60  thf(fact_8103_cos__add,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.60        = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_add
% 0.29/0.60  thf(fact_8104_cos__diff,axiom,
% 0.29/0.60      ! [X: complex,Y: complex] :
% 0.29/0.60        ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.29/0.60        = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_diff
% 0.29/0.60  thf(fact_8105_cos__diff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 0.29/0.60        = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_diff
% 0.29/0.60  thf(fact_8106_not__psubset__empty,axiom,
% 0.29/0.60      ! [A2: set_real] :
% 0.29/0.60        ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 0.29/0.60  
% 0.29/0.60  % not_psubset_empty
% 0.29/0.60  thf(fact_8107_not__psubset__empty,axiom,
% 0.29/0.60      ! [A2: set_o] :
% 0.29/0.60        ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% 0.29/0.60  
% 0.29/0.60  % not_psubset_empty
% 0.29/0.60  thf(fact_8108_not__psubset__empty,axiom,
% 0.29/0.60      ! [A2: set_nat] :
% 0.29/0.60        ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 0.29/0.60  
% 0.29/0.60  % not_psubset_empty
% 0.29/0.60  thf(fact_8109_not__psubset__empty,axiom,
% 0.29/0.60      ! [A2: set_int] :
% 0.29/0.60        ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 0.29/0.60  
% 0.29/0.60  % not_psubset_empty
% 0.29/0.60  thf(fact_8110_semiring__norm_I26_J,axiom,
% 0.29/0.60      ( ( bitM @ one )
% 0.29/0.60      = one ) ).
% 0.29/0.60  
% 0.29/0.60  % semiring_norm(26)
% 0.29/0.60  thf(fact_8111_sin__zero__norm__cos__one,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( sin_real @ X )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60       => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 0.29/0.60          = one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_zero_norm_cos_one
% 0.29/0.60  thf(fact_8112_sin__zero__norm__cos__one,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( ( sin_complex @ X )
% 0.29/0.60          = zero_zero_complex )
% 0.29/0.60       => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 0.29/0.60          = one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_zero_norm_cos_one
% 0.29/0.60  thf(fact_8113_sin__zero__abs__cos__one,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( sin_real @ X )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60       => ( ( abs_abs_real @ ( cos_real @ X ) )
% 0.29/0.60          = one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_zero_abs_cos_one
% 0.29/0.60  thf(fact_8114_sin__double,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_double
% 0.29/0.60  thf(fact_8115_sin__double,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_double
% 0.29/0.60  thf(fact_8116_sincos__principal__value,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60      ? [Y4: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 0.29/0.60        & ( ord_less_eq_real @ Y4 @ pi )
% 0.29/0.60        & ( ( sin_real @ Y4 )
% 0.29/0.60          = ( sin_real @ X ) )
% 0.29/0.60        & ( ( cos_real @ Y4 )
% 0.29/0.60          = ( cos_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sincos_principal_value
% 0.29/0.60  thf(fact_8117_diff__shunt__var,axiom,
% 0.29/0.60      ! [X: set_real,Y: set_real] :
% 0.29/0.60        ( ( ( minus_minus_set_real @ X @ Y )
% 0.29/0.60          = bot_bot_set_real )
% 0.29/0.60        = ( ord_less_eq_set_real @ X @ Y ) ) ).
% 0.29/0.60  
% 0.29/0.60  % diff_shunt_var
% 0.29/0.60  thf(fact_8118_diff__shunt__var,axiom,
% 0.29/0.60      ! [X: set_o,Y: set_o] :
% 0.29/0.60        ( ( ( minus_minus_set_o @ X @ Y )
% 0.29/0.60          = bot_bot_set_o )
% 0.29/0.60        = ( ord_less_eq_set_o @ X @ Y ) ) ).
% 0.29/0.60  
% 0.29/0.60  % diff_shunt_var
% 0.29/0.60  thf(fact_8119_diff__shunt__var,axiom,
% 0.29/0.60      ! [X: set_nat,Y: set_nat] :
% 0.29/0.60        ( ( ( minus_minus_set_nat @ X @ Y )
% 0.29/0.60          = bot_bot_set_nat )
% 0.29/0.60        = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.29/0.60  
% 0.29/0.60  % diff_shunt_var
% 0.29/0.60  thf(fact_8120_diff__shunt__var,axiom,
% 0.29/0.60      ! [X: set_int,Y: set_int] :
% 0.29/0.60        ( ( ( minus_minus_set_int @ X @ Y )
% 0.29/0.60          = bot_bot_set_int )
% 0.29/0.60        = ( ord_less_eq_set_int @ X @ Y ) ) ).
% 0.29/0.60  
% 0.29/0.60  % diff_shunt_var
% 0.29/0.60  thf(fact_8121_sin__x__le__x,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_x_le_x
% 0.29/0.60  thf(fact_8122_sin__le__one,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_le_one
% 0.29/0.60  thf(fact_8123_cos__le__one,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_le_one
% 0.29/0.60  thf(fact_8124_abs__sin__x__le__abs__x,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % abs_sin_x_le_abs_x
% 0.29/0.60  thf(fact_8125_cos__arctan__not__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cos_real @ ( arctan @ X ) )
% 0.29/0.60       != zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_arctan_not_zero
% 0.29/0.60  thf(fact_8126_subset__singletonD,axiom,
% 0.29/0.60      ! [A2: set_VEBT_VEBT,X: vEBT_VEBT] :
% 0.29/0.60        ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
% 0.29/0.60       => ( ( A2 = bot_bo8194388402131092736T_VEBT )
% 0.29/0.60          | ( A2
% 0.29/0.60            = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_singletonD
% 0.29/0.60  thf(fact_8127_subset__singletonD,axiom,
% 0.29/0.60      ! [A2: set_real,X: real] :
% 0.29/0.60        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) )
% 0.29/0.60       => ( ( A2 = bot_bot_set_real )
% 0.29/0.60          | ( A2
% 0.29/0.60            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_singletonD
% 0.29/0.60  thf(fact_8128_subset__singletonD,axiom,
% 0.29/0.60      ! [A2: set_o,X: $o] :
% 0.29/0.60        ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.29/0.60       => ( ( A2 = bot_bot_set_o )
% 0.29/0.60          | ( A2
% 0.29/0.60            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_singletonD
% 0.29/0.60  thf(fact_8129_subset__singletonD,axiom,
% 0.29/0.60      ! [A2: set_nat,X: nat] :
% 0.29/0.60        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.29/0.60       => ( ( A2 = bot_bot_set_nat )
% 0.29/0.60          | ( A2
% 0.29/0.60            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_singletonD
% 0.29/0.60  thf(fact_8130_subset__singletonD,axiom,
% 0.29/0.60      ! [A2: set_int,X: int] :
% 0.29/0.60        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.29/0.60       => ( ( A2 = bot_bot_set_int )
% 0.29/0.60          | ( A2
% 0.29/0.60            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_singletonD
% 0.29/0.60  thf(fact_8131_subset__singleton__iff,axiom,
% 0.29/0.60      ! [X8: set_VEBT_VEBT,A: vEBT_VEBT] :
% 0.29/0.60        ( ( ord_le4337996190870823476T_VEBT @ X8 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
% 0.29/0.60        = ( ( X8 = bot_bo8194388402131092736T_VEBT )
% 0.29/0.60          | ( X8
% 0.29/0.60            = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_singleton_iff
% 0.29/0.60  thf(fact_8132_subset__singleton__iff,axiom,
% 0.29/0.60      ! [X8: set_real,A: real] :
% 0.29/0.60        ( ( ord_less_eq_set_real @ X8 @ ( insert_real @ A @ bot_bot_set_real ) )
% 0.29/0.60        = ( ( X8 = bot_bot_set_real )
% 0.29/0.60          | ( X8
% 0.29/0.60            = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_singleton_iff
% 0.29/0.60  thf(fact_8133_subset__singleton__iff,axiom,
% 0.29/0.60      ! [X8: set_o,A: $o] :
% 0.29/0.60        ( ( ord_less_eq_set_o @ X8 @ ( insert_o @ A @ bot_bot_set_o ) )
% 0.29/0.60        = ( ( X8 = bot_bot_set_o )
% 0.29/0.60          | ( X8
% 0.29/0.60            = ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_singleton_iff
% 0.29/0.60  thf(fact_8134_subset__singleton__iff,axiom,
% 0.29/0.60      ! [X8: set_nat,A: nat] :
% 0.29/0.60        ( ( ord_less_eq_set_nat @ X8 @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 0.29/0.60        = ( ( X8 = bot_bot_set_nat )
% 0.29/0.60          | ( X8
% 0.29/0.60            = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_singleton_iff
% 0.29/0.60  thf(fact_8135_subset__singleton__iff,axiom,
% 0.29/0.60      ! [X8: set_int,A: int] :
% 0.29/0.60        ( ( ord_less_eq_set_int @ X8 @ ( insert_int @ A @ bot_bot_set_int ) )
% 0.29/0.60        = ( ( X8 = bot_bot_set_int )
% 0.29/0.60          | ( X8
% 0.29/0.60            = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_singleton_iff
% 0.29/0.60  thf(fact_8136_Diff__insert,axiom,
% 0.29/0.60      ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.29/0.60        ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 0.29/0.60        = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ B2 ) @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert
% 0.29/0.60  thf(fact_8137_Diff__insert,axiom,
% 0.29/0.60      ! [A2: set_real,A: real,B2: set_real] :
% 0.29/0.60        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 0.29/0.60        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert
% 0.29/0.60  thf(fact_8138_Diff__insert,axiom,
% 0.29/0.60      ! [A2: set_o,A: $o,B2: set_o] :
% 0.29/0.60        ( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
% 0.29/0.60        = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B2 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert
% 0.29/0.60  thf(fact_8139_Diff__insert,axiom,
% 0.29/0.60      ! [A2: set_int,A: int,B2: set_int] :
% 0.29/0.60        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 0.29/0.60        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert
% 0.29/0.60  thf(fact_8140_Diff__insert,axiom,
% 0.29/0.60      ! [A2: set_nat,A: nat,B2: set_nat] :
% 0.29/0.60        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 0.29/0.60        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert
% 0.29/0.60  thf(fact_8141_insert__Diff,axiom,
% 0.29/0.60      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.29/0.60        ( ( member_VEBT_VEBT @ A @ A2 )
% 0.29/0.60       => ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_Diff
% 0.29/0.60  thf(fact_8142_insert__Diff,axiom,
% 0.29/0.60      ! [A: complex,A2: set_complex] :
% 0.29/0.60        ( ( member_complex @ A @ A2 )
% 0.29/0.60       => ( ( insert_complex @ A @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_Diff
% 0.29/0.60  thf(fact_8143_insert__Diff,axiom,
% 0.29/0.60      ! [A: set_nat,A2: set_set_nat] :
% 0.29/0.60        ( ( member_set_nat @ A @ A2 )
% 0.29/0.60       => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_Diff
% 0.29/0.60  thf(fact_8144_insert__Diff,axiom,
% 0.29/0.60      ! [A: real,A2: set_real] :
% 0.29/0.60        ( ( member_real @ A @ A2 )
% 0.29/0.60       => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_Diff
% 0.29/0.60  thf(fact_8145_insert__Diff,axiom,
% 0.29/0.60      ! [A: $o,A2: set_o] :
% 0.29/0.60        ( ( member_o @ A @ A2 )
% 0.29/0.60       => ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_Diff
% 0.29/0.60  thf(fact_8146_insert__Diff,axiom,
% 0.29/0.60      ! [A: int,A2: set_int] :
% 0.29/0.60        ( ( member_int @ A @ A2 )
% 0.29/0.60       => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_Diff
% 0.29/0.60  thf(fact_8147_insert__Diff,axiom,
% 0.29/0.60      ! [A: nat,A2: set_nat] :
% 0.29/0.60        ( ( member_nat @ A @ A2 )
% 0.29/0.60       => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % insert_Diff
% 0.29/0.60  thf(fact_8148_Diff__insert2,axiom,
% 0.29/0.60      ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.29/0.60        ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 0.29/0.60        = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) @ B2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert2
% 0.29/0.60  thf(fact_8149_Diff__insert2,axiom,
% 0.29/0.60      ! [A2: set_real,A: real,B2: set_real] :
% 0.29/0.60        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 0.29/0.60        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert2
% 0.29/0.60  thf(fact_8150_Diff__insert2,axiom,
% 0.29/0.60      ! [A2: set_o,A: $o,B2: set_o] :
% 0.29/0.60        ( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
% 0.29/0.60        = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert2
% 0.29/0.60  thf(fact_8151_Diff__insert2,axiom,
% 0.29/0.60      ! [A2: set_int,A: int,B2: set_int] :
% 0.29/0.60        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 0.29/0.60        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert2
% 0.29/0.60  thf(fact_8152_Diff__insert2,axiom,
% 0.29/0.60      ! [A2: set_nat,A: nat,B2: set_nat] :
% 0.29/0.60        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 0.29/0.60        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert2
% 0.29/0.60  thf(fact_8153_Diff__insert__absorb,axiom,
% 0.29/0.60      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.29/0.60        ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 0.29/0.60       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert_absorb
% 0.29/0.60  thf(fact_8154_Diff__insert__absorb,axiom,
% 0.29/0.60      ! [X: complex,A2: set_complex] :
% 0.29/0.60        ( ~ ( member_complex @ X @ A2 )
% 0.29/0.60       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ ( insert_complex @ X @ bot_bot_set_complex ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert_absorb
% 0.29/0.60  thf(fact_8155_Diff__insert__absorb,axiom,
% 0.29/0.60      ! [X: set_nat,A2: set_set_nat] :
% 0.29/0.60        ( ~ ( member_set_nat @ X @ A2 )
% 0.29/0.60       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert_absorb
% 0.29/0.60  thf(fact_8156_Diff__insert__absorb,axiom,
% 0.29/0.60      ! [X: real,A2: set_real] :
% 0.29/0.60        ( ~ ( member_real @ X @ A2 )
% 0.29/0.60       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert_absorb
% 0.29/0.60  thf(fact_8157_Diff__insert__absorb,axiom,
% 0.29/0.60      ! [X: $o,A2: set_o] :
% 0.29/0.60        ( ~ ( member_o @ X @ A2 )
% 0.29/0.60       => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert_absorb
% 0.29/0.60  thf(fact_8158_Diff__insert__absorb,axiom,
% 0.29/0.60      ! [X: int,A2: set_int] :
% 0.29/0.60        ( ~ ( member_int @ X @ A2 )
% 0.29/0.60       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert_absorb
% 0.29/0.60  thf(fact_8159_Diff__insert__absorb,axiom,
% 0.29/0.60      ! [X: nat,A2: set_nat] :
% 0.29/0.60        ( ~ ( member_nat @ X @ A2 )
% 0.29/0.60       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.29/0.60          = A2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_insert_absorb
% 0.29/0.60  thf(fact_8160_in__image__insert__iff,axiom,
% 0.29/0.60      ! [B2: set_set_VEBT_VEBT,X: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.29/0.60        ( ! [C5: set_VEBT_VEBT] :
% 0.29/0.60            ( ( member_set_VEBT_VEBT @ C5 @ B2 )
% 0.29/0.60           => ~ ( member_VEBT_VEBT @ X @ C5 ) )
% 0.29/0.60       => ( ( member_set_VEBT_VEBT @ A2 @ ( image_1661326939266726661T_VEBT @ ( insert_VEBT_VEBT @ X ) @ B2 ) )
% 0.29/0.60          = ( ( member_VEBT_VEBT @ X @ A2 )
% 0.29/0.60            & ( member_set_VEBT_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % in_image_insert_iff
% 0.29/0.60  thf(fact_8161_in__image__insert__iff,axiom,
% 0.29/0.60      ! [B2: set_set_complex,X: complex,A2: set_complex] :
% 0.29/0.60        ( ! [C5: set_complex] :
% 0.29/0.60            ( ( member_set_complex @ C5 @ B2 )
% 0.29/0.60           => ~ ( member_complex @ X @ C5 ) )
% 0.29/0.60       => ( ( member_set_complex @ A2 @ ( image_7998606247489673935omplex @ ( insert_complex @ X ) @ B2 ) )
% 0.29/0.60          = ( ( member_complex @ X @ A2 )
% 0.29/0.60            & ( member_set_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % in_image_insert_iff
% 0.29/0.60  thf(fact_8162_in__image__insert__iff,axiom,
% 0.29/0.60      ! [B2: set_set_set_nat,X: set_nat,A2: set_set_nat] :
% 0.29/0.60        ( ! [C5: set_set_nat] :
% 0.29/0.60            ( ( member_set_set_nat @ C5 @ B2 )
% 0.29/0.60           => ~ ( member_set_nat @ X @ C5 ) )
% 0.29/0.60       => ( ( member_set_set_nat @ A2 @ ( image_7884819252390400639et_nat @ ( insert_set_nat @ X ) @ B2 ) )
% 0.29/0.60          = ( ( member_set_nat @ X @ A2 )
% 0.29/0.60            & ( member_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % in_image_insert_iff
% 0.29/0.60  thf(fact_8163_in__image__insert__iff,axiom,
% 0.29/0.60      ! [B2: set_set_real,X: real,A2: set_real] :
% 0.29/0.60        ( ! [C5: set_real] :
% 0.29/0.60            ( ( member_set_real @ C5 @ B2 )
% 0.29/0.60           => ~ ( member_real @ X @ C5 ) )
% 0.29/0.60       => ( ( member_set_real @ A2 @ ( image_2436557299294012491t_real @ ( insert_real @ X ) @ B2 ) )
% 0.29/0.60          = ( ( member_real @ X @ A2 )
% 0.29/0.60            & ( member_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % in_image_insert_iff
% 0.29/0.60  thf(fact_8164_in__image__insert__iff,axiom,
% 0.29/0.60      ! [B2: set_set_o,X: $o,A2: set_o] :
% 0.29/0.60        ( ! [C5: set_o] :
% 0.29/0.60            ( ( member_set_o @ C5 @ B2 )
% 0.29/0.60           => ~ ( member_o @ X @ C5 ) )
% 0.29/0.60       => ( ( member_set_o @ A2 @ ( image_set_o_set_o @ ( insert_o @ X ) @ B2 ) )
% 0.29/0.60          = ( ( member_o @ X @ A2 )
% 0.29/0.60            & ( member_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % in_image_insert_iff
% 0.29/0.60  thf(fact_8165_in__image__insert__iff,axiom,
% 0.29/0.60      ! [B2: set_set_int,X: int,A2: set_int] :
% 0.29/0.60        ( ! [C5: set_int] :
% 0.29/0.60            ( ( member_set_int @ C5 @ B2 )
% 0.29/0.60           => ~ ( member_int @ X @ C5 ) )
% 0.29/0.60       => ( ( member_set_int @ A2 @ ( image_524474410958335435et_int @ ( insert_int @ X ) @ B2 ) )
% 0.29/0.60          = ( ( member_int @ X @ A2 )
% 0.29/0.60            & ( member_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % in_image_insert_iff
% 0.29/0.60  thf(fact_8166_in__image__insert__iff,axiom,
% 0.29/0.60      ! [B2: set_set_nat,X: nat,A2: set_nat] :
% 0.29/0.60        ( ! [C5: set_nat] :
% 0.29/0.60            ( ( member_set_nat @ C5 @ B2 )
% 0.29/0.60           => ~ ( member_nat @ X @ C5 ) )
% 0.29/0.60       => ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B2 ) )
% 0.29/0.60          = ( ( member_nat @ X @ A2 )
% 0.29/0.60            & ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % in_image_insert_iff
% 0.29/0.60  thf(fact_8167_cos__int__times__real,axiom,
% 0.29/0.60      ! [M: int,X: real] :
% 0.29/0.60        ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 0.29/0.60        = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_int_times_real
% 0.29/0.60  thf(fact_8168_cos__int__times__real,axiom,
% 0.29/0.60      ! [M: int,X: real] :
% 0.29/0.60        ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 0.29/0.60        = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_int_times_real
% 0.29/0.60  thf(fact_8169_sin__cos__le1,axiom,
% 0.29/0.60      ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_cos_le1
% 0.29/0.60  thf(fact_8170_subset__Compl__self__eq,axiom,
% 0.29/0.60      ! [A2: set_real] :
% 0.29/0.60        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 0.29/0.60        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_Compl_self_eq
% 0.29/0.60  thf(fact_8171_subset__Compl__self__eq,axiom,
% 0.29/0.60      ! [A2: set_o] :
% 0.29/0.60        ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
% 0.29/0.60        = ( A2 = bot_bot_set_o ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_Compl_self_eq
% 0.29/0.60  thf(fact_8172_subset__Compl__self__eq,axiom,
% 0.29/0.60      ! [A2: set_nat] :
% 0.29/0.60        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.29/0.60        = ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_Compl_self_eq
% 0.29/0.60  thf(fact_8173_subset__Compl__self__eq,axiom,
% 0.29/0.60      ! [A2: set_int] :
% 0.29/0.60        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.29/0.60        = ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_Compl_self_eq
% 0.29/0.60  thf(fact_8174_sin__int__times__real,axiom,
% 0.29/0.60      ! [M: int,X: real] :
% 0.29/0.60        ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 0.29/0.60        = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_int_times_real
% 0.29/0.60  thf(fact_8175_sin__int__times__real,axiom,
% 0.29/0.60      ! [M: int,X: real] :
% 0.29/0.60        ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 0.29/0.60        = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_int_times_real
% 0.29/0.60  thf(fact_8176_semiring__norm_I28_J,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( bitM @ ( bit1 @ N ) )
% 0.29/0.60        = ( bit1 @ ( bit0 @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % semiring_norm(28)
% 0.29/0.60  thf(fact_8177_semiring__norm_I27_J,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( bitM @ ( bit0 @ N ) )
% 0.29/0.60        = ( bit1 @ ( bitM @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % semiring_norm(27)
% 0.29/0.60  thf(fact_8178_sin__squared__eq,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60        = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_squared_eq
% 0.29/0.60  thf(fact_8179_sin__squared__eq,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60        = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_squared_eq
% 0.29/0.60  thf(fact_8180_cos__squared__eq,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60        = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_squared_eq
% 0.29/0.60  thf(fact_8181_cos__squared__eq,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60        = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_squared_eq
% 0.29/0.60  thf(fact_8182_inc__BitM__eq,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( inc @ ( bitM @ N ) )
% 0.29/0.60        = ( bit0 @ N ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inc_BitM_eq
% 0.29/0.60  thf(fact_8183_BitM__inc__eq,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( bitM @ ( inc @ N ) )
% 0.29/0.60        = ( bit1 @ N ) ) ).
% 0.29/0.60  
% 0.29/0.60  % BitM_inc_eq
% 0.29/0.60  thf(fact_8184_sin__gt__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ pi )
% 0.29/0.60         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_gt_zero
% 0.29/0.60  thf(fact_8185_sin__x__ge__neg__x,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_x_ge_neg_x
% 0.29/0.60  thf(fact_8186_sin__ge__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ pi )
% 0.29/0.60         => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_ge_zero
% 0.29/0.60  thf(fact_8187_sin__ge__minus__one,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_ge_minus_one
% 0.29/0.60  thf(fact_8188_cos__monotone__0__pi__le,axiom,
% 0.29/0.60      ! [Y: real,X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ X )
% 0.29/0.60         => ( ( ord_less_eq_real @ X @ pi )
% 0.29/0.60           => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_monotone_0_pi_le
% 0.29/0.60  thf(fact_8189_cos__mono__le__eq,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ pi )
% 0.29/0.60         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60           => ( ( ord_less_eq_real @ Y @ pi )
% 0.29/0.60             => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
% 0.29/0.60                = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_mono_le_eq
% 0.29/0.60  thf(fact_8190_cos__inj__pi,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ pi )
% 0.29/0.60         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60           => ( ( ord_less_eq_real @ Y @ pi )
% 0.29/0.60             => ( ( ( cos_real @ X )
% 0.29/0.60                  = ( cos_real @ Y ) )
% 0.29/0.60               => ( X = Y ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_inj_pi
% 0.29/0.60  thf(fact_8191_cos__ge__minus__one,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_ge_minus_one
% 0.29/0.60  thf(fact_8192_abs__sin__le__one,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % abs_sin_le_one
% 0.29/0.60  thf(fact_8193_abs__cos__le__one,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % abs_cos_le_one
% 0.29/0.60  thf(fact_8194_and__nat__def,axiom,
% 0.29/0.60      ( bit_se727722235901077358nd_nat
% 0.29/0.60      = ( ^ [M6: nat,N3: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % and_nat_def
% 0.29/0.60  thf(fact_8195_cos__diff__cos,axiom,
% 0.29/0.60      ! [W: complex,Z: complex] :
% 0.29/0.60        ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_diff_cos
% 0.29/0.60  thf(fact_8196_cos__diff__cos,axiom,
% 0.29/0.60      ! [W: real,Z: real] :
% 0.29/0.60        ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_diff_cos
% 0.29/0.60  thf(fact_8197_sin__diff__sin,axiom,
% 0.29/0.60      ! [W: complex,Z: complex] :
% 0.29/0.60        ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_diff_sin
% 0.29/0.60  thf(fact_8198_sin__diff__sin,axiom,
% 0.29/0.60      ! [W: real,Z: real] :
% 0.29/0.60        ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_diff_sin
% 0.29/0.60  thf(fact_8199_sin__plus__sin,axiom,
% 0.29/0.60      ! [W: complex,Z: complex] :
% 0.29/0.60        ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_plus_sin
% 0.29/0.60  thf(fact_8200_sin__plus__sin,axiom,
% 0.29/0.60      ! [W: real,Z: real] :
% 0.29/0.60        ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_plus_sin
% 0.29/0.60  thf(fact_8201_cos__times__sin,axiom,
% 0.29/0.60      ! [W: complex,Z: complex] :
% 0.29/0.60        ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 0.29/0.60        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_times_sin
% 0.29/0.60  thf(fact_8202_cos__times__sin,axiom,
% 0.29/0.60      ! [W: real,Z: real] :
% 0.29/0.60        ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_times_sin
% 0.29/0.60  thf(fact_8203_sin__times__cos,axiom,
% 0.29/0.60      ! [W: complex,Z: complex] :
% 0.29/0.60        ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 0.29/0.60        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_times_cos
% 0.29/0.60  thf(fact_8204_sin__times__cos,axiom,
% 0.29/0.60      ! [W: real,Z: real] :
% 0.29/0.60        ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_times_cos
% 0.29/0.60  thf(fact_8205_sin__times__sin,axiom,
% 0.29/0.60      ! [W: complex,Z: complex] :
% 0.29/0.60        ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.29/0.60        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_times_sin
% 0.29/0.60  thf(fact_8206_sin__times__sin,axiom,
% 0.29/0.60      ! [W: real,Z: real] :
% 0.29/0.60        ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_times_sin
% 0.29/0.60  thf(fact_8207_cos__double,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60        = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_double
% 0.29/0.60  thf(fact_8208_cos__double,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60        = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_double
% 0.29/0.60  thf(fact_8209_cos__sin__eq,axiom,
% 0.29/0.60      ( cos_real
% 0.29/0.60      = ( ^ [X4: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_sin_eq
% 0.29/0.60  thf(fact_8210_cos__sin__eq,axiom,
% 0.29/0.60      ( cos_complex
% 0.29/0.60      = ( ^ [X4: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_sin_eq
% 0.29/0.60  thf(fact_8211_sin__cos__eq,axiom,
% 0.29/0.60      ( sin_real
% 0.29/0.60      = ( ^ [X4: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_cos_eq
% 0.29/0.60  thf(fact_8212_sin__cos__eq,axiom,
% 0.29/0.60      ( sin_complex
% 0.29/0.60      = ( ^ [X4: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_cos_eq
% 0.29/0.60  thf(fact_8213_subset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.29/0.60        ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_VEBT_VEBT @ X @ A2 )
% 0.29/0.60           => ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B2 ) )
% 0.29/0.60          & ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 0.29/0.60           => ( ord_le4337996190870823476T_VEBT @ A2 @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_insert_iff
% 0.29/0.60  thf(fact_8214_subset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_complex,X: complex,B2: set_complex] :
% 0.29/0.60        ( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_complex @ X @ A2 )
% 0.29/0.60           => ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) )
% 0.29/0.60          & ( ~ ( member_complex @ X @ A2 )
% 0.29/0.60           => ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_insert_iff
% 0.29/0.60  thf(fact_8215_subset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
% 0.29/0.60        ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_set_nat @ X @ A2 )
% 0.29/0.60           => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
% 0.29/0.60          & ( ~ ( member_set_nat @ X @ A2 )
% 0.29/0.60           => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_insert_iff
% 0.29/0.60  thf(fact_8216_subset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_real,X: real,B2: set_real] :
% 0.29/0.60        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_real @ X @ A2 )
% 0.29/0.60           => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
% 0.29/0.60          & ( ~ ( member_real @ X @ A2 )
% 0.29/0.60           => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_insert_iff
% 0.29/0.60  thf(fact_8217_subset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_o,X: $o,B2: set_o] :
% 0.29/0.60        ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_o @ X @ A2 )
% 0.29/0.60           => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
% 0.29/0.60          & ( ~ ( member_o @ X @ A2 )
% 0.29/0.60           => ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_insert_iff
% 0.29/0.60  thf(fact_8218_subset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_nat,X: nat,B2: set_nat] :
% 0.29/0.60        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_nat @ X @ A2 )
% 0.29/0.60           => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
% 0.29/0.60          & ( ~ ( member_nat @ X @ A2 )
% 0.29/0.60           => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_insert_iff
% 0.29/0.60  thf(fact_8219_subset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_int,X: int,B2: set_int] :
% 0.29/0.60        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_int @ X @ A2 )
% 0.29/0.60           => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
% 0.29/0.60          & ( ~ ( member_int @ X @ A2 )
% 0.29/0.60           => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % subset_insert_iff
% 0.29/0.60  thf(fact_8220_Diff__single__insert,axiom,
% 0.29/0.60      ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.29/0.60        ( ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B2 )
% 0.29/0.60       => ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B2 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_single_insert
% 0.29/0.60  thf(fact_8221_Diff__single__insert,axiom,
% 0.29/0.60      ! [A2: set_real,X: real,B2: set_real] :
% 0.29/0.60        ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 )
% 0.29/0.60       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_single_insert
% 0.29/0.60  thf(fact_8222_Diff__single__insert,axiom,
% 0.29/0.60      ! [A2: set_o,X: $o,B2: set_o] :
% 0.29/0.60        ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 )
% 0.29/0.60       => ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_single_insert
% 0.29/0.60  thf(fact_8223_Diff__single__insert,axiom,
% 0.29/0.60      ! [A2: set_nat,X: nat,B2: set_nat] :
% 0.29/0.60        ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 )
% 0.29/0.60       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_single_insert
% 0.29/0.60  thf(fact_8224_Diff__single__insert,axiom,
% 0.29/0.60      ! [A2: set_int,X: int,B2: set_int] :
% 0.29/0.60        ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 )
% 0.29/0.60       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Diff_single_insert
% 0.29/0.60  thf(fact_8225_Compl__insert,axiom,
% 0.29/0.60      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 0.29/0.60        ( ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) )
% 0.29/0.60        = ( minus_5127226145743854075T_VEBT @ ( uminus8041839845116263051T_VEBT @ A2 ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Compl_insert
% 0.29/0.60  thf(fact_8226_Compl__insert,axiom,
% 0.29/0.60      ! [X: real,A2: set_real] :
% 0.29/0.60        ( ( uminus612125837232591019t_real @ ( insert_real @ X @ A2 ) )
% 0.29/0.60        = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Compl_insert
% 0.29/0.60  thf(fact_8227_Compl__insert,axiom,
% 0.29/0.60      ! [X: $o,A2: set_o] :
% 0.29/0.60        ( ( uminus_uminus_set_o @ ( insert_o @ X @ A2 ) )
% 0.29/0.60        = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Compl_insert
% 0.29/0.60  thf(fact_8228_Compl__insert,axiom,
% 0.29/0.60      ! [X: int,A2: set_int] :
% 0.29/0.60        ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
% 0.29/0.60        = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Compl_insert
% 0.29/0.60  thf(fact_8229_Compl__insert,axiom,
% 0.29/0.60      ! [X: nat,A2: set_nat] :
% 0.29/0.60        ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
% 0.29/0.60        = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Compl_insert
% 0.29/0.60  thf(fact_8230_cos__double__sin,axiom,
% 0.29/0.60      ! [W: complex] :
% 0.29/0.60        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.29/0.60        = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_double_sin
% 0.29/0.60  thf(fact_8231_cos__double__sin,axiom,
% 0.29/0.60      ! [W: real] :
% 0.29/0.60        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.29/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_double_sin
% 0.29/0.60  thf(fact_8232_eval__nat__numeral_I2_J,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.29/0.60        = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % eval_nat_numeral(2)
% 0.29/0.60  thf(fact_8233_minus__sin__cos__eq,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( uminus_uminus_real @ ( sin_real @ X ) )
% 0.29/0.60        = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_sin_cos_eq
% 0.29/0.60  thf(fact_8234_minus__sin__cos__eq,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
% 0.29/0.60        = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_sin_cos_eq
% 0.29/0.60  thf(fact_8235_BitM__plus__one,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 0.29/0.60        = ( bit0 @ N ) ) ).
% 0.29/0.60  
% 0.29/0.60  % BitM_plus_one
% 0.29/0.60  thf(fact_8236_one__plus__BitM,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 0.29/0.60        = ( bit0 @ N ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_plus_BitM
% 0.29/0.60  thf(fact_8237_cos__two__neq__zero,axiom,
% 0.29/0.60      ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.60     != zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_two_neq_zero
% 0.29/0.60  thf(fact_8238_cos__monotone__0__pi,axiom,
% 0.29/0.60      ! [Y: real,X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60       => ( ( ord_less_real @ Y @ X )
% 0.29/0.60         => ( ( ord_less_eq_real @ X @ pi )
% 0.29/0.60           => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_monotone_0_pi
% 0.29/0.60  thf(fact_8239_cos__mono__less__eq,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ pi )
% 0.29/0.60         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60           => ( ( ord_less_eq_real @ Y @ pi )
% 0.29/0.60             => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
% 0.29/0.60                = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_mono_less_eq
% 0.29/0.60  thf(fact_8240_sin__eq__0__pi,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ pi )
% 0.29/0.60         => ( ( ( sin_real @ X )
% 0.29/0.60              = zero_zero_real )
% 0.29/0.60           => ( X = zero_zero_real ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_eq_0_pi
% 0.29/0.60  thf(fact_8241_sin__zero__pi__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
% 0.29/0.60       => ( ( ( sin_real @ X )
% 0.29/0.60            = zero_zero_real )
% 0.29/0.60          = ( X = zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_zero_pi_iff
% 0.29/0.60  thf(fact_8242_cos__monotone__minus__pi__0_H,axiom,
% 0.29/0.60      ! [Y: real,X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ X )
% 0.29/0.60         => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.60           => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_monotone_minus_pi_0'
% 0.29/0.60  thf(fact_8243_sin__zero__iff__int2,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( sin_real @ X )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60        = ( ? [I3: int] :
% 0.29/0.60              ( X
% 0.29/0.60              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_zero_iff_int2
% 0.29/0.60  thf(fact_8244_sincos__total__pi,axiom,
% 0.29/0.60      ! [Y: real,X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60       => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.60            = one_one_real )
% 0.29/0.60         => ? [T3: real] :
% 0.29/0.60              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.29/0.60              & ( ord_less_eq_real @ T3 @ pi )
% 0.29/0.60              & ( X
% 0.29/0.60                = ( cos_real @ T3 ) )
% 0.29/0.60              & ( Y
% 0.29/0.60                = ( sin_real @ T3 ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sincos_total_pi
% 0.29/0.60  thf(fact_8245_sin__cos__sqrt,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
% 0.29/0.60       => ( ( sin_real @ X )
% 0.29/0.60          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_cos_sqrt
% 0.29/0.60  thf(fact_8246_sin__expansion__lemma,axiom,
% 0.29/0.60      ! [X: real,M: nat] :
% 0.29/0.60        ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.29/0.60        = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_expansion_lemma
% 0.29/0.60  thf(fact_8247_psubset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.29/0.60        ( ( ord_le3480810397992357184T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_VEBT_VEBT @ X @ B2 )
% 0.29/0.60           => ( ord_le3480810397992357184T_VEBT @ A2 @ B2 ) )
% 0.29/0.60          & ( ~ ( member_VEBT_VEBT @ X @ B2 )
% 0.29/0.60           => ( ( ( member_VEBT_VEBT @ X @ A2 )
% 0.29/0.60               => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B2 ) )
% 0.29/0.60              & ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 0.29/0.60               => ( ord_le4337996190870823476T_VEBT @ A2 @ B2 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % psubset_insert_iff
% 0.29/0.60  thf(fact_8248_psubset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_complex,X: complex,B2: set_complex] :
% 0.29/0.60        ( ( ord_less_set_complex @ A2 @ ( insert_complex @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_complex @ X @ B2 )
% 0.29/0.60           => ( ord_less_set_complex @ A2 @ B2 ) )
% 0.29/0.60          & ( ~ ( member_complex @ X @ B2 )
% 0.29/0.60           => ( ( ( member_complex @ X @ A2 )
% 0.29/0.60               => ( ord_less_set_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) )
% 0.29/0.60              & ( ~ ( member_complex @ X @ A2 )
% 0.29/0.60               => ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % psubset_insert_iff
% 0.29/0.60  thf(fact_8249_psubset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
% 0.29/0.60        ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_set_nat @ X @ B2 )
% 0.29/0.60           => ( ord_less_set_set_nat @ A2 @ B2 ) )
% 0.29/0.60          & ( ~ ( member_set_nat @ X @ B2 )
% 0.29/0.60           => ( ( ( member_set_nat @ X @ A2 )
% 0.29/0.60               => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
% 0.29/0.60              & ( ~ ( member_set_nat @ X @ A2 )
% 0.29/0.60               => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % psubset_insert_iff
% 0.29/0.60  thf(fact_8250_psubset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_real,X: real,B2: set_real] :
% 0.29/0.60        ( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_real @ X @ B2 )
% 0.29/0.60           => ( ord_less_set_real @ A2 @ B2 ) )
% 0.29/0.60          & ( ~ ( member_real @ X @ B2 )
% 0.29/0.60           => ( ( ( member_real @ X @ A2 )
% 0.29/0.60               => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
% 0.29/0.60              & ( ~ ( member_real @ X @ A2 )
% 0.29/0.60               => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % psubset_insert_iff
% 0.29/0.60  thf(fact_8251_psubset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_o,X: $o,B2: set_o] :
% 0.29/0.60        ( ( ord_less_set_o @ A2 @ ( insert_o @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_o @ X @ B2 )
% 0.29/0.60           => ( ord_less_set_o @ A2 @ B2 ) )
% 0.29/0.60          & ( ~ ( member_o @ X @ B2 )
% 0.29/0.60           => ( ( ( member_o @ X @ A2 )
% 0.29/0.60               => ( ord_less_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
% 0.29/0.60              & ( ~ ( member_o @ X @ A2 )
% 0.29/0.60               => ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % psubset_insert_iff
% 0.29/0.60  thf(fact_8252_psubset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_nat,X: nat,B2: set_nat] :
% 0.29/0.60        ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_nat @ X @ B2 )
% 0.29/0.60           => ( ord_less_set_nat @ A2 @ B2 ) )
% 0.29/0.60          & ( ~ ( member_nat @ X @ B2 )
% 0.29/0.60           => ( ( ( member_nat @ X @ A2 )
% 0.29/0.60               => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
% 0.29/0.60              & ( ~ ( member_nat @ X @ A2 )
% 0.29/0.60               => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % psubset_insert_iff
% 0.29/0.60  thf(fact_8253_psubset__insert__iff,axiom,
% 0.29/0.60      ! [A2: set_int,X: int,B2: set_int] :
% 0.29/0.60        ( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 0.29/0.60        = ( ( ( member_int @ X @ B2 )
% 0.29/0.60           => ( ord_less_set_int @ A2 @ B2 ) )
% 0.29/0.60          & ( ~ ( member_int @ X @ B2 )
% 0.29/0.60           => ( ( ( member_int @ X @ A2 )
% 0.29/0.60               => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
% 0.29/0.60              & ( ~ ( member_int @ X @ A2 )
% 0.29/0.60               => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % psubset_insert_iff
% 0.29/0.60  thf(fact_8254_numeral__BitM,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
% 0.29/0.60        = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % numeral_BitM
% 0.29/0.60  thf(fact_8255_numeral__BitM,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( numeral_numeral_real @ ( bitM @ N ) )
% 0.29/0.60        = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % numeral_BitM
% 0.29/0.60  thf(fact_8256_numeral__BitM,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( numeral_numeral_rat @ ( bitM @ N ) )
% 0.29/0.60        = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % numeral_BitM
% 0.29/0.60  thf(fact_8257_numeral__BitM,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( numeral_numeral_int @ ( bitM @ N ) )
% 0.29/0.60        = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % numeral_BitM
% 0.29/0.60  thf(fact_8258_odd__numeral__BitM,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % odd_numeral_BitM
% 0.29/0.60  thf(fact_8259_odd__numeral__BitM,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % odd_numeral_BitM
% 0.29/0.60  thf(fact_8260_odd__numeral__BitM,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % odd_numeral_BitM
% 0.29/0.60  thf(fact_8261_cos__expansion__lemma,axiom,
% 0.29/0.60      ! [X: real,M: nat] :
% 0.29/0.60        ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.29/0.60        = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_expansion_lemma
% 0.29/0.60  thf(fact_8262_sin__gt__zero__02,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.60         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_gt_zero_02
% 0.29/0.60  thf(fact_8263_cos__two__less__zero,axiom,
% 0.29/0.60      ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.29/0.60  
% 0.29/0.60  % cos_two_less_zero
% 0.29/0.60  thf(fact_8264_cos__is__zero,axiom,
% 0.29/0.60      ? [X3: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.60        & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.60        & ( ( cos_real @ X3 )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60        & ! [Y5: real] :
% 0.29/0.60            ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.29/0.60              & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.60              & ( ( cos_real @ Y5 )
% 0.29/0.60                = zero_zero_real ) )
% 0.29/0.60           => ( Y5 = X3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_is_zero
% 0.29/0.60  thf(fact_8265_cos__two__le__zero,axiom,
% 0.29/0.60      ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.29/0.60  
% 0.29/0.60  % cos_two_le_zero
% 0.29/0.60  thf(fact_8266_cos__monotone__minus__pi__0,axiom,
% 0.29/0.60      ! [Y: real,X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
% 0.29/0.60       => ( ( ord_less_real @ Y @ X )
% 0.29/0.60         => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.60           => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_monotone_minus_pi_0
% 0.29/0.60  thf(fact_8267_cos__total,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ? [X3: real] :
% 0.29/0.60              ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.60              & ( ord_less_eq_real @ X3 @ pi )
% 0.29/0.60              & ( ( cos_real @ X3 )
% 0.29/0.60                = Y )
% 0.29/0.60              & ! [Y5: real] :
% 0.29/0.60                  ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.29/0.60                    & ( ord_less_eq_real @ Y5 @ pi )
% 0.29/0.60                    & ( ( cos_real @ Y5 )
% 0.29/0.60                      = Y ) )
% 0.29/0.60                 => ( Y5 = X3 ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_total
% 0.29/0.60  thf(fact_8268_sincos__total__pi__half,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60         => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.60              = one_one_real )
% 0.29/0.60           => ? [T3: real] :
% 0.29/0.60                ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.29/0.60                & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60                & ( X
% 0.29/0.60                  = ( cos_real @ T3 ) )
% 0.29/0.60                & ( Y
% 0.29/0.60                  = ( sin_real @ T3 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sincos_total_pi_half
% 0.29/0.60  thf(fact_8269_sincos__total__2pi__le,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.60          = one_one_real )
% 0.29/0.60       => ? [T3: real] :
% 0.29/0.60            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.29/0.60            & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.29/0.60            & ( X
% 0.29/0.60              = ( cos_real @ T3 ) )
% 0.29/0.60            & ( Y
% 0.29/0.60              = ( sin_real @ T3 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sincos_total_2pi_le
% 0.29/0.60  thf(fact_8270_sincos__total__2pi,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.60          = one_one_real )
% 0.29/0.60       => ~ ! [T3: real] :
% 0.29/0.60              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.29/0.60             => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.29/0.60               => ( ( X
% 0.29/0.60                    = ( cos_real @ T3 ) )
% 0.29/0.60                 => ( Y
% 0.29/0.60                   != ( sin_real @ T3 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sincos_total_2pi
% 0.29/0.60  thf(fact_8271_sin__pi__divide__n__ge__0,axiom,
% 0.29/0.60      ! [N: nat] :
% 0.29/0.60        ( ( N != zero_zero_nat )
% 0.29/0.60       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_pi_divide_n_ge_0
% 0.29/0.60  thf(fact_8272_sin__45,axiom,
% 0.29/0.60      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.29/0.60      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_45
% 0.29/0.60  thf(fact_8273_cos__45,axiom,
% 0.29/0.60      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.29/0.60      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_45
% 0.29/0.60  thf(fact_8274_cos__plus__cos,axiom,
% 0.29/0.60      ! [W: complex,Z: complex] :
% 0.29/0.60        ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_plus_cos
% 0.29/0.60  thf(fact_8275_cos__plus__cos,axiom,
% 0.29/0.60      ! [W: real,Z: real] :
% 0.29/0.60        ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_plus_cos
% 0.29/0.60  thf(fact_8276_cos__times__cos,axiom,
% 0.29/0.60      ! [W: complex,Z: complex] :
% 0.29/0.60        ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.29/0.60        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_times_cos
% 0.29/0.60  thf(fact_8277_cos__times__cos,axiom,
% 0.29/0.60      ! [W: real,Z: real] :
% 0.29/0.60        ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.29/0.60        = ( 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 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_times_cos
% 0.29/0.60  thf(fact_8278_sin__gt__zero2,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_gt_zero2
% 0.29/0.60  thf(fact_8279_sin__lt__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ pi @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.29/0.60         => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_lt_zero
% 0.29/0.60  thf(fact_8280_cos__double__less__one,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.60         => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_double_less_one
% 0.29/0.60  thf(fact_8281_sin__30,axiom,
% 0.29/0.60      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.29/0.60      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_30
% 0.29/0.60  thf(fact_8282_cos__gt__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_gt_zero
% 0.29/0.60  thf(fact_8283_sin__monotone__2pi__le,axiom,
% 0.29/0.60      ! [Y: real,X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ X )
% 0.29/0.60         => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60           => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_monotone_2pi_le
% 0.29/0.60  thf(fact_8284_sin__mono__le__eq,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.29/0.60           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60             => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 0.29/0.60                = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_mono_le_eq
% 0.29/0.60  thf(fact_8285_sin__inj__pi,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.29/0.60           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60             => ( ( ( sin_real @ X )
% 0.29/0.60                  = ( sin_real @ Y ) )
% 0.29/0.60               => ( X = Y ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_inj_pi
% 0.29/0.60  thf(fact_8286_cos__60,axiom,
% 0.29/0.60      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.29/0.60      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_60
% 0.29/0.60  thf(fact_8287_sin__60,axiom,
% 0.29/0.60      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.29/0.60      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_60
% 0.29/0.60  thf(fact_8288_cos__30,axiom,
% 0.29/0.60      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.29/0.60      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_30
% 0.29/0.60  thf(fact_8289_cos__one__2pi__int,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( cos_real @ X )
% 0.29/0.60          = one_one_real )
% 0.29/0.60        = ( ? [X4: int] :
% 0.29/0.60              ( X
% 0.29/0.60              = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_one_2pi_int
% 0.29/0.60  thf(fact_8290_cos__double__cos,axiom,
% 0.29/0.60      ! [W: complex] :
% 0.29/0.60        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.29/0.60        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_double_cos
% 0.29/0.60  thf(fact_8291_cos__double__cos,axiom,
% 0.29/0.60      ! [W: real] :
% 0.29/0.60        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.29/0.60        = ( 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 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_double_cos
% 0.29/0.60  thf(fact_8292_cos__treble__cos,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
% 0.29/0.60        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_treble_cos
% 0.29/0.60  thf(fact_8293_cos__treble__cos,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
% 0.29/0.60        = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_treble_cos
% 0.29/0.60  thf(fact_8294_and__nat__unfold,axiom,
% 0.29/0.60      ( bit_se727722235901077358nd_nat
% 0.29/0.60      = ( ^ [M6: nat,N3: nat] :
% 0.29/0.60            ( if_nat
% 0.29/0.60            @ ( ( M6 = zero_zero_nat )
% 0.29/0.60              | ( N3 = zero_zero_nat ) )
% 0.29/0.60            @ zero_zero_nat
% 0.29/0.60            @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % and_nat_unfold
% 0.29/0.60  thf(fact_8295_sin__le__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ pi @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.29/0.60         => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_le_zero
% 0.29/0.60  thf(fact_8296_sin__less__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ zero_zero_real )
% 0.29/0.60         => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_less_zero
% 0.29/0.60  thf(fact_8297_sin__monotone__2pi,axiom,
% 0.29/0.60      ! [Y: real,X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.29/0.60       => ( ( ord_less_real @ Y @ X )
% 0.29/0.60         => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60           => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_monotone_2pi
% 0.29/0.60  thf(fact_8298_sin__mono__less__eq,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.29/0.60           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60             => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 0.29/0.60                = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_mono_less_eq
% 0.29/0.60  thf(fact_8299_sin__total,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ? [X3: real] :
% 0.29/0.60              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.29/0.60              & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60              & ( ( sin_real @ X3 )
% 0.29/0.60                = Y )
% 0.29/0.60              & ! [Y5: real] :
% 0.29/0.60                  ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.29/0.60                    & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60                    & ( ( sin_real @ Y5 )
% 0.29/0.60                      = Y ) )
% 0.29/0.60                 => ( Y5 = X3 ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_total
% 0.29/0.60  thf(fact_8300_and__nat__rec,axiom,
% 0.29/0.60      ( bit_se727722235901077358nd_nat
% 0.29/0.60      = ( ^ [M6: nat,N3: nat] :
% 0.29/0.60            ( plus_plus_nat
% 0.29/0.60            @ ( zero_n2687167440665602831ol_nat
% 0.29/0.60              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 0.29/0.60                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.29/0.60            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % and_nat_rec
% 0.29/0.60  thf(fact_8301_cos__gt__zero__pi,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_gt_zero_pi
% 0.29/0.60  thf(fact_8302_cos__ge__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_ge_zero
% 0.29/0.60  thf(fact_8303_cos__one__2pi,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( cos_real @ X )
% 0.29/0.60          = one_one_real )
% 0.29/0.60        = ( ? [X4: nat] :
% 0.29/0.60              ( X
% 0.29/0.60              = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.29/0.60          | ? [X4: nat] :
% 0.29/0.60              ( X
% 0.29/0.60              = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_one_2pi
% 0.29/0.60  thf(fact_8304_sin__pi__divide__n__gt__0,axiom,
% 0.29/0.60      ! [N: nat] :
% 0.29/0.60        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.60       => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_pi_divide_n_gt_0
% 0.29/0.60  thf(fact_8305_sin__arctan,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( sin_real @ ( arctan @ X ) )
% 0.29/0.60        = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_arctan
% 0.29/0.60  thf(fact_8306_cos__arctan,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cos_real @ ( arctan @ X ) )
% 0.29/0.60        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_arctan
% 0.29/0.60  thf(fact_8307_sin__zero__iff__int,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( sin_real @ X )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60        = ( ? [I3: int] :
% 0.29/0.60              ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
% 0.29/0.60              & ( X
% 0.29/0.60                = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_zero_iff_int
% 0.29/0.60  thf(fact_8308_cos__zero__iff__int,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( cos_real @ X )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60        = ( ? [I3: int] :
% 0.29/0.60              ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
% 0.29/0.60              & ( X
% 0.29/0.60                = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_zero_iff_int
% 0.29/0.60  thf(fact_8309_sin__zero__lemma,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ( sin_real @ X )
% 0.29/0.60            = zero_zero_real )
% 0.29/0.60         => ? [N2: nat] :
% 0.29/0.60              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.29/0.60              & ( X
% 0.29/0.60                = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_zero_lemma
% 0.29/0.60  thf(fact_8310_sin__zero__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( sin_real @ X )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60        = ( ? [N3: nat] :
% 0.29/0.60              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.29/0.60              & ( X
% 0.29/0.60                = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.60          | ? [N3: nat] :
% 0.29/0.60              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.29/0.60              & ( X
% 0.29/0.60                = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_zero_iff
% 0.29/0.60  thf(fact_8311_cos__zero__lemma,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ( cos_real @ X )
% 0.29/0.60            = zero_zero_real )
% 0.29/0.60         => ? [N2: nat] :
% 0.29/0.60              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.29/0.60              & ( X
% 0.29/0.60                = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_zero_lemma
% 0.29/0.60  thf(fact_8312_cos__zero__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( cos_real @ X )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60        = ( ? [N3: nat] :
% 0.29/0.60              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.29/0.60              & ( X
% 0.29/0.60                = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.60          | ? [N3: nat] :
% 0.29/0.60              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.29/0.60              & ( X
% 0.29/0.60                = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_zero_iff
% 0.29/0.60  thf(fact_8313_tan__double,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( ( cos_complex @ X )
% 0.29/0.60         != zero_zero_complex )
% 0.29/0.60       => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60           != zero_zero_complex )
% 0.29/0.60         => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_double
% 0.29/0.60  thf(fact_8314_tan__double,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( cos_real @ X )
% 0.29/0.60         != zero_zero_real )
% 0.29/0.60       => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60           != zero_zero_real )
% 0.29/0.60         => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60            = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_double
% 0.29/0.60  thf(fact_8315_sin__tan,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60       => ( ( sin_real @ X )
% 0.29/0.60          = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_tan
% 0.29/0.60  thf(fact_8316_cos__tan,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60       => ( ( cos_real @ X )
% 0.29/0.60          = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_tan
% 0.29/0.60  thf(fact_8317_complex__unimodular__polar,axiom,
% 0.29/0.60      ! [Z: complex] :
% 0.29/0.60        ( ( ( real_V1022390504157884413omplex @ Z )
% 0.29/0.60          = one_one_real )
% 0.29/0.60       => ~ ! [T3: real] :
% 0.29/0.60              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.29/0.60             => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.29/0.60               => ( Z
% 0.29/0.60                 != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % complex_unimodular_polar
% 0.29/0.60  thf(fact_8318_and__int_Opelims,axiom,
% 0.29/0.60      ! [X: int,Xa: int,Y: int] :
% 0.29/0.60        ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 0.29/0.60          = Y )
% 0.29/0.60       => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 0.29/0.60         => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.29/0.60                    & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.29/0.60                 => ( Y
% 0.29/0.60                    = ( uminus_uminus_int
% 0.29/0.60                      @ ( zero_n2684676970156552555ol_int
% 0.29/0.60                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.29/0.60                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.29/0.60                & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.29/0.60                      & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.29/0.60                 => ( Y
% 0.29/0.60                    = ( plus_plus_int
% 0.29/0.60                      @ ( zero_n2684676970156552555ol_int
% 0.29/0.60                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.29/0.60                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.29/0.60                      @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.29/0.60             => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % and_int.pelims
% 0.29/0.60  thf(fact_8319_bit__0__eq,axiom,
% 0.29/0.60      ( ( bit_se1146084159140164899it_int @ zero_zero_int )
% 0.29/0.60      = bot_bot_nat_o ) ).
% 0.29/0.60  
% 0.29/0.60  % bit_0_eq
% 0.29/0.60  thf(fact_8320_bit__0__eq,axiom,
% 0.29/0.60      ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
% 0.29/0.60      = bot_bot_nat_o ) ).
% 0.29/0.60  
% 0.29/0.60  % bit_0_eq
% 0.29/0.60  thf(fact_8321_tan__zero,axiom,
% 0.29/0.60      ( ( tan_complex @ zero_zero_complex )
% 0.29/0.60      = zero_zero_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_zero
% 0.29/0.60  thf(fact_8322_tan__zero,axiom,
% 0.29/0.60      ( ( tan_real @ zero_zero_real )
% 0.29/0.60      = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_zero
% 0.29/0.60  thf(fact_8323_tan__minus,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( tan_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.60        = ( uminus_uminus_real @ ( tan_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_minus
% 0.29/0.60  thf(fact_8324_tan__minus,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( tan_complex @ ( uminus1482373934393186551omplex @ X ) )
% 0.29/0.60        = ( uminus1482373934393186551omplex @ ( tan_complex @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_minus
% 0.29/0.60  thf(fact_8325_set__decode__zero,axiom,
% 0.29/0.60      ( ( nat_set_decode @ zero_zero_nat )
% 0.29/0.60      = bot_bot_set_nat ) ).
% 0.29/0.60  
% 0.29/0.60  % set_decode_zero
% 0.29/0.60  thf(fact_8326_tan__pi,axiom,
% 0.29/0.60      ( ( tan_real @ pi )
% 0.29/0.60      = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_pi
% 0.29/0.60  thf(fact_8327_tan__periodic__pi,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
% 0.29/0.60        = ( tan_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_periodic_pi
% 0.29/0.60  thf(fact_8328_tan__npi,axiom,
% 0.29/0.60      ! [N: nat] :
% 0.29/0.60        ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.29/0.60        = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_npi
% 0.29/0.60  thf(fact_8329_tan__periodic__n,axiom,
% 0.29/0.60      ! [X: real,N: num] :
% 0.29/0.60        ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 0.29/0.60        = ( tan_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_periodic_n
% 0.29/0.60  thf(fact_8330_tan__periodic__nat,axiom,
% 0.29/0.60      ! [X: real,N: nat] :
% 0.29/0.60        ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 0.29/0.60        = ( tan_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_periodic_nat
% 0.29/0.60  thf(fact_8331_tan__periodic__int,axiom,
% 0.29/0.60      ! [X: real,I: int] :
% 0.29/0.60        ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 0.29/0.60        = ( tan_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_periodic_int
% 0.29/0.60  thf(fact_8332_norm__cos__sin,axiom,
% 0.29/0.60      ! [T: real] :
% 0.29/0.60        ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 0.29/0.60        = one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % norm_cos_sin
% 0.29/0.60  thf(fact_8333_tan__periodic,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.29/0.60        = ( tan_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_periodic
% 0.29/0.60  thf(fact_8334_bot__nat__def,axiom,
% 0.29/0.60      bot_bot_nat = zero_zero_nat ).
% 0.29/0.60  
% 0.29/0.60  % bot_nat_def
% 0.29/0.60  thf(fact_8335_bot__set__def,axiom,
% 0.29/0.60      ( bot_bot_set_complex
% 0.29/0.60      = ( collect_complex @ bot_bot_complex_o ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot_set_def
% 0.29/0.60  thf(fact_8336_bot__set__def,axiom,
% 0.29/0.60      ( bot_bot_set_set_nat
% 0.29/0.60      = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot_set_def
% 0.29/0.60  thf(fact_8337_bot__set__def,axiom,
% 0.29/0.60      ( bot_bot_set_real
% 0.29/0.60      = ( collect_real @ bot_bot_real_o ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot_set_def
% 0.29/0.60  thf(fact_8338_bot__set__def,axiom,
% 0.29/0.60      ( bot_bot_set_o
% 0.29/0.60      = ( collect_o @ bot_bot_o_o ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot_set_def
% 0.29/0.60  thf(fact_8339_bot__set__def,axiom,
% 0.29/0.60      ( bot_bot_set_nat
% 0.29/0.60      = ( collect_nat @ bot_bot_nat_o ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot_set_def
% 0.29/0.60  thf(fact_8340_bot__set__def,axiom,
% 0.29/0.60      ( bot_bot_set_int
% 0.29/0.60      = ( collect_int @ bot_bot_int_o ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bot_set_def
% 0.29/0.60  thf(fact_8341_tan__of__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( real_V1803761363581548252l_real @ ( tan_real @ X ) )
% 0.29/0.60        = ( tan_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_of_real
% 0.29/0.60  thf(fact_8342_tan__of__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( real_V4546457046886955230omplex @ ( tan_real @ X ) )
% 0.29/0.60        = ( tan_complex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_of_real
% 0.29/0.60  thf(fact_8343_zero__complex_Ocode,axiom,
% 0.29/0.60      ( zero_zero_complex
% 0.29/0.60      = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % zero_complex.code
% 0.29/0.60  thf(fact_8344_Complex__eq__0,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ( complex2 @ A @ B )
% 0.29/0.60          = zero_zero_complex )
% 0.29/0.60        = ( ( A = zero_zero_real )
% 0.29/0.60          & ( B = zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Complex_eq_0
% 0.29/0.60  thf(fact_8345_tan__arctan,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( tan_real @ ( arctan @ Y ) )
% 0.29/0.60        = Y ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_arctan
% 0.29/0.60  thf(fact_8346_bot__enat__def,axiom,
% 0.29/0.60      bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 0.29/0.60  
% 0.29/0.60  % bot_enat_def
% 0.29/0.60  thf(fact_8347_one__complex_Ocode,axiom,
% 0.29/0.60      ( one_one_complex
% 0.29/0.60      = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_complex.code
% 0.29/0.60  thf(fact_8348_Complex__eq__1,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ( complex2 @ A @ B )
% 0.29/0.60          = one_one_complex )
% 0.29/0.60        = ( ( A = one_one_real )
% 0.29/0.60          & ( B = zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Complex_eq_1
% 0.29/0.60  thf(fact_8349_Complex__eq__numeral,axiom,
% 0.29/0.60      ! [A: real,B: real,W: num] :
% 0.29/0.60        ( ( ( complex2 @ A @ B )
% 0.29/0.60          = ( numera6690914467698888265omplex @ W ) )
% 0.29/0.60        = ( ( A
% 0.29/0.60            = ( numeral_numeral_real @ W ) )
% 0.29/0.60          & ( B = zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Complex_eq_numeral
% 0.29/0.60  thf(fact_8350_complex__eq__cancel__iff2,axiom,
% 0.29/0.60      ! [X: real,Y: real,Xa: real] :
% 0.29/0.60        ( ( ( complex2 @ X @ Y )
% 0.29/0.60          = ( real_V4546457046886955230omplex @ Xa ) )
% 0.29/0.60        = ( ( X = Xa )
% 0.29/0.60          & ( Y = zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % complex_eq_cancel_iff2
% 0.29/0.60  thf(fact_8351_complex__of__real__code,axiom,
% 0.29/0.60      ( real_V4546457046886955230omplex
% 0.29/0.60      = ( ^ [X4: real] : ( complex2 @ X4 @ zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % complex_of_real_code
% 0.29/0.60  thf(fact_8352_complex__of__real__def,axiom,
% 0.29/0.60      ( real_V4546457046886955230omplex
% 0.29/0.60      = ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % complex_of_real_def
% 0.29/0.60  thf(fact_8353_Complex__eq__neg__1,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ( complex2 @ A @ B )
% 0.29/0.60          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.29/0.60        = ( ( A
% 0.29/0.60            = ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.60          & ( B = zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Complex_eq_neg_1
% 0.29/0.60  thf(fact_8354_Complex__eq__neg__numeral,axiom,
% 0.29/0.60      ! [A: real,B: real,W: num] :
% 0.29/0.60        ( ( ( complex2 @ A @ B )
% 0.29/0.60          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.60        = ( ( A
% 0.29/0.60            = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.60          & ( B = zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Complex_eq_neg_numeral
% 0.29/0.60  thf(fact_8355_tan__def,axiom,
% 0.29/0.60      ( tan_complex
% 0.29/0.60      = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X4 ) @ ( cos_complex @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_def
% 0.29/0.60  thf(fact_8356_tan__def,axiom,
% 0.29/0.60      ( tan_real
% 0.29/0.60      = ( ^ [X4: real] : ( divide_divide_real @ ( sin_real @ X4 ) @ ( cos_real @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_def
% 0.29/0.60  thf(fact_8357_tan__45,axiom,
% 0.29/0.60      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.29/0.60      = one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_45
% 0.29/0.60  thf(fact_8358_tan__60,axiom,
% 0.29/0.60      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.29/0.60      = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_60
% 0.29/0.60  thf(fact_8359_tan__gt__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_gt_zero
% 0.29/0.60  thf(fact_8360_lemma__tan__total,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.29/0.60       => ? [X3: real] :
% 0.29/0.60            ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.29/0.60            & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60            & ( ord_less_real @ Y @ ( tan_real @ X3 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % lemma_tan_total
% 0.29/0.60  thf(fact_8361_tan__total,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60      ? [X3: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.29/0.60        & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60        & ( ( tan_real @ X3 )
% 0.29/0.60          = Y )
% 0.29/0.60        & ! [Y5: real] :
% 0.29/0.60            ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.29/0.60              & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60              & ( ( tan_real @ Y5 )
% 0.29/0.60                = Y ) )
% 0.29/0.60           => ( Y5 = X3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_total
% 0.29/0.60  thf(fact_8362_tan__monotone,axiom,
% 0.29/0.60      ! [Y: real,X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.29/0.60       => ( ( ord_less_real @ Y @ X )
% 0.29/0.60         => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60           => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_monotone
% 0.29/0.60  thf(fact_8363_tan__monotone_H,axiom,
% 0.29/0.60      ! [Y: real,X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.29/0.60       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60           => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60             => ( ( ord_less_real @ Y @ X )
% 0.29/0.60                = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_monotone'
% 0.29/0.60  thf(fact_8364_tan__mono__lt__eq,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.29/0.60           => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60             => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.29/0.60                = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_mono_lt_eq
% 0.29/0.60  thf(fact_8365_lemma__tan__total1,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60      ? [X3: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.29/0.60        & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60        & ( ( tan_real @ X3 )
% 0.29/0.60          = Y ) ) ).
% 0.29/0.60  
% 0.29/0.60  % lemma_tan_total1
% 0.29/0.60  thf(fact_8366_tan__minus__45,axiom,
% 0.29/0.60      ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.60      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_minus_45
% 0.29/0.60  thf(fact_8367_tan__inverse,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
% 0.29/0.60        = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_inverse
% 0.29/0.60  thf(fact_8368_complex__norm,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
% 0.29/0.60        = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % complex_norm
% 0.29/0.60  thf(fact_8369_add__tan__eq,axiom,
% 0.29/0.60      ! [X: complex,Y: complex] :
% 0.29/0.60        ( ( ( cos_complex @ X )
% 0.29/0.60         != zero_zero_complex )
% 0.29/0.60       => ( ( ( cos_complex @ Y )
% 0.29/0.60           != zero_zero_complex )
% 0.29/0.60         => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
% 0.29/0.60            = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % add_tan_eq
% 0.29/0.60  thf(fact_8370_add__tan__eq,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( cos_real @ X )
% 0.29/0.60         != zero_zero_real )
% 0.29/0.60       => ( ( ( cos_real @ Y )
% 0.29/0.60           != zero_zero_real )
% 0.29/0.60         => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.29/0.60            = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % add_tan_eq
% 0.29/0.60  thf(fact_8371_tan__total__pos,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60       => ? [X3: real] :
% 0.29/0.60            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.60            & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60            & ( ( tan_real @ X3 )
% 0.29/0.60              = Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_total_pos
% 0.29/0.60  thf(fact_8372_tan__pos__pi2__le,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_pos_pi2_le
% 0.29/0.60  thf(fact_8373_tan__less__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ zero_zero_real )
% 0.29/0.60         => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_less_zero
% 0.29/0.60  thf(fact_8374_tan__mono__le,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.60         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60           => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_mono_le
% 0.29/0.60  thf(fact_8375_tan__mono__le__eq,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.29/0.60           => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60             => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.29/0.60                = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_mono_le_eq
% 0.29/0.60  thf(fact_8376_tan__bound__pi2,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.29/0.60       => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_bound_pi2
% 0.29/0.60  thf(fact_8377_tan__30,axiom,
% 0.29/0.60      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.29/0.60      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_30
% 0.29/0.60  thf(fact_8378_arctan,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 0.29/0.60        & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60        & ( ( tan_real @ ( arctan @ Y ) )
% 0.29/0.60          = Y ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arctan
% 0.29/0.60  thf(fact_8379_arctan__tan,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ( arctan @ ( tan_real @ X ) )
% 0.29/0.60            = X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arctan_tan
% 0.29/0.60  thf(fact_8380_arctan__unique,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ( ( tan_real @ X )
% 0.29/0.60              = Y )
% 0.29/0.60           => ( ( arctan @ Y )
% 0.29/0.60              = X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arctan_unique
% 0.29/0.60  thf(fact_8381_and__int_Opinduct,axiom,
% 0.29/0.60      ! [A0: int,A1: int,P: int > int > $o] :
% 0.29/0.60        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 0.29/0.60       => ( ! [K3: int,L2: int] :
% 0.29/0.60              ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L2 ) )
% 0.29/0.60             => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.29/0.60                      & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.29/0.60                 => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.29/0.60               => ( P @ K3 @ L2 ) ) )
% 0.29/0.60         => ( P @ A0 @ A1 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % and_int.pinduct
% 0.29/0.60  thf(fact_8382_tan__add,axiom,
% 0.29/0.60      ! [X: complex,Y: complex] :
% 0.29/0.60        ( ( ( cos_complex @ X )
% 0.29/0.60         != zero_zero_complex )
% 0.29/0.60       => ( ( ( cos_complex @ Y )
% 0.29/0.60           != zero_zero_complex )
% 0.29/0.60         => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.29/0.60             != zero_zero_complex )
% 0.29/0.60           => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.29/0.60              = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_add
% 0.29/0.60  thf(fact_8383_tan__add,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( cos_real @ X )
% 0.29/0.60         != zero_zero_real )
% 0.29/0.60       => ( ( ( cos_real @ Y )
% 0.29/0.60           != zero_zero_real )
% 0.29/0.60         => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.60             != zero_zero_real )
% 0.29/0.60           => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.60              = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_add
% 0.29/0.60  thf(fact_8384_tan__diff,axiom,
% 0.29/0.60      ! [X: complex,Y: complex] :
% 0.29/0.60        ( ( ( cos_complex @ X )
% 0.29/0.60         != zero_zero_complex )
% 0.29/0.60       => ( ( ( cos_complex @ Y )
% 0.29/0.60           != zero_zero_complex )
% 0.29/0.60         => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.29/0.60             != zero_zero_complex )
% 0.29/0.60           => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.29/0.60              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_diff
% 0.29/0.60  thf(fact_8385_tan__diff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( cos_real @ X )
% 0.29/0.60         != zero_zero_real )
% 0.29/0.60       => ( ( ( cos_real @ Y )
% 0.29/0.60           != zero_zero_real )
% 0.29/0.60         => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 0.29/0.60             != zero_zero_real )
% 0.29/0.60           => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
% 0.29/0.60              = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_diff
% 0.29/0.60  thf(fact_8386_lemma__tan__add1,axiom,
% 0.29/0.60      ! [X: complex,Y: complex] :
% 0.29/0.60        ( ( ( cos_complex @ X )
% 0.29/0.60         != zero_zero_complex )
% 0.29/0.60       => ( ( ( cos_complex @ Y )
% 0.29/0.60           != zero_zero_complex )
% 0.29/0.60         => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
% 0.29/0.60            = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % lemma_tan_add1
% 0.29/0.60  thf(fact_8387_lemma__tan__add1,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( cos_real @ X )
% 0.29/0.60         != zero_zero_real )
% 0.29/0.60       => ( ( ( cos_real @ Y )
% 0.29/0.60           != zero_zero_real )
% 0.29/0.60         => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
% 0.29/0.60            = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % lemma_tan_add1
% 0.29/0.60  thf(fact_8388_tan__total__pi4,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.60       => ? [Z3: real] :
% 0.29/0.60            ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
% 0.29/0.60            & ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.29/0.60            & ( ( tan_real @ Z3 )
% 0.29/0.60              = X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_total_pi4
% 0.29/0.60  thf(fact_8389_tan__half,axiom,
% 0.29/0.60      ( tan_complex
% 0.29/0.60      = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_complex ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_half
% 0.29/0.60  thf(fact_8390_tan__half,axiom,
% 0.29/0.60      ( tan_real
% 0.29/0.60      = ( ^ [X4: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_half
% 0.29/0.60  thf(fact_8391_and__int_Opsimps,axiom,
% 0.29/0.60      ! [K: int,L: int] :
% 0.29/0.60        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 0.29/0.60       => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.29/0.60              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.29/0.60           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.29/0.60              = ( uminus_uminus_int
% 0.29/0.60                @ ( zero_n2684676970156552555ol_int
% 0.29/0.60                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.29/0.60                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 0.29/0.60          & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.29/0.60                & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.29/0.60           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.29/0.60              = ( plus_plus_int
% 0.29/0.60                @ ( zero_n2684676970156552555ol_int
% 0.29/0.60                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.29/0.60                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.29/0.60                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % and_int.psimps
% 0.29/0.60  thf(fact_8392_upto_Opinduct,axiom,
% 0.29/0.60      ! [A0: int,A1: int,P: int > int > $o] :
% 0.29/0.60        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 0.29/0.60       => ( ! [I2: int,J3: int] :
% 0.29/0.60              ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J3 ) )
% 0.29/0.60             => ( ( ( ord_less_eq_int @ I2 @ J3 )
% 0.29/0.60                 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) )
% 0.29/0.60               => ( P @ I2 @ J3 ) ) )
% 0.29/0.60         => ( P @ A0 @ A1 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % upto.pinduct
% 0.29/0.60  thf(fact_8393_cos__arcsin,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.60         => ( ( cos_real @ ( arcsin @ X ) )
% 0.29/0.60            = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_arcsin
% 0.29/0.60  thf(fact_8394_sin__arccos__abs,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.29/0.60       => ( ( sin_real @ ( arccos @ Y ) )
% 0.29/0.60          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_arccos_abs
% 0.29/0.60  thf(fact_8395_sin__arccos,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.60         => ( ( sin_real @ ( arccos @ X ) )
% 0.29/0.60            = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_arccos
% 0.29/0.60  thf(fact_8396_cot__less__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ zero_zero_real )
% 0.29/0.60         => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_less_zero
% 0.29/0.60  thf(fact_8397_arcsin__0,axiom,
% 0.29/0.60      ( ( arcsin @ zero_zero_real )
% 0.29/0.60      = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_0
% 0.29/0.60  thf(fact_8398_cot__zero,axiom,
% 0.29/0.60      ( ( cot_complex @ zero_zero_complex )
% 0.29/0.60      = zero_zero_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_zero
% 0.29/0.60  thf(fact_8399_cot__zero,axiom,
% 0.29/0.60      ( ( cot_real @ zero_zero_real )
% 0.29/0.60      = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_zero
% 0.29/0.60  thf(fact_8400_cot__minus,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cot_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.60        = ( uminus_uminus_real @ ( cot_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_minus
% 0.29/0.60  thf(fact_8401_cot__minus,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( cot_complex @ ( uminus1482373934393186551omplex @ X ) )
% 0.29/0.60        = ( uminus1482373934393186551omplex @ ( cot_complex @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_minus
% 0.29/0.60  thf(fact_8402_arccos__1,axiom,
% 0.29/0.60      ( ( arccos @ one_one_real )
% 0.29/0.60      = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_1
% 0.29/0.60  thf(fact_8403_cot__pi,axiom,
% 0.29/0.60      ( ( cot_real @ pi )
% 0.29/0.60      = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_pi
% 0.29/0.60  thf(fact_8404_arccos__minus__1,axiom,
% 0.29/0.60      ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.60      = pi ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_minus_1
% 0.29/0.60  thf(fact_8405_cos__arccos,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ( cos_real @ ( arccos @ Y ) )
% 0.29/0.60            = Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_arccos
% 0.29/0.60  thf(fact_8406_sin__arcsin,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ( sin_real @ ( arcsin @ Y ) )
% 0.29/0.60            = Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_arcsin
% 0.29/0.60  thf(fact_8407_cot__npi,axiom,
% 0.29/0.60      ! [N: nat] :
% 0.29/0.60        ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.29/0.60        = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_npi
% 0.29/0.60  thf(fact_8408_arccos__0,axiom,
% 0.29/0.60      ( ( arccos @ zero_zero_real )
% 0.29/0.60      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_0
% 0.29/0.60  thf(fact_8409_arcsin__1,axiom,
% 0.29/0.60      ( ( arcsin @ one_one_real )
% 0.29/0.60      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_1
% 0.29/0.60  thf(fact_8410_cot__periodic,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.29/0.60        = ( cot_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_periodic
% 0.29/0.60  thf(fact_8411_arcsin__minus__1,axiom,
% 0.29/0.60      ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 0.29/0.60      = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_minus_1
% 0.29/0.60  thf(fact_8412_cot__of__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( real_V1803761363581548252l_real @ ( cot_real @ X ) )
% 0.29/0.60        = ( cot_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_of_real
% 0.29/0.60  thf(fact_8413_cot__of__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( real_V4546457046886955230omplex @ ( cot_real @ X ) )
% 0.29/0.60        = ( cot_complex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_of_real
% 0.29/0.60  thf(fact_8414_arccos__le__arccos,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.60         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60           => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_le_arccos
% 0.29/0.60  thf(fact_8415_arccos__le__mono,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.60       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 0.29/0.60            = ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_le_mono
% 0.29/0.60  thf(fact_8416_arccos__eq__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.60          & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
% 0.29/0.60       => ( ( ( arccos @ X )
% 0.29/0.60            = ( arccos @ Y ) )
% 0.29/0.60          = ( X = Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_eq_iff
% 0.29/0.60  thf(fact_8417_arcsin__le__arcsin,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.60         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60           => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_le_arcsin
% 0.29/0.60  thf(fact_8418_arcsin__minus,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.60         => ( ( arcsin @ ( uminus_uminus_real @ X ) )
% 0.29/0.60            = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_minus
% 0.29/0.60  thf(fact_8419_arcsin__le__mono,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.60       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 0.29/0.60            = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_le_mono
% 0.29/0.60  thf(fact_8420_arcsin__eq__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.60       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.29/0.60         => ( ( ( arcsin @ X )
% 0.29/0.60              = ( arcsin @ Y ) )
% 0.29/0.60            = ( X = Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_eq_iff
% 0.29/0.60  thf(fact_8421_arccos__lbound,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_lbound
% 0.29/0.60  thf(fact_8422_arccos__less__arccos,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ Y )
% 0.29/0.60         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60           => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_less_arccos
% 0.29/0.60  thf(fact_8423_arccos__less__mono,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.60       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.29/0.60         => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 0.29/0.60            = ( ord_less_real @ Y @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_less_mono
% 0.29/0.60  thf(fact_8424_arccos__ubound,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_ubound
% 0.29/0.60  thf(fact_8425_arccos__cos,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ pi )
% 0.29/0.60         => ( ( arccos @ ( cos_real @ X ) )
% 0.29/0.60            = X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_cos
% 0.29/0.60  thf(fact_8426_arcsin__less__arcsin,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ Y )
% 0.29/0.60         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60           => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_less_arcsin
% 0.29/0.60  thf(fact_8427_arcsin__less__mono,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.60       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.29/0.60         => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 0.29/0.60            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_less_mono
% 0.29/0.60  thf(fact_8428_cos__arccos__abs,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.29/0.60       => ( ( cos_real @ ( arccos @ Y ) )
% 0.29/0.60          = Y ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_arccos_abs
% 0.29/0.60  thf(fact_8429_arccos__cos__eq__abs,axiom,
% 0.29/0.60      ! [Theta: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 0.29/0.60       => ( ( arccos @ ( cos_real @ Theta ) )
% 0.29/0.60          = ( abs_abs_real @ Theta ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_cos_eq_abs
% 0.29/0.60  thf(fact_8430_cot__def,axiom,
% 0.29/0.60      ( cot_complex
% 0.29/0.60      = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X4 ) @ ( sin_complex @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_def
% 0.29/0.60  thf(fact_8431_cot__def,axiom,
% 0.29/0.60      ( cot_real
% 0.29/0.60      = ( ^ [X4: real] : ( divide_divide_real @ ( cos_real @ X4 ) @ ( sin_real @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_def
% 0.29/0.60  thf(fact_8432_arccos__lt__bounded,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_real @ Y @ one_one_real )
% 0.29/0.60         => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
% 0.29/0.60            & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_lt_bounded
% 0.29/0.60  thf(fact_8433_arccos__bounded,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 0.29/0.60            & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_bounded
% 0.29/0.60  thf(fact_8434_sin__arccos__nonzero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.60         => ( ( sin_real @ ( arccos @ X ) )
% 0.29/0.60           != zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sin_arccos_nonzero
% 0.29/0.60  thf(fact_8435_arccos__cos2,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.60       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.29/0.60         => ( ( arccos @ ( cos_real @ X ) )
% 0.29/0.60            = ( uminus_uminus_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_cos2
% 0.29/0.60  thf(fact_8436_arccos__minus,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.60         => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 0.29/0.60            = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_minus
% 0.29/0.60  thf(fact_8437_cos__arcsin__nonzero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.60         => ( ( cos_real @ ( arcsin @ X ) )
% 0.29/0.60           != zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cos_arcsin_nonzero
% 0.29/0.60  thf(fact_8438_arccos,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 0.29/0.60            & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
% 0.29/0.60            & ( ( cos_real @ ( arccos @ Y ) )
% 0.29/0.60              = Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos
% 0.29/0.60  thf(fact_8439_arccos__minus__abs,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.60       => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 0.29/0.60          = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_minus_abs
% 0.29/0.60  thf(fact_8440_arccos__le__pi2,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_le_pi2
% 0.29/0.60  thf(fact_8441_arcsin__lt__bounded,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_real @ Y @ one_one_real )
% 0.29/0.60         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.29/0.60            & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_lt_bounded
% 0.29/0.60  thf(fact_8442_arcsin__bounded,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.29/0.60            & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_bounded
% 0.29/0.60  thf(fact_8443_arcsin__ubound,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_ubound
% 0.29/0.60  thf(fact_8444_arcsin__lbound,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_lbound
% 0.29/0.60  thf(fact_8445_cot__gt__zero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_gt_zero
% 0.29/0.60  thf(fact_8446_arcsin__sin,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60         => ( ( arcsin @ ( sin_real @ X ) )
% 0.29/0.60            = X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_sin
% 0.29/0.60  thf(fact_8447_tan__cot_H,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 0.29/0.60        = ( cot_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_cot'
% 0.29/0.60  thf(fact_8448_le__arcsin__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 0.29/0.60           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60             => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
% 0.29/0.60                = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % le_arcsin_iff
% 0.29/0.60  thf(fact_8449_arcsin__le__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 0.29/0.60           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60             => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
% 0.29/0.60                = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_le_iff
% 0.29/0.60  thf(fact_8450_arcsin__pi,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.29/0.60            & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
% 0.29/0.60            & ( ( sin_real @ ( arcsin @ Y ) )
% 0.29/0.60              = Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin_pi
% 0.29/0.60  thf(fact_8451_arcsin,axiom,
% 0.29/0.60      ! [Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.29/0.60            & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60            & ( ( sin_real @ ( arcsin @ Y ) )
% 0.29/0.60              = Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcsin
% 0.29/0.60  thf(fact_8452_arccos__cos__eq__abs__2pi,axiom,
% 0.29/0.60      ! [Theta: real] :
% 0.29/0.60        ~ ! [K3: int] :
% 0.29/0.60            ( ( arccos @ ( cos_real @ Theta ) )
% 0.29/0.60           != ( 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 ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arccos_cos_eq_abs_2pi
% 0.29/0.60  thf(fact_8453_exp__two__pi__i_H,axiom,
% 0.29/0.60      ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 0.29/0.60      = one_one_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % exp_two_pi_i'
% 0.29/0.60  thf(fact_8454_exp__two__pi__i,axiom,
% 0.29/0.60      ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 0.29/0.60      = one_one_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % exp_two_pi_i
% 0.29/0.60  thf(fact_8455_powr__real__of__int,axiom,
% 0.29/0.60      ! [X: real,N: int] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.29/0.60           => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.29/0.60              = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
% 0.29/0.60          & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 0.29/0.60           => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.29/0.60              = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % powr_real_of_int
% 0.29/0.60  thf(fact_8456_cis__2pi,axiom,
% 0.29/0.60      ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.29/0.60      = one_one_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % cis_2pi
% 0.29/0.60  thf(fact_8457_Suc__0__xor__eq,axiom,
% 0.29/0.60      ! [N: nat] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.60        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.60          @ ( zero_n2687167440665602831ol_nat
% 0.29/0.60            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Suc_0_xor_eq
% 0.29/0.60  thf(fact_8458_inverse__nonzero__iff__nonzero,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ( inverse_inverse_real @ A )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60        = ( A = zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_nonzero_iff_nonzero
% 0.29/0.60  thf(fact_8459_inverse__nonzero__iff__nonzero,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( ( invers8013647133539491842omplex @ A )
% 0.29/0.60          = zero_zero_complex )
% 0.29/0.60        = ( A = zero_zero_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_nonzero_iff_nonzero
% 0.29/0.60  thf(fact_8460_inverse__nonzero__iff__nonzero,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ( inverse_inverse_rat @ A )
% 0.29/0.60          = zero_zero_rat )
% 0.29/0.60        = ( A = zero_zero_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_nonzero_iff_nonzero
% 0.29/0.60  thf(fact_8461_inverse__zero,axiom,
% 0.29/0.60      ( ( inverse_inverse_real @ zero_zero_real )
% 0.29/0.60      = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_zero
% 0.29/0.60  thf(fact_8462_inverse__zero,axiom,
% 0.29/0.60      ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 0.29/0.60      = zero_zero_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_zero
% 0.29/0.60  thf(fact_8463_inverse__zero,axiom,
% 0.29/0.60      ( ( inverse_inverse_rat @ zero_zero_rat )
% 0.29/0.60      = zero_zero_rat ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_zero
% 0.29/0.60  thf(fact_8464_inverse__1,axiom,
% 0.29/0.60      ( ( inverse_inverse_real @ one_one_real )
% 0.29/0.60      = one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_1
% 0.29/0.60  thf(fact_8465_inverse__1,axiom,
% 0.29/0.60      ( ( invers8013647133539491842omplex @ one_one_complex )
% 0.29/0.60      = one_one_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_1
% 0.29/0.60  thf(fact_8466_inverse__1,axiom,
% 0.29/0.60      ( ( inverse_inverse_rat @ one_one_rat )
% 0.29/0.60      = one_one_rat ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_1
% 0.29/0.60  thf(fact_8467_inverse__eq__1__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( inverse_inverse_real @ X )
% 0.29/0.60          = one_one_real )
% 0.29/0.60        = ( X = one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_1_iff
% 0.29/0.60  thf(fact_8468_inverse__eq__1__iff,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( ( invers8013647133539491842omplex @ X )
% 0.29/0.60          = one_one_complex )
% 0.29/0.60        = ( X = one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_1_iff
% 0.29/0.60  thf(fact_8469_inverse__eq__1__iff,axiom,
% 0.29/0.60      ! [X: rat] :
% 0.29/0.60        ( ( ( inverse_inverse_rat @ X )
% 0.29/0.60          = one_one_rat )
% 0.29/0.60        = ( X = one_one_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_1_iff
% 0.29/0.60  thf(fact_8470_inverse__divide,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.60        = ( divide_divide_real @ B @ A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_divide
% 0.29/0.60  thf(fact_8471_inverse__divide,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.60        = ( divide1717551699836669952omplex @ B @ A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_divide
% 0.29/0.60  thf(fact_8472_inverse__divide,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
% 0.29/0.60        = ( divide_divide_rat @ B @ A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_divide
% 0.29/0.60  thf(fact_8473_bit_Oxor__self,axiom,
% 0.29/0.60      ! [X: int] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ X @ X )
% 0.29/0.60        = zero_zero_int ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.xor_self
% 0.29/0.60  thf(fact_8474_xor__self__eq,axiom,
% 0.29/0.60      ! [A: nat] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ A @ A )
% 0.29/0.60        = zero_zero_nat ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_self_eq
% 0.29/0.60  thf(fact_8475_xor__self__eq,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ A @ A )
% 0.29/0.60        = zero_zero_int ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_self_eq
% 0.29/0.60  thf(fact_8476_xor_Oleft__neutral,axiom,
% 0.29/0.60      ! [A: nat] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 0.29/0.60        = A ) ).
% 0.29/0.60  
% 0.29/0.60  % xor.left_neutral
% 0.29/0.60  thf(fact_8477_xor_Oleft__neutral,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 0.29/0.60        = A ) ).
% 0.29/0.60  
% 0.29/0.60  % xor.left_neutral
% 0.29/0.60  thf(fact_8478_xor_Oright__neutral,axiom,
% 0.29/0.60      ! [A: nat] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 0.29/0.60        = A ) ).
% 0.29/0.60  
% 0.29/0.60  % xor.right_neutral
% 0.29/0.60  thf(fact_8479_xor_Oright__neutral,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 0.29/0.60        = A ) ).
% 0.29/0.60  
% 0.29/0.60  % xor.right_neutral
% 0.29/0.60  thf(fact_8480_inverse__nonnegative__iff__nonnegative,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.29/0.60        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_nonnegative_iff_nonnegative
% 0.29/0.60  thf(fact_8481_inverse__nonnegative__iff__nonnegative,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.29/0.60        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_nonnegative_iff_nonnegative
% 0.29/0.60  thf(fact_8482_inverse__nonpositive__iff__nonpositive,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.29/0.60        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_nonpositive_iff_nonpositive
% 0.29/0.60  thf(fact_8483_inverse__nonpositive__iff__nonpositive,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.29/0.60        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_nonpositive_iff_nonpositive
% 0.29/0.60  thf(fact_8484_inverse__less__iff__less,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.60       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.60         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60            = ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_iff_less
% 0.29/0.60  thf(fact_8485_inverse__less__iff__less,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.60       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.60         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60            = ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_iff_less
% 0.29/0.60  thf(fact_8486_inverse__less__iff__less__neg,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.60       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.60         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60            = ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_iff_less_neg
% 0.29/0.60  thf(fact_8487_inverse__less__iff__less__neg,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.60       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.60         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60            = ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_iff_less_neg
% 0.29/0.60  thf(fact_8488_inverse__negative__iff__negative,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.29/0.60        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_negative_iff_negative
% 0.29/0.60  thf(fact_8489_inverse__negative__iff__negative,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.29/0.60        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_negative_iff_negative
% 0.29/0.60  thf(fact_8490_inverse__positive__iff__positive,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.29/0.60        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_positive_iff_positive
% 0.29/0.60  thf(fact_8491_inverse__positive__iff__positive,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.29/0.60        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_positive_iff_positive
% 0.29/0.60  thf(fact_8492_norm__ii,axiom,
% 0.29/0.60      ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 0.29/0.60      = one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % norm_ii
% 0.29/0.60  thf(fact_8493_norm__cis,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 0.29/0.60        = one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % norm_cis
% 0.29/0.60  thf(fact_8494_cis__zero,axiom,
% 0.29/0.60      ( ( cis @ zero_zero_real )
% 0.29/0.60      = one_one_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % cis_zero
% 0.29/0.60  thf(fact_8495_inverse__le__iff__le,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.60       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60            = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_iff_le
% 0.29/0.60  thf(fact_8496_inverse__le__iff__le,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.60       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.60         => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60            = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_iff_le
% 0.29/0.60  thf(fact_8497_inverse__le__iff__le__neg,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.60       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60            = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_iff_le_neg
% 0.29/0.60  thf(fact_8498_inverse__le__iff__le__neg,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.60       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.60         => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60            = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_iff_le_neg
% 0.29/0.60  thf(fact_8499_right__inverse,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
% 0.29/0.60          = one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % right_inverse
% 0.29/0.60  thf(fact_8500_right__inverse,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
% 0.29/0.60          = one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % right_inverse
% 0.29/0.60  thf(fact_8501_right__inverse,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
% 0.29/0.60          = one_one_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % right_inverse
% 0.29/0.60  thf(fact_8502_left__inverse,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 0.29/0.60          = one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % left_inverse
% 0.29/0.60  thf(fact_8503_left__inverse,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 0.29/0.60          = one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % left_inverse
% 0.29/0.60  thf(fact_8504_left__inverse,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 0.29/0.60          = one_one_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % left_inverse
% 0.29/0.60  thf(fact_8505_inverse__eq__divide__numeral,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
% 0.29/0.60        = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_divide_numeral
% 0.29/0.60  thf(fact_8506_inverse__eq__divide__numeral,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.29/0.60        = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_divide_numeral
% 0.29/0.60  thf(fact_8507_inverse__eq__divide__numeral,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
% 0.29/0.60        = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_divide_numeral
% 0.29/0.60  thf(fact_8508_cis__pi,axiom,
% 0.29/0.60      ( ( cis @ pi )
% 0.29/0.60      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cis_pi
% 0.29/0.60  thf(fact_8509_divide__i,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
% 0.29/0.60        = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % divide_i
% 0.29/0.60  thf(fact_8510_i__squared,axiom,
% 0.29/0.60      ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 0.29/0.60      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % i_squared
% 0.29/0.60  thf(fact_8511_inverse__eq__divide__neg__numeral,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.29/0.60        = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_divide_neg_numeral
% 0.29/0.60  thf(fact_8512_inverse__eq__divide__neg__numeral,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.60        = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_divide_neg_numeral
% 0.29/0.60  thf(fact_8513_inverse__eq__divide__neg__numeral,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.29/0.60        = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_divide_neg_numeral
% 0.29/0.60  thf(fact_8514_xor__numerals_I3_J,axiom,
% 0.29/0.60      ! [X: num,Y: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.29/0.60        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(3)
% 0.29/0.60  thf(fact_8515_xor__numerals_I3_J,axiom,
% 0.29/0.60      ! [X: num,Y: num] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.29/0.60        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(3)
% 0.29/0.60  thf(fact_8516_xor__numerals_I1_J,axiom,
% 0.29/0.60      ! [Y: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.29/0.60        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(1)
% 0.29/0.60  thf(fact_8517_xor__numerals_I1_J,axiom,
% 0.29/0.60      ! [Y: num] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.29/0.60        = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(1)
% 0.29/0.60  thf(fact_8518_xor__numerals_I2_J,axiom,
% 0.29/0.60      ! [Y: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.29/0.60        = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(2)
% 0.29/0.60  thf(fact_8519_xor__numerals_I2_J,axiom,
% 0.29/0.60      ! [Y: num] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.29/0.60        = ( numeral_numeral_int @ ( bit0 @ Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(2)
% 0.29/0.60  thf(fact_8520_xor__numerals_I5_J,axiom,
% 0.29/0.60      ! [X: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 0.29/0.60        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(5)
% 0.29/0.60  thf(fact_8521_xor__numerals_I5_J,axiom,
% 0.29/0.60      ! [X: num] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 0.29/0.60        = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(5)
% 0.29/0.60  thf(fact_8522_xor__numerals_I8_J,axiom,
% 0.29/0.60      ! [X: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 0.29/0.60        = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(8)
% 0.29/0.60  thf(fact_8523_xor__numerals_I8_J,axiom,
% 0.29/0.60      ! [X: num] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 0.29/0.60        = ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(8)
% 0.29/0.60  thf(fact_8524_xor__numerals_I7_J,axiom,
% 0.29/0.60      ! [X: num,Y: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.29/0.60        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(7)
% 0.29/0.60  thf(fact_8525_xor__numerals_I7_J,axiom,
% 0.29/0.60      ! [X: num,Y: num] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.29/0.60        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(7)
% 0.29/0.60  thf(fact_8526_divide__numeral__i,axiom,
% 0.29/0.60      ! [Z: complex,N: num] :
% 0.29/0.60        ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 0.29/0.60        = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % divide_numeral_i
% 0.29/0.60  thf(fact_8527_xor__nat__numerals_I1_J,axiom,
% 0.29/0.60      ! [Y: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.29/0.60        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_nat_numerals(1)
% 0.29/0.60  thf(fact_8528_xor__nat__numerals_I2_J,axiom,
% 0.29/0.60      ! [Y: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.29/0.60        = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_nat_numerals(2)
% 0.29/0.60  thf(fact_8529_xor__nat__numerals_I3_J,axiom,
% 0.29/0.60      ! [X: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.29/0.60        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_nat_numerals(3)
% 0.29/0.60  thf(fact_8530_xor__nat__numerals_I4_J,axiom,
% 0.29/0.60      ! [X: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.29/0.60        = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_nat_numerals(4)
% 0.29/0.60  thf(fact_8531_power2__i,axiom,
% 0.29/0.60      ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power2_i
% 0.29/0.60  thf(fact_8532_cis__pi__half,axiom,
% 0.29/0.60      ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.60      = imaginary_unit ) ).
% 0.29/0.60  
% 0.29/0.60  % cis_pi_half
% 0.29/0.60  thf(fact_8533_i__even__power,axiom,
% 0.29/0.60      ! [N: nat] :
% 0.29/0.60        ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.60        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 0.29/0.60  
% 0.29/0.60  % i_even_power
% 0.29/0.60  thf(fact_8534_xor__numerals_I4_J,axiom,
% 0.29/0.60      ! [X: num,Y: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.29/0.60        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(4)
% 0.29/0.60  thf(fact_8535_xor__numerals_I4_J,axiom,
% 0.29/0.60      ! [X: num,Y: num] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.29/0.60        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(4)
% 0.29/0.60  thf(fact_8536_xor__numerals_I6_J,axiom,
% 0.29/0.60      ! [X: num,Y: num] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.29/0.60        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(6)
% 0.29/0.60  thf(fact_8537_xor__numerals_I6_J,axiom,
% 0.29/0.60      ! [X: num,Y: num] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.29/0.60        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_numerals(6)
% 0.29/0.60  thf(fact_8538_exp__pi__i_H,axiom,
% 0.29/0.60      ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 0.29/0.60      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % exp_pi_i'
% 0.29/0.60  thf(fact_8539_exp__pi__i,axiom,
% 0.29/0.60      ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 0.29/0.60      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % exp_pi_i
% 0.29/0.60  thf(fact_8540_cis__minus__pi__half,axiom,
% 0.29/0.60      ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.29/0.60      = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cis_minus_pi_half
% 0.29/0.60  thf(fact_8541_field__class_Ofield__inverse__zero,axiom,
% 0.29/0.60      ( ( inverse_inverse_real @ zero_zero_real )
% 0.29/0.60      = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % field_class.field_inverse_zero
% 0.29/0.60  thf(fact_8542_field__class_Ofield__inverse__zero,axiom,
% 0.29/0.60      ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 0.29/0.60      = zero_zero_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % field_class.field_inverse_zero
% 0.29/0.60  thf(fact_8543_field__class_Ofield__inverse__zero,axiom,
% 0.29/0.60      ( ( inverse_inverse_rat @ zero_zero_rat )
% 0.29/0.60      = zero_zero_rat ) ).
% 0.29/0.60  
% 0.29/0.60  % field_class.field_inverse_zero
% 0.29/0.60  thf(fact_8544_inverse__zero__imp__zero,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ( inverse_inverse_real @ A )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60       => ( A = zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_zero_imp_zero
% 0.29/0.60  thf(fact_8545_inverse__zero__imp__zero,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( ( invers8013647133539491842omplex @ A )
% 0.29/0.60          = zero_zero_complex )
% 0.29/0.60       => ( A = zero_zero_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_zero_imp_zero
% 0.29/0.60  thf(fact_8546_inverse__zero__imp__zero,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ( inverse_inverse_rat @ A )
% 0.29/0.60          = zero_zero_rat )
% 0.29/0.60       => ( A = zero_zero_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_zero_imp_zero
% 0.29/0.60  thf(fact_8547_nonzero__inverse__eq__imp__eq,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ( inverse_inverse_real @ A )
% 0.29/0.60          = ( inverse_inverse_real @ B ) )
% 0.29/0.60       => ( ( A != zero_zero_real )
% 0.29/0.60         => ( ( B != zero_zero_real )
% 0.29/0.60           => ( A = B ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_eq_imp_eq
% 0.29/0.60  thf(fact_8548_nonzero__inverse__eq__imp__eq,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( ( invers8013647133539491842omplex @ A )
% 0.29/0.60          = ( invers8013647133539491842omplex @ B ) )
% 0.29/0.60       => ( ( A != zero_zero_complex )
% 0.29/0.60         => ( ( B != zero_zero_complex )
% 0.29/0.60           => ( A = B ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_eq_imp_eq
% 0.29/0.60  thf(fact_8549_nonzero__inverse__eq__imp__eq,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ( inverse_inverse_rat @ A )
% 0.29/0.60          = ( inverse_inverse_rat @ B ) )
% 0.29/0.60       => ( ( A != zero_zero_rat )
% 0.29/0.60         => ( ( B != zero_zero_rat )
% 0.29/0.60           => ( A = B ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_eq_imp_eq
% 0.29/0.60  thf(fact_8550_nonzero__inverse__inverse__eq,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 0.29/0.60          = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_inverse_eq
% 0.29/0.60  thf(fact_8551_nonzero__inverse__inverse__eq,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.29/0.60          = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_inverse_eq
% 0.29/0.60  thf(fact_8552_nonzero__inverse__inverse__eq,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 0.29/0.60          = A ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_inverse_eq
% 0.29/0.60  thf(fact_8553_nonzero__imp__inverse__nonzero,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( inverse_inverse_real @ A )
% 0.29/0.60         != zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_imp_inverse_nonzero
% 0.29/0.60  thf(fact_8554_nonzero__imp__inverse__nonzero,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( invers8013647133539491842omplex @ A )
% 0.29/0.60         != zero_zero_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_imp_inverse_nonzero
% 0.29/0.60  thf(fact_8555_nonzero__imp__inverse__nonzero,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( inverse_inverse_rat @ A )
% 0.29/0.60         != zero_zero_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_imp_inverse_nonzero
% 0.29/0.60  thf(fact_8556_nonzero__norm__inverse,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
% 0.29/0.60          = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_norm_inverse
% 0.29/0.60  thf(fact_8557_nonzero__norm__inverse,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.29/0.60          = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_norm_inverse
% 0.29/0.60  thf(fact_8558_nonzero__of__real__inverse,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( X != zero_zero_real )
% 0.29/0.60       => ( ( real_V1803761363581548252l_real @ ( inverse_inverse_real @ X ) )
% 0.29/0.60          = ( inverse_inverse_real @ ( real_V1803761363581548252l_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_of_real_inverse
% 0.29/0.60  thf(fact_8559_nonzero__of__real__inverse,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( X != zero_zero_real )
% 0.29/0.60       => ( ( real_V4546457046886955230omplex @ ( inverse_inverse_real @ X ) )
% 0.29/0.60          = ( invers8013647133539491842omplex @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_of_real_inverse
% 0.29/0.60  thf(fact_8560_power__inverse,axiom,
% 0.29/0.60      ! [A: real,N: nat] :
% 0.29/0.60        ( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
% 0.29/0.60        = ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_inverse
% 0.29/0.60  thf(fact_8561_power__inverse,axiom,
% 0.29/0.60      ! [A: complex,N: nat] :
% 0.29/0.60        ( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N )
% 0.29/0.60        = ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_inverse
% 0.29/0.60  thf(fact_8562_power__inverse,axiom,
% 0.29/0.60      ! [A: rat,N: nat] :
% 0.29/0.60        ( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N )
% 0.29/0.60        = ( inverse_inverse_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_inverse
% 0.29/0.60  thf(fact_8563_of__int__xor__eq,axiom,
% 0.29/0.60      ! [K: int,L: int] :
% 0.29/0.60        ( ( ring_18347121197199848620nteger @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 0.29/0.60        = ( bit_se3222712562003087583nteger @ ( ring_18347121197199848620nteger @ K ) @ ( ring_18347121197199848620nteger @ L ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_xor_eq
% 0.29/0.60  thf(fact_8564_of__int__xor__eq,axiom,
% 0.29/0.60      ! [K: int,L: int] :
% 0.29/0.60        ( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 0.29/0.60        = ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_xor_eq
% 0.29/0.60  thf(fact_8565_of__nat__xor__eq,axiom,
% 0.29/0.60      ! [M: nat,N: nat] :
% 0.29/0.60        ( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 0.29/0.60        = ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_nat_xor_eq
% 0.29/0.60  thf(fact_8566_of__nat__xor__eq,axiom,
% 0.29/0.60      ! [M: nat,N: nat] :
% 0.29/0.60        ( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 0.29/0.60        = ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_nat_xor_eq
% 0.29/0.60  thf(fact_8567_complex__i__not__zero,axiom,
% 0.29/0.60      imaginary_unit != zero_zero_complex ).
% 0.29/0.60  
% 0.29/0.60  % complex_i_not_zero
% 0.29/0.60  thf(fact_8568_complex__i__not__one,axiom,
% 0.29/0.60      imaginary_unit != one_one_complex ).
% 0.29/0.60  
% 0.29/0.60  % complex_i_not_one
% 0.29/0.60  thf(fact_8569_cis__neq__zero,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( cis @ A )
% 0.29/0.60       != zero_zero_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % cis_neq_zero
% 0.29/0.60  thf(fact_8570_complex__i__not__numeral,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ( imaginary_unit
% 0.29/0.60       != ( numera6690914467698888265omplex @ W ) ) ).
% 0.29/0.60  
% 0.29/0.60  % complex_i_not_numeral
% 0.29/0.60  thf(fact_8571_norm__inverse__le__norm,axiom,
% 0.29/0.60      ! [R2: real,X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X ) )
% 0.29/0.60       => ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.29/0.60         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % norm_inverse_le_norm
% 0.29/0.60  thf(fact_8572_norm__inverse__le__norm,axiom,
% 0.29/0.60      ! [R2: real,X: complex] :
% 0.29/0.60        ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X ) )
% 0.29/0.60       => ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.29/0.60         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % norm_inverse_le_norm
% 0.29/0.60  thf(fact_8573_inverse__less__imp__less,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.60         => ( ord_less_real @ B @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_imp_less
% 0.29/0.60  thf(fact_8574_inverse__less__imp__less,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.60         => ( ord_less_rat @ B @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_imp_less
% 0.29/0.60  thf(fact_8575_less__imp__inverse__less,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_real @ A @ B )
% 0.29/0.60       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.60         => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % less_imp_inverse_less
% 0.29/0.60  thf(fact_8576_less__imp__inverse__less,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_rat @ A @ B )
% 0.29/0.60       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.60         => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % less_imp_inverse_less
% 0.29/0.60  thf(fact_8577_inverse__less__imp__less__neg,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.60         => ( ord_less_real @ B @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_imp_less_neg
% 0.29/0.60  thf(fact_8578_inverse__less__imp__less__neg,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.60         => ( ord_less_rat @ B @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_imp_less_neg
% 0.29/0.60  thf(fact_8579_less__imp__inverse__less__neg,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_real @ A @ B )
% 0.29/0.60       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.60         => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % less_imp_inverse_less_neg
% 0.29/0.60  thf(fact_8580_less__imp__inverse__less__neg,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_rat @ A @ B )
% 0.29/0.60       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.60         => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % less_imp_inverse_less_neg
% 0.29/0.60  thf(fact_8581_inverse__negative__imp__negative,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.29/0.60       => ( ( A != zero_zero_real )
% 0.29/0.60         => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_negative_imp_negative
% 0.29/0.60  thf(fact_8582_inverse__negative__imp__negative,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.29/0.60       => ( ( A != zero_zero_rat )
% 0.29/0.60         => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_negative_imp_negative
% 0.29/0.60  thf(fact_8583_inverse__positive__imp__positive,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.29/0.60       => ( ( A != zero_zero_real )
% 0.29/0.60         => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_positive_imp_positive
% 0.29/0.60  thf(fact_8584_inverse__positive__imp__positive,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.29/0.60       => ( ( A != zero_zero_rat )
% 0.29/0.60         => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_positive_imp_positive
% 0.29/0.60  thf(fact_8585_negative__imp__inverse__negative,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.60       => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % negative_imp_inverse_negative
% 0.29/0.60  thf(fact_8586_negative__imp__inverse__negative,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.60       => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % negative_imp_inverse_negative
% 0.29/0.60  thf(fact_8587_positive__imp__inverse__positive,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.60       => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % positive_imp_inverse_positive
% 0.29/0.60  thf(fact_8588_positive__imp__inverse__positive,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.60       => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % positive_imp_inverse_positive
% 0.29/0.60  thf(fact_8589_nonzero__inverse__mult__distrib,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( B != zero_zero_real )
% 0.29/0.60         => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 0.29/0.60            = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_mult_distrib
% 0.29/0.60  thf(fact_8590_nonzero__inverse__mult__distrib,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( B != zero_zero_complex )
% 0.29/0.60         => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 0.29/0.60            = ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_mult_distrib
% 0.29/0.60  thf(fact_8591_nonzero__inverse__mult__distrib,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( B != zero_zero_rat )
% 0.29/0.60         => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.60            = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_mult_distrib
% 0.29/0.60  thf(fact_8592_nonzero__inverse__minus__eq,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 0.29/0.60          = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_minus_eq
% 0.29/0.60  thf(fact_8593_nonzero__inverse__minus__eq,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.29/0.60          = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_minus_eq
% 0.29/0.60  thf(fact_8594_nonzero__inverse__minus__eq,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 0.29/0.60          = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_minus_eq
% 0.29/0.60  thf(fact_8595_inverse__unique,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ( times_times_real @ A @ B )
% 0.29/0.60          = one_one_real )
% 0.29/0.60       => ( ( inverse_inverse_real @ A )
% 0.29/0.60          = B ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_unique
% 0.29/0.60  thf(fact_8596_inverse__unique,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( ( times_times_complex @ A @ B )
% 0.29/0.60          = one_one_complex )
% 0.29/0.60       => ( ( invers8013647133539491842omplex @ A )
% 0.29/0.60          = B ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_unique
% 0.29/0.60  thf(fact_8597_inverse__unique,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ( times_times_rat @ A @ B )
% 0.29/0.60          = one_one_rat )
% 0.29/0.60       => ( ( inverse_inverse_rat @ A )
% 0.29/0.60          = B ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_unique
% 0.29/0.60  thf(fact_8598_inverse__numeral__1,axiom,
% 0.29/0.60      ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
% 0.29/0.60      = ( numeral_numeral_real @ one ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_numeral_1
% 0.29/0.60  thf(fact_8599_inverse__numeral__1,axiom,
% 0.29/0.60      ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.29/0.60      = ( numera6690914467698888265omplex @ one ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_numeral_1
% 0.29/0.60  thf(fact_8600_inverse__numeral__1,axiom,
% 0.29/0.60      ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
% 0.29/0.60      = ( numeral_numeral_rat @ one ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_numeral_1
% 0.29/0.60  thf(fact_8601_divide__inverse__commute,axiom,
% 0.29/0.60      ( divide_divide_real
% 0.29/0.60      = ( ^ [A3: real,B4: real] : ( times_times_real @ ( inverse_inverse_real @ B4 ) @ A3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % divide_inverse_commute
% 0.29/0.60  thf(fact_8602_divide__inverse__commute,axiom,
% 0.29/0.60      ( divide1717551699836669952omplex
% 0.29/0.60      = ( ^ [A3: complex,B4: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B4 ) @ A3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % divide_inverse_commute
% 0.29/0.60  thf(fact_8603_divide__inverse__commute,axiom,
% 0.29/0.60      ( divide_divide_rat
% 0.29/0.60      = ( ^ [A3: rat,B4: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B4 ) @ A3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % divide_inverse_commute
% 0.29/0.60  thf(fact_8604_divide__inverse,axiom,
% 0.29/0.60      ( divide_divide_real
% 0.29/0.60      = ( ^ [A3: real,B4: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % divide_inverse
% 0.29/0.60  thf(fact_8605_divide__inverse,axiom,
% 0.29/0.60      ( divide1717551699836669952omplex
% 0.29/0.60      = ( ^ [A3: complex,B4: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % divide_inverse
% 0.29/0.60  thf(fact_8606_divide__inverse,axiom,
% 0.29/0.60      ( divide_divide_rat
% 0.29/0.60      = ( ^ [A3: rat,B4: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % divide_inverse
% 0.29/0.60  thf(fact_8607_field__class_Ofield__divide__inverse,axiom,
% 0.29/0.60      ( divide_divide_real
% 0.29/0.60      = ( ^ [A3: real,B4: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % field_class.field_divide_inverse
% 0.29/0.60  thf(fact_8608_field__class_Ofield__divide__inverse,axiom,
% 0.29/0.60      ( divide1717551699836669952omplex
% 0.29/0.60      = ( ^ [A3: complex,B4: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % field_class.field_divide_inverse
% 0.29/0.60  thf(fact_8609_field__class_Ofield__divide__inverse,axiom,
% 0.29/0.60      ( divide_divide_rat
% 0.29/0.60      = ( ^ [A3: rat,B4: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % field_class.field_divide_inverse
% 0.29/0.60  thf(fact_8610_inverse__eq__divide,axiom,
% 0.29/0.60      ( inverse_inverse_real
% 0.29/0.60      = ( divide_divide_real @ one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_divide
% 0.29/0.60  thf(fact_8611_inverse__eq__divide,axiom,
% 0.29/0.60      ( invers8013647133539491842omplex
% 0.29/0.60      = ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_divide
% 0.29/0.60  thf(fact_8612_inverse__eq__divide,axiom,
% 0.29/0.60      ( inverse_inverse_rat
% 0.29/0.60      = ( divide_divide_rat @ one_one_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_eq_divide
% 0.29/0.60  thf(fact_8613_power__mult__inverse__distrib,axiom,
% 0.29/0.60      ! [X: real,M: nat] :
% 0.29/0.60        ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( inverse_inverse_real @ X ) )
% 0.29/0.60        = ( times_times_real @ ( inverse_inverse_real @ X ) @ ( power_power_real @ X @ M ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_mult_inverse_distrib
% 0.29/0.60  thf(fact_8614_power__mult__inverse__distrib,axiom,
% 0.29/0.60      ! [X: complex,M: nat] :
% 0.29/0.60        ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( invers8013647133539491842omplex @ X ) )
% 0.29/0.60        = ( times_times_complex @ ( invers8013647133539491842omplex @ X ) @ ( power_power_complex @ X @ M ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_mult_inverse_distrib
% 0.29/0.60  thf(fact_8615_power__mult__inverse__distrib,axiom,
% 0.29/0.60      ! [X: rat,M: nat] :
% 0.29/0.60        ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( inverse_inverse_rat @ X ) )
% 0.29/0.60        = ( times_times_rat @ ( inverse_inverse_rat @ X ) @ ( power_power_rat @ X @ M ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_mult_inverse_distrib
% 0.29/0.60  thf(fact_8616_power__mult__power__inverse__commute,axiom,
% 0.29/0.60      ! [X: real,M: nat,N: nat] :
% 0.29/0.60        ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) )
% 0.29/0.60        = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) @ ( power_power_real @ X @ M ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_mult_power_inverse_commute
% 0.29/0.60  thf(fact_8617_power__mult__power__inverse__commute,axiom,
% 0.29/0.60      ! [X: complex,M: nat,N: nat] :
% 0.29/0.60        ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) )
% 0.29/0.60        = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) @ ( power_power_complex @ X @ M ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_mult_power_inverse_commute
% 0.29/0.60  thf(fact_8618_power__mult__power__inverse__commute,axiom,
% 0.29/0.60      ! [X: rat,M: nat,N: nat] :
% 0.29/0.60        ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) )
% 0.29/0.60        = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) @ ( power_power_rat @ X @ M ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_mult_power_inverse_commute
% 0.29/0.60  thf(fact_8619_mult__inverse__of__nat__commute,axiom,
% 0.29/0.60      ! [Xa: nat,X: real] :
% 0.29/0.60        ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X )
% 0.29/0.60        = ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % mult_inverse_of_nat_commute
% 0.29/0.60  thf(fact_8620_mult__inverse__of__nat__commute,axiom,
% 0.29/0.60      ! [Xa: nat,X: complex] :
% 0.29/0.60        ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X )
% 0.29/0.60        = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % mult_inverse_of_nat_commute
% 0.29/0.60  thf(fact_8621_mult__inverse__of__nat__commute,axiom,
% 0.29/0.60      ! [Xa: nat,X: rat] :
% 0.29/0.60        ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X )
% 0.29/0.60        = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % mult_inverse_of_nat_commute
% 0.29/0.60  thf(fact_8622_nonzero__abs__inverse,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 0.29/0.60          = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_abs_inverse
% 0.29/0.60  thf(fact_8623_nonzero__abs__inverse,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 0.29/0.60          = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_abs_inverse
% 0.29/0.60  thf(fact_8624_mult__inverse__of__int__commute,axiom,
% 0.29/0.60      ! [Xa: int,X: real] :
% 0.29/0.60        ( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) @ X )
% 0.29/0.60        = ( times_times_real @ X @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % mult_inverse_of_int_commute
% 0.29/0.60  thf(fact_8625_mult__inverse__of__int__commute,axiom,
% 0.29/0.60      ! [Xa: int,X: complex] :
% 0.29/0.60        ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) @ X )
% 0.29/0.60        = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % mult_inverse_of_int_commute
% 0.29/0.60  thf(fact_8626_mult__inverse__of__int__commute,axiom,
% 0.29/0.60      ! [Xa: int,X: rat] :
% 0.29/0.60        ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X )
% 0.29/0.60        = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % mult_inverse_of_int_commute
% 0.29/0.60  thf(fact_8627_exp__minus,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( exp_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.60        = ( inverse_inverse_real @ ( exp_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % exp_minus
% 0.29/0.60  thf(fact_8628_exp__minus,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( exp_complex @ ( uminus1482373934393186551omplex @ X ) )
% 0.29/0.60        = ( invers8013647133539491842omplex @ ( exp_complex @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % exp_minus
% 0.29/0.60  thf(fact_8629_powr__minus,axiom,
% 0.29/0.60      ! [X: real,A: real] :
% 0.29/0.60        ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
% 0.29/0.60        = ( inverse_inverse_real @ ( powr_real @ X @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % powr_minus
% 0.29/0.60  thf(fact_8630_divide__real__def,axiom,
% 0.29/0.60      ( divide_divide_real
% 0.29/0.60      = ( ^ [X4: real,Y6: real] : ( times_times_real @ X4 @ ( inverse_inverse_real @ Y6 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % divide_real_def
% 0.29/0.60  thf(fact_8631_cot__altdef,axiom,
% 0.29/0.60      ( cot_real
% 0.29/0.60      = ( ^ [X4: real] : ( inverse_inverse_real @ ( tan_real @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_altdef
% 0.29/0.60  thf(fact_8632_cot__altdef,axiom,
% 0.29/0.60      ( cot_complex
% 0.29/0.60      = ( ^ [X4: complex] : ( invers8013647133539491842omplex @ ( tan_complex @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cot_altdef
% 0.29/0.60  thf(fact_8633_tan__altdef,axiom,
% 0.29/0.60      ( tan_real
% 0.29/0.60      = ( ^ [X4: real] : ( inverse_inverse_real @ ( cot_real @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_altdef
% 0.29/0.60  thf(fact_8634_tan__altdef,axiom,
% 0.29/0.60      ( tan_complex
% 0.29/0.60      = ( ^ [X4: complex] : ( invers8013647133539491842omplex @ ( cot_complex @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_altdef
% 0.29/0.60  thf(fact_8635_DeMoivre,axiom,
% 0.29/0.60      ! [A: real,N: nat] :
% 0.29/0.60        ( ( power_power_complex @ ( cis @ A ) @ N )
% 0.29/0.60        = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % DeMoivre
% 0.29/0.60  thf(fact_8636_cis__divide,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 0.29/0.60        = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cis_divide
% 0.29/0.60  thf(fact_8637_inverse__le__imp__le,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.60         => ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_imp_le
% 0.29/0.60  thf(fact_8638_inverse__le__imp__le,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.60         => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_imp_le
% 0.29/0.60  thf(fact_8639_le__imp__inverse__le,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.60       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.60         => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % le_imp_inverse_le
% 0.29/0.60  thf(fact_8640_le__imp__inverse__le,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.60       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.60         => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % le_imp_inverse_le
% 0.29/0.60  thf(fact_8641_inverse__le__imp__le__neg,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.60         => ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_imp_le_neg
% 0.29/0.60  thf(fact_8642_inverse__le__imp__le__neg,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.60         => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_imp_le_neg
% 0.29/0.60  thf(fact_8643_le__imp__inverse__le__neg,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.60       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.60         => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % le_imp_inverse_le_neg
% 0.29/0.60  thf(fact_8644_le__imp__inverse__le__neg,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.60       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.60         => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % le_imp_inverse_le_neg
% 0.29/0.60  thf(fact_8645_inverse__le__1__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 0.29/0.60        = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.60          | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_1_iff
% 0.29/0.60  thf(fact_8646_inverse__le__1__iff,axiom,
% 0.29/0.60      ! [X: rat] :
% 0.29/0.60        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 0.29/0.60        = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.29/0.60          | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_1_iff
% 0.29/0.60  thf(fact_8647_one__less__inverse__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 0.29/0.60        = ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60          & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_less_inverse_iff
% 0.29/0.60  thf(fact_8648_one__less__inverse__iff,axiom,
% 0.29/0.60      ! [X: rat] :
% 0.29/0.60        ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 0.29/0.60        = ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.29/0.60          & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_less_inverse_iff
% 0.29/0.60  thf(fact_8649_one__less__inverse,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.60       => ( ( ord_less_real @ A @ one_one_real )
% 0.29/0.60         => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_less_inverse
% 0.29/0.60  thf(fact_8650_one__less__inverse,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.60       => ( ( ord_less_rat @ A @ one_one_rat )
% 0.29/0.60         => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_less_inverse
% 0.29/0.60  thf(fact_8651_field__class_Ofield__inverse,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 0.29/0.60          = one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % field_class.field_inverse
% 0.29/0.60  thf(fact_8652_field__class_Ofield__inverse,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 0.29/0.60          = one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % field_class.field_inverse
% 0.29/0.60  thf(fact_8653_field__class_Ofield__inverse,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 0.29/0.60          = one_one_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % field_class.field_inverse
% 0.29/0.60  thf(fact_8654_division__ring__inverse__add,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( B != zero_zero_real )
% 0.29/0.60         => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60            = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % division_ring_inverse_add
% 0.29/0.60  thf(fact_8655_division__ring__inverse__add,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( B != zero_zero_complex )
% 0.29/0.60         => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.29/0.60            = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % division_ring_inverse_add
% 0.29/0.60  thf(fact_8656_division__ring__inverse__add,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( B != zero_zero_rat )
% 0.29/0.60         => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60            = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % division_ring_inverse_add
% 0.29/0.60  thf(fact_8657_inverse__add,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( B != zero_zero_real )
% 0.29/0.60         => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60            = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_add
% 0.29/0.60  thf(fact_8658_inverse__add,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( B != zero_zero_complex )
% 0.29/0.60         => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.29/0.60            = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_add
% 0.29/0.60  thf(fact_8659_inverse__add,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( B != zero_zero_rat )
% 0.29/0.60         => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60            = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_add
% 0.29/0.60  thf(fact_8660_division__ring__inverse__diff,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( B != zero_zero_real )
% 0.29/0.60         => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60            = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % division_ring_inverse_diff
% 0.29/0.60  thf(fact_8661_division__ring__inverse__diff,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( B != zero_zero_complex )
% 0.29/0.60         => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.29/0.60            = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % division_ring_inverse_diff
% 0.29/0.60  thf(fact_8662_division__ring__inverse__diff,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( B != zero_zero_rat )
% 0.29/0.60         => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60            = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % division_ring_inverse_diff
% 0.29/0.60  thf(fact_8663_nonzero__inverse__eq__divide,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( inverse_inverse_real @ A )
% 0.29/0.60          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_eq_divide
% 0.29/0.60  thf(fact_8664_nonzero__inverse__eq__divide,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( invers8013647133539491842omplex @ A )
% 0.29/0.60          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_eq_divide
% 0.29/0.60  thf(fact_8665_nonzero__inverse__eq__divide,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( inverse_inverse_rat @ A )
% 0.29/0.60          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % nonzero_inverse_eq_divide
% 0.29/0.60  thf(fact_8666_complex__i__not__neg__numeral,axiom,
% 0.29/0.60      ! [W: num] :
% 0.29/0.60        ( imaginary_unit
% 0.29/0.60       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % complex_i_not_neg_numeral
% 0.29/0.60  thf(fact_8667_inverse__powr,axiom,
% 0.29/0.60      ! [Y: real,A: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60       => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
% 0.29/0.60          = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_powr
% 0.29/0.60  thf(fact_8668_Complex__eq__i,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( complex2 @ X @ Y )
% 0.29/0.60          = imaginary_unit )
% 0.29/0.60        = ( ( X = zero_zero_real )
% 0.29/0.60          & ( Y = one_one_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Complex_eq_i
% 0.29/0.60  thf(fact_8669_imaginary__unit_Ocode,axiom,
% 0.29/0.60      ( imaginary_unit
% 0.29/0.60      = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % imaginary_unit.code
% 0.29/0.60  thf(fact_8670_inverse__le__iff,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60        = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.60           => ( ord_less_eq_real @ B @ A ) )
% 0.29/0.60          & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.29/0.60           => ( ord_less_eq_real @ A @ B ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_iff
% 0.29/0.60  thf(fact_8671_inverse__le__iff,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60        = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.60           => ( ord_less_eq_rat @ B @ A ) )
% 0.29/0.60          & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.29/0.60           => ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_le_iff
% 0.29/0.60  thf(fact_8672_inverse__less__iff,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60        = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.60           => ( ord_less_real @ B @ A ) )
% 0.29/0.60          & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.29/0.60           => ( ord_less_real @ A @ B ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_iff
% 0.29/0.60  thf(fact_8673_inverse__less__iff,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60        = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.60           => ( ord_less_rat @ B @ A ) )
% 0.29/0.60          & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.29/0.60           => ( ord_less_rat @ A @ B ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_iff
% 0.29/0.60  thf(fact_8674_one__le__inverse__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 0.29/0.60        = ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60          & ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_le_inverse_iff
% 0.29/0.60  thf(fact_8675_one__le__inverse__iff,axiom,
% 0.29/0.60      ! [X: rat] :
% 0.29/0.60        ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 0.29/0.60        = ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.29/0.60          & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_le_inverse_iff
% 0.29/0.60  thf(fact_8676_inverse__less__1__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 0.29/0.60        = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.60          | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_1_iff
% 0.29/0.60  thf(fact_8677_inverse__less__1__iff,axiom,
% 0.29/0.60      ! [X: rat] :
% 0.29/0.60        ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 0.29/0.60        = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.29/0.60          | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_less_1_iff
% 0.29/0.60  thf(fact_8678_one__le__inverse,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.60       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.29/0.60         => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_le_inverse
% 0.29/0.60  thf(fact_8679_one__le__inverse,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.60       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.29/0.60         => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_le_inverse
% 0.29/0.60  thf(fact_8680_inverse__diff__inverse,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( A != zero_zero_real )
% 0.29/0.60       => ( ( B != zero_zero_real )
% 0.29/0.60         => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.29/0.60            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_diff_inverse
% 0.29/0.60  thf(fact_8681_inverse__diff__inverse,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( A != zero_zero_complex )
% 0.29/0.60       => ( ( B != zero_zero_complex )
% 0.29/0.60         => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.29/0.60            = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_diff_inverse
% 0.29/0.60  thf(fact_8682_inverse__diff__inverse,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( A != zero_zero_rat )
% 0.29/0.60       => ( ( B != zero_zero_rat )
% 0.29/0.60         => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.29/0.60            = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % inverse_diff_inverse
% 0.29/0.60  thf(fact_8683_reals__Archimedean,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ? [N2: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % reals_Archimedean
% 0.29/0.60  thf(fact_8684_reals__Archimedean,axiom,
% 0.29/0.60      ! [X: rat] :
% 0.29/0.60        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.29/0.60       => ? [N2: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % reals_Archimedean
% 0.29/0.60  thf(fact_8685_i__complex__of__real,axiom,
% 0.29/0.60      ! [R2: real] :
% 0.29/0.60        ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
% 0.29/0.60        = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % i_complex_of_real
% 0.29/0.60  thf(fact_8686_complex__of__real__i,axiom,
% 0.29/0.60      ! [R2: real] :
% 0.29/0.60        ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
% 0.29/0.60        = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 0.29/0.60  
% 0.29/0.60  % complex_of_real_i
% 0.29/0.60  thf(fact_8687_forall__pos__mono__1,axiom,
% 0.29/0.60      ! [P: real > $o,E2: real] :
% 0.29/0.60        ( ! [D5: real,E: real] :
% 0.29/0.60            ( ( ord_less_real @ D5 @ E )
% 0.29/0.60           => ( ( P @ D5 )
% 0.29/0.60             => ( P @ E ) ) )
% 0.29/0.60       => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 0.29/0.60         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.29/0.60           => ( P @ E2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % forall_pos_mono_1
% 0.29/0.60  thf(fact_8688_forall__pos__mono,axiom,
% 0.29/0.60      ! [P: real > $o,E2: real] :
% 0.29/0.60        ( ! [D5: real,E: real] :
% 0.29/0.60            ( ( ord_less_real @ D5 @ E )
% 0.29/0.60           => ( ( P @ D5 )
% 0.29/0.60             => ( P @ E ) ) )
% 0.29/0.60       => ( ! [N2: nat] :
% 0.29/0.60              ( ( N2 != zero_zero_nat )
% 0.29/0.60             => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
% 0.29/0.60         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.29/0.60           => ( P @ E2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % forall_pos_mono
% 0.29/0.60  thf(fact_8689_real__arch__inverse,axiom,
% 0.29/0.60      ! [E2: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.29/0.60        = ( ? [N3: nat] :
% 0.29/0.60              ( ( N3 != zero_zero_nat )
% 0.29/0.60              & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 0.29/0.60              & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E2 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % real_arch_inverse
% 0.29/0.60  thf(fact_8690_even__xor__iff,axiom,
% 0.29/0.60      ! [A: code_integer,B: code_integer] :
% 0.29/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B ) )
% 0.29/0.60        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.29/0.60          = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % even_xor_iff
% 0.29/0.60  thf(fact_8691_even__xor__iff,axiom,
% 0.29/0.60      ! [A: nat,B: nat] :
% 0.29/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 0.29/0.60        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.29/0.60          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % even_xor_iff
% 0.29/0.60  thf(fact_8692_even__xor__iff,axiom,
% 0.29/0.60      ! [A: int,B: int] :
% 0.29/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 0.29/0.60        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.29/0.60          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % even_xor_iff
% 0.29/0.60  thf(fact_8693_sqrt__divide__self__eq,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
% 0.29/0.60          = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sqrt_divide_self_eq
% 0.29/0.60  thf(fact_8694_ln__inverse,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
% 0.29/0.60          = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % ln_inverse
% 0.29/0.60  thf(fact_8695_ex__inverse__of__nat__less,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ? [N2: nat] :
% 0.29/0.60            ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.60            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % ex_inverse_of_nat_less
% 0.29/0.60  thf(fact_8696_ex__inverse__of__nat__less,axiom,
% 0.29/0.60      ! [X: rat] :
% 0.29/0.60        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.29/0.60       => ? [N2: nat] :
% 0.29/0.60            ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.60            & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % ex_inverse_of_nat_less
% 0.29/0.60  thf(fact_8697_power__diff__conv__inverse,axiom,
% 0.29/0.60      ! [X: real,M: nat,N: nat] :
% 0.29/0.60        ( ( X != zero_zero_real )
% 0.29/0.60       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.60         => ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M ) )
% 0.29/0.60            = ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_diff_conv_inverse
% 0.29/0.60  thf(fact_8698_power__diff__conv__inverse,axiom,
% 0.29/0.60      ! [X: complex,M: nat,N: nat] :
% 0.29/0.60        ( ( X != zero_zero_complex )
% 0.29/0.60       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.60         => ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M ) )
% 0.29/0.60            = ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_diff_conv_inverse
% 0.29/0.60  thf(fact_8699_power__diff__conv__inverse,axiom,
% 0.29/0.60      ! [X: rat,M: nat,N: nat] :
% 0.29/0.60        ( ( X != zero_zero_rat )
% 0.29/0.60       => ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.60         => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M ) )
% 0.29/0.60            = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % power_diff_conv_inverse
% 0.29/0.60  thf(fact_8700_log__inverse,axiom,
% 0.29/0.60      ! [A: real,X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.60       => ( ( A != one_one_real )
% 0.29/0.60         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60           => ( ( log @ A @ ( inverse_inverse_real @ X ) )
% 0.29/0.60              = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % log_inverse
% 0.29/0.60  thf(fact_8701_exp__plus__inverse__exp,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % exp_plus_inverse_exp
% 0.29/0.60  thf(fact_8702_plus__inverse__ge__2,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % plus_inverse_ge_2
% 0.29/0.60  thf(fact_8703_real__inv__sqrt__pow2,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60          = ( inverse_inverse_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % real_inv_sqrt_pow2
% 0.29/0.60  thf(fact_8704_tan__cot,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 0.29/0.60        = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_cot
% 0.29/0.60  thf(fact_8705_cmod__unit__one,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 0.29/0.60        = one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % cmod_unit_one
% 0.29/0.60  thf(fact_8706_xor__nat__unfold,axiom,
% 0.29/0.60      ( bit_se6528837805403552850or_nat
% 0.29/0.60      = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_nat_unfold
% 0.29/0.60  thf(fact_8707_real__le__x__sinh,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % real_le_x_sinh
% 0.29/0.60  thf(fact_8708_xor__nat__rec,axiom,
% 0.29/0.60      ( bit_se6528837805403552850or_nat
% 0.29/0.60      = ( ^ [M6: nat,N3: nat] :
% 0.29/0.60            ( plus_plus_nat
% 0.29/0.60            @ ( zero_n2687167440665602831ol_nat
% 0.29/0.60              @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
% 0.29/0.60               != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.29/0.60            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_nat_rec
% 0.29/0.60  thf(fact_8709_xor__one__eq,axiom,
% 0.29/0.60      ! [A: code_integer] :
% 0.29/0.60        ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
% 0.29/0.60        = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.29/0.60          @ ( zero_n356916108424825756nteger
% 0.29/0.60            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_one_eq
% 0.29/0.60  thf(fact_8710_xor__one__eq,axiom,
% 0.29/0.60      ! [A: nat] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
% 0.29/0.60        = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.29/0.60          @ ( zero_n2687167440665602831ol_nat
% 0.29/0.60            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_one_eq
% 0.29/0.60  thf(fact_8711_xor__one__eq,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
% 0.29/0.60        = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.29/0.60          @ ( zero_n2684676970156552555ol_int
% 0.29/0.60            @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_one_eq
% 0.29/0.60  thf(fact_8712_one__xor__eq,axiom,
% 0.29/0.60      ! [A: code_integer] :
% 0.29/0.60        ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
% 0.29/0.60        = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.29/0.60          @ ( zero_n356916108424825756nteger
% 0.29/0.60            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_xor_eq
% 0.29/0.60  thf(fact_8713_one__xor__eq,axiom,
% 0.29/0.60      ! [A: nat] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
% 0.29/0.60        = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.29/0.60          @ ( zero_n2687167440665602831ol_nat
% 0.29/0.60            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_xor_eq
% 0.29/0.60  thf(fact_8714_one__xor__eq,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
% 0.29/0.60        = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.29/0.60          @ ( zero_n2684676970156552555ol_int
% 0.29/0.60            @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % one_xor_eq
% 0.29/0.60  thf(fact_8715_real__le__abs__sinh,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % real_le_abs_sinh
% 0.29/0.60  thf(fact_8716_tan__sec,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( cos_real @ X )
% 0.29/0.60         != zero_zero_real )
% 0.29/0.60       => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.60          = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_sec
% 0.29/0.60  thf(fact_8717_tan__sec,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( ( cos_complex @ X )
% 0.29/0.60         != zero_zero_complex )
% 0.29/0.60       => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.60          = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tan_sec
% 0.29/0.60  thf(fact_8718_xor__Suc__0__eq,axiom,
% 0.29/0.60      ! [N: nat] :
% 0.29/0.60        ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.60        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.60          @ ( zero_n2687167440665602831ol_nat
% 0.29/0.60            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_Suc_0_eq
% 0.29/0.60  thf(fact_8719_csqrt__ii,axiom,
% 0.29/0.60      ( ( csqrt @ imaginary_unit )
% 0.29/0.60      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % csqrt_ii
% 0.29/0.60  thf(fact_8720_Arg__minus__ii,axiom,
% 0.29/0.60      ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 0.29/0.60      = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Arg_minus_ii
% 0.29/0.60  thf(fact_8721_Arg__ii,axiom,
% 0.29/0.60      ( ( arg @ imaginary_unit )
% 0.29/0.60      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Arg_ii
% 0.29/0.60  thf(fact_8722_Arg__correct,axiom,
% 0.29/0.60      ! [Z: complex] :
% 0.29/0.60        ( ( Z != zero_zero_complex )
% 0.29/0.60       => ( ( ( sgn_sgn_complex @ Z )
% 0.29/0.60            = ( cis @ ( arg @ Z ) ) )
% 0.29/0.60          & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.29/0.60          & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Arg_correct
% 0.29/0.60  thf(fact_8723_cis__Arg__unique,axiom,
% 0.29/0.60      ! [Z: complex,X: real] :
% 0.29/0.60        ( ( ( sgn_sgn_complex @ Z )
% 0.29/0.60          = ( cis @ X ) )
% 0.29/0.60       => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.29/0.60         => ( ( ord_less_eq_real @ X @ pi )
% 0.29/0.60           => ( ( arg @ Z )
% 0.29/0.60              = X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cis_Arg_unique
% 0.29/0.60  thf(fact_8724_csqrt__eq__0,axiom,
% 0.29/0.60      ! [Z: complex] :
% 0.29/0.60        ( ( ( csqrt @ Z )
% 0.29/0.60          = zero_zero_complex )
% 0.29/0.60        = ( Z = zero_zero_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % csqrt_eq_0
% 0.29/0.60  thf(fact_8725_csqrt__0,axiom,
% 0.29/0.60      ( ( csqrt @ zero_zero_complex )
% 0.29/0.60      = zero_zero_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % csqrt_0
% 0.29/0.60  thf(fact_8726_csqrt__eq__1,axiom,
% 0.29/0.60      ! [Z: complex] :
% 0.29/0.60        ( ( ( csqrt @ Z )
% 0.29/0.60          = one_one_complex )
% 0.29/0.60        = ( Z = one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % csqrt_eq_1
% 0.29/0.60  thf(fact_8727_csqrt__1,axiom,
% 0.29/0.60      ( ( csqrt @ one_one_complex )
% 0.29/0.60      = one_one_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % csqrt_1
% 0.29/0.60  thf(fact_8728_xor__nonnegative__int__iff,axiom,
% 0.29/0.60      ! [K: int,L: int] :
% 0.29/0.60        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 0.29/0.60        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.60          = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_nonnegative_int_iff
% 0.29/0.60  thf(fact_8729_xor__negative__int__iff,axiom,
% 0.29/0.60      ! [K: int,L: int] :
% 0.29/0.60        ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 0.29/0.60        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.29/0.60         != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_negative_int_iff
% 0.29/0.60  thf(fact_8730_power2__csqrt,axiom,
% 0.29/0.60      ! [Z: complex] :
% 0.29/0.60        ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60        = Z ) ).
% 0.29/0.60  
% 0.29/0.60  % power2_csqrt
% 0.29/0.60  thf(fact_8731_XOR__lower,axiom,
% 0.29/0.60      ! [X: int,Y: int] :
% 0.29/0.60        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.60       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.60         => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % XOR_lower
% 0.29/0.60  thf(fact_8732_divide__complex__def,axiom,
% 0.29/0.60      ( divide1717551699836669952omplex
% 0.29/0.60      = ( ^ [X4: complex,Y6: complex] : ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ Y6 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % divide_complex_def
% 0.29/0.60  thf(fact_8733_xor__nat__def,axiom,
% 0.29/0.60      ( bit_se6528837805403552850or_nat
% 0.29/0.60      = ( ^ [M6: nat,N3: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_nat_def
% 0.29/0.60  thf(fact_8734_Arg__zero,axiom,
% 0.29/0.60      ( ( arg @ zero_zero_complex )
% 0.29/0.60      = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % Arg_zero
% 0.29/0.60  thf(fact_8735_of__real__sqrt,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
% 0.29/0.60          = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_real_sqrt
% 0.29/0.60  thf(fact_8736_cis__Arg,axiom,
% 0.29/0.60      ! [Z: complex] :
% 0.29/0.60        ( ( Z != zero_zero_complex )
% 0.29/0.60       => ( ( cis @ ( arg @ Z ) )
% 0.29/0.60          = ( sgn_sgn_complex @ Z ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cis_Arg
% 0.29/0.60  thf(fact_8737_Arg__bounded,axiom,
% 0.29/0.60      ! [Z: complex] :
% 0.29/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.29/0.60        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Arg_bounded
% 0.29/0.60  thf(fact_8738_XOR__upper,axiom,
% 0.29/0.60      ! [X: int,N: nat,Y: int] :
% 0.29/0.60        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.60       => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.60         => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.60           => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % XOR_upper
% 0.29/0.60  thf(fact_8739_complex__inverse,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 0.29/0.60        = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % complex_inverse
% 0.29/0.60  thf(fact_8740_xor__int__rec,axiom,
% 0.29/0.60      ( bit_se6526347334894502574or_int
% 0.29/0.60      = ( ^ [K2: int,L3: int] :
% 0.29/0.60            ( plus_plus_int
% 0.29/0.60            @ ( zero_n2684676970156552555ol_int
% 0.29/0.60              @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
% 0.29/0.60               != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
% 0.29/0.60            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_int_rec
% 0.29/0.60  thf(fact_8741_xor__int__unfold,axiom,
% 0.29/0.60      ( bit_se6526347334894502574or_int
% 0.29/0.60      = ( ^ [K2: int,L3: int] :
% 0.29/0.60            ( if_int
% 0.29/0.60            @ ( K2
% 0.29/0.60              = ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.60            @ ( bit_ri7919022796975470100ot_int @ L3 )
% 0.29/0.60            @ ( if_int
% 0.29/0.60              @ ( L3
% 0.29/0.60                = ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.60              @ ( bit_ri7919022796975470100ot_int @ K2 )
% 0.29/0.60              @ ( if_int @ ( K2 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = 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 @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % xor_int_unfold
% 0.29/0.60  thf(fact_8742_sinh__ln__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 0.29/0.60          = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_ln_real
% 0.29/0.60  thf(fact_8743_cis__multiple__2pi,axiom,
% 0.29/0.60      ! [N: real] :
% 0.29/0.60        ( ( member_real @ N @ ring_1_Ints_real )
% 0.29/0.60       => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.29/0.60          = one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cis_multiple_2pi
% 0.29/0.60  thf(fact_8744_cosh__ln__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 0.29/0.60          = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_ln_real
% 0.29/0.60  thf(fact_8745_sinh__real__eq__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( sinh_real @ X )
% 0.29/0.60          = ( sinh_real @ Y ) )
% 0.29/0.60        = ( X = Y ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_real_eq_iff
% 0.29/0.60  thf(fact_8746_sinh__0,axiom,
% 0.29/0.60      ( ( sinh_complex @ zero_zero_complex )
% 0.29/0.60      = zero_zero_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_0
% 0.29/0.60  thf(fact_8747_sinh__0,axiom,
% 0.29/0.60      ( ( sinh_real @ zero_zero_real )
% 0.29/0.60      = zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_0
% 0.29/0.60  thf(fact_8748_cosh__minus,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cosh_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.60        = ( cosh_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_minus
% 0.29/0.60  thf(fact_8749_cosh__minus,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( cosh_complex @ ( uminus1482373934393186551omplex @ X ) )
% 0.29/0.60        = ( cosh_complex @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_minus
% 0.29/0.60  thf(fact_8750_sinh__minus,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( sinh_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.60        = ( uminus_uminus_real @ ( sinh_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_minus
% 0.29/0.60  thf(fact_8751_sinh__minus,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( sinh_complex @ ( uminus1482373934393186551omplex @ X ) )
% 0.29/0.60        = ( uminus1482373934393186551omplex @ ( sinh_complex @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_minus
% 0.29/0.60  thf(fact_8752_sinh__real__zero__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( sinh_real @ X )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60        = ( X = zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_real_zero_iff
% 0.29/0.60  thf(fact_8753_sinh__real__less__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
% 0.29/0.60        = ( ord_less_real @ X @ Y ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_real_less_iff
% 0.29/0.60  thf(fact_8754_sinh__real__le__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
% 0.29/0.60        = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_real_le_iff
% 0.29/0.60  thf(fact_8755_cosh__real__eq__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( cosh_real @ X )
% 0.29/0.60          = ( cosh_real @ Y ) )
% 0.29/0.60        = ( ( abs_abs_real @ X )
% 0.29/0.60          = ( abs_abs_real @ Y ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_eq_iff
% 0.29/0.60  thf(fact_8756_cosh__real__abs,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cosh_real @ ( abs_abs_real @ X ) )
% 0.29/0.60        = ( cosh_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_abs
% 0.29/0.60  thf(fact_8757_sinh__real__abs,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( sinh_real @ ( abs_abs_real @ X ) )
% 0.29/0.60        = ( abs_abs_real @ ( sinh_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_real_abs
% 0.29/0.60  thf(fact_8758_frac__in__Ints__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( member_real @ ( archim2898591450579166408c_real @ X ) @ ring_1_Ints_real )
% 0.29/0.60        = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % frac_in_Ints_iff
% 0.29/0.60  thf(fact_8759_cosh__0,axiom,
% 0.29/0.60      ( ( cosh_complex @ zero_zero_complex )
% 0.29/0.60      = one_one_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_0
% 0.29/0.60  thf(fact_8760_cosh__0,axiom,
% 0.29/0.60      ( ( cosh_real @ zero_zero_real )
% 0.29/0.60      = one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_0
% 0.29/0.60  thf(fact_8761_bit_Oconj__cancel__right,axiom,
% 0.29/0.60      ! [X: int] :
% 0.29/0.60        ( ( bit_se725231765392027082nd_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 0.29/0.60        = zero_zero_int ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.conj_cancel_right
% 0.29/0.60  thf(fact_8762_bit_Oconj__cancel__left,axiom,
% 0.29/0.60      ! [X: int] :
% 0.29/0.60        ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 0.29/0.60        = zero_zero_int ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.conj_cancel_left
% 0.29/0.60  thf(fact_8763_sinh__real__pos__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
% 0.29/0.60        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_real_pos_iff
% 0.29/0.60  thf(fact_8764_sinh__real__neg__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
% 0.29/0.60        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_real_neg_iff
% 0.29/0.60  thf(fact_8765_sinh__real__nonpos__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
% 0.29/0.60        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_real_nonpos_iff
% 0.29/0.60  thf(fact_8766_sinh__real__nonneg__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
% 0.29/0.60        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_real_nonneg_iff
% 0.29/0.60  thf(fact_8767_frac__eq__0__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ( archim2898591450579166408c_real @ X )
% 0.29/0.60          = zero_zero_real )
% 0.29/0.60        = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % frac_eq_0_iff
% 0.29/0.60  thf(fact_8768_frac__eq__0__iff,axiom,
% 0.29/0.60      ! [X: rat] :
% 0.29/0.60        ( ( ( archimedean_frac_rat @ X )
% 0.29/0.60          = zero_zero_rat )
% 0.29/0.60        = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % frac_eq_0_iff
% 0.29/0.60  thf(fact_8769_bit_Ocompl__zero,axiom,
% 0.29/0.60      ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
% 0.29/0.60      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.compl_zero
% 0.29/0.60  thf(fact_8770_bit_Ocompl__zero,axiom,
% 0.29/0.60      ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
% 0.29/0.60      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.compl_zero
% 0.29/0.60  thf(fact_8771_bit_Ocompl__one,axiom,
% 0.29/0.60      ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.60      = zero_z3403309356797280102nteger ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.compl_one
% 0.29/0.60  thf(fact_8772_bit_Ocompl__one,axiom,
% 0.29/0.60      ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.60      = zero_zero_int ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.compl_one
% 0.29/0.60  thf(fact_8773_bit_Oxor__one__left,axiom,
% 0.29/0.60      ! [X: code_integer] :
% 0.29/0.60        ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.29/0.60        = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.xor_one_left
% 0.29/0.60  thf(fact_8774_bit_Oxor__one__left,axiom,
% 0.29/0.60      ! [X: int] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.29/0.60        = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.xor_one_left
% 0.29/0.60  thf(fact_8775_bit_Oxor__one__right,axiom,
% 0.29/0.60      ! [X: code_integer] :
% 0.29/0.60        ( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.60        = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.xor_one_right
% 0.29/0.60  thf(fact_8776_bit_Oxor__one__right,axiom,
% 0.29/0.60      ! [X: int] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.60        = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.xor_one_right
% 0.29/0.60  thf(fact_8777_bit_Oxor__cancel__left,axiom,
% 0.29/0.60      ! [X: code_integer] :
% 0.29/0.60        ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
% 0.29/0.60        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.xor_cancel_left
% 0.29/0.60  thf(fact_8778_bit_Oxor__cancel__left,axiom,
% 0.29/0.60      ! [X: int] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 0.29/0.60        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.xor_cancel_left
% 0.29/0.60  thf(fact_8779_bit_Oxor__cancel__right,axiom,
% 0.29/0.60      ! [X: code_integer] :
% 0.29/0.60        ( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
% 0.29/0.60        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.xor_cancel_right
% 0.29/0.60  thf(fact_8780_bit_Oxor__cancel__right,axiom,
% 0.29/0.60      ! [X: int] :
% 0.29/0.60        ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 0.29/0.60        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit.xor_cancel_right
% 0.29/0.60  thf(fact_8781_not__nonnegative__int__iff,axiom,
% 0.29/0.60      ! [K: int] :
% 0.29/0.60        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.29/0.60        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_nonnegative_int_iff
% 0.29/0.60  thf(fact_8782_not__negative__int__iff,axiom,
% 0.29/0.60      ! [K: int] :
% 0.29/0.60        ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.29/0.60        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_negative_int_iff
% 0.29/0.60  thf(fact_8783_floor__add2,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ( member_real @ X @ ring_1_Ints_real )
% 0.29/0.60          | ( member_real @ Y @ ring_1_Ints_real ) )
% 0.29/0.60       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.60          = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % floor_add2
% 0.29/0.60  thf(fact_8784_floor__add2,axiom,
% 0.29/0.60      ! [X: rat,Y: rat] :
% 0.29/0.60        ( ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.29/0.60          | ( member_rat @ Y @ ring_1_Ints_rat ) )
% 0.29/0.60       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.29/0.60          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % floor_add2
% 0.29/0.60  thf(fact_8785_minus__not__numeral__eq,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.29/0.60        = ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_not_numeral_eq
% 0.29/0.60  thf(fact_8786_minus__not__numeral__eq,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.60        = ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_not_numeral_eq
% 0.29/0.60  thf(fact_8787_frac__gt__0__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
% 0.29/0.60        = ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % frac_gt_0_iff
% 0.29/0.60  thf(fact_8788_frac__gt__0__iff,axiom,
% 0.29/0.60      ! [X: rat] :
% 0.29/0.60        ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
% 0.29/0.60        = ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % frac_gt_0_iff
% 0.29/0.60  thf(fact_8789_even__not__iff,axiom,
% 0.29/0.60      ! [A: code_integer] :
% 0.29/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
% 0.29/0.60        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % even_not_iff
% 0.29/0.60  thf(fact_8790_even__not__iff,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
% 0.29/0.60        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % even_not_iff
% 0.29/0.60  thf(fact_8791_not__one__eq,axiom,
% 0.29/0.60      ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
% 0.29/0.60      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_one_eq
% 0.29/0.60  thf(fact_8792_not__one__eq,axiom,
% 0.29/0.60      ( ( bit_ri7919022796975470100ot_int @ one_one_int )
% 0.29/0.60      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_one_eq
% 0.29/0.60  thf(fact_8793_Ints__power,axiom,
% 0.29/0.60      ! [A: real,N: nat] :
% 0.29/0.60        ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.60       => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_power
% 0.29/0.60  thf(fact_8794_Ints__power,axiom,
% 0.29/0.60      ! [A: int,N: nat] :
% 0.29/0.60        ( ( member_int @ A @ ring_1_Ints_int )
% 0.29/0.60       => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_power
% 0.29/0.60  thf(fact_8795_Ints__power,axiom,
% 0.29/0.60      ! [A: complex,N: nat] :
% 0.29/0.60        ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.29/0.60       => ( member_complex @ ( power_power_complex @ A @ N ) @ ring_1_Ints_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_power
% 0.29/0.60  thf(fact_8796_Ints__1,axiom,
% 0.29/0.60      member_rat @ one_one_rat @ ring_1_Ints_rat ).
% 0.29/0.60  
% 0.29/0.60  % Ints_1
% 0.29/0.60  thf(fact_8797_Ints__1,axiom,
% 0.29/0.60      member_int @ one_one_int @ ring_1_Ints_int ).
% 0.29/0.60  
% 0.29/0.60  % Ints_1
% 0.29/0.60  thf(fact_8798_Ints__1,axiom,
% 0.29/0.60      member_real @ one_one_real @ ring_1_Ints_real ).
% 0.29/0.60  
% 0.29/0.60  % Ints_1
% 0.29/0.60  thf(fact_8799_Ints__1,axiom,
% 0.29/0.60      member_complex @ one_one_complex @ ring_1_Ints_complex ).
% 0.29/0.60  
% 0.29/0.60  % Ints_1
% 0.29/0.60  thf(fact_8800_minus__in__Ints__iff,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
% 0.29/0.60        = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_in_Ints_iff
% 0.29/0.60  thf(fact_8801_minus__in__Ints__iff,axiom,
% 0.29/0.60      ! [X: int] :
% 0.29/0.60        ( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
% 0.29/0.60        = ( member_int @ X @ ring_1_Ints_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_in_Ints_iff
% 0.29/0.60  thf(fact_8802_minus__in__Ints__iff,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( member_complex @ ( uminus1482373934393186551omplex @ X ) @ ring_1_Ints_complex )
% 0.29/0.60        = ( member_complex @ X @ ring_1_Ints_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_in_Ints_iff
% 0.29/0.60  thf(fact_8803_minus__in__Ints__iff,axiom,
% 0.29/0.60      ! [X: rat] :
% 0.29/0.60        ( ( member_rat @ ( uminus_uminus_rat @ X ) @ ring_1_Ints_rat )
% 0.29/0.60        = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_in_Ints_iff
% 0.29/0.60  thf(fact_8804_minus__in__Ints__iff,axiom,
% 0.29/0.60      ! [X: code_integer] :
% 0.29/0.60        ( ( member_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ring_11222124179247155820nteger )
% 0.29/0.60        = ( member_Code_integer @ X @ ring_11222124179247155820nteger ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_in_Ints_iff
% 0.29/0.60  thf(fact_8805_Ints__minus,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.60       => ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_minus
% 0.29/0.60  thf(fact_8806_Ints__minus,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( member_int @ A @ ring_1_Ints_int )
% 0.29/0.60       => ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_minus
% 0.29/0.60  thf(fact_8807_Ints__minus,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.29/0.60       => ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_minus
% 0.29/0.60  thf(fact_8808_Ints__minus,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.60       => ( member_rat @ ( uminus_uminus_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_minus
% 0.29/0.60  thf(fact_8809_Ints__minus,axiom,
% 0.29/0.60      ! [A: code_integer] :
% 0.29/0.60        ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 0.29/0.60       => ( member_Code_integer @ ( uminus1351360451143612070nteger @ A ) @ ring_11222124179247155820nteger ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_minus
% 0.29/0.60  thf(fact_8810_cosh__real__nonzero,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cosh_real @ X )
% 0.29/0.60       != zero_zero_real ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_nonzero
% 0.29/0.60  thf(fact_8811_Ints__numeral,axiom,
% 0.29/0.60      ! [N: num] : ( member_complex @ ( numera6690914467698888265omplex @ N ) @ ring_1_Ints_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_numeral
% 0.29/0.60  thf(fact_8812_Ints__numeral,axiom,
% 0.29/0.60      ! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_numeral
% 0.29/0.60  thf(fact_8813_Ints__numeral,axiom,
% 0.29/0.60      ! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_numeral
% 0.29/0.60  thf(fact_8814_Ints__numeral,axiom,
% 0.29/0.60      ! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_numeral
% 0.29/0.60  thf(fact_8815_Ints__0,axiom,
% 0.29/0.60      member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% 0.29/0.60  
% 0.29/0.60  % Ints_0
% 0.29/0.60  thf(fact_8816_Ints__0,axiom,
% 0.29/0.60      member_real @ zero_zero_real @ ring_1_Ints_real ).
% 0.29/0.60  
% 0.29/0.60  % Ints_0
% 0.29/0.60  thf(fact_8817_Ints__0,axiom,
% 0.29/0.60      member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% 0.29/0.60  
% 0.29/0.60  % Ints_0
% 0.29/0.60  thf(fact_8818_Ints__0,axiom,
% 0.29/0.60      member_int @ zero_zero_int @ ring_1_Ints_int ).
% 0.29/0.60  
% 0.29/0.60  % Ints_0
% 0.29/0.60  thf(fact_8819_Ints__cases,axiom,
% 0.29/0.60      ! [Q3: int] :
% 0.29/0.60        ( ( member_int @ Q3 @ ring_1_Ints_int )
% 0.29/0.60       => ~ ! [Z3: int] :
% 0.29/0.60              ( Q3
% 0.29/0.60             != ( ring_1_of_int_int @ Z3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_cases
% 0.29/0.60  thf(fact_8820_Ints__cases,axiom,
% 0.29/0.60      ! [Q3: real] :
% 0.29/0.60        ( ( member_real @ Q3 @ ring_1_Ints_real )
% 0.29/0.60       => ~ ! [Z3: int] :
% 0.29/0.60              ( Q3
% 0.29/0.60             != ( ring_1_of_int_real @ Z3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_cases
% 0.29/0.60  thf(fact_8821_Ints__cases,axiom,
% 0.29/0.60      ! [Q3: rat] :
% 0.29/0.60        ( ( member_rat @ Q3 @ ring_1_Ints_rat )
% 0.29/0.60       => ~ ! [Z3: int] :
% 0.29/0.60              ( Q3
% 0.29/0.60             != ( ring_1_of_int_rat @ Z3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_cases
% 0.29/0.60  thf(fact_8822_Ints__cases,axiom,
% 0.29/0.60      ! [Q3: code_integer] :
% 0.29/0.60        ( ( member_Code_integer @ Q3 @ ring_11222124179247155820nteger )
% 0.29/0.60       => ~ ! [Z3: int] :
% 0.29/0.60              ( Q3
% 0.29/0.60             != ( ring_18347121197199848620nteger @ Z3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_cases
% 0.29/0.60  thf(fact_8823_Ints__cases,axiom,
% 0.29/0.60      ! [Q3: complex] :
% 0.29/0.60        ( ( member_complex @ Q3 @ ring_1_Ints_complex )
% 0.29/0.60       => ~ ! [Z3: int] :
% 0.29/0.60              ( Q3
% 0.29/0.60             != ( ring_17405671764205052669omplex @ Z3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_cases
% 0.29/0.60  thf(fact_8824_Ints__induct,axiom,
% 0.29/0.60      ! [Q3: int,P: int > $o] :
% 0.29/0.60        ( ( member_int @ Q3 @ ring_1_Ints_int )
% 0.29/0.60       => ( ! [Z3: int] : ( P @ ( ring_1_of_int_int @ Z3 ) )
% 0.29/0.60         => ( P @ Q3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_induct
% 0.29/0.60  thf(fact_8825_Ints__induct,axiom,
% 0.29/0.60      ! [Q3: real,P: real > $o] :
% 0.29/0.60        ( ( member_real @ Q3 @ ring_1_Ints_real )
% 0.29/0.60       => ( ! [Z3: int] : ( P @ ( ring_1_of_int_real @ Z3 ) )
% 0.29/0.60         => ( P @ Q3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_induct
% 0.29/0.60  thf(fact_8826_Ints__induct,axiom,
% 0.29/0.60      ! [Q3: rat,P: rat > $o] :
% 0.29/0.60        ( ( member_rat @ Q3 @ ring_1_Ints_rat )
% 0.29/0.60       => ( ! [Z3: int] : ( P @ ( ring_1_of_int_rat @ Z3 ) )
% 0.29/0.60         => ( P @ Q3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_induct
% 0.29/0.60  thf(fact_8827_Ints__induct,axiom,
% 0.29/0.60      ! [Q3: code_integer,P: code_integer > $o] :
% 0.29/0.60        ( ( member_Code_integer @ Q3 @ ring_11222124179247155820nteger )
% 0.29/0.60       => ( ! [Z3: int] : ( P @ ( ring_18347121197199848620nteger @ Z3 ) )
% 0.29/0.60         => ( P @ Q3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_induct
% 0.29/0.60  thf(fact_8828_Ints__induct,axiom,
% 0.29/0.60      ! [Q3: complex,P: complex > $o] :
% 0.29/0.60        ( ( member_complex @ Q3 @ ring_1_Ints_complex )
% 0.29/0.60       => ( ! [Z3: int] : ( P @ ( ring_17405671764205052669omplex @ Z3 ) )
% 0.29/0.60         => ( P @ Q3 ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_induct
% 0.29/0.60  thf(fact_8829_Ints__of__int,axiom,
% 0.29/0.60      ! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_of_int
% 0.29/0.60  thf(fact_8830_Ints__of__int,axiom,
% 0.29/0.60      ! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_of_int
% 0.29/0.60  thf(fact_8831_Ints__of__int,axiom,
% 0.29/0.60      ! [Z: int] : ( member_rat @ ( ring_1_of_int_rat @ Z ) @ ring_1_Ints_rat ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_of_int
% 0.29/0.60  thf(fact_8832_Ints__of__int,axiom,
% 0.29/0.60      ! [Z: int] : ( member_Code_integer @ ( ring_18347121197199848620nteger @ Z ) @ ring_11222124179247155820nteger ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_of_int
% 0.29/0.60  thf(fact_8833_Ints__of__int,axiom,
% 0.29/0.60      ! [Z: int] : ( member_complex @ ( ring_17405671764205052669omplex @ Z ) @ ring_1_Ints_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_of_int
% 0.29/0.60  thf(fact_8834_of__int__not__eq,axiom,
% 0.29/0.60      ! [K: int] :
% 0.29/0.60        ( ( ring_18347121197199848620nteger @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.29/0.60        = ( bit_ri7632146776885996613nteger @ ( ring_18347121197199848620nteger @ K ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_not_eq
% 0.29/0.60  thf(fact_8835_of__int__not__eq,axiom,
% 0.29/0.60      ! [K: int] :
% 0.29/0.60        ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.29/0.60        = ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_not_eq
% 0.29/0.60  thf(fact_8836_sinh__less__cosh__real,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_less_cosh_real
% 0.29/0.60  thf(fact_8837_sinh__le__cosh__real,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_le_cosh_real
% 0.29/0.60  thf(fact_8838_Ints__abs,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( member_int @ A @ ring_1_Ints_int )
% 0.29/0.60       => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_abs
% 0.29/0.60  thf(fact_8839_Ints__abs,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.60       => ( member_rat @ ( abs_abs_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_abs
% 0.29/0.60  thf(fact_8840_Ints__abs,axiom,
% 0.29/0.60      ! [A: code_integer] :
% 0.29/0.60        ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 0.29/0.60       => ( member_Code_integer @ ( abs_abs_Code_integer @ A ) @ ring_11222124179247155820nteger ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_abs
% 0.29/0.60  thf(fact_8841_Ints__abs,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.60       => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_abs
% 0.29/0.60  thf(fact_8842_Ints__of__nat,axiom,
% 0.29/0.60      ! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ ring_1_Ints_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_of_nat
% 0.29/0.60  thf(fact_8843_Ints__of__nat,axiom,
% 0.29/0.60      ! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ ring_1_Ints_real ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_of_nat
% 0.29/0.60  thf(fact_8844_Ints__of__nat,axiom,
% 0.29/0.60      ! [N: nat] : ( member_rat @ ( semiri681578069525770553at_rat @ N ) @ ring_1_Ints_rat ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_of_nat
% 0.29/0.60  thf(fact_8845_Ints__of__nat,axiom,
% 0.29/0.60      ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ring_1_Ints_int ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_of_nat
% 0.29/0.60  thf(fact_8846_tanh__def,axiom,
% 0.29/0.60      ( tanh_complex
% 0.29/0.60      = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tanh_def
% 0.29/0.60  thf(fact_8847_tanh__def,axiom,
% 0.29/0.60      ( tanh_real
% 0.29/0.60      = ( ^ [X4: real] : ( divide_divide_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % tanh_def
% 0.29/0.60  thf(fact_8848_Ints__diff,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.29/0.60       => ( ( member_complex @ B @ ring_1_Ints_complex )
% 0.29/0.60         => ( member_complex @ ( minus_minus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_diff
% 0.29/0.60  thf(fact_8849_Ints__diff,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.60       => ( ( member_real @ B @ ring_1_Ints_real )
% 0.29/0.60         => ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_diff
% 0.29/0.60  thf(fact_8850_Ints__diff,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.60       => ( ( member_rat @ B @ ring_1_Ints_rat )
% 0.29/0.60         => ( member_rat @ ( minus_minus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_diff
% 0.29/0.60  thf(fact_8851_Ints__diff,axiom,
% 0.29/0.60      ! [A: int,B: int] :
% 0.29/0.60        ( ( member_int @ A @ ring_1_Ints_int )
% 0.29/0.60       => ( ( member_int @ B @ ring_1_Ints_int )
% 0.29/0.60         => ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_diff
% 0.29/0.60  thf(fact_8852_Ints__add,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.60       => ( ( member_real @ B @ ring_1_Ints_real )
% 0.29/0.60         => ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_add
% 0.29/0.60  thf(fact_8853_Ints__add,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.60       => ( ( member_rat @ B @ ring_1_Ints_rat )
% 0.29/0.60         => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_add
% 0.29/0.60  thf(fact_8854_Ints__add,axiom,
% 0.29/0.60      ! [A: int,B: int] :
% 0.29/0.60        ( ( member_int @ A @ ring_1_Ints_int )
% 0.29/0.60       => ( ( member_int @ B @ ring_1_Ints_int )
% 0.29/0.60         => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_add
% 0.29/0.60  thf(fact_8855_Ints__add,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.29/0.60       => ( ( member_complex @ B @ ring_1_Ints_complex )
% 0.29/0.60         => ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_add
% 0.29/0.60  thf(fact_8856_Ints__mult,axiom,
% 0.29/0.60      ! [A: complex,B: complex] :
% 0.29/0.60        ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.29/0.60       => ( ( member_complex @ B @ ring_1_Ints_complex )
% 0.29/0.60         => ( member_complex @ ( times_times_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_mult
% 0.29/0.60  thf(fact_8857_Ints__mult,axiom,
% 0.29/0.60      ! [A: real,B: real] :
% 0.29/0.60        ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.60       => ( ( member_real @ B @ ring_1_Ints_real )
% 0.29/0.60         => ( member_real @ ( times_times_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_mult
% 0.29/0.60  thf(fact_8858_Ints__mult,axiom,
% 0.29/0.60      ! [A: rat,B: rat] :
% 0.29/0.60        ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.60       => ( ( member_rat @ B @ ring_1_Ints_rat )
% 0.29/0.60         => ( member_rat @ ( times_times_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_mult
% 0.29/0.60  thf(fact_8859_Ints__mult,axiom,
% 0.29/0.60      ! [A: int,B: int] :
% 0.29/0.60        ( ( member_int @ A @ ring_1_Ints_int )
% 0.29/0.60       => ( ( member_int @ B @ ring_1_Ints_int )
% 0.29/0.60         => ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_mult
% 0.29/0.60  thf(fact_8860_sinh__plus__cosh,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( plus_plus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
% 0.29/0.60        = ( exp_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_plus_cosh
% 0.29/0.60  thf(fact_8861_sinh__plus__cosh,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( plus_plus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
% 0.29/0.60        = ( exp_complex @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_plus_cosh
% 0.29/0.60  thf(fact_8862_cosh__plus__sinh,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( plus_plus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
% 0.29/0.60        = ( exp_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_plus_sinh
% 0.29/0.60  thf(fact_8863_cosh__plus__sinh,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( plus_plus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
% 0.29/0.60        = ( exp_complex @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_plus_sinh
% 0.29/0.60  thf(fact_8864_sinh__add,axiom,
% 0.29/0.60      ! [X: complex,Y: complex] :
% 0.29/0.60        ( ( sinh_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.29/0.60        = ( plus_plus_complex @ ( times_times_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ Y ) ) @ ( times_times_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_add
% 0.29/0.60  thf(fact_8865_sinh__add,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( sinh_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.60        = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_add
% 0.29/0.60  thf(fact_8866_cosh__add,axiom,
% 0.29/0.60      ! [X: complex,Y: complex] :
% 0.29/0.60        ( ( cosh_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.29/0.60        = ( plus_plus_complex @ ( times_times_complex @ ( cosh_complex @ X ) @ ( cosh_complex @ Y ) ) @ ( times_times_complex @ ( sinh_complex @ X ) @ ( sinh_complex @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_add
% 0.29/0.60  thf(fact_8867_cosh__add,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( cosh_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.60        = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_add
% 0.29/0.60  thf(fact_8868_sinh__diff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( sinh_real @ ( minus_minus_real @ X @ Y ) )
% 0.29/0.60        = ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_diff
% 0.29/0.60  thf(fact_8869_cosh__diff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( cosh_real @ ( minus_minus_real @ X @ Y ) )
% 0.29/0.60        = ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_diff
% 0.29/0.60  thf(fact_8870_of__int__not__numeral,axiom,
% 0.29/0.60      ! [K: num] :
% 0.29/0.60        ( ( ring_18347121197199848620nteger @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.60        = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_not_numeral
% 0.29/0.60  thf(fact_8871_of__int__not__numeral,axiom,
% 0.29/0.60      ! [K: num] :
% 0.29/0.60        ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.60        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_not_numeral
% 0.29/0.60  thf(fact_8872_arsinh__sinh__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( arsinh_real @ ( sinh_real @ X ) )
% 0.29/0.60        = X ) ).
% 0.29/0.60  
% 0.29/0.60  % arsinh_sinh_real
% 0.29/0.60  thf(fact_8873_sinh__minus__cosh,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( minus_minus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
% 0.29/0.60        = ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_minus_cosh
% 0.29/0.60  thf(fact_8874_sinh__minus__cosh,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( minus_minus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
% 0.29/0.60        = ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_minus_cosh
% 0.29/0.60  thf(fact_8875_cosh__minus__sinh,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( minus_minus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
% 0.29/0.60        = ( exp_real @ ( uminus_uminus_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_minus_sinh
% 0.29/0.60  thf(fact_8876_cosh__minus__sinh,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( minus_minus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
% 0.29/0.60        = ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_minus_sinh
% 0.29/0.60  thf(fact_8877_not__add__distrib,axiom,
% 0.29/0.60      ! [A: int,B: int] :
% 0.29/0.60        ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
% 0.29/0.60        = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_add_distrib
% 0.29/0.60  thf(fact_8878_not__diff__distrib,axiom,
% 0.29/0.60      ! [A: int,B: int] :
% 0.29/0.60        ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
% 0.29/0.60        = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_diff_distrib
% 0.29/0.60  thf(fact_8879_cosh__real__pos,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_pos
% 0.29/0.60  thf(fact_8880_cosh__real__nonpos__le__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 0.29/0.60            = ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_nonpos_le_iff
% 0.29/0.60  thf(fact_8881_cosh__real__nonneg__le__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60         => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 0.29/0.60            = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_nonneg_le_iff
% 0.29/0.60  thf(fact_8882_cosh__real__nonneg,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_nonneg
% 0.29/0.60  thf(fact_8883_cosh__real__ge__1,axiom,
% 0.29/0.60      ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_ge_1
% 0.29/0.60  thf(fact_8884_sinh__double,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X ) ) @ ( cosh_complex @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_double
% 0.29/0.60  thf(fact_8885_sinh__double,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X ) ) @ ( cosh_real @ X ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_double
% 0.29/0.60  thf(fact_8886_Ints__double__eq__0__iff,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.29/0.60       => ( ( ( plus_plus_complex @ A @ A )
% 0.29/0.60            = zero_zero_complex )
% 0.29/0.60          = ( A = zero_zero_complex ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_double_eq_0_iff
% 0.29/0.60  thf(fact_8887_Ints__double__eq__0__iff,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.60       => ( ( ( plus_plus_real @ A @ A )
% 0.29/0.60            = zero_zero_real )
% 0.29/0.60          = ( A = zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_double_eq_0_iff
% 0.29/0.60  thf(fact_8888_Ints__double__eq__0__iff,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.60       => ( ( ( plus_plus_rat @ A @ A )
% 0.29/0.60            = zero_zero_rat )
% 0.29/0.60          = ( A = zero_zero_rat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_double_eq_0_iff
% 0.29/0.60  thf(fact_8889_Ints__double__eq__0__iff,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( member_int @ A @ ring_1_Ints_int )
% 0.29/0.60       => ( ( ( plus_plus_int @ A @ A )
% 0.29/0.60            = zero_zero_int )
% 0.29/0.60          = ( A = zero_zero_int ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_double_eq_0_iff
% 0.29/0.60  thf(fact_8890_minus__eq__not__plus__1,axiom,
% 0.29/0.60      ( uminus1351360451143612070nteger
% 0.29/0.60      = ( ^ [A3: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_eq_not_plus_1
% 0.29/0.60  thf(fact_8891_minus__eq__not__plus__1,axiom,
% 0.29/0.60      ( uminus_uminus_int
% 0.29/0.60      = ( ^ [A3: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ one_one_int ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_eq_not_plus_1
% 0.29/0.60  thf(fact_8892_minus__eq__not__minus__1,axiom,
% 0.29/0.60      ( uminus1351360451143612070nteger
% 0.29/0.60      = ( ^ [A3: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A3 @ one_one_Code_integer ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_eq_not_minus_1
% 0.29/0.60  thf(fact_8893_minus__eq__not__minus__1,axiom,
% 0.29/0.60      ( uminus_uminus_int
% 0.29/0.60      = ( ^ [A3: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A3 @ one_one_int ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_eq_not_minus_1
% 0.29/0.60  thf(fact_8894_not__eq__complement,axiom,
% 0.29/0.60      ( bit_ri7632146776885996613nteger
% 0.29/0.60      = ( ^ [A3: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_eq_complement
% 0.29/0.60  thf(fact_8895_not__eq__complement,axiom,
% 0.29/0.60      ( bit_ri7919022796975470100ot_int
% 0.29/0.60      = ( ^ [A3: int] : ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ one_one_int ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_eq_complement
% 0.29/0.60  thf(fact_8896_not__int__def,axiom,
% 0.29/0.60      ( bit_ri7919022796975470100ot_int
% 0.29/0.60      = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_int_def
% 0.29/0.60  thf(fact_8897_and__not__numerals_I1_J,axiom,
% 0.29/0.60      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.29/0.60      = zero_zero_int ) ).
% 0.29/0.60  
% 0.29/0.60  % and_not_numerals(1)
% 0.29/0.60  thf(fact_8898_cosh__real__nonpos__less__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.60       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.29/0.60         => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 0.29/0.60            = ( ord_less_real @ Y @ X ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_nonpos_less_iff
% 0.29/0.60  thf(fact_8899_cosh__real__nonneg__less__iff,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.60         => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 0.29/0.60            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_nonneg_less_iff
% 0.29/0.60  thf(fact_8900_cosh__real__strict__mono,axiom,
% 0.29/0.60      ! [X: real,Y: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( ord_less_real @ X @ Y )
% 0.29/0.60         => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_real_strict_mono
% 0.29/0.60  thf(fact_8901_Ints__odd__nonzero,axiom,
% 0.29/0.60      ! [A: complex] :
% 0.29/0.60        ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.29/0.60       => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
% 0.29/0.60         != zero_zero_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_odd_nonzero
% 0.29/0.60  thf(fact_8902_Ints__odd__nonzero,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.60       => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
% 0.29/0.60         != zero_zero_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_odd_nonzero
% 0.29/0.60  thf(fact_8903_Ints__odd__nonzero,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.60       => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
% 0.29/0.60         != zero_zero_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_odd_nonzero
% 0.29/0.60  thf(fact_8904_Ints__odd__nonzero,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( member_int @ A @ ring_1_Ints_int )
% 0.29/0.60       => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
% 0.29/0.60         != zero_zero_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_odd_nonzero
% 0.29/0.60  thf(fact_8905_minus__numeral__inc__eq,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
% 0.29/0.60        = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_numeral_inc_eq
% 0.29/0.60  thf(fact_8906_minus__numeral__inc__eq,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
% 0.29/0.60        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % minus_numeral_inc_eq
% 0.29/0.60  thf(fact_8907_cosh__square__eq,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60        = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_square_eq
% 0.29/0.60  thf(fact_8908_cosh__square__eq,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60        = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_square_eq
% 0.29/0.60  thf(fact_8909_hyperbolic__pythagoras,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.60        = one_one_complex ) ).
% 0.29/0.60  
% 0.29/0.60  % hyperbolic_pythagoras
% 0.29/0.60  thf(fact_8910_hyperbolic__pythagoras,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.60        = one_one_real ) ).
% 0.29/0.60  
% 0.29/0.60  % hyperbolic_pythagoras
% 0.29/0.60  thf(fact_8911_sinh__square__eq,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60        = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_square_eq
% 0.29/0.60  thf(fact_8912_sinh__square__eq,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.60        = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % sinh_square_eq
% 0.29/0.60  thf(fact_8913_of__int__divide__in__Ints,axiom,
% 0.29/0.60      ! [B: int,A: int] :
% 0.29/0.60        ( ( dvd_dvd_int @ B @ A )
% 0.29/0.60       => ( member_Code_integer @ ( divide6298287555418463151nteger @ ( ring_18347121197199848620nteger @ A ) @ ( ring_18347121197199848620nteger @ B ) ) @ ring_11222124179247155820nteger ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_divide_in_Ints
% 0.29/0.60  thf(fact_8914_of__int__divide__in__Ints,axiom,
% 0.29/0.60      ! [B: int,A: int] :
% 0.29/0.60        ( ( dvd_dvd_int @ B @ A )
% 0.29/0.60       => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B ) ) @ ring_1_Ints_complex ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_divide_in_Ints
% 0.29/0.60  thf(fact_8915_of__int__divide__in__Ints,axiom,
% 0.29/0.60      ! [B: int,A: int] :
% 0.29/0.60        ( ( dvd_dvd_int @ B @ A )
% 0.29/0.60       => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) @ ring_1_Ints_real ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_divide_in_Ints
% 0.29/0.60  thf(fact_8916_of__int__divide__in__Ints,axiom,
% 0.29/0.60      ! [B: int,A: int] :
% 0.29/0.60        ( ( dvd_dvd_int @ B @ A )
% 0.29/0.60       => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) @ ring_1_Ints_rat ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_divide_in_Ints
% 0.29/0.60  thf(fact_8917_of__int__divide__in__Ints,axiom,
% 0.29/0.60      ! [B: int,A: int] :
% 0.29/0.60        ( ( dvd_dvd_int @ B @ A )
% 0.29/0.60       => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B ) ) @ ring_1_Ints_int ) ) ).
% 0.29/0.60  
% 0.29/0.60  % of_int_divide_in_Ints
% 0.29/0.60  thf(fact_8918_arcosh__cosh__real,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.60       => ( ( arcosh_real @ ( cosh_real @ X ) )
% 0.29/0.60          = X ) ) ).
% 0.29/0.60  
% 0.29/0.60  % arcosh_cosh_real
% 0.29/0.60  thf(fact_8919_cosh__double,axiom,
% 0.29/0.60      ! [X: complex] :
% 0.29/0.60        ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60        = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_double
% 0.29/0.60  thf(fact_8920_cosh__double,axiom,
% 0.29/0.60      ! [X: real] :
% 0.29/0.60        ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.29/0.60        = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % cosh_double
% 0.29/0.60  thf(fact_8921_not__int__div__2,axiom,
% 0.29/0.60      ! [K: int] :
% 0.29/0.60        ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.60        = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_int_div_2
% 0.29/0.60  thf(fact_8922_even__not__iff__int,axiom,
% 0.29/0.60      ! [K: int] :
% 0.29/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.29/0.60        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % even_not_iff_int
% 0.29/0.60  thf(fact_8923_and__not__numerals_I2_J,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.29/0.60        = one_one_int ) ).
% 0.29/0.60  
% 0.29/0.60  % and_not_numerals(2)
% 0.29/0.60  thf(fact_8924_and__not__numerals_I4_J,axiom,
% 0.29/0.60      ! [M: num] :
% 0.29/0.60        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.29/0.60        = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % and_not_numerals(4)
% 0.29/0.60  thf(fact_8925_not__numeral__Bit0__eq,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
% 0.29/0.60        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_numeral_Bit0_eq
% 0.29/0.60  thf(fact_8926_not__numeral__Bit0__eq,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
% 0.29/0.60        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_numeral_Bit0_eq
% 0.29/0.60  thf(fact_8927_bit__minus__int__iff,axiom,
% 0.29/0.60      ! [K: int,N: nat] :
% 0.29/0.60        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 0.29/0.60        = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 0.29/0.60  
% 0.29/0.60  % bit_minus_int_iff
% 0.29/0.60  thf(fact_8928_not__numeral__BitM__eq,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bitM @ N ) ) )
% 0.29/0.60        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_numeral_BitM_eq
% 0.29/0.60  thf(fact_8929_not__numeral__BitM__eq,axiom,
% 0.29/0.60      ! [N: num] :
% 0.29/0.60        ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bitM @ N ) ) )
% 0.29/0.60        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % not_numeral_BitM_eq
% 0.29/0.60  thf(fact_8930_Ints__odd__less__0,axiom,
% 0.29/0.60      ! [A: real] :
% 0.29/0.60        ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.60       => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
% 0.29/0.60          = ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_odd_less_0
% 0.29/0.60  thf(fact_8931_Ints__odd__less__0,axiom,
% 0.29/0.60      ! [A: rat] :
% 0.29/0.60        ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.60       => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
% 0.29/0.60          = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.29/0.60  
% 0.29/0.60  % Ints_odd_less_0
% 0.29/0.60  thf(fact_8932_Ints__odd__less__0,axiom,
% 0.29/0.60      ! [A: int] :
% 0.29/0.60        ( ( member_int @ A @ ring_1_Ints_int )
% 0.29/0.60       => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
% 0.29/0.61          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_odd_less_0
% 0.29/0.61  thf(fact_8933_take__bit__not__mask__eq__0,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.61       => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) )
% 0.29/0.61          = zero_zero_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % take_bit_not_mask_eq_0
% 0.29/0.61  thf(fact_8934_Ints__nonzero__abs__ge1,axiom,
% 0.29/0.61      ! [X: code_integer] :
% 0.29/0.61        ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 0.29/0.61       => ( ( X != zero_z3403309356797280102nteger )
% 0.29/0.61         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_nonzero_abs_ge1
% 0.29/0.61  thf(fact_8935_Ints__nonzero__abs__ge1,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( member_real @ X @ ring_1_Ints_real )
% 0.29/0.61       => ( ( X != zero_zero_real )
% 0.29/0.61         => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_nonzero_abs_ge1
% 0.29/0.61  thf(fact_8936_Ints__nonzero__abs__ge1,axiom,
% 0.29/0.61      ! [X: rat] :
% 0.29/0.61        ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.29/0.61       => ( ( X != zero_zero_rat )
% 0.29/0.61         => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_nonzero_abs_ge1
% 0.29/0.61  thf(fact_8937_Ints__nonzero__abs__ge1,axiom,
% 0.29/0.61      ! [X: int] :
% 0.29/0.61        ( ( member_int @ X @ ring_1_Ints_int )
% 0.29/0.61       => ( ( X != zero_zero_int )
% 0.29/0.61         => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_nonzero_abs_ge1
% 0.29/0.61  thf(fact_8938_Ints__nonzero__abs__less1,axiom,
% 0.29/0.61      ! [X: code_integer] :
% 0.29/0.61        ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 0.29/0.61       => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
% 0.29/0.61         => ( X = zero_z3403309356797280102nteger ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_nonzero_abs_less1
% 0.29/0.61  thf(fact_8939_Ints__nonzero__abs__less1,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( member_real @ X @ ring_1_Ints_real )
% 0.29/0.61       => ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.61         => ( X = zero_zero_real ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_nonzero_abs_less1
% 0.29/0.61  thf(fact_8940_Ints__nonzero__abs__less1,axiom,
% 0.29/0.61      ! [X: rat] :
% 0.29/0.61        ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.29/0.61       => ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
% 0.29/0.61         => ( X = zero_zero_rat ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_nonzero_abs_less1
% 0.29/0.61  thf(fact_8941_Ints__nonzero__abs__less1,axiom,
% 0.29/0.61      ! [X: int] :
% 0.29/0.61        ( ( member_int @ X @ ring_1_Ints_int )
% 0.29/0.61       => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
% 0.29/0.61         => ( X = zero_zero_int ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_nonzero_abs_less1
% 0.29/0.61  thf(fact_8942_Ints__eq__abs__less1,axiom,
% 0.29/0.61      ! [X: code_integer,Y: code_integer] :
% 0.29/0.61        ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 0.29/0.61       => ( ( member_Code_integer @ Y @ ring_11222124179247155820nteger )
% 0.29/0.61         => ( ( X = Y )
% 0.29/0.61            = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y ) ) @ one_one_Code_integer ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_eq_abs_less1
% 0.29/0.61  thf(fact_8943_Ints__eq__abs__less1,axiom,
% 0.29/0.61      ! [X: real,Y: real] :
% 0.29/0.61        ( ( member_real @ X @ ring_1_Ints_real )
% 0.29/0.61       => ( ( member_real @ Y @ ring_1_Ints_real )
% 0.29/0.61         => ( ( X = Y )
% 0.29/0.61            = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_eq_abs_less1
% 0.29/0.61  thf(fact_8944_Ints__eq__abs__less1,axiom,
% 0.29/0.61      ! [X: rat,Y: rat] :
% 0.29/0.61        ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.29/0.61       => ( ( member_rat @ Y @ ring_1_Ints_rat )
% 0.29/0.61         => ( ( X = Y )
% 0.29/0.61            = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y ) ) @ one_one_rat ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_eq_abs_less1
% 0.29/0.61  thf(fact_8945_Ints__eq__abs__less1,axiom,
% 0.29/0.61      ! [X: int,Y: int] :
% 0.29/0.61        ( ( member_int @ X @ ring_1_Ints_int )
% 0.29/0.61       => ( ( member_int @ Y @ ring_1_Ints_int )
% 0.29/0.61         => ( ( X = Y )
% 0.29/0.61            = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Ints_eq_abs_less1
% 0.29/0.61  thf(fact_8946_sin__times__pi__eq__0,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61        = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sin_times_pi_eq_0
% 0.29/0.61  thf(fact_8947_and__not__numerals_I5_J,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.29/0.61        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % and_not_numerals(5)
% 0.29/0.61  thf(fact_8948_and__not__numerals_I7_J,axiom,
% 0.29/0.61      ! [M: num] :
% 0.29/0.61        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.29/0.61        = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % and_not_numerals(7)
% 0.29/0.61  thf(fact_8949_and__not__numerals_I3_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.29/0.61        = zero_zero_int ) ).
% 0.29/0.61  
% 0.29/0.61  % and_not_numerals(3)
% 0.29/0.61  thf(fact_8950_frac__neg,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( ( member_real @ X @ ring_1_Ints_real )
% 0.29/0.61         => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.61            = zero_zero_real ) )
% 0.29/0.61        & ( ~ ( member_real @ X @ ring_1_Ints_real )
% 0.29/0.61         => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 0.29/0.61            = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % frac_neg
% 0.29/0.61  thf(fact_8951_frac__neg,axiom,
% 0.29/0.61      ! [X: rat] :
% 0.29/0.61        ( ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.29/0.61         => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 0.29/0.61            = zero_zero_rat ) )
% 0.29/0.61        & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
% 0.29/0.61         => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 0.29/0.61            = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % frac_neg
% 0.29/0.61  thf(fact_8952_and__not__numerals_I9_J,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.29/0.61        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % and_not_numerals(9)
% 0.29/0.61  thf(fact_8953_and__not__numerals_I6_J,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.29/0.61        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % and_not_numerals(6)
% 0.29/0.61  thf(fact_8954_bit__not__iff__eq,axiom,
% 0.29/0.61      ! [A: int,N: nat] :
% 0.29/0.61        ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N )
% 0.29/0.61        = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.29/0.61           != zero_zero_int )
% 0.29/0.61          & ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bit_not_iff_eq
% 0.29/0.61  thf(fact_8955_minus__exp__eq__not__mask,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.61        = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % minus_exp_eq_not_mask
% 0.29/0.61  thf(fact_8956_minus__exp__eq__not__mask,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.61        = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % minus_exp_eq_not_mask
% 0.29/0.61  thf(fact_8957_le__mult__floor__Ints,axiom,
% 0.29/0.61      ! [A: real,B: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.61       => ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.61         => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % le_mult_floor_Ints
% 0.29/0.61  thf(fact_8958_le__mult__floor__Ints,axiom,
% 0.29/0.61      ! [A: real,B: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.61       => ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.61         => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % le_mult_floor_Ints
% 0.29/0.61  thf(fact_8959_le__mult__floor__Ints,axiom,
% 0.29/0.61      ! [A: real,B: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.61       => ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.61         => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % le_mult_floor_Ints
% 0.29/0.61  thf(fact_8960_le__mult__floor__Ints,axiom,
% 0.29/0.61      ! [A: real,B: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.61       => ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.61         => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % le_mult_floor_Ints
% 0.29/0.61  thf(fact_8961_le__mult__floor__Ints,axiom,
% 0.29/0.61      ! [A: rat,B: rat] :
% 0.29/0.61        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.61       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.61         => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % le_mult_floor_Ints
% 0.29/0.61  thf(fact_8962_le__mult__floor__Ints,axiom,
% 0.29/0.61      ! [A: rat,B: rat] :
% 0.29/0.61        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.61       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.61         => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % le_mult_floor_Ints
% 0.29/0.61  thf(fact_8963_le__mult__floor__Ints,axiom,
% 0.29/0.61      ! [A: rat,B: rat] :
% 0.29/0.61        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.61       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.61         => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % le_mult_floor_Ints
% 0.29/0.61  thf(fact_8964_le__mult__floor__Ints,axiom,
% 0.29/0.61      ! [A: rat,B: rat] :
% 0.29/0.61        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.61       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.61         => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % le_mult_floor_Ints
% 0.29/0.61  thf(fact_8965_frac__unique__iff,axiom,
% 0.29/0.61      ! [X: real,A: real] :
% 0.29/0.61        ( ( ( archim2898591450579166408c_real @ X )
% 0.29/0.61          = A )
% 0.29/0.61        = ( ( member_real @ ( minus_minus_real @ X @ A ) @ ring_1_Ints_real )
% 0.29/0.61          & ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.61          & ( ord_less_real @ A @ one_one_real ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % frac_unique_iff
% 0.29/0.61  thf(fact_8966_frac__unique__iff,axiom,
% 0.29/0.61      ! [X: rat,A: rat] :
% 0.29/0.61        ( ( ( archimedean_frac_rat @ X )
% 0.29/0.61          = A )
% 0.29/0.61        = ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
% 0.29/0.61          & ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.61          & ( ord_less_rat @ A @ one_one_rat ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % frac_unique_iff
% 0.29/0.61  thf(fact_8967_mult__ceiling__le__Ints,axiom,
% 0.29/0.61      ! [A: real,B: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.61       => ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.61         => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % mult_ceiling_le_Ints
% 0.29/0.61  thf(fact_8968_mult__ceiling__le__Ints,axiom,
% 0.29/0.61      ! [A: real,B: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.61       => ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.61         => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % mult_ceiling_le_Ints
% 0.29/0.61  thf(fact_8969_mult__ceiling__le__Ints,axiom,
% 0.29/0.61      ! [A: real,B: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.61       => ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.61         => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % mult_ceiling_le_Ints
% 0.29/0.61  thf(fact_8970_mult__ceiling__le__Ints,axiom,
% 0.29/0.61      ! [A: real,B: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.61       => ( ( member_real @ A @ ring_1_Ints_real )
% 0.29/0.61         => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % mult_ceiling_le_Ints
% 0.29/0.61  thf(fact_8971_mult__ceiling__le__Ints,axiom,
% 0.29/0.61      ! [A: rat,B: rat] :
% 0.29/0.61        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.61       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.61         => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % mult_ceiling_le_Ints
% 0.29/0.61  thf(fact_8972_mult__ceiling__le__Ints,axiom,
% 0.29/0.61      ! [A: rat,B: rat] :
% 0.29/0.61        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.61       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.61         => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % mult_ceiling_le_Ints
% 0.29/0.61  thf(fact_8973_mult__ceiling__le__Ints,axiom,
% 0.29/0.61      ! [A: rat,B: rat] :
% 0.29/0.61        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.61       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.61         => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % mult_ceiling_le_Ints
% 0.29/0.61  thf(fact_8974_mult__ceiling__le__Ints,axiom,
% 0.29/0.61      ! [A: rat,B: rat] :
% 0.29/0.61        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.61       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.29/0.61         => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % mult_ceiling_le_Ints
% 0.29/0.61  thf(fact_8975_tanh__add,axiom,
% 0.29/0.61      ! [X: complex,Y: complex] :
% 0.29/0.61        ( ( ( cosh_complex @ X )
% 0.29/0.61         != zero_zero_complex )
% 0.29/0.61       => ( ( ( cosh_complex @ Y )
% 0.29/0.61           != zero_zero_complex )
% 0.29/0.61         => ( ( tanh_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.29/0.61            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % tanh_add
% 0.29/0.61  thf(fact_8976_tanh__add,axiom,
% 0.29/0.61      ! [X: real,Y: real] :
% 0.29/0.61        ( ( ( cosh_real @ X )
% 0.29/0.61         != zero_zero_real )
% 0.29/0.61       => ( ( ( cosh_real @ Y )
% 0.29/0.61           != zero_zero_real )
% 0.29/0.61         => ( ( tanh_real @ ( plus_plus_real @ X @ Y ) )
% 0.29/0.61            = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % tanh_add
% 0.29/0.61  thf(fact_8977_sinh__zero__iff,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( ( sinh_real @ X )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61        = ( member_real @ ( exp_real @ X ) @ ( insert_real @ one_one_real @ ( insert_real @ ( uminus_uminus_real @ one_one_real ) @ bot_bot_set_real ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sinh_zero_iff
% 0.29/0.61  thf(fact_8978_sinh__zero__iff,axiom,
% 0.29/0.61      ! [X: complex] :
% 0.29/0.61        ( ( ( sinh_complex @ X )
% 0.29/0.61          = zero_zero_complex )
% 0.29/0.61        = ( member_complex @ ( exp_complex @ X ) @ ( insert_complex @ one_one_complex @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ bot_bot_set_complex ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sinh_zero_iff
% 0.29/0.61  thf(fact_8979_sin__integer__2pi,axiom,
% 0.29/0.61      ! [N: real] :
% 0.29/0.61        ( ( member_real @ N @ ring_1_Ints_real )
% 0.29/0.61       => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.29/0.61          = zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sin_integer_2pi
% 0.29/0.61  thf(fact_8980_cos__integer__2pi,axiom,
% 0.29/0.61      ! [N: real] :
% 0.29/0.61        ( ( member_real @ N @ ring_1_Ints_real )
% 0.29/0.61       => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.29/0.61          = one_one_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cos_integer_2pi
% 0.29/0.61  thf(fact_8981_cosh__field__def,axiom,
% 0.29/0.61      ( cosh_real
% 0.29/0.61      = ( ^ [Z4: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z4 ) @ ( exp_real @ ( uminus_uminus_real @ Z4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cosh_field_def
% 0.29/0.61  thf(fact_8982_cosh__field__def,axiom,
% 0.29/0.61      ( cosh_complex
% 0.29/0.61      = ( ^ [Z4: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z4 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cosh_field_def
% 0.29/0.61  thf(fact_8983_and__not__numerals_I8_J,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % and_not_numerals(8)
% 0.29/0.61  thf(fact_8984_not__int__rec,axiom,
% 0.29/0.61      ( bit_ri7919022796975470100ot_int
% 0.29/0.61      = ( ^ [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 ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % not_int_rec
% 0.29/0.61  thf(fact_8985_sinh__field__def,axiom,
% 0.29/0.61      ( sinh_real
% 0.29/0.61      = ( ^ [Z4: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z4 ) @ ( exp_real @ ( uminus_uminus_real @ Z4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sinh_field_def
% 0.29/0.61  thf(fact_8986_sinh__field__def,axiom,
% 0.29/0.61      ( sinh_complex
% 0.29/0.61      = ( ^ [Z4: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z4 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sinh_field_def
% 0.29/0.61  thf(fact_8987_cosh__zero__iff,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( ( cosh_real @ X )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61        = ( ( power_power_real @ ( exp_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.61          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cosh_zero_iff
% 0.29/0.61  thf(fact_8988_cosh__zero__iff,axiom,
% 0.29/0.61      ! [X: complex] :
% 0.29/0.61        ( ( ( cosh_complex @ X )
% 0.29/0.61          = zero_zero_complex )
% 0.29/0.61        = ( ( power_power_complex @ ( exp_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.61          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cosh_zero_iff
% 0.29/0.61  thf(fact_8989_gbinomial__absorption_H,axiom,
% 0.29/0.61      ! [K: nat,A: complex] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.61       => ( ( gbinomial_complex @ A @ K )
% 0.29/0.61          = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_absorption'
% 0.29/0.61  thf(fact_8990_gbinomial__absorption_H,axiom,
% 0.29/0.61      ! [K: nat,A: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.61       => ( ( gbinomial_real @ A @ K )
% 0.29/0.61          = ( 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 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_absorption'
% 0.29/0.61  thf(fact_8991_gbinomial__absorption_H,axiom,
% 0.29/0.61      ! [K: nat,A: rat] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.61       => ( ( gbinomial_rat @ A @ K )
% 0.29/0.61          = ( 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 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_absorption'
% 0.29/0.61  thf(fact_8992_horner__sum__of__bool__2__less,axiom,
% 0.29/0.61      ! [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 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % horner_sum_of_bool_2_less
% 0.29/0.61  thf(fact_8993_the__elem__eq,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT] :
% 0.29/0.61        ( ( the_elem_VEBT_VEBT @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
% 0.29/0.61        = X ) ).
% 0.29/0.61  
% 0.29/0.61  % the_elem_eq
% 0.29/0.61  thf(fact_8994_the__elem__eq,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( the_elem_real @ ( insert_real @ X @ bot_bot_set_real ) )
% 0.29/0.61        = X ) ).
% 0.29/0.61  
% 0.29/0.61  % the_elem_eq
% 0.29/0.61  thf(fact_8995_the__elem__eq,axiom,
% 0.29/0.61      ! [X: $o] :
% 0.29/0.61        ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.29/0.61        = X ) ).
% 0.29/0.61  
% 0.29/0.61  % the_elem_eq
% 0.29/0.61  thf(fact_8996_the__elem__eq,axiom,
% 0.29/0.61      ! [X: nat] :
% 0.29/0.61        ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.29/0.61        = X ) ).
% 0.29/0.61  
% 0.29/0.61  % the_elem_eq
% 0.29/0.61  thf(fact_8997_the__elem__eq,axiom,
% 0.29/0.61      ! [X: int] :
% 0.29/0.61        ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.29/0.61        = X ) ).
% 0.29/0.61  
% 0.29/0.61  % the_elem_eq
% 0.29/0.61  thf(fact_8998_push__bit__numeral__minus__1,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.61        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_numeral_minus_1
% 0.29/0.61  thf(fact_8999_push__bit__numeral__minus__1,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_numeral_minus_1
% 0.29/0.61  thf(fact_9000_push__bit__nonnegative__int__iff,axiom,
% 0.29/0.61      ! [N: nat,K: int] :
% 0.29/0.61        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.29/0.61        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_nonnegative_int_iff
% 0.29/0.61  thf(fact_9001_push__bit__negative__int__iff,axiom,
% 0.29/0.61      ! [N: nat,K: int] :
% 0.29/0.61        ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 0.29/0.61        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_negative_int_iff
% 0.29/0.61  thf(fact_9002_push__bit__eq__0__iff,axiom,
% 0.29/0.61      ! [N: nat,A: int] :
% 0.29/0.61        ( ( ( bit_se545348938243370406it_int @ N @ A )
% 0.29/0.61          = zero_zero_int )
% 0.29/0.61        = ( A = zero_zero_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_eq_0_iff
% 0.29/0.61  thf(fact_9003_push__bit__eq__0__iff,axiom,
% 0.29/0.61      ! [N: nat,A: nat] :
% 0.29/0.61        ( ( ( bit_se547839408752420682it_nat @ N @ A )
% 0.29/0.61          = zero_zero_nat )
% 0.29/0.61        = ( A = zero_zero_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_eq_0_iff
% 0.29/0.61  thf(fact_9004_push__bit__of__0,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
% 0.29/0.61        = zero_zero_int ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_of_0
% 0.29/0.61  thf(fact_9005_push__bit__of__0,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
% 0.29/0.61        = zero_zero_nat ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_of_0
% 0.29/0.61  thf(fact_9006_push__bit__push__bit,axiom,
% 0.29/0.61      ! [M: nat,N: nat,A: int] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.29/0.61        = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_push_bit
% 0.29/0.61  thf(fact_9007_push__bit__push__bit,axiom,
% 0.29/0.61      ! [M: nat,N: nat,A: nat] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.29/0.61        = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_push_bit
% 0.29/0.61  thf(fact_9008_concat__bit__of__zero__1,axiom,
% 0.29/0.61      ! [N: nat,L: int] :
% 0.29/0.61        ( ( bit_concat_bit @ N @ zero_zero_int @ L )
% 0.29/0.61        = ( bit_se545348938243370406it_int @ N @ L ) ) ).
% 0.29/0.61  
% 0.29/0.61  % concat_bit_of_zero_1
% 0.29/0.61  thf(fact_9009_gbinomial__0_I2_J,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
% 0.29/0.61        = zero_zero_complex ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_0(2)
% 0.29/0.61  thf(fact_9010_gbinomial__0_I2_J,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 0.29/0.61        = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_0(2)
% 0.29/0.61  thf(fact_9011_gbinomial__0_I2_J,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 0.29/0.61        = zero_zero_rat ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_0(2)
% 0.29/0.61  thf(fact_9012_gbinomial__0_I2_J,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 0.29/0.61        = zero_zero_nat ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_0(2)
% 0.29/0.61  thf(fact_9013_gbinomial__0_I2_J,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 0.29/0.61        = zero_zero_int ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_0(2)
% 0.29/0.61  thf(fact_9014_gbinomial__0_I1_J,axiom,
% 0.29/0.61      ! [A: complex] :
% 0.29/0.61        ( ( gbinomial_complex @ A @ zero_zero_nat )
% 0.29/0.61        = one_one_complex ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_0(1)
% 0.29/0.61  thf(fact_9015_gbinomial__0_I1_J,axiom,
% 0.29/0.61      ! [A: real] :
% 0.29/0.61        ( ( gbinomial_real @ A @ zero_zero_nat )
% 0.29/0.61        = one_one_real ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_0(1)
% 0.29/0.61  thf(fact_9016_gbinomial__0_I1_J,axiom,
% 0.29/0.61      ! [A: rat] :
% 0.29/0.61        ( ( gbinomial_rat @ A @ zero_zero_nat )
% 0.29/0.61        = one_one_rat ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_0(1)
% 0.29/0.61  thf(fact_9017_gbinomial__0_I1_J,axiom,
% 0.29/0.61      ! [A: nat] :
% 0.29/0.61        ( ( gbinomial_nat @ A @ zero_zero_nat )
% 0.29/0.61        = one_one_nat ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_0(1)
% 0.29/0.61  thf(fact_9018_gbinomial__0_I1_J,axiom,
% 0.29/0.61      ! [A: int] :
% 0.29/0.61        ( ( gbinomial_int @ A @ zero_zero_nat )
% 0.29/0.61        = one_one_int ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_0(1)
% 0.29/0.61  thf(fact_9019_push__bit__Suc__numeral,axiom,
% 0.29/0.61      ! [N: nat,K: num] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
% 0.29/0.61        = ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_Suc_numeral
% 0.29/0.61  thf(fact_9020_push__bit__Suc__numeral,axiom,
% 0.29/0.61      ! [N: nat,K: num] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.29/0.61        = ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_Suc_numeral
% 0.29/0.61  thf(fact_9021_push__bit__Suc__minus__numeral,axiom,
% 0.29/0.61      ! [N: nat,K: num] :
% 0.29/0.61        ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.29/0.61        = ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_Suc_minus_numeral
% 0.29/0.61  thf(fact_9022_push__bit__Suc__minus__numeral,axiom,
% 0.29/0.61      ! [N: nat,K: num] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.61        = ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_Suc_minus_numeral
% 0.29/0.61  thf(fact_9023_push__bit__numeral,axiom,
% 0.29/0.61      ! [L: num,K: num] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ K ) )
% 0.29/0.61        = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_numeral
% 0.29/0.61  thf(fact_9024_push__bit__numeral,axiom,
% 0.29/0.61      ! [L: num,K: num] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ K ) )
% 0.29/0.61        = ( bit_se547839408752420682it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_numeral
% 0.29/0.61  thf(fact_9025_push__bit__minus__one__eq__not__mask,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.29/0.61        = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_minus_one_eq_not_mask
% 0.29/0.61  thf(fact_9026_push__bit__minus__one__eq__not__mask,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.61        = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_minus_one_eq_not_mask
% 0.29/0.61  thf(fact_9027_push__bit__Suc,axiom,
% 0.29/0.61      ! [N: nat,A: int] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
% 0.29/0.61        = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_Suc
% 0.29/0.61  thf(fact_9028_push__bit__Suc,axiom,
% 0.29/0.61      ! [N: nat,A: nat] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
% 0.29/0.61        = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_Suc
% 0.29/0.61  thf(fact_9029_push__bit__of__1,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ N @ one_one_int )
% 0.29/0.61        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_of_1
% 0.29/0.61  thf(fact_9030_push__bit__of__1,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
% 0.29/0.61        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_of_1
% 0.29/0.61  thf(fact_9031_push__bit__of__Suc__0,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.61        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_of_Suc_0
% 0.29/0.61  thf(fact_9032_even__push__bit__iff,axiom,
% 0.29/0.61      ! [N: nat,A: code_integer] :
% 0.29/0.61        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 0.29/0.61        = ( ( N != zero_zero_nat )
% 0.29/0.61          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % even_push_bit_iff
% 0.29/0.61  thf(fact_9033_even__push__bit__iff,axiom,
% 0.29/0.61      ! [N: nat,A: int] :
% 0.29/0.61        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.29/0.61        = ( ( N != zero_zero_nat )
% 0.29/0.61          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % even_push_bit_iff
% 0.29/0.61  thf(fact_9034_even__push__bit__iff,axiom,
% 0.29/0.61      ! [N: nat,A: nat] :
% 0.29/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.29/0.61        = ( ( N != zero_zero_nat )
% 0.29/0.61          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % even_push_bit_iff
% 0.29/0.61  thf(fact_9035_push__bit__minus__numeral,axiom,
% 0.29/0.61      ! [L: num,K: num] :
% 0.29/0.61        ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.29/0.61        = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_minus_numeral
% 0.29/0.61  thf(fact_9036_push__bit__minus__numeral,axiom,
% 0.29/0.61      ! [L: num,K: num] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.61        = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_minus_numeral
% 0.29/0.61  thf(fact_9037_of__nat__gbinomial,axiom,
% 0.29/0.61      ! [N: nat,K: nat] :
% 0.29/0.61        ( ( semiri8010041392384452111omplex @ ( gbinomial_nat @ N @ K ) )
% 0.29/0.61        = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 0.29/0.61  
% 0.29/0.61  % of_nat_gbinomial
% 0.29/0.61  thf(fact_9038_of__nat__gbinomial,axiom,
% 0.29/0.61      ! [N: nat,K: nat] :
% 0.29/0.61        ( ( semiri5074537144036343181t_real @ ( gbinomial_nat @ N @ K ) )
% 0.29/0.61        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 0.29/0.61  
% 0.29/0.61  % of_nat_gbinomial
% 0.29/0.61  thf(fact_9039_of__nat__gbinomial,axiom,
% 0.29/0.61      ! [N: nat,K: nat] :
% 0.29/0.61        ( ( semiri681578069525770553at_rat @ ( gbinomial_nat @ N @ K ) )
% 0.29/0.61        = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 0.29/0.61  
% 0.29/0.61  % of_nat_gbinomial
% 0.29/0.61  thf(fact_9040_push__bit__of__nat,axiom,
% 0.29/0.61      ! [N: nat,M: nat] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.29/0.61        = ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_of_nat
% 0.29/0.61  thf(fact_9041_push__bit__of__nat,axiom,
% 0.29/0.61      ! [N: nat,M: nat] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.29/0.61        = ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_of_nat
% 0.29/0.61  thf(fact_9042_of__nat__push__bit,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 0.29/0.61        = ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % of_nat_push_bit
% 0.29/0.61  thf(fact_9043_of__nat__push__bit,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 0.29/0.61        = ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % of_nat_push_bit
% 0.29/0.61  thf(fact_9044_push__bit__of__int,axiom,
% 0.29/0.61      ! [N: nat,K: int] :
% 0.29/0.61        ( ( bit_se7788150548672797655nteger @ N @ ( ring_18347121197199848620nteger @ K ) )
% 0.29/0.61        = ( ring_18347121197199848620nteger @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_of_int
% 0.29/0.61  thf(fact_9045_push__bit__of__int,axiom,
% 0.29/0.61      ! [N: nat,K: int] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ N @ ( ring_1_of_int_int @ K ) )
% 0.29/0.61        = ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_of_int
% 0.29/0.61  thf(fact_9046_push__bit__add,axiom,
% 0.29/0.61      ! [N: nat,A: int,B: int] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B ) )
% 0.29/0.61        = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_add
% 0.29/0.61  thf(fact_9047_push__bit__add,axiom,
% 0.29/0.61      ! [N: nat,A: nat,B: nat] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.61        = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_add
% 0.29/0.61  thf(fact_9048_binomial__gbinomial,axiom,
% 0.29/0.61      ! [N: nat,K: nat] :
% 0.29/0.61        ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 0.29/0.61        = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 0.29/0.61  
% 0.29/0.61  % binomial_gbinomial
% 0.29/0.61  thf(fact_9049_binomial__gbinomial,axiom,
% 0.29/0.61      ! [N: nat,K: nat] :
% 0.29/0.61        ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 0.29/0.61        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 0.29/0.61  
% 0.29/0.61  % binomial_gbinomial
% 0.29/0.61  thf(fact_9050_binomial__gbinomial,axiom,
% 0.29/0.61      ! [N: nat,K: nat] :
% 0.29/0.61        ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 0.29/0.61        = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 0.29/0.61  
% 0.29/0.61  % binomial_gbinomial
% 0.29/0.61  thf(fact_9051_push__bit__take__bit,axiom,
% 0.29/0.61      ! [M: nat,N: nat,A: int] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.29/0.61        = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_take_bit
% 0.29/0.61  thf(fact_9052_push__bit__take__bit,axiom,
% 0.29/0.61      ! [M: nat,N: nat,A: nat] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.29/0.61        = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_take_bit
% 0.29/0.61  thf(fact_9053_flip__bit__nat__def,axiom,
% 0.29/0.61      ( bit_se2161824704523386999it_nat
% 0.29/0.61      = ( ^ [M6: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % flip_bit_nat_def
% 0.29/0.61  thf(fact_9054_gbinomial__Suc__Suc,axiom,
% 0.29/0.61      ! [A: real,K: nat] :
% 0.29/0.61        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.29/0.61        = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_Suc_Suc
% 0.29/0.61  thf(fact_9055_gbinomial__Suc__Suc,axiom,
% 0.29/0.61      ! [A: rat,K: nat] :
% 0.29/0.61        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.29/0.61        = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_Suc_Suc
% 0.29/0.61  thf(fact_9056_gbinomial__Suc__Suc,axiom,
% 0.29/0.61      ! [A: complex,K: nat] :
% 0.29/0.61        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.29/0.61        = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_Suc_Suc
% 0.29/0.61  thf(fact_9057_gbinomial__of__nat__symmetric,axiom,
% 0.29/0.61      ! [K: nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.61       => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K )
% 0.29/0.61          = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_of_nat_symmetric
% 0.29/0.61  thf(fact_9058_gbinomial__of__nat__symmetric,axiom,
% 0.29/0.61      ! [K: nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.61       => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 0.29/0.61          = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_of_nat_symmetric
% 0.29/0.61  thf(fact_9059_gbinomial__of__nat__symmetric,axiom,
% 0.29/0.61      ! [K: nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.61       => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
% 0.29/0.61          = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_of_nat_symmetric
% 0.29/0.61  thf(fact_9060_bit__push__bit__iff__int,axiom,
% 0.29/0.61      ! [M: nat,K: int,N: nat] :
% 0.29/0.61        ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 0.29/0.61        = ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.61          & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bit_push_bit_iff_int
% 0.29/0.61  thf(fact_9061_bit__push__bit__iff__nat,axiom,
% 0.29/0.61      ! [M: nat,Q3: nat,N: nat] :
% 0.29/0.61        ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N )
% 0.29/0.61        = ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.61          & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bit_push_bit_iff_nat
% 0.29/0.61  thf(fact_9062_concat__bit__eq,axiom,
% 0.29/0.61      ( bit_concat_bit
% 0.29/0.61      = ( ^ [N3: nat,K2: int,L3: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L3 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % concat_bit_eq
% 0.29/0.61  thf(fact_9063_flip__bit__eq__xor,axiom,
% 0.29/0.61      ( bit_se2159334234014336723it_int
% 0.29/0.61      = ( ^ [N3: nat,A3: int] : ( bit_se6526347334894502574or_int @ A3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % flip_bit_eq_xor
% 0.29/0.61  thf(fact_9064_flip__bit__eq__xor,axiom,
% 0.29/0.61      ( bit_se2161824704523386999it_nat
% 0.29/0.61      = ( ^ [N3: nat,A3: nat] : ( bit_se6528837805403552850or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % flip_bit_eq_xor
% 0.29/0.61  thf(fact_9065_flip__bit__int__def,axiom,
% 0.29/0.61      ( bit_se2159334234014336723it_int
% 0.29/0.61      = ( ^ [N3: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % flip_bit_int_def
% 0.29/0.61  thf(fact_9066_gbinomial__addition__formula,axiom,
% 0.29/0.61      ! [A: complex,K: nat] :
% 0.29/0.61        ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 0.29/0.61        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_addition_formula
% 0.29/0.61  thf(fact_9067_gbinomial__addition__formula,axiom,
% 0.29/0.61      ! [A: real,K: nat] :
% 0.29/0.61        ( ( gbinomial_real @ A @ ( suc @ K ) )
% 0.29/0.61        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_addition_formula
% 0.29/0.61  thf(fact_9068_gbinomial__addition__formula,axiom,
% 0.29/0.61      ! [A: rat,K: nat] :
% 0.29/0.61        ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 0.29/0.61        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_addition_formula
% 0.29/0.61  thf(fact_9069_gbinomial__absorb__comp,axiom,
% 0.29/0.61      ! [A: complex,K: nat] :
% 0.29/0.61        ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
% 0.29/0.61        = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_absorb_comp
% 0.29/0.61  thf(fact_9070_gbinomial__absorb__comp,axiom,
% 0.29/0.61      ! [A: real,K: nat] :
% 0.29/0.61        ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
% 0.29/0.61        = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_absorb_comp
% 0.29/0.61  thf(fact_9071_gbinomial__absorb__comp,axiom,
% 0.29/0.61      ! [A: rat,K: nat] :
% 0.29/0.61        ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
% 0.29/0.61        = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_absorb_comp
% 0.29/0.61  thf(fact_9072_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.29/0.61      ! [K: nat,A: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 0.29/0.61       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_ge_n_over_k_pow_k
% 0.29/0.61  thf(fact_9073_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.29/0.61      ! [K: nat,A: rat] :
% 0.29/0.61        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 0.29/0.61       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_ge_n_over_k_pow_k
% 0.29/0.61  thf(fact_9074_gbinomial__mult__1,axiom,
% 0.29/0.61      ! [A: complex,K: nat] :
% 0.29/0.61        ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
% 0.29/0.61        = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_mult_1
% 0.29/0.61  thf(fact_9075_gbinomial__mult__1,axiom,
% 0.29/0.61      ! [A: real,K: nat] :
% 0.29/0.61        ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_mult_1
% 0.29/0.61  thf(fact_9076_gbinomial__mult__1,axiom,
% 0.29/0.61      ! [A: rat,K: nat] :
% 0.29/0.61        ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_mult_1
% 0.29/0.61  thf(fact_9077_gbinomial__mult__1_H,axiom,
% 0.29/0.61      ! [A: complex,K: nat] :
% 0.29/0.61        ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
% 0.29/0.61        = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_mult_1'
% 0.29/0.61  thf(fact_9078_gbinomial__mult__1_H,axiom,
% 0.29/0.61      ! [A: real,K: nat] :
% 0.29/0.61        ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 0.29/0.61        = ( 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 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_mult_1'
% 0.29/0.61  thf(fact_9079_gbinomial__mult__1_H,axiom,
% 0.29/0.61      ! [A: rat,K: nat] :
% 0.29/0.61        ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 0.29/0.61        = ( 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 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_mult_1'
% 0.29/0.61  thf(fact_9080_push__bit__double,axiom,
% 0.29/0.61      ! [N: nat,A: int] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.29/0.61        = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_double
% 0.29/0.61  thf(fact_9081_push__bit__double,axiom,
% 0.29/0.61      ! [N: nat,A: nat] :
% 0.29/0.61        ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.61        = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_double
% 0.29/0.61  thf(fact_9082_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.29/0.61      ( bit_se1146084159140164899it_int
% 0.29/0.61      = ( ^ [A3: int,N3: nat] :
% 0.29/0.61            ( ( bit_se725231765392027082nd_int @ A3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) )
% 0.29/0.61           != zero_zero_int ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bit_iff_and_push_bit_not_eq_0
% 0.29/0.61  thf(fact_9083_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.29/0.61      ( bit_se1148574629649215175it_nat
% 0.29/0.61      = ( ^ [A3: nat,N3: nat] :
% 0.29/0.61            ( ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) )
% 0.29/0.61           != zero_zero_nat ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bit_iff_and_push_bit_not_eq_0
% 0.29/0.61  thf(fact_9084_push__bit__mask__eq,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.29/0.61        = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_mask_eq
% 0.29/0.61  thf(fact_9085_unset__bit__eq__and__not,axiom,
% 0.29/0.61      ( bit_se4203085406695923979it_int
% 0.29/0.61      = ( ^ [N3: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % unset_bit_eq_and_not
% 0.29/0.61  thf(fact_9086_unset__bit__int__def,axiom,
% 0.29/0.61      ( bit_se4203085406695923979it_int
% 0.29/0.61      = ( ^ [N3: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % unset_bit_int_def
% 0.29/0.61  thf(fact_9087_Suc__times__gbinomial,axiom,
% 0.29/0.61      ! [K: nat,A: complex] :
% 0.29/0.61        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 0.29/0.61        = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Suc_times_gbinomial
% 0.29/0.61  thf(fact_9088_Suc__times__gbinomial,axiom,
% 0.29/0.61      ! [K: nat,A: real] :
% 0.29/0.61        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 0.29/0.61        = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Suc_times_gbinomial
% 0.29/0.61  thf(fact_9089_Suc__times__gbinomial,axiom,
% 0.29/0.61      ! [K: nat,A: rat] :
% 0.29/0.61        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 0.29/0.61        = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Suc_times_gbinomial
% 0.29/0.61  thf(fact_9090_gbinomial__absorption,axiom,
% 0.29/0.61      ! [K: nat,A: complex] :
% 0.29/0.61        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 0.29/0.61        = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_absorption
% 0.29/0.61  thf(fact_9091_gbinomial__absorption,axiom,
% 0.29/0.61      ! [K: nat,A: real] :
% 0.29/0.61        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 0.29/0.61        = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_absorption
% 0.29/0.61  thf(fact_9092_gbinomial__absorption,axiom,
% 0.29/0.61      ! [K: nat,A: rat] :
% 0.29/0.61        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 0.29/0.61        = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_absorption
% 0.29/0.61  thf(fact_9093_gbinomial__trinomial__revision,axiom,
% 0.29/0.61      ! [K: nat,M: nat,A: complex] :
% 0.29/0.61        ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.61       => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 0.29/0.61          = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_trinomial_revision
% 0.29/0.61  thf(fact_9094_gbinomial__trinomial__revision,axiom,
% 0.29/0.61      ! [K: nat,M: nat,A: real] :
% 0.29/0.61        ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.61       => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 0.29/0.61          = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_trinomial_revision
% 0.29/0.61  thf(fact_9095_gbinomial__trinomial__revision,axiom,
% 0.29/0.61      ! [K: nat,M: nat,A: rat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.61       => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 0.29/0.61          = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_trinomial_revision
% 0.29/0.61  thf(fact_9096_push__bit__int__def,axiom,
% 0.29/0.61      ( bit_se545348938243370406it_int
% 0.29/0.61      = ( ^ [N3: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_int_def
% 0.29/0.61  thf(fact_9097_push__bit__nat__def,axiom,
% 0.29/0.61      ( bit_se547839408752420682it_nat
% 0.29/0.61      = ( ^ [N3: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_nat_def
% 0.29/0.61  thf(fact_9098_push__bit__eq__mult,axiom,
% 0.29/0.61      ( bit_se545348938243370406it_int
% 0.29/0.61      = ( ^ [N3: nat,A3: int] : ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_eq_mult
% 0.29/0.61  thf(fact_9099_push__bit__eq__mult,axiom,
% 0.29/0.61      ( bit_se547839408752420682it_nat
% 0.29/0.61      = ( ^ [N3: nat,A3: nat] : ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_eq_mult
% 0.29/0.61  thf(fact_9100_exp__dvdE,axiom,
% 0.29/0.61      ! [N: nat,A: code_integer] :
% 0.29/0.61        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A )
% 0.29/0.61       => ~ ! [B6: code_integer] :
% 0.29/0.61              ( A
% 0.29/0.61             != ( bit_se7788150548672797655nteger @ N @ B6 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % exp_dvdE
% 0.29/0.61  thf(fact_9101_exp__dvdE,axiom,
% 0.29/0.61      ! [N: nat,A: int] :
% 0.29/0.61        ( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A )
% 0.29/0.61       => ~ ! [B6: int] :
% 0.29/0.61              ( A
% 0.29/0.61             != ( bit_se545348938243370406it_int @ N @ B6 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % exp_dvdE
% 0.29/0.61  thf(fact_9102_exp__dvdE,axiom,
% 0.29/0.61      ! [N: nat,A: nat] :
% 0.29/0.61        ( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A )
% 0.29/0.61       => ~ ! [B6: nat] :
% 0.29/0.61              ( A
% 0.29/0.61             != ( bit_se547839408752420682it_nat @ N @ B6 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % exp_dvdE
% 0.29/0.61  thf(fact_9103_the__elem__image__unique,axiom,
% 0.29/0.61      ! [A2: set_nat,F: nat > set_nat,X: nat] :
% 0.29/0.61        ( ( A2 != bot_bot_set_nat )
% 0.29/0.61       => ( ! [Y4: nat] :
% 0.29/0.61              ( ( member_nat @ Y4 @ A2 )
% 0.29/0.61             => ( ( F @ Y4 )
% 0.29/0.61                = ( F @ X ) ) )
% 0.29/0.61         => ( ( the_elem_set_nat @ ( image_nat_set_nat @ F @ A2 ) )
% 0.29/0.61            = ( F @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % the_elem_image_unique
% 0.29/0.61  thf(fact_9104_the__elem__image__unique,axiom,
% 0.29/0.61      ! [A2: set_nat,F: nat > nat,X: nat] :
% 0.29/0.61        ( ( A2 != bot_bot_set_nat )
% 0.29/0.61       => ( ! [Y4: nat] :
% 0.29/0.61              ( ( member_nat @ Y4 @ A2 )
% 0.29/0.61             => ( ( F @ Y4 )
% 0.29/0.61                = ( F @ X ) ) )
% 0.29/0.61         => ( ( the_elem_nat @ ( image_nat_nat @ F @ A2 ) )
% 0.29/0.61            = ( F @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % the_elem_image_unique
% 0.29/0.61  thf(fact_9105_the__elem__image__unique,axiom,
% 0.29/0.61      ! [A2: set_nat,F: nat > int,X: nat] :
% 0.29/0.61        ( ( A2 != bot_bot_set_nat )
% 0.29/0.61       => ( ! [Y4: nat] :
% 0.29/0.61              ( ( member_nat @ Y4 @ A2 )
% 0.29/0.61             => ( ( F @ Y4 )
% 0.29/0.61                = ( F @ X ) ) )
% 0.29/0.61         => ( ( the_elem_int @ ( image_nat_int @ F @ A2 ) )
% 0.29/0.61            = ( F @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % the_elem_image_unique
% 0.29/0.61  thf(fact_9106_the__elem__image__unique,axiom,
% 0.29/0.61      ! [A2: set_int,F: int > nat,X: int] :
% 0.29/0.61        ( ( A2 != bot_bot_set_int )
% 0.29/0.61       => ( ! [Y4: int] :
% 0.29/0.61              ( ( member_int @ Y4 @ A2 )
% 0.29/0.61             => ( ( F @ Y4 )
% 0.29/0.61                = ( F @ X ) ) )
% 0.29/0.61         => ( ( the_elem_nat @ ( image_int_nat @ F @ A2 ) )
% 0.29/0.61            = ( F @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % the_elem_image_unique
% 0.29/0.61  thf(fact_9107_the__elem__image__unique,axiom,
% 0.29/0.61      ! [A2: set_int,F: int > int,X: int] :
% 0.29/0.61        ( ( A2 != bot_bot_set_int )
% 0.29/0.61       => ( ! [Y4: int] :
% 0.29/0.61              ( ( member_int @ Y4 @ A2 )
% 0.29/0.61             => ( ( F @ Y4 )
% 0.29/0.61                = ( F @ X ) ) )
% 0.29/0.61         => ( ( the_elem_int @ ( image_int_int @ F @ A2 ) )
% 0.29/0.61            = ( F @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % the_elem_image_unique
% 0.29/0.61  thf(fact_9108_gbinomial__rec,axiom,
% 0.29/0.61      ! [A: complex,K: nat] :
% 0.29/0.61        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.29/0.61        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_rec
% 0.29/0.61  thf(fact_9109_gbinomial__rec,axiom,
% 0.29/0.61      ! [A: real,K: nat] :
% 0.29/0.61        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.29/0.61        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_rec
% 0.29/0.61  thf(fact_9110_gbinomial__rec,axiom,
% 0.29/0.61      ! [A: rat,K: nat] :
% 0.29/0.61        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.29/0.61        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_rec
% 0.29/0.61  thf(fact_9111_gbinomial__factors,axiom,
% 0.29/0.61      ! [A: complex,K: nat] :
% 0.29/0.61        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.29/0.61        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_factors
% 0.29/0.61  thf(fact_9112_gbinomial__factors,axiom,
% 0.29/0.61      ! [A: real,K: nat] :
% 0.29/0.61        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.29/0.61        = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_factors
% 0.29/0.61  thf(fact_9113_gbinomial__factors,axiom,
% 0.29/0.61      ! [A: rat,K: nat] :
% 0.29/0.61        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.29/0.61        = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_factors
% 0.29/0.61  thf(fact_9114_gbinomial__negated__upper,axiom,
% 0.29/0.61      ( gbinomial_complex
% 0.29/0.61      = ( ^ [A3: complex,K2: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A3 ) @ one_one_complex ) @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_negated_upper
% 0.29/0.61  thf(fact_9115_gbinomial__negated__upper,axiom,
% 0.29/0.61      ( gbinomial_real
% 0.29/0.61      = ( ^ [A3: 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 ) @ A3 ) @ one_one_real ) @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_negated_upper
% 0.29/0.61  thf(fact_9116_gbinomial__negated__upper,axiom,
% 0.29/0.61      ( gbinomial_rat
% 0.29/0.61      = ( ^ [A3: 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 ) @ A3 ) @ one_one_rat ) @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_negated_upper
% 0.29/0.61  thf(fact_9117_gbinomial__index__swap,axiom,
% 0.29/0.61      ! [K: nat,N: nat] :
% 0.29/0.61        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
% 0.29/0.61        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_index_swap
% 0.29/0.61  thf(fact_9118_gbinomial__index__swap,axiom,
% 0.29/0.61      ! [K: nat,N: nat] :
% 0.29/0.61        ( ( 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 ) )
% 0.29/0.61        = ( 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 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_index_swap
% 0.29/0.61  thf(fact_9119_gbinomial__index__swap,axiom,
% 0.29/0.61      ! [K: nat,N: nat] :
% 0.29/0.61        ( ( 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 ) )
% 0.29/0.61        = ( 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 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_index_swap
% 0.29/0.61  thf(fact_9120_push__bit__minus__one,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % push_bit_minus_one
% 0.29/0.61  thf(fact_9121_gbinomial__minus,axiom,
% 0.29/0.61      ! [A: complex,K: nat] :
% 0.29/0.61        ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 0.29/0.61        = ( 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 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_minus
% 0.29/0.61  thf(fact_9122_gbinomial__minus,axiom,
% 0.29/0.61      ! [A: real,K: nat] :
% 0.29/0.61        ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 0.29/0.61        = ( 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 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_minus
% 0.29/0.61  thf(fact_9123_gbinomial__minus,axiom,
% 0.29/0.61      ! [A: rat,K: nat] :
% 0.29/0.61        ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 0.29/0.61        = ( 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 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_minus
% 0.29/0.61  thf(fact_9124_gbinomial__reduce__nat,axiom,
% 0.29/0.61      ! [K: nat,A: complex] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.61       => ( ( gbinomial_complex @ A @ K )
% 0.29/0.61          = ( 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 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_reduce_nat
% 0.29/0.61  thf(fact_9125_gbinomial__reduce__nat,axiom,
% 0.29/0.61      ! [K: nat,A: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.61       => ( ( gbinomial_real @ A @ K )
% 0.29/0.61          = ( 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 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_reduce_nat
% 0.29/0.61  thf(fact_9126_gbinomial__reduce__nat,axiom,
% 0.29/0.61      ! [K: nat,A: rat] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.61       => ( ( gbinomial_rat @ A @ K )
% 0.29/0.61          = ( 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 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_reduce_nat
% 0.29/0.61  thf(fact_9127_gbinomial__pochhammer,axiom,
% 0.29/0.61      ( gbinomial_rat
% 0.29/0.61      = ( ^ [A3: 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 @ A3 ) @ K2 ) ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_pochhammer
% 0.29/0.61  thf(fact_9128_gbinomial__pochhammer,axiom,
% 0.29/0.61      ( gbinomial_real
% 0.29/0.61      = ( ^ [A3: 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 @ A3 ) @ K2 ) ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_pochhammer
% 0.29/0.61  thf(fact_9129_gbinomial__pochhammer,axiom,
% 0.29/0.61      ( gbinomial_complex
% 0.29/0.61      = ( ^ [A3: complex,K2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A3 ) @ K2 ) ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_pochhammer
% 0.29/0.61  thf(fact_9130_gbinomial__pochhammer_H,axiom,
% 0.29/0.61      ( gbinomial_rat
% 0.29/0.61      = ( ^ [A3: rat,K2: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_pochhammer'
% 0.29/0.61  thf(fact_9131_gbinomial__pochhammer_H,axiom,
% 0.29/0.61      ( gbinomial_real
% 0.29/0.61      = ( ^ [A3: real,K2: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_pochhammer'
% 0.29/0.61  thf(fact_9132_gbinomial__pochhammer_H,axiom,
% 0.29/0.61      ( gbinomial_complex
% 0.29/0.61      = ( ^ [A3: complex,K2: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gbinomial_pochhammer'
% 0.29/0.61  thf(fact_9133_bit__horner__sum__bit__iff,axiom,
% 0.29/0.61      ! [Bs: list_o,N: nat] :
% 0.29/0.61        ( ( bit_se9216721137139052372nteger @ ( groups3417619833198082522nteger @ zero_n356916108424825756nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Bs ) @ N )
% 0.29/0.61        = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 0.29/0.61          & ( nth_o @ Bs @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bit_horner_sum_bit_iff
% 0.29/0.61  thf(fact_9134_bit__horner__sum__bit__iff,axiom,
% 0.29/0.61      ! [Bs: list_o,N: nat] :
% 0.29/0.61        ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N )
% 0.29/0.61        = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 0.29/0.61          & ( nth_o @ Bs @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bit_horner_sum_bit_iff
% 0.29/0.61  thf(fact_9135_bit__horner__sum__bit__iff,axiom,
% 0.29/0.61      ! [Bs: list_o,N: nat] :
% 0.29/0.61        ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N )
% 0.29/0.61        = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 0.29/0.61          & ( nth_o @ Bs @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bit_horner_sum_bit_iff
% 0.29/0.61  thf(fact_9136_Cauchy__iff2,axiom,
% 0.29/0.61      ( topolo4055970368930404560y_real
% 0.29/0.61      = ( ^ [X6: nat > real] :
% 0.29/0.61          ! [J2: nat] :
% 0.29/0.61          ? [M8: nat] :
% 0.29/0.61          ! [M6: nat] :
% 0.29/0.61            ( ( ord_less_eq_nat @ M8 @ M6 )
% 0.29/0.61           => ! [N3: nat] :
% 0.29/0.61                ( ( ord_less_eq_nat @ M8 @ N3 )
% 0.29/0.61               => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M6 ) @ ( X6 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J2 ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Cauchy_iff2
% 0.29/0.61  thf(fact_9137_is__singleton__the__elem,axiom,
% 0.29/0.61      ( is_sin24926331636114728T_VEBT
% 0.29/0.61      = ( ^ [A4: set_VEBT_VEBT] :
% 0.29/0.61            ( A4
% 0.29/0.61            = ( insert_VEBT_VEBT @ ( the_elem_VEBT_VEBT @ A4 ) @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % is_singleton_the_elem
% 0.29/0.61  thf(fact_9138_is__singleton__the__elem,axiom,
% 0.29/0.61      ( is_singleton_real
% 0.29/0.61      = ( ^ [A4: set_real] :
% 0.29/0.61            ( A4
% 0.29/0.61            = ( insert_real @ ( the_elem_real @ A4 ) @ bot_bot_set_real ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % is_singleton_the_elem
% 0.29/0.61  thf(fact_9139_is__singleton__the__elem,axiom,
% 0.29/0.61      ( is_singleton_o
% 0.29/0.61      = ( ^ [A4: set_o] :
% 0.29/0.61            ( A4
% 0.29/0.61            = ( insert_o @ ( the_elem_o @ A4 ) @ bot_bot_set_o ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % is_singleton_the_elem
% 0.29/0.61  thf(fact_9140_is__singleton__the__elem,axiom,
% 0.29/0.61      ( is_singleton_nat
% 0.29/0.61      = ( ^ [A4: set_nat] :
% 0.29/0.61            ( A4
% 0.29/0.61            = ( insert_nat @ ( the_elem_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % is_singleton_the_elem
% 0.29/0.61  thf(fact_9141_is__singleton__the__elem,axiom,
% 0.29/0.61      ( is_singleton_int
% 0.29/0.61      = ( ^ [A4: set_int] :
% 0.29/0.61            ( A4
% 0.29/0.61            = ( insert_int @ ( the_elem_int @ A4 ) @ bot_bot_set_int ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % is_singleton_the_elem
% 0.29/0.61  thf(fact_9142_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.29/0.61        ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.29/0.61          = Y )
% 0.29/0.61       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.29/0.61                ( X
% 0.29/0.61                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.29/0.61           => Y )
% 0.29/0.61         => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.29/0.61                  ( X
% 0.29/0.61                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.29/0.61             => Y )
% 0.29/0.61           => ( ! [Mi: nat,Ma2: nat] :
% 0.29/0.61                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.29/0.61                      ( X
% 0.29/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.29/0.61                 => ( Y
% 0.29/0.61                    = ( ~ ( ( Xa = Mi )
% 0.29/0.61                          | ( Xa = Ma2 ) ) ) ) )
% 0.29/0.61             => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.29/0.61                    ( ? [Vc: vEBT_VEBT] :
% 0.29/0.61                        ( X
% 0.29/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 0.29/0.61                   => ( Y
% 0.29/0.61                      = ( ~ ( ( Xa = Mi )
% 0.29/0.61                            | ( Xa = Ma2 )
% 0.29/0.61                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 0.29/0.61               => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.29/0.61                      ( ? [Vd: vEBT_VEBT] :
% 0.29/0.61                          ( X
% 0.29/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 0.29/0.61                     => ( Y
% 0.29/0.61                        = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.membermima.elims(1)
% 0.29/0.61  thf(fact_9143_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.61        ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 0.29/0.61       => ( ! [Uu2: $o,Uv2: $o] :
% 0.29/0.61              ( X
% 0.29/0.61             != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.29/0.61         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.29/0.61                ( X
% 0.29/0.61               != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.29/0.61           => ( ! [Mi: nat,Ma2: nat] :
% 0.29/0.61                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.29/0.61                      ( X
% 0.29/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.29/0.61                 => ( ( Xa = Mi )
% 0.29/0.61                    | ( Xa = Ma2 ) ) )
% 0.29/0.61             => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.29/0.61                    ( ? [Vc: vEBT_VEBT] :
% 0.29/0.61                        ( X
% 0.29/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 0.29/0.61                   => ( ( Xa = Mi )
% 0.29/0.61                      | ( Xa = Ma2 )
% 0.29/0.61                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.29/0.61               => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.29/0.61                      ( ? [Vd: vEBT_VEBT] :
% 0.29/0.61                          ( X
% 0.29/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 0.29/0.61                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.membermima.elims(3)
% 0.29/0.61  thf(fact_9144_image__ident,axiom,
% 0.29/0.61      ! [Y7: set_nat] :
% 0.29/0.61        ( ( image_nat_nat
% 0.29/0.61          @ ^ [X4: nat] : X4
% 0.29/0.61          @ Y7 )
% 0.29/0.61        = Y7 ) ).
% 0.29/0.61  
% 0.29/0.61  % image_ident
% 0.29/0.61  thf(fact_9145_image__ident,axiom,
% 0.29/0.61      ! [Y7: set_int] :
% 0.29/0.61        ( ( image_int_int
% 0.29/0.61          @ ^ [X4: int] : X4
% 0.29/0.61          @ Y7 )
% 0.29/0.61        = Y7 ) ).
% 0.29/0.61  
% 0.29/0.61  % image_ident
% 0.29/0.61  thf(fact_9146_singleton__conv,axiom,
% 0.29/0.61      ! [A: complex] :
% 0.29/0.61        ( ( collect_complex
% 0.29/0.61          @ ^ [X4: complex] : ( X4 = A ) )
% 0.29/0.61        = ( insert_complex @ A @ bot_bot_set_complex ) ) ).
% 0.29/0.61  
% 0.29/0.61  % singleton_conv
% 0.29/0.61  thf(fact_9147_singleton__conv,axiom,
% 0.29/0.61      ! [A: set_nat] :
% 0.29/0.61        ( ( collect_set_nat
% 0.29/0.61          @ ^ [X4: set_nat] : ( X4 = A ) )
% 0.29/0.61        = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % singleton_conv
% 0.29/0.61  thf(fact_9148_singleton__conv,axiom,
% 0.29/0.61      ! [A: real] :
% 0.29/0.61        ( ( collect_real
% 0.29/0.61          @ ^ [X4: real] : ( X4 = A ) )
% 0.29/0.61        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % singleton_conv
% 0.29/0.61  thf(fact_9149_singleton__conv,axiom,
% 0.29/0.61      ! [A: $o] :
% 0.29/0.61        ( ( collect_o
% 0.29/0.61          @ ^ [X4: $o] : ( X4 = A ) )
% 0.29/0.61        = ( insert_o @ A @ bot_bot_set_o ) ) ).
% 0.29/0.61  
% 0.29/0.61  % singleton_conv
% 0.29/0.61  thf(fact_9150_singleton__conv,axiom,
% 0.29/0.61      ! [A: nat] :
% 0.29/0.61        ( ( collect_nat
% 0.29/0.61          @ ^ [X4: nat] : ( X4 = A ) )
% 0.29/0.61        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % singleton_conv
% 0.29/0.61  thf(fact_9151_singleton__conv,axiom,
% 0.29/0.61      ! [A: int] :
% 0.29/0.61        ( ( collect_int
% 0.29/0.61          @ ^ [X4: int] : ( X4 = A ) )
% 0.29/0.61        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % singleton_conv
% 0.29/0.61  thf(fact_9152_Max__divisors__self__int,axiom,
% 0.29/0.61      ! [N: int] :
% 0.29/0.61        ( ( N != zero_zero_int )
% 0.29/0.61       => ( ( lattic8263393255366662781ax_int
% 0.29/0.61            @ ( collect_int
% 0.29/0.61              @ ^ [D3: int] : ( dvd_dvd_int @ D3 @ N ) ) )
% 0.29/0.61          = ( abs_abs_int @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Max_divisors_self_int
% 0.29/0.61  thf(fact_9153_Max__divisors__self__nat,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( N != zero_zero_nat )
% 0.29/0.61       => ( ( lattic8265883725875713057ax_nat
% 0.29/0.61            @ ( collect_nat
% 0.29/0.61              @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ N ) ) )
% 0.29/0.61          = N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Max_divisors_self_nat
% 0.29/0.61  thf(fact_9154_nat__less__as__int,axiom,
% 0.29/0.61      ( ord_less_nat
% 0.29/0.61      = ( ^ [A3: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % nat_less_as_int
% 0.29/0.61  thf(fact_9155_nat__leq__as__int,axiom,
% 0.29/0.61      ( ord_less_eq_nat
% 0.29/0.61      = ( ^ [A3: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % nat_leq_as_int
% 0.29/0.61  thf(fact_9156_nat__plus__as__int,axiom,
% 0.29/0.61      ( plus_plus_nat
% 0.29/0.61      = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % nat_plus_as_int
% 0.29/0.61  thf(fact_9157_nat__times__as__int,axiom,
% 0.29/0.61      ( times_times_nat
% 0.29/0.61      = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % nat_times_as_int
% 0.29/0.61  thf(fact_9158_nat__minus__as__int,axiom,
% 0.29/0.61      ( minus_minus_nat
% 0.29/0.61      = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % nat_minus_as_int
% 0.29/0.61  thf(fact_9159_nat__div__as__int,axiom,
% 0.29/0.61      ( divide_divide_nat
% 0.29/0.61      = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % nat_div_as_int
% 0.29/0.61  thf(fact_9160_nat__mod__as__int,axiom,
% 0.29/0.61      ( modulo_modulo_nat
% 0.29/0.61      = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % nat_mod_as_int
% 0.29/0.61  thf(fact_9161_divide__nat__def,axiom,
% 0.29/0.61      ( divide_divide_nat
% 0.29/0.61      = ( ^ [M6: nat,N3: nat] :
% 0.29/0.61            ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
% 0.29/0.61            @ ( lattic8265883725875713057ax_nat
% 0.29/0.61              @ ( collect_nat
% 0.29/0.61                @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N3 ) @ M6 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divide_nat_def
% 0.29/0.61  thf(fact_9162_diff__nat__eq__if,axiom,
% 0.29/0.61      ! [Z6: int,Z: int] :
% 0.29/0.61        ( ( ( ord_less_int @ Z6 @ zero_zero_int )
% 0.29/0.61         => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 0.29/0.61            = ( nat2 @ Z ) ) )
% 0.29/0.61        & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
% 0.29/0.61         => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 0.29/0.61            = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % diff_nat_eq_if
% 0.29/0.61  thf(fact_9163_set__decode__def,axiom,
% 0.29/0.61      ( nat_set_decode
% 0.29/0.61      = ( ^ [X4: nat] :
% 0.29/0.61            ( collect_nat
% 0.29/0.61            @ ^ [N3: nat] :
% 0.29/0.61                ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % set_decode_def
% 0.29/0.61  thf(fact_9164_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.29/0.61      ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 0.29/0.61        ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
% 0.29/0.61        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.29/0.61           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.naive_member.simps(3)
% 0.29/0.61  thf(fact_9165_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.29/0.61      ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
% 0.29/0.61        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd2 ) @ X )
% 0.29/0.61        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.29/0.61           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.membermima.simps(5)
% 0.29/0.61  thf(fact_9166_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.29/0.61      ! [Mi2: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
% 0.29/0.61        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ X )
% 0.29/0.61        = ( ( X = Mi2 )
% 0.29/0.61          | ( X = Ma )
% 0.29/0.61          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.29/0.61             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.membermima.simps(4)
% 0.29/0.61  thf(fact_9167_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.61        ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.29/0.61       => ( ! [A5: $o,B6: $o] :
% 0.29/0.61              ( ( X
% 0.29/0.61                = ( vEBT_Leaf @ A5 @ B6 ) )
% 0.29/0.61             => ( ( ( Xa = zero_zero_nat )
% 0.29/0.61                 => A5 )
% 0.29/0.61                & ( ( Xa != zero_zero_nat )
% 0.29/0.61                 => ( ( ( Xa = one_one_nat )
% 0.29/0.61                     => B6 )
% 0.29/0.61                    & ( Xa = one_one_nat ) ) ) ) )
% 0.29/0.61         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.29/0.61                ( X
% 0.29/0.61               != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.29/0.61           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.29/0.61                  ( ? [S3: vEBT_VEBT] :
% 0.29/0.61                      ( X
% 0.29/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.29/0.61                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                     => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.naive_member.elims(3)
% 0.29/0.61  thf(fact_9168_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.61        ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.29/0.61       => ( ! [A5: $o,B6: $o] :
% 0.29/0.61              ( ( X
% 0.29/0.61                = ( vEBT_Leaf @ A5 @ B6 ) )
% 0.29/0.61             => ~ ( ( ( Xa = zero_zero_nat )
% 0.29/0.61                   => A5 )
% 0.29/0.61                  & ( ( Xa != zero_zero_nat )
% 0.29/0.61                   => ( ( ( Xa = one_one_nat )
% 0.29/0.61                       => B6 )
% 0.29/0.61                      & ( Xa = one_one_nat ) ) ) ) )
% 0.29/0.61         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.29/0.61                ( ? [S3: vEBT_VEBT] :
% 0.29/0.61                    ( X
% 0.29/0.61                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.29/0.61               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                     => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.naive_member.elims(2)
% 0.29/0.61  thf(fact_9169_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.29/0.61        ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.29/0.61          = Y )
% 0.29/0.61       => ( ! [A5: $o,B6: $o] :
% 0.29/0.61              ( ( X
% 0.29/0.61                = ( vEBT_Leaf @ A5 @ B6 ) )
% 0.29/0.61             => ( Y
% 0.29/0.61                = ( ~ ( ( ( Xa = zero_zero_nat )
% 0.29/0.61                       => A5 )
% 0.29/0.61                      & ( ( Xa != zero_zero_nat )
% 0.29/0.61                       => ( ( ( Xa = one_one_nat )
% 0.29/0.61                           => B6 )
% 0.29/0.61                          & ( Xa = one_one_nat ) ) ) ) ) ) )
% 0.29/0.61         => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.29/0.61                  ( X
% 0.29/0.61                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.29/0.61             => Y )
% 0.29/0.61           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.29/0.61                  ( ? [S3: vEBT_VEBT] :
% 0.29/0.61                      ( X
% 0.29/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.29/0.61                 => ( Y
% 0.29/0.61                    = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.naive_member.elims(1)
% 0.29/0.61  thf(fact_9170_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.61        ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.29/0.61       => ( ! [Mi: nat,Ma2: nat] :
% 0.29/0.61              ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.29/0.61                  ( X
% 0.29/0.61                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.29/0.61             => ~ ( ( Xa = Mi )
% 0.29/0.61                  | ( Xa = Ma2 ) ) )
% 0.29/0.61         => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.29/0.61                ( ? [Vc: vEBT_VEBT] :
% 0.29/0.61                    ( X
% 0.29/0.61                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 0.29/0.61               => ~ ( ( Xa = Mi )
% 0.29/0.61                    | ( Xa = Ma2 )
% 0.29/0.61                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.29/0.61           => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.29/0.61                  ( ? [Vd: vEBT_VEBT] :
% 0.29/0.61                      ( X
% 0.29/0.61                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 0.29/0.61                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.membermima.elims(2)
% 0.29/0.61  thf(fact_9171_monoseq__arctan__series,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.61       => ( topolo6980174941875973593q_real
% 0.29/0.61          @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % monoseq_arctan_series
% 0.29/0.61  thf(fact_9172_ln__series,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.61       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.61         => ( ( ln_ln_real @ X )
% 0.29/0.61            = ( suminf_real
% 0.29/0.61              @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % ln_series
% 0.29/0.61  thf(fact_9173_arctan__series,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.61       => ( ( arctan @ X )
% 0.29/0.61          = ( suminf_real
% 0.29/0.61            @ ^ [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 @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % arctan_series
% 0.29/0.61  thf(fact_9174_monoseq__realpow,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.61       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.61         => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % monoseq_realpow
% 0.29/0.61  thf(fact_9175_pi__series,axiom,
% 0.29/0.61      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.29/0.61      = ( suminf_real
% 0.29/0.61        @ ^ [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 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % pi_series
% 0.29/0.61  thf(fact_9176_summable__arctan__series,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.61       => ( summable_real
% 0.29/0.61          @ ^ [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 @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % summable_arctan_series
% 0.29/0.61  thf(fact_9177_vebt__buildup_Oelims,axiom,
% 0.29/0.61      ! [X: nat,Y: vEBT_VEBT] :
% 0.29/0.61        ( ( ( vEBT_vebt_buildup @ X )
% 0.29/0.61          = Y )
% 0.29/0.61       => ( ( ( X = zero_zero_nat )
% 0.29/0.61           => ( Y
% 0.29/0.61             != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.29/0.61         => ( ( ( X
% 0.29/0.61                = ( suc @ zero_zero_nat ) )
% 0.29/0.61             => ( Y
% 0.29/0.61               != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.29/0.61           => ~ ! [Va: nat] :
% 0.29/0.61                  ( ( X
% 0.29/0.61                    = ( suc @ ( suc @ Va ) ) )
% 0.29/0.61                 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.29/0.61                       => ( Y
% 0.29/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.29/0.61                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.29/0.61                       => ( Y
% 0.29/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % vebt_buildup.elims
% 0.29/0.61  thf(fact_9178_summable__rabs__comparison__test,axiom,
% 0.29/0.61      ! [F: nat > real,G: nat > real] :
% 0.29/0.61        ( ? [N6: nat] :
% 0.29/0.61          ! [N2: nat] :
% 0.29/0.61            ( ( ord_less_eq_nat @ N6 @ N2 )
% 0.29/0.61           => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 0.29/0.61       => ( ( summable_real @ G )
% 0.29/0.61         => ( summable_real
% 0.29/0.61            @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % summable_rabs_comparison_test
% 0.29/0.61  thf(fact_9179_summable__rabs,axiom,
% 0.29/0.61      ! [F: nat > real] :
% 0.29/0.61        ( ( summable_real
% 0.29/0.61          @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) )
% 0.29/0.61       => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 0.29/0.61          @ ( suminf_real
% 0.29/0.61            @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % summable_rabs
% 0.29/0.61  thf(fact_9180_summable__power__series,axiom,
% 0.29/0.61      ! [F: nat > real,Z: real] :
% 0.29/0.61        ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
% 0.29/0.61       => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.29/0.61         => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 0.29/0.61           => ( ( ord_less_real @ Z @ one_one_real )
% 0.29/0.61             => ( summable_real
% 0.29/0.61                @ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % summable_power_series
% 0.29/0.61  thf(fact_9181_vebt__buildup_Osimps_I3_J,axiom,
% 0.29/0.61      ! [Va2: nat] :
% 0.29/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.29/0.61         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 0.29/0.61            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.29/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.29/0.61         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 0.29/0.61            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % vebt_buildup.simps(3)
% 0.29/0.61  thf(fact_9182_sin__paired,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( sums_real
% 0.29/0.61        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.29/0.61        @ ( sin_real @ X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sin_paired
% 0.29/0.61  thf(fact_9183_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.61        ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 0.29/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.29/0.61         => ( ! [Uu2: $o,Uv2: $o] :
% 0.29/0.61                ( ( X
% 0.29/0.61                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.29/0.61               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
% 0.29/0.61           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.29/0.61                  ( ( X
% 0.29/0.61                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.29/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
% 0.29/0.61             => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.29/0.61                    ( ( X
% 0.29/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.29/0.61                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
% 0.29/0.61                     => ( ( Xa = Mi )
% 0.29/0.61                        | ( Xa = Ma2 ) ) ) )
% 0.29/0.61               => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.29/0.61                      ( ( X
% 0.29/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 0.29/0.61                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) @ Xa ) )
% 0.29/0.61                       => ( ( Xa = Mi )
% 0.29/0.61                          | ( Xa = Ma2 )
% 0.29/0.61                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.29/0.61                 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 0.29/0.61                        ( ( X
% 0.29/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 0.29/0.61                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa ) )
% 0.29/0.61                         => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.membermima.pelims(3)
% 0.29/0.61  thf(fact_9184_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.29/0.61        ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.29/0.61          = Y )
% 0.29/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.29/0.61         => ( ! [Uu2: $o,Uv2: $o] :
% 0.29/0.61                ( ( X
% 0.29/0.61                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.29/0.61               => ( ~ Y
% 0.29/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 0.29/0.61           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.29/0.61                  ( ( X
% 0.29/0.61                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.29/0.61                 => ( ~ Y
% 0.29/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
% 0.29/0.61             => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.29/0.61                    ( ( X
% 0.29/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.29/0.61                   => ( ( Y
% 0.29/0.61                        = ( ( Xa = Mi )
% 0.29/0.61                          | ( Xa = Ma2 ) ) )
% 0.29/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) ) ) )
% 0.29/0.61               => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.29/0.61                      ( ( X
% 0.29/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 0.29/0.61                     => ( ( Y
% 0.29/0.61                          = ( ( Xa = Mi )
% 0.29/0.61                            | ( Xa = Ma2 )
% 0.29/0.61                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.29/0.61                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) @ Xa ) ) ) )
% 0.29/0.61                 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 0.29/0.61                        ( ( X
% 0.29/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 0.29/0.61                       => ( ( Y
% 0.29/0.61                            = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.29/0.61                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.membermima.pelims(1)
% 0.29/0.61  thf(fact_9185_power__half__series,axiom,
% 0.29/0.61      ( sums_real
% 0.29/0.61      @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
% 0.29/0.61      @ one_one_real ) ).
% 0.29/0.61  
% 0.29/0.61  % power_half_series
% 0.29/0.61  thf(fact_9186_sums__if_H,axiom,
% 0.29/0.61      ! [G: nat > real,X: real] :
% 0.29/0.61        ( ( sums_real @ G @ X )
% 0.29/0.61       => ( sums_real
% 0.29/0.61          @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.61          @ X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sums_if'
% 0.29/0.61  thf(fact_9187_sums__if,axiom,
% 0.29/0.61      ! [G: nat > real,X: real,F: nat > real,Y: real] :
% 0.29/0.61        ( ( sums_real @ G @ X )
% 0.29/0.61       => ( ( sums_real @ F @ Y )
% 0.29/0.61         => ( sums_real
% 0.29/0.61            @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.61            @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sums_if
% 0.29/0.61  thf(fact_9188_cos__paired,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( sums_real
% 0.29/0.61        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.29/0.61        @ ( cos_real @ X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cos_paired
% 0.29/0.61  thf(fact_9189_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.61        ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.29/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.29/0.61         => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.29/0.61                ( ( X
% 0.29/0.61                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.29/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
% 0.29/0.61                 => ~ ( ( Xa = Mi )
% 0.29/0.61                      | ( Xa = Ma2 ) ) ) )
% 0.29/0.61           => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.29/0.61                  ( ( X
% 0.29/0.61                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 0.29/0.61                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) @ Xa ) )
% 0.29/0.61                   => ~ ( ( Xa = Mi )
% 0.29/0.61                        | ( Xa = Ma2 )
% 0.29/0.61                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.29/0.61             => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 0.29/0.61                    ( ( X
% 0.29/0.61                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 0.29/0.61                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa ) )
% 0.29/0.61                     => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.membermima.pelims(2)
% 0.29/0.61  thf(fact_9190_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.29/0.61        ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.29/0.61          = Y )
% 0.29/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.29/0.61         => ( ! [A5: $o,B6: $o] :
% 0.29/0.61                ( ( X
% 0.29/0.61                  = ( vEBT_Leaf @ A5 @ B6 ) )
% 0.29/0.61               => ( ( Y
% 0.29/0.61                    = ( ( ( Xa = zero_zero_nat )
% 0.29/0.61                       => A5 )
% 0.29/0.61                      & ( ( Xa != zero_zero_nat )
% 0.29/0.61                       => ( ( ( Xa = one_one_nat )
% 0.29/0.61                           => B6 )
% 0.29/0.61                          & ( Xa = one_one_nat ) ) ) ) )
% 0.29/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa ) ) ) )
% 0.29/0.61           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.29/0.61                  ( ( X
% 0.29/0.61                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.29/0.61                 => ( ~ Y
% 0.29/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
% 0.29/0.61             => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.29/0.61                    ( ( X
% 0.29/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.29/0.61                   => ( ( Y
% 0.29/0.61                        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.29/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.naive_member.pelims(1)
% 0.29/0.61  thf(fact_9191_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.61        ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.29/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.29/0.61         => ( ! [A5: $o,B6: $o] :
% 0.29/0.61                ( ( X
% 0.29/0.61                  = ( vEBT_Leaf @ A5 @ B6 ) )
% 0.29/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa ) )
% 0.29/0.61                 => ~ ( ( ( Xa = zero_zero_nat )
% 0.29/0.61                       => A5 )
% 0.29/0.61                      & ( ( Xa != zero_zero_nat )
% 0.29/0.61                       => ( ( ( Xa = one_one_nat )
% 0.29/0.61                           => B6 )
% 0.29/0.61                          & ( Xa = one_one_nat ) ) ) ) ) )
% 0.29/0.61           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.29/0.61                  ( ( X
% 0.29/0.61                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.29/0.61                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 0.29/0.61                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.naive_member.pelims(2)
% 0.29/0.61  thf(fact_9192_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.29/0.61      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.61        ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.29/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.29/0.61         => ( ! [A5: $o,B6: $o] :
% 0.29/0.61                ( ( X
% 0.29/0.61                  = ( vEBT_Leaf @ A5 @ B6 ) )
% 0.29/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa ) )
% 0.29/0.61                 => ( ( ( Xa = zero_zero_nat )
% 0.29/0.61                     => A5 )
% 0.29/0.61                    & ( ( Xa != zero_zero_nat )
% 0.29/0.61                     => ( ( ( Xa = one_one_nat )
% 0.29/0.61                         => B6 )
% 0.29/0.61                        & ( Xa = one_one_nat ) ) ) ) ) )
% 0.29/0.61           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.29/0.61                  ( ( X
% 0.29/0.61                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.29/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
% 0.29/0.61             => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.29/0.61                    ( ( X
% 0.29/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.29/0.61                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 0.29/0.61                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.29/0.61                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT_internal.naive_member.pelims(3)
% 0.29/0.61  thf(fact_9193_diffs__sin__coeff,axiom,
% 0.29/0.61      ( ( diffs_real @ sin_coeff )
% 0.29/0.61      = cos_coeff ) ).
% 0.29/0.61  
% 0.29/0.61  % diffs_sin_coeff
% 0.29/0.61  thf(fact_9194_diffs__cos__coeff,axiom,
% 0.29/0.61      ( ( diffs_real @ cos_coeff )
% 0.29/0.61      = ( ^ [N3: nat] : ( uminus_uminus_real @ ( sin_coeff @ N3 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % diffs_cos_coeff
% 0.29/0.61  thf(fact_9195_Arg__def,axiom,
% 0.29/0.61      ( arg
% 0.29/0.61      = ( ^ [Z4: complex] :
% 0.29/0.61            ( if_real @ ( Z4 = zero_zero_complex ) @ zero_zero_real
% 0.29/0.61            @ ( fChoice_real
% 0.29/0.61              @ ^ [A3: real] :
% 0.29/0.61                  ( ( ( sgn_sgn_complex @ Z4 )
% 0.29/0.61                    = ( cis @ A3 ) )
% 0.29/0.61                  & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
% 0.29/0.61                  & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Arg_def
% 0.29/0.61  thf(fact_9196_set__vebt__def,axiom,
% 0.29/0.61      ( vEBT_set_vebt
% 0.29/0.61      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % set_vebt_def
% 0.29/0.61  thf(fact_9197_atMost__0,axiom,
% 0.29/0.61      ( ( set_ord_atMost_nat @ zero_zero_nat )
% 0.29/0.61      = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atMost_0
% 0.29/0.61  thf(fact_9198_sum__choose__upper,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [K2: nat] : ( binomial @ K2 @ M )
% 0.29/0.61          @ ( set_ord_atMost_nat @ N ) )
% 0.29/0.61        = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sum_choose_upper
% 0.29/0.61  thf(fact_9199_sum__choose__lower,axiom,
% 0.29/0.61      ! [R2: nat,N: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K2 ) @ K2 )
% 0.29/0.61          @ ( set_ord_atMost_nat @ N ) )
% 0.29/0.61        = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sum_choose_lower
% 0.29/0.61  thf(fact_9200_choose__rising__sum_I1_J,axiom,
% 0.29/0.61      ! [N: nat,M: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N @ J2 ) @ N )
% 0.29/0.61          @ ( set_ord_atMost_nat @ M ) )
% 0.29/0.61        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % choose_rising_sum(1)
% 0.29/0.61  thf(fact_9201_choose__rising__sum_I2_J,axiom,
% 0.29/0.61      ! [N: nat,M: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N @ J2 ) @ N )
% 0.29/0.61          @ ( set_ord_atMost_nat @ M ) )
% 0.29/0.61        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 0.29/0.61  
% 0.29/0.61  % choose_rising_sum(2)
% 0.29/0.61  thf(fact_9202_sum__choose__diagonal,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.61       => ( ( groups3542108847815614940at_nat
% 0.29/0.61            @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
% 0.29/0.61            @ ( set_ord_atMost_nat @ M ) )
% 0.29/0.61          = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sum_choose_diagonal
% 0.29/0.61  thf(fact_9203_vandermonde,axiom,
% 0.29/0.61      ! [M: nat,N: nat,R2: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K2 ) ) )
% 0.29/0.61          @ ( set_ord_atMost_nat @ R2 ) )
% 0.29/0.61        = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % vandermonde
% 0.29/0.61  thf(fact_9204_atMost__Suc,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 0.29/0.61        = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atMost_Suc
% 0.29/0.61  thf(fact_9205_choose__row__sum,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 0.29/0.61        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % choose_row_sum
% 0.29/0.61  thf(fact_9206_binomial,axiom,
% 0.29/0.61      ! [A: nat,B: nat,N: nat] :
% 0.29/0.61        ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 0.29/0.61        = ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.29/0.61          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % binomial
% 0.29/0.61  thf(fact_9207_polynomial__product__nat,axiom,
% 0.29/0.61      ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
% 0.29/0.61        ( ! [I2: nat] :
% 0.29/0.61            ( ( ord_less_nat @ M @ I2 )
% 0.29/0.61           => ( ( A @ I2 )
% 0.29/0.61              = zero_zero_nat ) )
% 0.29/0.61       => ( ! [J3: nat] :
% 0.29/0.61              ( ( ord_less_nat @ N @ J3 )
% 0.29/0.61             => ( ( B @ J3 )
% 0.29/0.61                = zero_zero_nat ) )
% 0.29/0.61         => ( ( times_times_nat
% 0.29/0.61              @ ( groups3542108847815614940at_nat
% 0.29/0.61                @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
% 0.29/0.61                @ ( set_ord_atMost_nat @ M ) )
% 0.29/0.61              @ ( groups3542108847815614940at_nat
% 0.29/0.61                @ ^ [J2: nat] : ( times_times_nat @ ( B @ J2 ) @ ( power_power_nat @ X @ J2 ) )
% 0.29/0.61                @ ( set_ord_atMost_nat @ N ) ) )
% 0.29/0.61            = ( groups3542108847815614940at_nat
% 0.29/0.61              @ ^ [R5: nat] :
% 0.29/0.61                  ( times_times_nat
% 0.29/0.61                  @ ( groups3542108847815614940at_nat
% 0.29/0.61                    @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 0.29/0.61                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.29/0.61                  @ ( power_power_nat @ X @ R5 ) )
% 0.29/0.61              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % polynomial_product_nat
% 0.29/0.61  thf(fact_9208_choose__square__sum,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.61          @ ( set_ord_atMost_nat @ N ) )
% 0.29/0.61        = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % choose_square_sum
% 0.29/0.61  thf(fact_9209_atMost__nat__numeral,axiom,
% 0.29/0.61      ! [K: num] :
% 0.29/0.61        ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 0.29/0.61        = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atMost_nat_numeral
% 0.29/0.61  thf(fact_9210_binomial__r__part__sum,axiom,
% 0.29/0.61      ! [M: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.29/0.61        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % binomial_r_part_sum
% 0.29/0.61  thf(fact_9211_choose__linear__sum,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N @ I3 ) )
% 0.29/0.61          @ ( set_ord_atMost_nat @ N ) )
% 0.29/0.61        = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % choose_linear_sum
% 0.29/0.61  thf(fact_9212_atMost__Suc__eq__insert__0,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 0.29/0.61        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atMost_Suc_eq_insert_0
% 0.29/0.61  thf(fact_9213_mask__eq__sum__exp__nat,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 0.29/0.61        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.61          @ ( collect_nat
% 0.29/0.61            @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % mask_eq_sum_exp_nat
% 0.29/0.61  thf(fact_9214_sum__nth__roots,axiom,
% 0.29/0.61      ! [N: nat,C: complex] :
% 0.29/0.61        ( ( ord_less_nat @ one_one_nat @ N )
% 0.29/0.61       => ( ( groups7754918857620584856omplex
% 0.29/0.61            @ ^ [X4: complex] : X4
% 0.29/0.61            @ ( collect_complex
% 0.29/0.61              @ ^ [Z4: complex] :
% 0.29/0.61                  ( ( power_power_complex @ Z4 @ N )
% 0.29/0.61                  = C ) ) )
% 0.29/0.61          = zero_zero_complex ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sum_nth_roots
% 0.29/0.61  thf(fact_9215_sum__roots__unity,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( ord_less_nat @ one_one_nat @ N )
% 0.29/0.61       => ( ( groups7754918857620584856omplex
% 0.29/0.61            @ ^ [X4: complex] : X4
% 0.29/0.61            @ ( collect_complex
% 0.29/0.61              @ ^ [Z4: complex] :
% 0.29/0.61                  ( ( power_power_complex @ Z4 @ N )
% 0.29/0.61                  = one_one_complex ) ) )
% 0.29/0.61          = zero_zero_complex ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sum_roots_unity
% 0.29/0.61  thf(fact_9216_Maclaurin__minus__cos__expansion,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ X @ zero_zero_real )
% 0.29/0.61         => ? [T3: real] :
% 0.29/0.61              ( ( ord_less_real @ X @ T3 )
% 0.29/0.61              & ( ord_less_real @ T3 @ zero_zero_real )
% 0.29/0.61              & ( ( cos_real @ X )
% 0.29/0.61                = ( plus_plus_real
% 0.29/0.61                  @ ( groups6591440286371151544t_real
% 0.29/0.61                    @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.61                    @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_minus_cos_expansion
% 0.29/0.61  thf(fact_9217_Maclaurin__cos__expansion2,axiom,
% 0.29/0.61      ! [X: real,N: nat] :
% 0.29/0.61        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.61       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61         => ? [T3: real] :
% 0.29/0.61              ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.29/0.61              & ( ord_less_real @ T3 @ X )
% 0.29/0.61              & ( ( cos_real @ X )
% 0.29/0.61                = ( plus_plus_real
% 0.29/0.61                  @ ( groups6591440286371151544t_real
% 0.29/0.61                    @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.61                    @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_cos_expansion2
% 0.29/0.61  thf(fact_9218_Maclaurin__sin__expansion3,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.61         => ? [T3: real] :
% 0.29/0.61              ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.29/0.61              & ( ord_less_real @ T3 @ X )
% 0.29/0.61              & ( ( sin_real @ X )
% 0.29/0.61                = ( plus_plus_real
% 0.29/0.61                  @ ( groups6591440286371151544t_real
% 0.29/0.61                    @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.61                    @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61                  @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_sin_expansion3
% 0.29/0.61  thf(fact_9219_Maclaurin__sin__expansion4,axiom,
% 0.29/0.61      ! [X: real,N: nat] :
% 0.29/0.61        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.61       => ? [T3: real] :
% 0.29/0.61            ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.29/0.61            & ( ord_less_eq_real @ T3 @ X )
% 0.29/0.61            & ( ( sin_real @ X )
% 0.29/0.61              = ( plus_plus_real
% 0.29/0.61                @ ( groups6591440286371151544t_real
% 0.29/0.61                  @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.61                  @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_sin_expansion4
% 0.29/0.61  thf(fact_9220_lessThan__0,axiom,
% 0.29/0.61      ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 0.29/0.61      = bot_bot_set_nat ) ).
% 0.29/0.61  
% 0.29/0.61  % lessThan_0
% 0.29/0.61  thf(fact_9221_sumr__cos__zero__one,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( groups6591440286371151544t_real
% 0.29/0.61          @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
% 0.29/0.61          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.29/0.61        = one_one_real ) ).
% 0.29/0.61  
% 0.29/0.61  % sumr_cos_zero_one
% 0.29/0.61  thf(fact_9222_lessThan__Suc__atMost,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.29/0.61        = ( set_ord_atMost_nat @ K ) ) ).
% 0.29/0.61  
% 0.29/0.61  % lessThan_Suc_atMost
% 0.29/0.61  thf(fact_9223_lessThan__Suc,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.29/0.61        = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % lessThan_Suc
% 0.29/0.61  thf(fact_9224_lessThan__empty__iff,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( ( set_ord_lessThan_nat @ N )
% 0.29/0.61          = bot_bot_set_nat )
% 0.29/0.61        = ( N = zero_zero_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % lessThan_empty_iff
% 0.29/0.61  thf(fact_9225_lessThan__nat__numeral,axiom,
% 0.29/0.61      ! [K: num] :
% 0.29/0.61        ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 0.29/0.61        = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % lessThan_nat_numeral
% 0.29/0.61  thf(fact_9226_lessThan__Suc__eq__insert__0,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 0.29/0.61        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % lessThan_Suc_eq_insert_0
% 0.29/0.61  thf(fact_9227_Maclaurin__lemma,axiom,
% 0.29/0.61      ! [H: real,F: real > real,J: nat > real,N: nat] :
% 0.29/0.61        ( ( ord_less_real @ zero_zero_real @ H )
% 0.29/0.61       => ? [B3: real] :
% 0.29/0.61            ( ( F @ H )
% 0.29/0.61            = ( plus_plus_real
% 0.29/0.61              @ ( groups6591440286371151544t_real
% 0.29/0.61                @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
% 0.29/0.61                @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61              @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ H @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_lemma
% 0.29/0.61  thf(fact_9228_sum__split__even__odd,axiom,
% 0.29/0.61      ! [F: nat > real,G: nat > real,N: nat] :
% 0.29/0.61        ( ( groups6591440286371151544t_real
% 0.29/0.61          @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
% 0.29/0.61          @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.61        = ( plus_plus_real
% 0.29/0.61          @ ( groups6591440286371151544t_real
% 0.29/0.61            @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
% 0.29/0.61            @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61          @ ( groups6591440286371151544t_real
% 0.29/0.61            @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
% 0.29/0.61            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sum_split_even_odd
% 0.29/0.61  thf(fact_9229_Maclaurin__exp__le,axiom,
% 0.29/0.61      ! [X: real,N: nat] :
% 0.29/0.61      ? [T3: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.29/0.61        & ( ( exp_real @ X )
% 0.29/0.61          = ( plus_plus_real
% 0.29/0.61            @ ( groups6591440286371151544t_real
% 0.29/0.61              @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 0.29/0.61              @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61            @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_exp_le
% 0.29/0.61  thf(fact_9230_Maclaurin__sin__bound,axiom,
% 0.29/0.61      ! [X: real,N: nat] :
% 0.29/0.61        ( ord_less_eq_real
% 0.29/0.61        @ ( abs_abs_real
% 0.29/0.61          @ ( minus_minus_real @ ( sin_real @ X )
% 0.29/0.61            @ ( groups6591440286371151544t_real
% 0.29/0.61              @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.61              @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.29/0.61        @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_sin_bound
% 0.29/0.61  thf(fact_9231_sum__pos__lt__pair,axiom,
% 0.29/0.61      ! [F: nat > real,K: nat] :
% 0.29/0.61        ( ( summable_real @ F )
% 0.29/0.61       => ( ! [D5: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) @ one_one_nat ) ) ) ) )
% 0.29/0.61         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sum_pos_lt_pair
% 0.29/0.61  thf(fact_9232_Maclaurin__exp__lt,axiom,
% 0.29/0.61      ! [X: real,N: nat] :
% 0.29/0.61        ( ( X != zero_zero_real )
% 0.29/0.61       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61         => ? [T3: real] :
% 0.29/0.61              ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 0.29/0.61              & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.29/0.61              & ( ( exp_real @ X )
% 0.29/0.61                = ( plus_plus_real
% 0.29/0.61                  @ ( groups6591440286371151544t_real
% 0.29/0.61                    @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 0.29/0.61                    @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61                  @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_exp_lt
% 0.29/0.61  thf(fact_9233_Maclaurin__sin__expansion,axiom,
% 0.29/0.61      ! [X: real,N: nat] :
% 0.29/0.61      ? [T3: real] :
% 0.29/0.61        ( ( sin_real @ X )
% 0.29/0.61        = ( plus_plus_real
% 0.29/0.61          @ ( groups6591440286371151544t_real
% 0.29/0.61            @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.61            @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_sin_expansion
% 0.29/0.61  thf(fact_9234_Maclaurin__sin__expansion2,axiom,
% 0.29/0.61      ! [X: real,N: nat] :
% 0.29/0.61      ? [T3: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.29/0.61        & ( ( sin_real @ X )
% 0.29/0.61          = ( plus_plus_real
% 0.29/0.61            @ ( groups6591440286371151544t_real
% 0.29/0.61              @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.61              @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61            @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_sin_expansion2
% 0.29/0.61  thf(fact_9235_Maclaurin__cos__expansion,axiom,
% 0.29/0.61      ! [X: real,N: nat] :
% 0.29/0.61      ? [T3: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.29/0.61        & ( ( cos_real @ X )
% 0.29/0.61          = ( plus_plus_real
% 0.29/0.61            @ ( groups6591440286371151544t_real
% 0.29/0.61              @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.61              @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61            @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Maclaurin_cos_expansion
% 0.29/0.61  thf(fact_9236_bij__betw__roots__unity,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( bij_betw_nat_complex
% 0.29/0.61          @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.29/0.61          @ ( set_ord_lessThan_nat @ N )
% 0.29/0.61          @ ( collect_complex
% 0.29/0.61            @ ^ [Z4: complex] :
% 0.29/0.61                ( ( power_power_complex @ Z4 @ N )
% 0.29/0.61                = one_one_complex ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bij_betw_roots_unity
% 0.29/0.61  thf(fact_9237_image__Suc__atLeastAtMost,axiom,
% 0.29/0.61      ! [I: nat,J: nat] :
% 0.29/0.61        ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.29/0.61        = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % image_Suc_atLeastAtMost
% 0.29/0.61  thf(fact_9238_all__nat__less,axiom,
% 0.29/0.61      ! [N: nat,P: nat > $o] :
% 0.29/0.61        ( ( ! [M6: nat] :
% 0.29/0.61              ( ( ord_less_eq_nat @ M6 @ N )
% 0.29/0.61             => ( P @ M6 ) ) )
% 0.29/0.61        = ( ! [X4: nat] :
% 0.29/0.61              ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.29/0.61             => ( P @ X4 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % all_nat_less
% 0.29/0.61  thf(fact_9239_ex__nat__less,axiom,
% 0.29/0.61      ! [N: nat,P: nat > $o] :
% 0.29/0.61        ( ( ? [M6: nat] :
% 0.29/0.61              ( ( ord_less_eq_nat @ M6 @ N )
% 0.29/0.61              & ( P @ M6 ) ) )
% 0.29/0.61        = ( ? [X4: nat] :
% 0.29/0.61              ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.29/0.61              & ( P @ X4 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % ex_nat_less
% 0.29/0.61  thf(fact_9240_atMost__atLeast0,axiom,
% 0.29/0.61      ( set_ord_atMost_nat
% 0.29/0.61      = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atMost_atLeast0
% 0.29/0.61  thf(fact_9241_atLeast0__atMost__Suc,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.29/0.61        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeast0_atMost_Suc
% 0.29/0.61  thf(fact_9242_atLeastAtMost__insertL,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.61       => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.29/0.61          = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeastAtMost_insertL
% 0.29/0.61  thf(fact_9243_atLeastAtMostSuc__conv,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.29/0.61       => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 0.29/0.61          = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeastAtMostSuc_conv
% 0.29/0.61  thf(fact_9244_Icc__eq__insert__lb__nat,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.61       => ( ( set_or1269000886237332187st_nat @ M @ N )
% 0.29/0.61          = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Icc_eq_insert_lb_nat
% 0.29/0.61  thf(fact_9245_image__Suc__lessThan,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.61        = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % image_Suc_lessThan
% 0.29/0.61  thf(fact_9246_image__Suc__atMost,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 0.29/0.61        = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % image_Suc_atMost
% 0.29/0.61  thf(fact_9247_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.29/0.61        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeast0_atMost_Suc_eq_insert_0
% 0.29/0.61  thf(fact_9248_atLeast1__atMost__eq__remove0,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.61        = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeast1_atMost_eq_remove0
% 0.29/0.61  thf(fact_9249_gauss__sum__nat,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [X4: nat] : X4
% 0.29/0.61          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.29/0.61        = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % gauss_sum_nat
% 0.29/0.61  thf(fact_9250_arith__series__nat,axiom,
% 0.29/0.61      ! [A: nat,D: nat,N: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
% 0.29/0.61          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.29/0.61        = ( 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 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % arith_series_nat
% 0.29/0.61  thf(fact_9251_Sum__Icc__nat,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [X4: nat] : X4
% 0.29/0.61          @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.29/0.61        = ( 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 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Sum_Icc_nat
% 0.29/0.61  thf(fact_9252_vebt__buildup_Opelims,axiom,
% 0.29/0.61      ! [X: nat,Y: vEBT_VEBT] :
% 0.29/0.61        ( ( ( vEBT_vebt_buildup @ X )
% 0.29/0.61          = Y )
% 0.29/0.61       => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 0.29/0.61         => ( ( ( X = zero_zero_nat )
% 0.29/0.61             => ( ( Y
% 0.29/0.61                  = ( vEBT_Leaf @ $false @ $false ) )
% 0.29/0.61               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.29/0.61           => ( ( ( X
% 0.29/0.61                  = ( suc @ zero_zero_nat ) )
% 0.29/0.61               => ( ( Y
% 0.29/0.61                    = ( vEBT_Leaf @ $false @ $false ) )
% 0.29/0.61                 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.29/0.61             => ~ ! [Va: nat] :
% 0.29/0.61                    ( ( X
% 0.29/0.61                      = ( suc @ ( suc @ Va ) ) )
% 0.29/0.61                   => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.29/0.61                         => ( Y
% 0.29/0.61                            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.29/0.61                        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.29/0.61                         => ( Y
% 0.29/0.61                            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.29/0.61                     => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % vebt_buildup.pelims
% 0.29/0.61  thf(fact_9253_arctan__def,axiom,
% 0.29/0.61      ( arctan
% 0.29/0.61      = ( ^ [Y6: real] :
% 0.29/0.61            ( the_real
% 0.29/0.61            @ ^ [X4: real] :
% 0.29/0.61                ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.29/0.61                & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.61                & ( ( tan_real @ X4 )
% 0.29/0.61                  = Y6 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % arctan_def
% 0.29/0.61  thf(fact_9254_bij__betw__Suc,axiom,
% 0.29/0.61      ! [M2: set_nat,N4: set_nat] :
% 0.29/0.61        ( ( bij_betw_nat_nat @ suc @ M2 @ N4 )
% 0.29/0.61        = ( ( image_nat_nat @ suc @ M2 )
% 0.29/0.61          = N4 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bij_betw_Suc
% 0.29/0.61  thf(fact_9255_image__int__atLeastAtMost,axiom,
% 0.29/0.61      ! [A: nat,B: nat] :
% 0.29/0.61        ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.29/0.61        = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % image_int_atLeastAtMost
% 0.29/0.61  thf(fact_9256_ln__real__def,axiom,
% 0.29/0.61      ( ln_ln_real
% 0.29/0.61      = ( ^ [X4: real] :
% 0.29/0.61            ( the_real
% 0.29/0.61            @ ^ [U2: real] :
% 0.29/0.61                ( ( exp_real @ U2 )
% 0.29/0.61                = X4 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % ln_real_def
% 0.29/0.61  thf(fact_9257_bset_I1_J,axiom,
% 0.29/0.61      ! [D2: int,B2: set_int,P: int > $o,Q: int > $o] :
% 0.29/0.61        ( ! [X3: int] :
% 0.29/0.61            ( ! [Xa2: int] :
% 0.29/0.61                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb2: int] :
% 0.29/0.61                    ( ( member_int @ Xb2 @ B2 )
% 0.29/0.61                   => ( X3
% 0.29/0.61                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.29/0.61           => ( ( P @ X3 )
% 0.29/0.61             => ( P @ ( minus_minus_int @ X3 @ D2 ) ) ) )
% 0.29/0.61       => ( ! [X3: int] :
% 0.29/0.61              ( ! [Xa2: int] :
% 0.29/0.61                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb2: int] :
% 0.29/0.61                      ( ( member_int @ Xb2 @ B2 )
% 0.29/0.61                     => ( X3
% 0.29/0.61                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.29/0.61             => ( ( Q @ X3 )
% 0.29/0.61               => ( Q @ ( minus_minus_int @ X3 @ D2 ) ) ) )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ B2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( ( P @ X2 )
% 0.29/0.61                  & ( Q @ X2 ) )
% 0.29/0.61               => ( ( P @ ( minus_minus_int @ X2 @ D2 ) )
% 0.29/0.61                  & ( Q @ ( minus_minus_int @ X2 @ D2 ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bset(1)
% 0.29/0.61  thf(fact_9258_bset_I2_J,axiom,
% 0.29/0.61      ! [D2: int,B2: set_int,P: int > $o,Q: int > $o] :
% 0.29/0.61        ( ! [X3: int] :
% 0.29/0.61            ( ! [Xa2: int] :
% 0.29/0.61                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb2: int] :
% 0.29/0.61                    ( ( member_int @ Xb2 @ B2 )
% 0.29/0.61                   => ( X3
% 0.29/0.61                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.29/0.61           => ( ( P @ X3 )
% 0.29/0.61             => ( P @ ( minus_minus_int @ X3 @ D2 ) ) ) )
% 0.29/0.61       => ( ! [X3: int] :
% 0.29/0.61              ( ! [Xa2: int] :
% 0.29/0.61                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb2: int] :
% 0.29/0.61                      ( ( member_int @ Xb2 @ B2 )
% 0.29/0.61                     => ( X3
% 0.29/0.61                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.29/0.61             => ( ( Q @ X3 )
% 0.29/0.61               => ( Q @ ( minus_minus_int @ X3 @ D2 ) ) ) )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ B2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( ( P @ X2 )
% 0.29/0.61                  | ( Q @ X2 ) )
% 0.29/0.61               => ( ( P @ ( minus_minus_int @ X2 @ D2 ) )
% 0.29/0.61                  | ( Q @ ( minus_minus_int @ X2 @ D2 ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bset(2)
% 0.29/0.61  thf(fact_9259_aset_I1_J,axiom,
% 0.29/0.61      ! [D2: int,A2: set_int,P: int > $o,Q: int > $o] :
% 0.29/0.61        ( ! [X3: int] :
% 0.29/0.61            ( ! [Xa2: int] :
% 0.29/0.61                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb2: int] :
% 0.29/0.61                    ( ( member_int @ Xb2 @ A2 )
% 0.29/0.61                   => ( X3
% 0.29/0.61                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.29/0.61           => ( ( P @ X3 )
% 0.29/0.61             => ( P @ ( plus_plus_int @ X3 @ D2 ) ) ) )
% 0.29/0.61       => ( ! [X3: int] :
% 0.29/0.61              ( ! [Xa2: int] :
% 0.29/0.61                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb2: int] :
% 0.29/0.61                      ( ( member_int @ Xb2 @ A2 )
% 0.29/0.61                     => ( X3
% 0.29/0.61                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.29/0.61             => ( ( Q @ X3 )
% 0.29/0.61               => ( Q @ ( plus_plus_int @ X3 @ D2 ) ) ) )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ A2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( ( P @ X2 )
% 0.29/0.61                  & ( Q @ X2 ) )
% 0.29/0.61               => ( ( P @ ( plus_plus_int @ X2 @ D2 ) )
% 0.29/0.61                  & ( Q @ ( plus_plus_int @ X2 @ D2 ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % aset(1)
% 0.29/0.61  thf(fact_9260_aset_I2_J,axiom,
% 0.29/0.61      ! [D2: int,A2: set_int,P: int > $o,Q: int > $o] :
% 0.29/0.61        ( ! [X3: int] :
% 0.29/0.61            ( ! [Xa2: int] :
% 0.29/0.61                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb2: int] :
% 0.29/0.61                    ( ( member_int @ Xb2 @ A2 )
% 0.29/0.61                   => ( X3
% 0.29/0.61                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.29/0.61           => ( ( P @ X3 )
% 0.29/0.61             => ( P @ ( plus_plus_int @ X3 @ D2 ) ) ) )
% 0.29/0.61       => ( ! [X3: int] :
% 0.29/0.61              ( ! [Xa2: int] :
% 0.29/0.61                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb2: int] :
% 0.29/0.61                      ( ( member_int @ Xb2 @ A2 )
% 0.29/0.61                     => ( X3
% 0.29/0.61                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.29/0.61             => ( ( Q @ X3 )
% 0.29/0.61               => ( Q @ ( plus_plus_int @ X3 @ D2 ) ) ) )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ A2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( ( P @ X2 )
% 0.29/0.61                  | ( Q @ X2 ) )
% 0.29/0.61               => ( ( P @ ( plus_plus_int @ X2 @ D2 ) )
% 0.29/0.61                  | ( Q @ ( plus_plus_int @ X2 @ D2 ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % aset(2)
% 0.29/0.61  thf(fact_9261_ln__neg__is__const,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.61       => ( ( ln_ln_real @ X )
% 0.29/0.61          = ( the_real
% 0.29/0.61            @ ^ [X4: real] : $false ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % ln_neg_is_const
% 0.29/0.61  thf(fact_9262_bset_I9_J,axiom,
% 0.29/0.61      ! [D: int,D2: int,B2: set_int,T: int] :
% 0.29/0.61        ( ( dvd_dvd_int @ D @ D2 )
% 0.29/0.61       => ! [X2: int] :
% 0.29/0.61            ( ! [Xa3: int] :
% 0.29/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb3: int] :
% 0.29/0.61                    ( ( member_int @ Xb3 @ B2 )
% 0.29/0.61                   => ( X2
% 0.29/0.61                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61           => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 0.29/0.61             => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bset(9)
% 0.29/0.61  thf(fact_9263_bset_I10_J,axiom,
% 0.29/0.61      ! [D: int,D2: int,B2: set_int,T: int] :
% 0.29/0.61        ( ( dvd_dvd_int @ D @ D2 )
% 0.29/0.61       => ! [X2: int] :
% 0.29/0.61            ( ! [Xa3: int] :
% 0.29/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb3: int] :
% 0.29/0.61                    ( ( member_int @ Xb3 @ B2 )
% 0.29/0.61                   => ( X2
% 0.29/0.61                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61           => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 0.29/0.61             => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bset(10)
% 0.29/0.61  thf(fact_9264_aset_I9_J,axiom,
% 0.29/0.61      ! [D: int,D2: int,A2: set_int,T: int] :
% 0.29/0.61        ( ( dvd_dvd_int @ D @ D2 )
% 0.29/0.61       => ! [X2: int] :
% 0.29/0.61            ( ! [Xa3: int] :
% 0.29/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb3: int] :
% 0.29/0.61                    ( ( member_int @ Xb3 @ A2 )
% 0.29/0.61                   => ( X2
% 0.29/0.61                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61           => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 0.29/0.61             => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % aset(9)
% 0.29/0.61  thf(fact_9265_aset_I10_J,axiom,
% 0.29/0.61      ! [D: int,D2: int,A2: set_int,T: int] :
% 0.29/0.61        ( ( dvd_dvd_int @ D @ D2 )
% 0.29/0.61       => ! [X2: int] :
% 0.29/0.61            ( ! [Xa3: int] :
% 0.29/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb3: int] :
% 0.29/0.61                    ( ( member_int @ Xb3 @ A2 )
% 0.29/0.61                   => ( X2
% 0.29/0.61                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61           => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 0.29/0.61             => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % aset(10)
% 0.29/0.61  thf(fact_9266_atLeastAtMostPlus1__int__conv,axiom,
% 0.29/0.61      ! [M: int,N: int] :
% 0.29/0.61        ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.29/0.61       => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.29/0.61          = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeastAtMostPlus1_int_conv
% 0.29/0.61  thf(fact_9267_periodic__finite__ex,axiom,
% 0.29/0.61      ! [D: int,P: int > $o] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.61       => ( ! [X3: int,K3: int] :
% 0.29/0.61              ( ( P @ X3 )
% 0.29/0.61              = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.29/0.61         => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.29/0.61            = ( ? [X4: int] :
% 0.29/0.61                  ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.29/0.61                  & ( P @ X4 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % periodic_finite_ex
% 0.29/0.61  thf(fact_9268_aset_I7_J,axiom,
% 0.29/0.61      ! [D2: int,A2: set_int,T: int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ! [X2: int] :
% 0.29/0.61            ( ! [Xa3: int] :
% 0.29/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb3: int] :
% 0.29/0.61                    ( ( member_int @ Xb3 @ A2 )
% 0.29/0.61                   => ( X2
% 0.29/0.61                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61           => ( ( ord_less_int @ T @ X2 )
% 0.29/0.61             => ( ord_less_int @ T @ ( plus_plus_int @ X2 @ D2 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % aset(7)
% 0.29/0.61  thf(fact_9269_aset_I5_J,axiom,
% 0.29/0.61      ! [D2: int,T: int,A2: set_int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ( ( member_int @ T @ A2 )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ A2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( ord_less_int @ X2 @ T )
% 0.29/0.61               => ( ord_less_int @ ( plus_plus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % aset(5)
% 0.29/0.61  thf(fact_9270_aset_I4_J,axiom,
% 0.29/0.61      ! [D2: int,T: int,A2: set_int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ( ( member_int @ T @ A2 )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ A2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( X2 != T )
% 0.29/0.61               => ( ( plus_plus_int @ X2 @ D2 )
% 0.29/0.61                 != T ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % aset(4)
% 0.29/0.61  thf(fact_9271_aset_I3_J,axiom,
% 0.29/0.61      ! [D2: int,T: int,A2: set_int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ A2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( X2 = T )
% 0.29/0.61               => ( ( plus_plus_int @ X2 @ D2 )
% 0.29/0.61                  = T ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % aset(3)
% 0.29/0.61  thf(fact_9272_bset_I7_J,axiom,
% 0.29/0.61      ! [D2: int,T: int,B2: set_int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ( ( member_int @ T @ B2 )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ B2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( ord_less_int @ T @ X2 )
% 0.29/0.61               => ( ord_less_int @ T @ ( minus_minus_int @ X2 @ D2 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bset(7)
% 0.29/0.61  thf(fact_9273_bset_I5_J,axiom,
% 0.29/0.61      ! [D2: int,B2: set_int,T: int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ! [X2: int] :
% 0.29/0.61            ( ! [Xa3: int] :
% 0.29/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb3: int] :
% 0.29/0.61                    ( ( member_int @ Xb3 @ B2 )
% 0.29/0.61                   => ( X2
% 0.29/0.61                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61           => ( ( ord_less_int @ X2 @ T )
% 0.29/0.61             => ( ord_less_int @ ( minus_minus_int @ X2 @ D2 ) @ T ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bset(5)
% 0.29/0.61  thf(fact_9274_bset_I4_J,axiom,
% 0.29/0.61      ! [D2: int,T: int,B2: set_int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ( ( member_int @ T @ B2 )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ B2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( X2 != T )
% 0.29/0.61               => ( ( minus_minus_int @ X2 @ D2 )
% 0.29/0.61                 != T ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bset(4)
% 0.29/0.61  thf(fact_9275_bset_I3_J,axiom,
% 0.29/0.61      ! [D2: int,T: int,B2: set_int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ B2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( X2 = T )
% 0.29/0.61               => ( ( minus_minus_int @ X2 @ D2 )
% 0.29/0.61                  = T ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bset(3)
% 0.29/0.61  thf(fact_9276_fact__eq__fact__times,axiom,
% 0.29/0.61      ! [N: nat,M: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.61       => ( ( semiri1408675320244567234ct_nat @ M )
% 0.29/0.61          = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 0.29/0.61            @ ( groups708209901874060359at_nat
% 0.29/0.61              @ ^ [X4: nat] : X4
% 0.29/0.61              @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % fact_eq_fact_times
% 0.29/0.61  thf(fact_9277_simp__from__to,axiom,
% 0.29/0.61      ( set_or1266510415728281911st_int
% 0.29/0.61      = ( ^ [I3: int,J2: int] : ( if_set_int @ ( ord_less_int @ J2 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % simp_from_to
% 0.29/0.61  thf(fact_9278_bset_I6_J,axiom,
% 0.29/0.61      ! [D2: int,B2: set_int,T: int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ! [X2: int] :
% 0.29/0.61            ( ! [Xa3: int] :
% 0.29/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb3: int] :
% 0.29/0.61                    ( ( member_int @ Xb3 @ B2 )
% 0.29/0.61                   => ( X2
% 0.29/0.61                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61           => ( ( ord_less_eq_int @ X2 @ T )
% 0.29/0.61             => ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D2 ) @ T ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bset(6)
% 0.29/0.61  thf(fact_9279_bset_I8_J,axiom,
% 0.29/0.61      ! [D2: int,T: int,B2: set_int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ B2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( ord_less_eq_int @ T @ X2 )
% 0.29/0.61               => ( ord_less_eq_int @ T @ ( minus_minus_int @ X2 @ D2 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bset(8)
% 0.29/0.61  thf(fact_9280_aset_I6_J,axiom,
% 0.29/0.61      ! [D2: int,T: int,A2: set_int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.29/0.61         => ! [X2: int] :
% 0.29/0.61              ( ! [Xa3: int] :
% 0.29/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                 => ! [Xb3: int] :
% 0.29/0.61                      ( ( member_int @ Xb3 @ A2 )
% 0.29/0.61                     => ( X2
% 0.29/0.61                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61             => ( ( ord_less_eq_int @ X2 @ T )
% 0.29/0.61               => ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % aset(6)
% 0.29/0.61  thf(fact_9281_aset_I8_J,axiom,
% 0.29/0.61      ! [D2: int,A2: set_int,T: int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ! [X2: int] :
% 0.29/0.61            ( ! [Xa3: int] :
% 0.29/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61               => ! [Xb3: int] :
% 0.29/0.61                    ( ( member_int @ Xb3 @ A2 )
% 0.29/0.61                   => ( X2
% 0.29/0.61                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.29/0.61           => ( ( ord_less_eq_int @ T @ X2 )
% 0.29/0.61             => ( ord_less_eq_int @ T @ ( plus_plus_int @ X2 @ D2 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % aset(8)
% 0.29/0.61  thf(fact_9282_cpmi,axiom,
% 0.29/0.61      ! [D2: int,P: int > $o,P6: int > $o,B2: set_int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ( ? [Z5: int] :
% 0.29/0.61            ! [X3: int] :
% 0.29/0.61              ( ( ord_less_int @ X3 @ Z5 )
% 0.29/0.61             => ( ( P @ X3 )
% 0.29/0.61                = ( P6 @ X3 ) ) )
% 0.29/0.61         => ( ! [X3: int] :
% 0.29/0.61                ( ! [Xa2: int] :
% 0.29/0.61                    ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                   => ! [Xb2: int] :
% 0.29/0.61                        ( ( member_int @ Xb2 @ B2 )
% 0.29/0.61                       => ( X3
% 0.29/0.61                         != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.29/0.61               => ( ( P @ X3 )
% 0.29/0.61                 => ( P @ ( minus_minus_int @ X3 @ D2 ) ) ) )
% 0.29/0.61           => ( ! [X3: int,K3: int] :
% 0.29/0.61                  ( ( P6 @ X3 )
% 0.29/0.61                  = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D2 ) ) ) )
% 0.29/0.61             => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.29/0.61                = ( ? [X4: int] :
% 0.29/0.61                      ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                      & ( P6 @ X4 ) )
% 0.29/0.61                  | ? [X4: int] :
% 0.29/0.61                      ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                      & ? [Y6: int] :
% 0.29/0.61                          ( ( member_int @ Y6 @ B2 )
% 0.29/0.61                          & ( P @ ( plus_plus_int @ Y6 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cpmi
% 0.29/0.61  thf(fact_9283_cppi,axiom,
% 0.29/0.61      ! [D2: int,P: int > $o,P6: int > $o,A2: set_int] :
% 0.29/0.61        ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.29/0.61       => ( ? [Z5: int] :
% 0.29/0.61            ! [X3: int] :
% 0.29/0.61              ( ( ord_less_int @ Z5 @ X3 )
% 0.29/0.61             => ( ( P @ X3 )
% 0.29/0.61                = ( P6 @ X3 ) ) )
% 0.29/0.61         => ( ! [X3: int] :
% 0.29/0.61                ( ! [Xa2: int] :
% 0.29/0.61                    ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                   => ! [Xb2: int] :
% 0.29/0.61                        ( ( member_int @ Xb2 @ A2 )
% 0.29/0.61                       => ( X3
% 0.29/0.61                         != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.29/0.61               => ( ( P @ X3 )
% 0.29/0.61                 => ( P @ ( plus_plus_int @ X3 @ D2 ) ) ) )
% 0.29/0.61           => ( ! [X3: int,K3: int] :
% 0.29/0.61                  ( ( P6 @ X3 )
% 0.29/0.61                  = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D2 ) ) ) )
% 0.29/0.61             => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.29/0.61                = ( ? [X4: int] :
% 0.29/0.61                      ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                      & ( P6 @ X4 ) )
% 0.29/0.61                  | ? [X4: int] :
% 0.29/0.61                      ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.29/0.61                      & ? [Y6: int] :
% 0.29/0.61                          ( ( member_int @ Y6 @ A2 )
% 0.29/0.61                          & ( P @ ( minus_minus_int @ Y6 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cppi
% 0.29/0.61  thf(fact_9284_arccos__def,axiom,
% 0.29/0.61      ( arccos
% 0.29/0.61      = ( ^ [Y6: real] :
% 0.29/0.61            ( the_real
% 0.29/0.61            @ ^ [X4: real] :
% 0.29/0.61                ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.29/0.61                & ( ord_less_eq_real @ X4 @ pi )
% 0.29/0.61                & ( ( cos_real @ X4 )
% 0.29/0.61                  = Y6 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % arccos_def
% 0.29/0.61  thf(fact_9285_fact__div__fact,axiom,
% 0.29/0.61      ! [N: nat,M: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.61       => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.29/0.61          = ( groups708209901874060359at_nat
% 0.29/0.61            @ ^ [X4: nat] : X4
% 0.29/0.61            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % fact_div_fact
% 0.29/0.61  thf(fact_9286_divmod__step__nat__def,axiom,
% 0.29/0.61      ( unique5026877609467782581ep_nat
% 0.29/0.61      = ( ^ [L3: num] :
% 0.29/0.61            ( produc2626176000494625587at_nat
% 0.29/0.61            @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ 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 @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_step_nat_def
% 0.29/0.61  thf(fact_9287_divmod__step__int__def,axiom,
% 0.29/0.61      ( unique5024387138958732305ep_int
% 0.29/0.61      = ( ^ [L3: num] :
% 0.29/0.61            ( produc4245557441103728435nt_int
% 0.29/0.61            @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ 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 @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_step_int_def
% 0.29/0.61  thf(fact_9288_pi__half,axiom,
% 0.29/0.61      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.61      = ( the_real
% 0.29/0.61        @ ^ [X4: real] :
% 0.29/0.61            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.29/0.61            & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.61            & ( ( cos_real @ X4 )
% 0.29/0.61              = zero_zero_real ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % pi_half
% 0.29/0.61  thf(fact_9289_pi__def,axiom,
% 0.29/0.61      ( pi
% 0.29/0.61      = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.29/0.61        @ ( the_real
% 0.29/0.61          @ ^ [X4: real] :
% 0.29/0.61              ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.29/0.61              & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.61              & ( ( cos_real @ X4 )
% 0.29/0.61                = zero_zero_real ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % pi_def
% 0.29/0.61  thf(fact_9290_Sum__Icc__int,axiom,
% 0.29/0.61      ! [M: int,N: int] :
% 0.29/0.61        ( ( ord_less_eq_int @ M @ N )
% 0.29/0.61       => ( ( groups4538972089207619220nt_int
% 0.29/0.61            @ ^ [X4: int] : X4
% 0.29/0.61            @ ( set_or1266510415728281911st_int @ M @ N ) )
% 0.29/0.61          = ( 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 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Sum_Icc_int
% 0.29/0.61  thf(fact_9291_arcsin__def,axiom,
% 0.29/0.61      ( arcsin
% 0.29/0.61      = ( ^ [Y6: real] :
% 0.29/0.61            ( the_real
% 0.29/0.61            @ ^ [X4: real] :
% 0.29/0.61                ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.29/0.61                & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.61                & ( ( sin_real @ X4 )
% 0.29/0.61                  = Y6 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % arcsin_def
% 0.29/0.61  thf(fact_9292_bij__betw__nth__root__unity,axiom,
% 0.29/0.61      ! [C: complex,N: nat] :
% 0.29/0.61        ( ( C != zero_zero_complex )
% 0.29/0.61       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61         => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.29/0.61            @ ( collect_complex
% 0.29/0.61              @ ^ [Z4: complex] :
% 0.29/0.61                  ( ( power_power_complex @ Z4 @ N )
% 0.29/0.61                  = one_one_complex ) )
% 0.29/0.61            @ ( collect_complex
% 0.29/0.61              @ ^ [Z4: complex] :
% 0.29/0.61                  ( ( power_power_complex @ Z4 @ N )
% 0.29/0.61                  = C ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bij_betw_nth_root_unity
% 0.29/0.61  thf(fact_9293_divmod__nat__if,axiom,
% 0.29/0.61      ( divmod_nat
% 0.29/0.61      = ( ^ [M6: nat,N3: nat] :
% 0.29/0.61            ( if_Pro6206227464963214023at_nat
% 0.29/0.61            @ ( ( N3 = zero_zero_nat )
% 0.29/0.61              | ( ord_less_nat @ M6 @ N3 ) )
% 0.29/0.61            @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
% 0.29/0.61            @ ( produc2626176000494625587at_nat
% 0.29/0.61              @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
% 0.29/0.61              @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_nat_if
% 0.29/0.61  thf(fact_9294_set__encode__def,axiom,
% 0.29/0.61      ( nat_set_encode
% 0.29/0.61      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % set_encode_def
% 0.29/0.61  thf(fact_9295_real__root__zero,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( root @ N @ zero_zero_real )
% 0.29/0.61        = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_zero
% 0.29/0.61  thf(fact_9296_real__root__Suc__0,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 0.29/0.61        = X ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_Suc_0
% 0.29/0.61  thf(fact_9297_root__0,axiom,
% 0.29/0.61      ! [X: real] :
% 0.29/0.61        ( ( root @ zero_zero_nat @ X )
% 0.29/0.61        = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % root_0
% 0.29/0.61  thf(fact_9298_real__root__eq__iff,axiom,
% 0.29/0.61      ! [N: nat,X: real,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ( root @ N @ X )
% 0.29/0.61            = ( root @ N @ Y ) )
% 0.29/0.61          = ( X = Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_eq_iff
% 0.29/0.61  thf(fact_9299_set__encode__empty,axiom,
% 0.29/0.61      ( ( nat_set_encode @ bot_bot_set_nat )
% 0.29/0.61      = zero_zero_nat ) ).
% 0.29/0.61  
% 0.29/0.61  % set_encode_empty
% 0.29/0.61  thf(fact_9300_real__root__eq__0__iff,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ( root @ N @ X )
% 0.29/0.61            = zero_zero_real )
% 0.29/0.61          = ( X = zero_zero_real ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_eq_0_iff
% 0.29/0.61  thf(fact_9301_real__root__less__iff,axiom,
% 0.29/0.61      ! [N: nat,X: real,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 0.29/0.61          = ( ord_less_real @ X @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_less_iff
% 0.29/0.61  thf(fact_9302_real__root__le__iff,axiom,
% 0.29/0.61      ! [N: nat,X: real,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 0.29/0.61          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_le_iff
% 0.29/0.61  thf(fact_9303_real__root__one,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( root @ N @ one_one_real )
% 0.29/0.61          = one_one_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_one
% 0.29/0.61  thf(fact_9304_real__root__eq__1__iff,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ( root @ N @ X )
% 0.29/0.61            = one_one_real )
% 0.29/0.61          = ( X = one_one_real ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_eq_1_iff
% 0.29/0.61  thf(fact_9305_real__root__lt__0__iff,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
% 0.29/0.61          = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_lt_0_iff
% 0.29/0.61  thf(fact_9306_real__root__gt__0__iff,axiom,
% 0.29/0.61      ! [N: nat,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
% 0.29/0.61          = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_gt_0_iff
% 0.29/0.61  thf(fact_9307_real__root__le__0__iff,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
% 0.29/0.61          = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_le_0_iff
% 0.29/0.61  thf(fact_9308_real__root__ge__0__iff,axiom,
% 0.29/0.61      ! [N: nat,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
% 0.29/0.61          = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_ge_0_iff
% 0.29/0.61  thf(fact_9309_real__root__gt__1__iff,axiom,
% 0.29/0.61      ! [N: nat,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
% 0.29/0.61          = ( ord_less_real @ one_one_real @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_gt_1_iff
% 0.29/0.61  thf(fact_9310_real__root__lt__1__iff,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
% 0.29/0.61          = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_lt_1_iff
% 0.29/0.61  thf(fact_9311_real__root__ge__1__iff,axiom,
% 0.29/0.61      ! [N: nat,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
% 0.29/0.61          = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_ge_1_iff
% 0.29/0.61  thf(fact_9312_real__root__le__1__iff,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
% 0.29/0.61          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_le_1_iff
% 0.29/0.61  thf(fact_9313_real__root__pow__pos2,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.61         => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.29/0.61            = X ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_pow_pos2
% 0.29/0.61  thf(fact_9314_real__root__divide,axiom,
% 0.29/0.61      ! [N: nat,X: real,Y: real] :
% 0.29/0.61        ( ( root @ N @ ( divide_divide_real @ X @ Y ) )
% 0.29/0.61        = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_divide
% 0.29/0.61  thf(fact_9315_real__root__pos__pos__le,axiom,
% 0.29/0.61      ! [X: real,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.61       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_pos_pos_le
% 0.29/0.61  thf(fact_9316_prod__int__eq,axiom,
% 0.29/0.61      ! [I: nat,J: nat] :
% 0.29/0.61        ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.29/0.61        = ( groups1705073143266064639nt_int
% 0.29/0.61          @ ^ [X4: int] : X4
% 0.29/0.61          @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % prod_int_eq
% 0.29/0.61  thf(fact_9317_real__root__less__mono,axiom,
% 0.29/0.61      ! [N: nat,X: real,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ X @ Y )
% 0.29/0.61         => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_less_mono
% 0.29/0.61  thf(fact_9318_real__root__le__mono,axiom,
% 0.29/0.61      ! [N: nat,X: real,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.61         => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_le_mono
% 0.29/0.61  thf(fact_9319_real__root__power,axiom,
% 0.29/0.61      ! [N: nat,X: real,K: nat] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( root @ N @ ( power_power_real @ X @ K ) )
% 0.29/0.61          = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_power
% 0.29/0.61  thf(fact_9320_real__root__abs,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( root @ N @ ( abs_abs_real @ X ) )
% 0.29/0.61          = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_abs
% 0.29/0.61  thf(fact_9321_sgn__root,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( sgn_sgn_real @ ( root @ N @ X ) )
% 0.29/0.61          = ( sgn_sgn_real @ X ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sgn_root
% 0.29/0.61  thf(fact_9322_prod__int__plus__eq,axiom,
% 0.29/0.61      ! [I: nat,J: nat] :
% 0.29/0.61        ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 0.29/0.61        = ( groups1705073143266064639nt_int
% 0.29/0.61          @ ^ [X4: int] : X4
% 0.29/0.61          @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % prod_int_plus_eq
% 0.29/0.61  thf(fact_9323_real__root__gt__zero,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.61         => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_gt_zero
% 0.29/0.61  thf(fact_9324_real__root__strict__decreasing,axiom,
% 0.29/0.61      ! [N: nat,N4: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_nat @ N @ N4 )
% 0.29/0.61         => ( ( ord_less_real @ one_one_real @ X )
% 0.29/0.61           => ( ord_less_real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_strict_decreasing
% 0.29/0.61  thf(fact_9325_sqrt__def,axiom,
% 0.29/0.61      ( sqrt
% 0.29/0.61      = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sqrt_def
% 0.29/0.61  thf(fact_9326_root__abs__power,axiom,
% 0.29/0.61      ! [N: nat,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
% 0.29/0.61          = ( abs_abs_real @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % root_abs_power
% 0.29/0.61  thf(fact_9327_real__root__pos__pos,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.61         => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_pos_pos
% 0.29/0.61  thf(fact_9328_real__root__strict__increasing,axiom,
% 0.29/0.61      ! [N: nat,N4: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_nat @ N @ N4 )
% 0.29/0.61         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.61           => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.61             => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_strict_increasing
% 0.29/0.61  thf(fact_9329_real__root__decreasing,axiom,
% 0.29/0.61      ! [N: nat,N4: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_nat @ N @ N4 )
% 0.29/0.61         => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.29/0.61           => ( ord_less_eq_real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_decreasing
% 0.29/0.61  thf(fact_9330_real__root__pow__pos,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.61         => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.29/0.61            = X ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_pow_pos
% 0.29/0.61  thf(fact_9331_real__root__power__cancel,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.61         => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 0.29/0.61            = X ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_power_cancel
% 0.29/0.61  thf(fact_9332_real__root__pos__unique,axiom,
% 0.29/0.61      ! [N: nat,Y: real,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.29/0.61         => ( ( ( power_power_real @ Y @ N )
% 0.29/0.61              = X )
% 0.29/0.61           => ( ( root @ N @ X )
% 0.29/0.61              = Y ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_pos_unique
% 0.29/0.61  thf(fact_9333_odd__real__root__power__cancel,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.61       => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 0.29/0.61          = X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % odd_real_root_power_cancel
% 0.29/0.61  thf(fact_9334_odd__real__root__unique,axiom,
% 0.29/0.61      ! [N: nat,Y: real,X: real] :
% 0.29/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.61       => ( ( ( power_power_real @ Y @ N )
% 0.29/0.61            = X )
% 0.29/0.61         => ( ( root @ N @ X )
% 0.29/0.61            = Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % odd_real_root_unique
% 0.29/0.61  thf(fact_9335_odd__real__root__pow,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.61       => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.29/0.61          = X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % odd_real_root_pow
% 0.29/0.61  thf(fact_9336_real__root__increasing,axiom,
% 0.29/0.61      ! [N: nat,N4: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_eq_nat @ N @ N4 )
% 0.29/0.61         => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.61           => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.61             => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_root_increasing
% 0.29/0.61  thf(fact_9337_root__sgn__power,axiom,
% 0.29/0.61      ! [N: nat,Y: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
% 0.29/0.61          = Y ) ) ).
% 0.29/0.61  
% 0.29/0.61  % root_sgn_power
% 0.29/0.61  thf(fact_9338_sgn__power__root,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
% 0.29/0.61          = X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sgn_power_root
% 0.29/0.61  thf(fact_9339_log__root,axiom,
% 0.29/0.61      ! [N: nat,A: real,B: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.61         => ( ( log @ B @ ( root @ N @ A ) )
% 0.29/0.61            = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % log_root
% 0.29/0.61  thf(fact_9340_log__base__root,axiom,
% 0.29/0.61      ! [N: nat,B: real,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.61         => ( ( log @ ( root @ N @ B ) @ X )
% 0.29/0.61            = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % log_base_root
% 0.29/0.61  thf(fact_9341_ln__root,axiom,
% 0.29/0.61      ! [N: nat,B: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.61         => ( ( ln_ln_real @ ( root @ N @ B ) )
% 0.29/0.61            = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % ln_root
% 0.29/0.61  thf(fact_9342_split__root,axiom,
% 0.29/0.61      ! [P: real > $o,N: nat,X: real] :
% 0.29/0.61        ( ( P @ ( root @ N @ X ) )
% 0.29/0.61        = ( ( ( N = zero_zero_nat )
% 0.29/0.61           => ( P @ zero_zero_real ) )
% 0.29/0.61          & ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61           => ! [Y6: real] :
% 0.29/0.61                ( ( ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 0.29/0.61                  = X )
% 0.29/0.61               => ( P @ Y6 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % split_root
% 0.29/0.61  thf(fact_9343_divmod__nat__def,axiom,
% 0.29/0.61      ( divmod_nat
% 0.29/0.61      = ( ^ [M6: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N3 ) @ ( modulo_modulo_nat @ M6 @ N3 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_nat_def
% 0.29/0.61  thf(fact_9344_floor__real__def,axiom,
% 0.29/0.61      ( archim6058952711729229775r_real
% 0.29/0.61      = ( ^ [X4: real] :
% 0.29/0.61            ( the_int
% 0.29/0.61            @ ^ [Z4: int] :
% 0.29/0.61                ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X4 )
% 0.29/0.61                & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % floor_real_def
% 0.29/0.61  thf(fact_9345_root__powr__inverse,axiom,
% 0.29/0.61      ! [N: nat,X: real] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.61         => ( ( root @ N @ X )
% 0.29/0.61            = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % root_powr_inverse
% 0.29/0.61  thf(fact_9346_set__encode__insert,axiom,
% 0.29/0.61      ! [A2: set_nat,N: nat] :
% 0.29/0.61        ( ( finite_finite_nat @ A2 )
% 0.29/0.61       => ( ~ ( member_nat @ N @ A2 )
% 0.29/0.61         => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
% 0.29/0.61            = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % set_encode_insert
% 0.29/0.61  thf(fact_9347_floor__rat__def,axiom,
% 0.29/0.61      ( archim3151403230148437115or_rat
% 0.29/0.61      = ( ^ [X4: rat] :
% 0.29/0.61            ( the_int
% 0.29/0.61            @ ^ [Z4: int] :
% 0.29/0.61                ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X4 )
% 0.29/0.61                & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % floor_rat_def
% 0.29/0.61  thf(fact_9348_VEBT_Osize_I3_J,axiom,
% 0.29/0.61      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.29/0.61        ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.29/0.61        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT.size(3)
% 0.29/0.61  thf(fact_9349_finite__M__bounded__by__nat,axiom,
% 0.29/0.61      ! [P: nat > $o,I: nat] :
% 0.29/0.61        ( finite_finite_nat
% 0.29/0.61        @ ( collect_nat
% 0.29/0.61          @ ^ [K2: nat] :
% 0.29/0.61              ( ( P @ K2 )
% 0.29/0.61              & ( ord_less_nat @ K2 @ I ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_M_bounded_by_nat
% 0.29/0.61  thf(fact_9350_finite__less__ub,axiom,
% 0.29/0.61      ! [F: nat > nat,U: nat] :
% 0.29/0.61        ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
% 0.29/0.61       => ( finite_finite_nat
% 0.29/0.61          @ ( collect_nat
% 0.29/0.61            @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_less_ub
% 0.29/0.61  thf(fact_9351_less__eq__rat__def,axiom,
% 0.29/0.61      ( ord_less_eq_rat
% 0.29/0.61      = ( ^ [X4: rat,Y6: rat] :
% 0.29/0.61            ( ( ord_less_rat @ X4 @ Y6 )
% 0.29/0.61            | ( X4 = Y6 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % less_eq_rat_def
% 0.29/0.61  thf(fact_9352_finite__nat__set__iff__bounded__le,axiom,
% 0.29/0.61      ( finite_finite_nat
% 0.29/0.61      = ( ^ [N7: set_nat] :
% 0.29/0.61          ? [M6: nat] :
% 0.29/0.61          ! [X4: nat] :
% 0.29/0.61            ( ( member_nat @ X4 @ N7 )
% 0.29/0.61           => ( ord_less_eq_nat @ X4 @ M6 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_nat_set_iff_bounded_le
% 0.29/0.61  thf(fact_9353_finite__nat__set__iff__bounded,axiom,
% 0.29/0.61      ( finite_finite_nat
% 0.29/0.61      = ( ^ [N7: set_nat] :
% 0.29/0.61          ? [M6: nat] :
% 0.29/0.61          ! [X4: nat] :
% 0.29/0.61            ( ( member_nat @ X4 @ N7 )
% 0.29/0.61           => ( ord_less_nat @ X4 @ M6 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_nat_set_iff_bounded
% 0.29/0.61  thf(fact_9354_bounded__nat__set__is__finite,axiom,
% 0.29/0.61      ! [N4: set_nat,N: nat] :
% 0.29/0.61        ( ! [X3: nat] :
% 0.29/0.61            ( ( member_nat @ X3 @ N4 )
% 0.29/0.61           => ( ord_less_nat @ X3 @ N ) )
% 0.29/0.61       => ( finite_finite_nat @ N4 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bounded_nat_set_is_finite
% 0.29/0.61  thf(fact_9355_sgn__rat__def,axiom,
% 0.29/0.61      ( sgn_sgn_rat
% 0.29/0.61      = ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sgn_rat_def
% 0.29/0.61  thf(fact_9356_abs__rat__def,axiom,
% 0.29/0.61      ( abs_abs_rat
% 0.29/0.61      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % abs_rat_def
% 0.29/0.61  thf(fact_9357_obtain__pos__sum,axiom,
% 0.29/0.61      ! [R2: rat] :
% 0.29/0.61        ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.29/0.61       => ~ ! [S3: rat] :
% 0.29/0.61              ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 0.29/0.61             => ! [T3: rat] :
% 0.29/0.61                  ( ( ord_less_rat @ zero_zero_rat @ T3 )
% 0.29/0.61                 => ( R2
% 0.29/0.61                   != ( plus_plus_rat @ S3 @ T3 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % obtain_pos_sum
% 0.29/0.61  thf(fact_9358_set__encode__inf,axiom,
% 0.29/0.61      ! [A2: set_nat] :
% 0.29/0.61        ( ~ ( finite_finite_nat @ A2 )
% 0.29/0.61       => ( ( nat_set_encode @ A2 )
% 0.29/0.61          = zero_zero_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % set_encode_inf
% 0.29/0.61  thf(fact_9359_finite__divisors__nat,axiom,
% 0.29/0.61      ! [M: nat] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.61       => ( finite_finite_nat
% 0.29/0.61          @ ( collect_nat
% 0.29/0.61            @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_divisors_nat
% 0.29/0.61  thf(fact_9360_subset__eq__atLeast0__atMost__finite,axiom,
% 0.29/0.61      ! [N4: set_nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_set_nat @ N4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.29/0.61       => ( finite_finite_nat @ N4 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % subset_eq_atLeast0_atMost_finite
% 0.29/0.61  thf(fact_9361_even__set__encode__iff,axiom,
% 0.29/0.61      ! [A2: set_nat] :
% 0.29/0.61        ( ( finite_finite_nat @ A2 )
% 0.29/0.61       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 0.29/0.61          = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % even_set_encode_iff
% 0.29/0.61  thf(fact_9362_finite__Collect__le__nat,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( finite_finite_nat
% 0.29/0.61        @ ( collect_nat
% 0.29/0.61          @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_Collect_le_nat
% 0.29/0.61  thf(fact_9363_finite__Collect__less__nat,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( finite_finite_nat
% 0.29/0.61        @ ( collect_nat
% 0.29/0.61          @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_Collect_less_nat
% 0.29/0.61  thf(fact_9364_finite__interval__int1,axiom,
% 0.29/0.61      ! [A: int,B: int] :
% 0.29/0.61        ( finite_finite_int
% 0.29/0.61        @ ( collect_int
% 0.29/0.61          @ ^ [I3: int] :
% 0.29/0.61              ( ( ord_less_eq_int @ A @ I3 )
% 0.29/0.61              & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_interval_int1
% 0.29/0.61  thf(fact_9365_finite__interval__int4,axiom,
% 0.29/0.61      ! [A: int,B: int] :
% 0.29/0.61        ( finite_finite_int
% 0.29/0.61        @ ( collect_int
% 0.29/0.61          @ ^ [I3: int] :
% 0.29/0.61              ( ( ord_less_int @ A @ I3 )
% 0.29/0.61              & ( ord_less_int @ I3 @ B ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_interval_int4
% 0.29/0.61  thf(fact_9366_finite__interval__int3,axiom,
% 0.29/0.61      ! [A: int,B: int] :
% 0.29/0.61        ( finite_finite_int
% 0.29/0.61        @ ( collect_int
% 0.29/0.61          @ ^ [I3: int] :
% 0.29/0.61              ( ( ord_less_int @ A @ I3 )
% 0.29/0.61              & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_interval_int3
% 0.29/0.61  thf(fact_9367_finite__interval__int2,axiom,
% 0.29/0.61      ! [A: int,B: int] :
% 0.29/0.61        ( finite_finite_int
% 0.29/0.61        @ ( collect_int
% 0.29/0.61          @ ^ [I3: int] :
% 0.29/0.61              ( ( ord_less_eq_int @ A @ I3 )
% 0.29/0.61              & ( ord_less_int @ I3 @ B ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_interval_int2
% 0.29/0.61  thf(fact_9368_finite__nth__roots,axiom,
% 0.29/0.61      ! [N: nat,C: complex] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( finite3207457112153483333omplex
% 0.29/0.61          @ ( collect_complex
% 0.29/0.61            @ ^ [Z4: complex] :
% 0.29/0.61                ( ( power_power_complex @ Z4 @ N )
% 0.29/0.61                = C ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_nth_roots
% 0.29/0.61  thf(fact_9369_finite__divisors__int,axiom,
% 0.29/0.61      ! [I: int] :
% 0.29/0.61        ( ( I != zero_zero_int )
% 0.29/0.61       => ( finite_finite_int
% 0.29/0.61          @ ( collect_int
% 0.29/0.61            @ ^ [D3: int] : ( dvd_dvd_int @ D3 @ I ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_divisors_int
% 0.29/0.61  thf(fact_9370_VEBT_Osize__gen_I1_J,axiom,
% 0.29/0.61      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.29/0.61        ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.29/0.61        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT.size_gen(1)
% 0.29/0.61  thf(fact_9371_finite__nat__iff__bounded__le,axiom,
% 0.29/0.61      ( finite_finite_nat
% 0.29/0.61      = ( ^ [S4: set_nat] :
% 0.29/0.61          ? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_nat_iff_bounded_le
% 0.29/0.61  thf(fact_9372_finite__int__iff__bounded__le,axiom,
% 0.29/0.61      ( finite_finite_int
% 0.29/0.61      = ( ^ [S4: set_int] :
% 0.29/0.61          ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_int_iff_bounded_le
% 0.29/0.61  thf(fact_9373_finite__int__iff__bounded,axiom,
% 0.29/0.61      ( finite_finite_int
% 0.29/0.61      = ( ^ [S4: set_int] :
% 0.29/0.61          ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_int_iff_bounded
% 0.29/0.61  thf(fact_9374_VEBT_Osize__gen_I2_J,axiom,
% 0.29/0.61      ! [X21: $o,X222: $o] :
% 0.29/0.61        ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.29/0.61        = zero_zero_nat ) ).
% 0.29/0.61  
% 0.29/0.61  % VEBT.size_gen(2)
% 0.29/0.61  thf(fact_9375_infinite__int__iff__unbounded__le,axiom,
% 0.29/0.61      ! [S2: set_int] :
% 0.29/0.61        ( ( ~ ( finite_finite_int @ S2 ) )
% 0.29/0.61        = ( ! [M6: int] :
% 0.29/0.61            ? [N3: int] :
% 0.29/0.61              ( ( ord_less_eq_int @ M6 @ ( abs_abs_int @ N3 ) )
% 0.29/0.61              & ( member_int @ N3 @ S2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % infinite_int_iff_unbounded_le
% 0.29/0.61  thf(fact_9376_infinite__int__iff__unbounded,axiom,
% 0.29/0.61      ! [S2: set_int] :
% 0.29/0.61        ( ( ~ ( finite_finite_int @ S2 ) )
% 0.29/0.61        = ( ! [M6: int] :
% 0.29/0.61            ? [N3: int] :
% 0.29/0.61              ( ( ord_less_int @ M6 @ ( abs_abs_int @ N3 ) )
% 0.29/0.61              & ( member_int @ N3 @ S2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % infinite_int_iff_unbounded
% 0.29/0.61  thf(fact_9377_unbounded__k__infinite,axiom,
% 0.29/0.61      ! [K: nat,S2: set_nat] :
% 0.29/0.61        ( ! [M5: nat] :
% 0.29/0.61            ( ( ord_less_nat @ K @ M5 )
% 0.29/0.61           => ? [N8: nat] :
% 0.29/0.61                ( ( ord_less_nat @ M5 @ N8 )
% 0.29/0.61                & ( member_nat @ N8 @ S2 ) ) )
% 0.29/0.61       => ~ ( finite_finite_nat @ S2 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % unbounded_k_infinite
% 0.29/0.61  thf(fact_9378_infinite__nat__iff__unbounded,axiom,
% 0.29/0.61      ! [S2: set_nat] :
% 0.29/0.61        ( ( ~ ( finite_finite_nat @ S2 ) )
% 0.29/0.61        = ( ! [M6: nat] :
% 0.29/0.61            ? [N3: nat] :
% 0.29/0.61              ( ( ord_less_nat @ M6 @ N3 )
% 0.29/0.61              & ( member_nat @ N3 @ S2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % infinite_nat_iff_unbounded
% 0.29/0.61  thf(fact_9379_infinite__nat__iff__unbounded__le,axiom,
% 0.29/0.61      ! [S2: set_nat] :
% 0.29/0.61        ( ( ~ ( finite_finite_nat @ S2 ) )
% 0.29/0.61        = ( ! [M6: nat] :
% 0.29/0.61            ? [N3: nat] :
% 0.29/0.61              ( ( ord_less_eq_nat @ M6 @ N3 )
% 0.29/0.61              & ( member_nat @ N3 @ S2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % infinite_nat_iff_unbounded_le
% 0.29/0.61  thf(fact_9380_finite__nat__bounded,axiom,
% 0.29/0.61      ! [S2: set_nat] :
% 0.29/0.61        ( ( finite_finite_nat @ S2 )
% 0.29/0.61       => ? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_nat_bounded
% 0.29/0.61  thf(fact_9381_finite__nat__iff__bounded,axiom,
% 0.29/0.61      ( finite_finite_nat
% 0.29/0.61      = ( ^ [S4: set_nat] :
% 0.29/0.61          ? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_nat_iff_bounded
% 0.29/0.61  thf(fact_9382_int__ge__less__than2__def,axiom,
% 0.29/0.61      ( int_ge_less_than2
% 0.29/0.61      = ( ^ [D3: int] :
% 0.29/0.61            ( collec213857154873943460nt_int
% 0.29/0.61            @ ( produc4947309494688390418_int_o
% 0.29/0.61              @ ^ [Z7: int,Z4: int] :
% 0.29/0.61                  ( ( ord_less_eq_int @ D3 @ Z4 )
% 0.29/0.61                  & ( ord_less_int @ Z7 @ Z4 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % int_ge_less_than2_def
% 0.29/0.61  thf(fact_9383_int__ge__less__than__def,axiom,
% 0.29/0.61      ( int_ge_less_than
% 0.29/0.61      = ( ^ [D3: int] :
% 0.29/0.61            ( collec213857154873943460nt_int
% 0.29/0.61            @ ( produc4947309494688390418_int_o
% 0.29/0.61              @ ^ [Z7: int,Z4: int] :
% 0.29/0.61                  ( ( ord_less_eq_int @ D3 @ Z7 )
% 0.29/0.61                  & ( ord_less_int @ Z7 @ Z4 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % int_ge_less_than_def
% 0.29/0.61  thf(fact_9384_rat__inverse__code,axiom,
% 0.29/0.61      ! [P5: rat] :
% 0.29/0.61        ( ( quotient_of @ ( inverse_inverse_rat @ P5 ) )
% 0.29/0.61        = ( produc4245557441103728435nt_int
% 0.29/0.61          @ ^ [A3: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B4 ) @ ( abs_abs_int @ A3 ) ) )
% 0.29/0.61          @ ( quotient_of @ P5 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % rat_inverse_code
% 0.29/0.61  thf(fact_9385_normalize__negative,axiom,
% 0.29/0.61      ! [Q3: int,P5: int] :
% 0.29/0.61        ( ( ord_less_int @ Q3 @ zero_zero_int )
% 0.29/0.61       => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q3 ) )
% 0.29/0.61          = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P5 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % normalize_negative
% 0.29/0.61  thf(fact_9386_quotient__of__number_I3_J,axiom,
% 0.29/0.61      ! [K: num] :
% 0.29/0.61        ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 0.29/0.61        = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % quotient_of_number(3)
% 0.29/0.61  thf(fact_9387_normalize__denom__zero,axiom,
% 0.29/0.61      ! [P5: int] :
% 0.29/0.61        ( ( normalize @ ( product_Pair_int_int @ P5 @ zero_zero_int ) )
% 0.29/0.61        = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % normalize_denom_zero
% 0.29/0.61  thf(fact_9388_rat__one__code,axiom,
% 0.29/0.61      ( ( quotient_of @ one_one_rat )
% 0.29/0.61      = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % rat_one_code
% 0.29/0.61  thf(fact_9389_rat__zero__code,axiom,
% 0.29/0.61      ( ( quotient_of @ zero_zero_rat )
% 0.29/0.61      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % rat_zero_code
% 0.29/0.61  thf(fact_9390_quotient__of__number_I5_J,axiom,
% 0.29/0.61      ! [K: num] :
% 0.29/0.61        ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.29/0.61        = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % quotient_of_number(5)
% 0.29/0.61  thf(fact_9391_quotient__of__number_I4_J,axiom,
% 0.29/0.61      ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.29/0.61      = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % quotient_of_number(4)
% 0.29/0.61  thf(fact_9392_divide__rat__def,axiom,
% 0.29/0.61      ( divide_divide_rat
% 0.29/0.61      = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divide_rat_def
% 0.29/0.61  thf(fact_9393_rat__divide__code,axiom,
% 0.29/0.61      ! [P5: rat,Q3: rat] :
% 0.29/0.61        ( ( quotient_of @ ( divide_divide_rat @ P5 @ Q3 ) )
% 0.29/0.61        = ( produc4245557441103728435nt_int
% 0.29/0.61          @ ^ [A3: int,C3: int] :
% 0.29/0.61              ( produc4245557441103728435nt_int
% 0.29/0.61              @ ^ [B4: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ C3 @ B4 ) ) )
% 0.29/0.61              @ ( quotient_of @ Q3 ) )
% 0.29/0.61          @ ( quotient_of @ P5 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % rat_divide_code
% 0.29/0.61  thf(fact_9394_quotient__of__div,axiom,
% 0.29/0.61      ! [R2: rat,N: int,D: int] :
% 0.29/0.61        ( ( ( quotient_of @ R2 )
% 0.29/0.61          = ( product_Pair_int_int @ N @ D ) )
% 0.29/0.61       => ( R2
% 0.29/0.61          = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % quotient_of_div
% 0.29/0.61  thf(fact_9395_quotient__of__denom__pos,axiom,
% 0.29/0.61      ! [R2: rat,P5: int,Q3: int] :
% 0.29/0.61        ( ( ( quotient_of @ R2 )
% 0.29/0.61          = ( product_Pair_int_int @ P5 @ Q3 ) )
% 0.29/0.61       => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % quotient_of_denom_pos
% 0.29/0.61  thf(fact_9396_rat__plus__code,axiom,
% 0.29/0.61      ! [P5: rat,Q3: rat] :
% 0.29/0.61        ( ( quotient_of @ ( plus_plus_rat @ P5 @ Q3 ) )
% 0.29/0.61        = ( produc4245557441103728435nt_int
% 0.29/0.61          @ ^ [A3: int,C3: int] :
% 0.29/0.61              ( produc4245557441103728435nt_int
% 0.29/0.61              @ ^ [B4: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D3 ) ) )
% 0.29/0.61              @ ( quotient_of @ Q3 ) )
% 0.29/0.61          @ ( quotient_of @ P5 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % rat_plus_code
% 0.29/0.61  thf(fact_9397_normalize__denom__pos,axiom,
% 0.29/0.61      ! [R2: product_prod_int_int,P5: int,Q3: int] :
% 0.29/0.61        ( ( ( normalize @ R2 )
% 0.29/0.61          = ( product_Pair_int_int @ P5 @ Q3 ) )
% 0.29/0.61       => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % normalize_denom_pos
% 0.29/0.61  thf(fact_9398_normalize__crossproduct,axiom,
% 0.29/0.61      ! [Q3: int,S: int,P5: int,R2: int] :
% 0.29/0.61        ( ( Q3 != zero_zero_int )
% 0.29/0.61       => ( ( S != zero_zero_int )
% 0.29/0.61         => ( ( ( normalize @ ( product_Pair_int_int @ P5 @ Q3 ) )
% 0.29/0.61              = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
% 0.29/0.61           => ( ( times_times_int @ P5 @ S )
% 0.29/0.61              = ( times_times_int @ R2 @ Q3 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % normalize_crossproduct
% 0.29/0.61  thf(fact_9399_rat__less__code,axiom,
% 0.29/0.61      ( ord_less_rat
% 0.29/0.61      = ( ^ [P4: rat,Q4: rat] :
% 0.29/0.61            ( produc4947309494688390418_int_o
% 0.29/0.61            @ ^ [A3: int,C3: int] :
% 0.29/0.61                ( produc4947309494688390418_int_o
% 0.29/0.61                @ ^ [B4: int,D3: int] : ( ord_less_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ C3 @ B4 ) )
% 0.29/0.61                @ ( quotient_of @ Q4 ) )
% 0.29/0.61            @ ( quotient_of @ P4 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % rat_less_code
% 0.29/0.61  thf(fact_9400_rat__less__eq__code,axiom,
% 0.29/0.61      ( ord_less_eq_rat
% 0.29/0.61      = ( ^ [P4: rat,Q4: rat] :
% 0.29/0.61            ( produc4947309494688390418_int_o
% 0.29/0.61            @ ^ [A3: int,C3: int] :
% 0.29/0.61                ( produc4947309494688390418_int_o
% 0.29/0.61                @ ^ [B4: int,D3: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ C3 @ B4 ) )
% 0.29/0.61                @ ( quotient_of @ Q4 ) )
% 0.29/0.61            @ ( quotient_of @ P4 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % rat_less_eq_code
% 0.29/0.61  thf(fact_9401_quotient__of__int,axiom,
% 0.29/0.61      ! [A: int] :
% 0.29/0.61        ( ( quotient_of @ ( of_int @ A ) )
% 0.29/0.61        = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % quotient_of_int
% 0.29/0.61  thf(fact_9402_Sum__Ico__nat,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [X4: nat] : X4
% 0.29/0.61          @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.29/0.61        = ( 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 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Sum_Ico_nat
% 0.29/0.61  thf(fact_9403_sum__power2,axiom,
% 0.29/0.61      ! [K: nat] :
% 0.29/0.61        ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.29/0.61        = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sum_power2
% 0.29/0.61  thf(fact_9404_image__Suc__atLeastLessThan,axiom,
% 0.29/0.61      ! [I: nat,J: nat] :
% 0.29/0.61        ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 0.29/0.61        = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % image_Suc_atLeastLessThan
% 0.29/0.61  thf(fact_9405_atLeastLessThan__singleton,axiom,
% 0.29/0.61      ! [M: nat] :
% 0.29/0.61        ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 0.29/0.61        = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeastLessThan_singleton
% 0.29/0.61  thf(fact_9406_all__nat__less__eq,axiom,
% 0.29/0.61      ! [N: nat,P: nat > $o] :
% 0.29/0.61        ( ( ! [M6: nat] :
% 0.29/0.61              ( ( ord_less_nat @ M6 @ N )
% 0.29/0.61             => ( P @ M6 ) ) )
% 0.29/0.61        = ( ! [X4: nat] :
% 0.29/0.61              ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.29/0.61             => ( P @ X4 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % all_nat_less_eq
% 0.29/0.61  thf(fact_9407_ex__nat__less__eq,axiom,
% 0.29/0.61      ! [N: nat,P: nat > $o] :
% 0.29/0.61        ( ( ? [M6: nat] :
% 0.29/0.61              ( ( ord_less_nat @ M6 @ N )
% 0.29/0.61              & ( P @ M6 ) ) )
% 0.29/0.61        = ( ? [X4: nat] :
% 0.29/0.61              ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.29/0.61              & ( P @ X4 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % ex_nat_less_eq
% 0.29/0.61  thf(fact_9408_atLeastLessThanSuc__atLeastAtMost,axiom,
% 0.29/0.61      ! [L: nat,U: nat] :
% 0.29/0.61        ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 0.29/0.61        = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeastLessThanSuc_atLeastAtMost
% 0.29/0.61  thf(fact_9409_lessThan__atLeast0,axiom,
% 0.29/0.61      ( set_ord_lessThan_nat
% 0.29/0.61      = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % lessThan_atLeast0
% 0.29/0.61  thf(fact_9410_atLeastLessThan0,axiom,
% 0.29/0.61      ! [M: nat] :
% 0.29/0.61        ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 0.29/0.61        = bot_bot_set_nat ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeastLessThan0
% 0.29/0.61  thf(fact_9411_atLeast0__lessThan__Suc,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.29/0.61        = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeast0_lessThan_Suc
% 0.29/0.61  thf(fact_9412_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.29/0.61      ! [N4: set_nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.29/0.61       => ( finite_finite_nat @ N4 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % subset_eq_atLeast0_lessThan_finite
% 0.29/0.61  thf(fact_9413_atLeastLessThanSuc,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.61         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.29/0.61            = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 0.29/0.61        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.29/0.61         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.29/0.61            = bot_bot_set_nat ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeastLessThanSuc
% 0.29/0.61  thf(fact_9414_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.29/0.61        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeast0_lessThan_Suc_eq_insert_0
% 0.29/0.61  thf(fact_9415_prod__Suc__fact,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.29/0.61        = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % prod_Suc_fact
% 0.29/0.61  thf(fact_9416_prod__Suc__Suc__fact,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.29/0.61        = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % prod_Suc_Suc_fact
% 0.29/0.61  thf(fact_9417_atLeastLessThan__nat__numeral,axiom,
% 0.29/0.61      ! [M: nat,K: num] :
% 0.29/0.61        ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.29/0.61         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.29/0.61            = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.29/0.61        & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.29/0.61         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.29/0.61            = bot_bot_set_nat ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeastLessThan_nat_numeral
% 0.29/0.61  thf(fact_9418_Rat_Oof__int__def,axiom,
% 0.29/0.61      of_int = ring_1_of_int_rat ).
% 0.29/0.61  
% 0.29/0.61  % Rat.of_int_def
% 0.29/0.61  thf(fact_9419_image__minus__const__atLeastLessThan__nat,axiom,
% 0.29/0.61      ! [C: nat,Y: nat,X: nat] :
% 0.29/0.61        ( ( ( ord_less_nat @ C @ Y )
% 0.29/0.61         => ( ( image_nat_nat
% 0.29/0.61              @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 0.29/0.61              @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.29/0.61            = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
% 0.29/0.61        & ( ~ ( ord_less_nat @ C @ Y )
% 0.29/0.61         => ( ( ( ord_less_nat @ X @ Y )
% 0.29/0.61             => ( ( image_nat_nat
% 0.29/0.61                  @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 0.29/0.61                  @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.29/0.61                = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.29/0.61            & ( ~ ( ord_less_nat @ X @ Y )
% 0.29/0.61             => ( ( image_nat_nat
% 0.29/0.61                  @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 0.29/0.61                  @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.29/0.61                = bot_bot_set_nat ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % image_minus_const_atLeastLessThan_nat
% 0.29/0.61  thf(fact_9420_atLeast1__lessThan__eq__remove0,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.61        = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeast1_lessThan_eq_remove0
% 0.29/0.61  thf(fact_9421_Chebyshev__sum__upper__nat,axiom,
% 0.29/0.61      ! [N: nat,A: nat > nat,B: nat > nat] :
% 0.29/0.61        ( ! [I2: nat,J3: nat] :
% 0.29/0.61            ( ( ord_less_eq_nat @ I2 @ J3 )
% 0.29/0.61           => ( ( ord_less_nat @ J3 @ N )
% 0.29/0.61             => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J3 ) ) ) )
% 0.29/0.61       => ( ! [I2: nat,J3: nat] :
% 0.29/0.61              ( ( ord_less_eq_nat @ I2 @ J3 )
% 0.29/0.61             => ( ( ord_less_nat @ J3 @ N )
% 0.29/0.61               => ( ord_less_eq_nat @ ( B @ J3 ) @ ( B @ I2 ) ) ) )
% 0.29/0.61         => ( ord_less_eq_nat
% 0.29/0.61            @ ( times_times_nat @ N
% 0.29/0.61              @ ( groups3542108847815614940at_nat
% 0.29/0.61                @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
% 0.29/0.61                @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.29/0.61            @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Chebyshev_sum_upper_nat
% 0.29/0.61  thf(fact_9422_finite__atLeastZeroLessThan__int,axiom,
% 0.29/0.61      ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 0.29/0.61  
% 0.29/0.61  % finite_atLeastZeroLessThan_int
% 0.29/0.61  thf(fact_9423_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 0.29/0.61      ! [L: int,U: int] :
% 0.29/0.61        ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
% 0.29/0.61        = ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 0.29/0.61  
% 0.29/0.61  % atLeastLessThanPlusOne_atLeastAtMost_int
% 0.29/0.61  thf(fact_9424_image__int__atLeastLessThan,axiom,
% 0.29/0.61      ! [A: nat,B: nat] :
% 0.29/0.61        ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
% 0.29/0.61        = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % image_int_atLeastLessThan
% 0.29/0.61  thf(fact_9425_image__add__int__atLeastLessThan,axiom,
% 0.29/0.61      ! [L: int,U: int] :
% 0.29/0.61        ( ( image_int_int
% 0.29/0.61          @ ^ [X4: int] : ( plus_plus_int @ X4 @ L )
% 0.29/0.61          @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
% 0.29/0.61        = ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 0.29/0.61  
% 0.29/0.61  % image_add_int_atLeastLessThan
% 0.29/0.61  thf(fact_9426_image__atLeastZeroLessThan__int,axiom,
% 0.29/0.61      ! [U: int] :
% 0.29/0.61        ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.29/0.61       => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 0.29/0.61          = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % image_atLeastZeroLessThan_int
% 0.29/0.61  thf(fact_9427_Frct__code__post_I5_J,axiom,
% 0.29/0.61      ! [K: num] :
% 0.29/0.61        ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 0.29/0.61        = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Frct_code_post(5)
% 0.29/0.61  thf(fact_9428_int__of__nat__def,axiom,
% 0.29/0.61      code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).
% 0.29/0.61  
% 0.29/0.61  % int_of_nat_def
% 0.29/0.61  thf(fact_9429_Frct__code__post_I2_J,axiom,
% 0.29/0.61      ! [A: int] :
% 0.29/0.61        ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 0.29/0.61        = zero_zero_rat ) ).
% 0.29/0.61  
% 0.29/0.61  % Frct_code_post(2)
% 0.29/0.61  thf(fact_9430_Frct__code__post_I1_J,axiom,
% 0.29/0.61      ! [A: int] :
% 0.29/0.61        ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 0.29/0.61        = zero_zero_rat ) ).
% 0.29/0.61  
% 0.29/0.61  % Frct_code_post(1)
% 0.29/0.61  thf(fact_9431_Frct__code__post_I3_J,axiom,
% 0.29/0.61      ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 0.29/0.61      = one_one_rat ) ).
% 0.29/0.61  
% 0.29/0.61  % Frct_code_post(3)
% 0.29/0.61  thf(fact_9432_Frct__code__post_I4_J,axiom,
% 0.29/0.61      ! [K: num] :
% 0.29/0.61        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 0.29/0.61        = ( numeral_numeral_rat @ K ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Frct_code_post(4)
% 0.29/0.61  thf(fact_9433_Frct__code__post_I6_J,axiom,
% 0.29/0.61      ! [K: num,L: num] :
% 0.29/0.61        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 0.29/0.61        = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Frct_code_post(6)
% 0.29/0.61  thf(fact_9434_Code__Target__Int_Opositive__def,axiom,
% 0.29/0.61      code_Target_positive = numeral_numeral_int ).
% 0.29/0.61  
% 0.29/0.61  % Code_Target_Int.positive_def
% 0.29/0.61  thf(fact_9435_divmod__step__integer__def,axiom,
% 0.29/0.61      ( unique4921790084139445826nteger
% 0.29/0.61      = ( ^ [L3: num] :
% 0.29/0.61            ( produc6916734918728496179nteger
% 0.29/0.61            @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_step_integer_def
% 0.29/0.61  thf(fact_9436_times__integer__code_I2_J,axiom,
% 0.29/0.61      ! [L: code_integer] :
% 0.29/0.61        ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 0.29/0.61        = zero_z3403309356797280102nteger ) ).
% 0.29/0.61  
% 0.29/0.61  % times_integer_code(2)
% 0.29/0.61  thf(fact_9437_times__integer__code_I1_J,axiom,
% 0.29/0.61      ! [K: code_integer] :
% 0.29/0.61        ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 0.29/0.61        = zero_z3403309356797280102nteger ) ).
% 0.29/0.61  
% 0.29/0.61  % times_integer_code(1)
% 0.29/0.61  thf(fact_9438_divmod__integer_H__def,axiom,
% 0.29/0.61      ( unique3479559517661332726nteger
% 0.29/0.61      = ( ^ [M6: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_integer'_def
% 0.29/0.61  thf(fact_9439_less__eq__integer__code_I1_J,axiom,
% 0.29/0.61      ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 0.29/0.61  
% 0.29/0.61  % less_eq_integer_code(1)
% 0.29/0.61  thf(fact_9440_sgn__integer__code,axiom,
% 0.29/0.61      ( sgn_sgn_Code_integer
% 0.29/0.61      = ( ^ [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 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sgn_integer_code
% 0.29/0.61  thf(fact_9441_minus__integer__code_I2_J,axiom,
% 0.29/0.61      ! [L: code_integer] :
% 0.29/0.61        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
% 0.29/0.61        = ( uminus1351360451143612070nteger @ L ) ) ).
% 0.29/0.61  
% 0.29/0.61  % minus_integer_code(2)
% 0.29/0.61  thf(fact_9442_minus__integer__code_I1_J,axiom,
% 0.29/0.61      ! [K: code_integer] :
% 0.29/0.61        ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 0.29/0.61        = K ) ).
% 0.29/0.61  
% 0.29/0.61  % minus_integer_code(1)
% 0.29/0.61  thf(fact_9443_plus__integer__code_I2_J,axiom,
% 0.29/0.61      ! [L: code_integer] :
% 0.29/0.61        ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
% 0.29/0.61        = L ) ).
% 0.29/0.61  
% 0.29/0.61  % plus_integer_code(2)
% 0.29/0.61  thf(fact_9444_plus__integer__code_I1_J,axiom,
% 0.29/0.61      ! [K: code_integer] :
% 0.29/0.61        ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
% 0.29/0.61        = K ) ).
% 0.29/0.61  
% 0.29/0.61  % plus_integer_code(1)
% 0.29/0.61  thf(fact_9445_zero__natural_Orsp,axiom,
% 0.29/0.61      zero_zero_nat = zero_zero_nat ).
% 0.29/0.61  
% 0.29/0.61  % zero_natural.rsp
% 0.29/0.61  thf(fact_9446_zero__integer_Orsp,axiom,
% 0.29/0.61      zero_zero_int = zero_zero_int ).
% 0.29/0.61  
% 0.29/0.61  % zero_integer.rsp
% 0.29/0.61  thf(fact_9447_one__integer_Orsp,axiom,
% 0.29/0.61      one_one_int = one_one_int ).
% 0.29/0.61  
% 0.29/0.61  % one_integer.rsp
% 0.29/0.61  thf(fact_9448_one__natural_Orsp,axiom,
% 0.29/0.61      one_one_nat = one_one_nat ).
% 0.29/0.61  
% 0.29/0.61  % one_natural.rsp
% 0.29/0.61  thf(fact_9449_integer__of__int__code,axiom,
% 0.29/0.61      ( code_integer_of_int
% 0.29/0.61      = ( ^ [K2: int] :
% 0.29/0.61            ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
% 0.29/0.61            @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.29/0.61              @ ( if_Code_integer
% 0.29/0.61                @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.61                  = zero_zero_int )
% 0.29/0.61                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.29/0.61                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % integer_of_int_code
% 0.29/0.61  thf(fact_9450_Code__Numeral_Opositive__def,axiom,
% 0.29/0.61      code_positive = numera6620942414471956472nteger ).
% 0.29/0.61  
% 0.29/0.61  % Code_Numeral.positive_def
% 0.29/0.61  thf(fact_9451_integer__of__int__eq__of__int,axiom,
% 0.29/0.61      code_integer_of_int = ring_18347121197199848620nteger ).
% 0.29/0.61  
% 0.29/0.61  % integer_of_int_eq_of_int
% 0.29/0.61  thf(fact_9452_divide__integer_Oabs__eq,axiom,
% 0.29/0.61      ! [Xa: int,X: int] :
% 0.29/0.61        ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.29/0.61        = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divide_integer.abs_eq
% 0.29/0.61  thf(fact_9453_abs__integer__code,axiom,
% 0.29/0.61      ( abs_abs_Code_integer
% 0.29/0.61      = ( ^ [K2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K2 ) @ K2 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % abs_integer_code
% 0.29/0.61  thf(fact_9454_less__integer__code_I1_J,axiom,
% 0.29/0.61      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 0.29/0.61  
% 0.29/0.61  % less_integer_code(1)
% 0.29/0.61  thf(fact_9455_zero__integer__def,axiom,
% 0.29/0.61      ( zero_z3403309356797280102nteger
% 0.29/0.61      = ( code_integer_of_int @ zero_zero_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % zero_integer_def
% 0.29/0.61  thf(fact_9456_uminus__integer__code_I1_J,axiom,
% 0.29/0.61      ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.29/0.61      = zero_z3403309356797280102nteger ) ).
% 0.29/0.61  
% 0.29/0.61  % uminus_integer_code(1)
% 0.29/0.61  thf(fact_9457_less__integer_Oabs__eq,axiom,
% 0.29/0.61      ! [Xa: int,X: int] :
% 0.29/0.61        ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.29/0.61        = ( ord_less_int @ Xa @ X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % less_integer.abs_eq
% 0.29/0.61  thf(fact_9458_plus__integer_Oabs__eq,axiom,
% 0.29/0.61      ! [Xa: int,X: int] :
% 0.29/0.61        ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.29/0.61        = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % plus_integer.abs_eq
% 0.29/0.61  thf(fact_9459_one__integer__def,axiom,
% 0.29/0.61      ( one_one_Code_integer
% 0.29/0.61      = ( code_integer_of_int @ one_one_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % one_integer_def
% 0.29/0.61  thf(fact_9460_less__eq__integer_Oabs__eq,axiom,
% 0.29/0.61      ! [Xa: int,X: int] :
% 0.29/0.61        ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.29/0.61        = ( ord_less_eq_int @ Xa @ X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % less_eq_integer.abs_eq
% 0.29/0.61  thf(fact_9461_integer__of__num_I3_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( code_integer_of_num @ ( bit1 @ N ) )
% 0.29/0.61        = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 0.29/0.61  
% 0.29/0.61  % integer_of_num(3)
% 0.29/0.61  thf(fact_9462_bit__cut__integer__def,axiom,
% 0.29/0.61      ( code_bit_cut_integer
% 0.29/0.61      = ( ^ [K2: code_integer] :
% 0.29/0.61            ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.61            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bit_cut_integer_def
% 0.29/0.61  thf(fact_9463_integer__of__num__def,axiom,
% 0.29/0.61      code_integer_of_num = numera6620942414471956472nteger ).
% 0.29/0.61  
% 0.29/0.61  % integer_of_num_def
% 0.29/0.61  thf(fact_9464_integer__of__num__triv_I1_J,axiom,
% 0.29/0.61      ( ( code_integer_of_num @ one )
% 0.29/0.61      = one_one_Code_integer ) ).
% 0.29/0.61  
% 0.29/0.61  % integer_of_num_triv(1)
% 0.29/0.61  thf(fact_9465_integer__of__num_I2_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( code_integer_of_num @ ( bit0 @ N ) )
% 0.29/0.61        = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % integer_of_num(2)
% 0.29/0.61  thf(fact_9466_integer__of__num__triv_I2_J,axiom,
% 0.29/0.61      ( ( code_integer_of_num @ ( bit0 @ one ) )
% 0.29/0.61      = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % integer_of_num_triv(2)
% 0.29/0.61  thf(fact_9467_bit__cut__integer__code,axiom,
% 0.29/0.61      ( code_bit_cut_integer
% 0.29/0.61      = ( ^ [K2: code_integer] :
% 0.29/0.61            ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.29/0.61            @ ( produc9125791028180074456eger_o
% 0.29/0.61              @ ^ [R5: code_integer,S5: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S5 ) ) @ ( S5 = one_one_Code_integer ) )
% 0.29/0.61              @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % bit_cut_integer_code
% 0.29/0.61  thf(fact_9468_divmod__integer__def,axiom,
% 0.29/0.61      ( code_divmod_integer
% 0.29/0.61      = ( ^ [K2: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K2 @ L3 ) @ ( modulo364778990260209775nteger @ K2 @ L3 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_integer_def
% 0.29/0.61  thf(fact_9469_csqrt_Osimps_I1_J,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( re @ ( csqrt @ Z ) )
% 0.29/0.61        = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % csqrt.simps(1)
% 0.29/0.61  thf(fact_9470_complex__Re__of__nat,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( re @ ( semiri8010041392384452111omplex @ N ) )
% 0.29/0.61        = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_Re_of_nat
% 0.29/0.61  thf(fact_9471_complex__Re__of__int,axiom,
% 0.29/0.61      ! [Z: int] :
% 0.29/0.61        ( ( re @ ( ring_17405671764205052669omplex @ Z ) )
% 0.29/0.61        = ( ring_1_of_int_real @ Z ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_Re_of_int
% 0.29/0.61  thf(fact_9472_complex__Re__numeral,axiom,
% 0.29/0.61      ! [V: num] :
% 0.29/0.61        ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 0.29/0.61        = ( numeral_numeral_real @ V ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_Re_numeral
% 0.29/0.61  thf(fact_9473_Re__divide__of__nat,axiom,
% 0.29/0.61      ! [Z: complex,N: nat] :
% 0.29/0.61        ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.29/0.61        = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_divide_of_nat
% 0.29/0.61  thf(fact_9474_Re__divide__of__real,axiom,
% 0.29/0.61      ! [Z: complex,R2: real] :
% 0.29/0.61        ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 0.29/0.61        = ( divide_divide_real @ ( re @ Z ) @ R2 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_divide_of_real
% 0.29/0.61  thf(fact_9475_Re__sgn,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( re @ ( sgn_sgn_complex @ Z ) )
% 0.29/0.61        = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_sgn
% 0.29/0.61  thf(fact_9476_Re__divide__numeral,axiom,
% 0.29/0.61      ! [Z: complex,W: num] :
% 0.29/0.61        ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.61        = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_divide_numeral
% 0.29/0.61  thf(fact_9477_cos__Arg__i__mult__zero,axiom,
% 0.29/0.61      ! [Y: complex] :
% 0.29/0.61        ( ( Y != zero_zero_complex )
% 0.29/0.61       => ( ( ( re @ Y )
% 0.29/0.61            = zero_zero_real )
% 0.29/0.61         => ( ( cos_real @ ( arg @ Y ) )
% 0.29/0.61            = zero_zero_real ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cos_Arg_i_mult_zero
% 0.29/0.61  thf(fact_9478_imaginary__unit_Osimps_I1_J,axiom,
% 0.29/0.61      ( ( re @ imaginary_unit )
% 0.29/0.61      = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % imaginary_unit.simps(1)
% 0.29/0.61  thf(fact_9479_complex__Re__le__cmod,axiom,
% 0.29/0.61      ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_Re_le_cmod
% 0.29/0.61  thf(fact_9480_zero__complex_Osimps_I1_J,axiom,
% 0.29/0.61      ( ( re @ zero_zero_complex )
% 0.29/0.61      = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % zero_complex.simps(1)
% 0.29/0.61  thf(fact_9481_one__complex_Osimps_I1_J,axiom,
% 0.29/0.61      ( ( re @ one_one_complex )
% 0.29/0.61      = one_one_real ) ).
% 0.29/0.61  
% 0.29/0.61  % one_complex.simps(1)
% 0.29/0.61  thf(fact_9482_abs__Re__le__cmod,axiom,
% 0.29/0.61      ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % abs_Re_le_cmod
% 0.29/0.61  thf(fact_9483_Re__csqrt,axiom,
% 0.29/0.61      ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_csqrt
% 0.29/0.61  thf(fact_9484_divmod__abs__code_I6_J,axiom,
% 0.29/0.61      ! [J: code_integer] :
% 0.29/0.61        ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
% 0.29/0.61        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_abs_code(6)
% 0.29/0.61  thf(fact_9485_cmod__plus__Re__le__0__iff,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 0.29/0.61        = ( ( re @ Z )
% 0.29/0.61          = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cmod_plus_Re_le_0_iff
% 0.29/0.61  thf(fact_9486_cos__n__Re__cis__pow__n,axiom,
% 0.29/0.61      ! [N: nat,A: real] :
% 0.29/0.61        ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.29/0.61        = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cos_n_Re_cis_pow_n
% 0.29/0.61  thf(fact_9487_divmod__abs__code_I5_J,axiom,
% 0.29/0.61      ! [J: code_integer] :
% 0.29/0.61        ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
% 0.29/0.61        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_abs_code(5)
% 0.29/0.61  thf(fact_9488_divmod__abs__def,axiom,
% 0.29/0.61      ( code_divmod_abs
% 0.29/0.61      = ( ^ [K2: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L3 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L3 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_abs_def
% 0.29/0.61  thf(fact_9489_divmod__integer__code,axiom,
% 0.29/0.61      ( code_divmod_integer
% 0.29/0.61      = ( ^ [K2: code_integer,L3: code_integer] :
% 0.29/0.61            ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.29/0.61            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L3 )
% 0.29/0.61              @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L3 )
% 0.29/0.61                @ ( produc6916734918728496179nteger
% 0.29/0.61                  @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L3 @ S5 ) ) )
% 0.29/0.61                  @ ( code_divmod_abs @ K2 @ L3 ) ) )
% 0.29/0.61              @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 0.29/0.61                @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.29/0.61                  @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L3 )
% 0.29/0.61                    @ ( produc6916734918728496179nteger
% 0.29/0.61                      @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L3 ) @ S5 ) ) )
% 0.29/0.61                      @ ( code_divmod_abs @ K2 @ L3 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % divmod_integer_code
% 0.29/0.61  thf(fact_9490_csqrt_Ocode,axiom,
% 0.29/0.61      ( csqrt
% 0.29/0.61      = ( ^ [Z4: complex] :
% 0.29/0.61            ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.61            @ ( times_times_real
% 0.29/0.61              @ ( if_real
% 0.29/0.61                @ ( ( im @ Z4 )
% 0.29/0.61                  = zero_zero_real )
% 0.29/0.61                @ one_one_real
% 0.29/0.61                @ ( sgn_sgn_real @ ( im @ Z4 ) ) )
% 0.29/0.61              @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % csqrt.code
% 0.29/0.61  thf(fact_9491_csqrt_Osimps_I2_J,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( im @ ( csqrt @ Z ) )
% 0.29/0.61        = ( times_times_real
% 0.29/0.61          @ ( if_real
% 0.29/0.61            @ ( ( im @ Z )
% 0.29/0.61              = zero_zero_real )
% 0.29/0.61            @ one_one_real
% 0.29/0.61            @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 0.29/0.61          @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % csqrt.simps(2)
% 0.29/0.61  thf(fact_9492_complex__Im__fact,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( im @ ( semiri5044797733671781792omplex @ N ) )
% 0.29/0.61        = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_Im_fact
% 0.29/0.61  thf(fact_9493_Im__complex__of__real,axiom,
% 0.29/0.61      ! [Z: real] :
% 0.29/0.61        ( ( im @ ( real_V4546457046886955230omplex @ Z ) )
% 0.29/0.61        = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_complex_of_real
% 0.29/0.61  thf(fact_9494_Im__power__real,axiom,
% 0.29/0.61      ! [X: complex,N: nat] :
% 0.29/0.61        ( ( ( im @ X )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61       => ( ( im @ ( power_power_complex @ X @ N ) )
% 0.29/0.61          = zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_power_real
% 0.29/0.61  thf(fact_9495_complex__Im__numeral,axiom,
% 0.29/0.61      ! [V: num] :
% 0.29/0.61        ( ( im @ ( numera6690914467698888265omplex @ V ) )
% 0.29/0.61        = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_Im_numeral
% 0.29/0.61  thf(fact_9496_complex__Im__of__nat,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( im @ ( semiri8010041392384452111omplex @ N ) )
% 0.29/0.61        = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_Im_of_nat
% 0.29/0.61  thf(fact_9497_complex__Im__of__int,axiom,
% 0.29/0.61      ! [Z: int] :
% 0.29/0.61        ( ( im @ ( ring_17405671764205052669omplex @ Z ) )
% 0.29/0.61        = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_Im_of_int
% 0.29/0.61  thf(fact_9498_Im__divide__of__real,axiom,
% 0.29/0.61      ! [Z: complex,R2: real] :
% 0.29/0.61        ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 0.29/0.61        = ( divide_divide_real @ ( im @ Z ) @ R2 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_divide_of_real
% 0.29/0.61  thf(fact_9499_Im__sgn,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( im @ ( sgn_sgn_complex @ Z ) )
% 0.29/0.61        = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_sgn
% 0.29/0.61  thf(fact_9500_Re__power__real,axiom,
% 0.29/0.61      ! [X: complex,N: nat] :
% 0.29/0.61        ( ( ( im @ X )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61       => ( ( re @ ( power_power_complex @ X @ N ) )
% 0.29/0.61          = ( power_power_real @ ( re @ X ) @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_power_real
% 0.29/0.61  thf(fact_9501_Im__divide__numeral,axiom,
% 0.29/0.61      ! [Z: complex,W: num] :
% 0.29/0.61        ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.61        = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_divide_numeral
% 0.29/0.61  thf(fact_9502_Im__divide__of__nat,axiom,
% 0.29/0.61      ! [Z: complex,N: nat] :
% 0.29/0.61        ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.29/0.61        = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_divide_of_nat
% 0.29/0.61  thf(fact_9503_csqrt__of__real__nonneg,axiom,
% 0.29/0.61      ! [X: complex] :
% 0.29/0.61        ( ( ( im @ X )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61       => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
% 0.29/0.61         => ( ( csqrt @ X )
% 0.29/0.61            = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % csqrt_of_real_nonneg
% 0.29/0.61  thf(fact_9504_csqrt__minus,axiom,
% 0.29/0.61      ! [X: complex] :
% 0.29/0.61        ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
% 0.29/0.61          | ( ( ( im @ X )
% 0.29/0.61              = zero_zero_real )
% 0.29/0.61            & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
% 0.29/0.61       => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
% 0.29/0.61          = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % csqrt_minus
% 0.29/0.61  thf(fact_9505_csqrt__of__real__nonpos,axiom,
% 0.29/0.61      ! [X: complex] :
% 0.29/0.61        ( ( ( im @ X )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61       => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
% 0.29/0.61         => ( ( csqrt @ X )
% 0.29/0.61            = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % csqrt_of_real_nonpos
% 0.29/0.61  thf(fact_9506_imaginary__unit_Osimps_I2_J,axiom,
% 0.29/0.61      ( ( im @ imaginary_unit )
% 0.29/0.61      = one_one_real ) ).
% 0.29/0.61  
% 0.29/0.61  % imaginary_unit.simps(2)
% 0.29/0.61  thf(fact_9507_zero__complex_Osimps_I2_J,axiom,
% 0.29/0.61      ( ( im @ zero_zero_complex )
% 0.29/0.61      = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % zero_complex.simps(2)
% 0.29/0.61  thf(fact_9508_one__complex_Osimps_I2_J,axiom,
% 0.29/0.61      ( ( im @ one_one_complex )
% 0.29/0.61      = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % one_complex.simps(2)
% 0.29/0.61  thf(fact_9509_complex__is__Int__iff,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( member_complex @ Z @ ring_1_Ints_complex )
% 0.29/0.61        = ( ( ( im @ Z )
% 0.29/0.61            = zero_zero_real )
% 0.29/0.61          & ? [I3: int] :
% 0.29/0.61              ( ( re @ Z )
% 0.29/0.61              = ( ring_1_of_int_real @ I3 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_is_Int_iff
% 0.29/0.61  thf(fact_9510_abs__Im__le__cmod,axiom,
% 0.29/0.61      ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.29/0.61  
% 0.29/0.61  % abs_Im_le_cmod
% 0.29/0.61  thf(fact_9511_cmod__eq__Re,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( ( im @ Z )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61       => ( ( real_V1022390504157884413omplex @ Z )
% 0.29/0.61          = ( abs_abs_real @ ( re @ Z ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cmod_eq_Re
% 0.29/0.61  thf(fact_9512_cmod__eq__Im,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( ( re @ Z )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61       => ( ( real_V1022390504157884413omplex @ Z )
% 0.29/0.61          = ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cmod_eq_Im
% 0.29/0.61  thf(fact_9513_Im__eq__0,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( ( abs_abs_real @ ( re @ Z ) )
% 0.29/0.61          = ( real_V1022390504157884413omplex @ Z ) )
% 0.29/0.61       => ( ( im @ Z )
% 0.29/0.61          = zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_eq_0
% 0.29/0.61  thf(fact_9514_cmod__Im__le__iff,axiom,
% 0.29/0.61      ! [X: complex,Y: complex] :
% 0.29/0.61        ( ( ( re @ X )
% 0.29/0.61          = ( re @ Y ) )
% 0.29/0.61       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
% 0.29/0.61          = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cmod_Im_le_iff
% 0.29/0.61  thf(fact_9515_cmod__Re__le__iff,axiom,
% 0.29/0.61      ! [X: complex,Y: complex] :
% 0.29/0.61        ( ( ( im @ X )
% 0.29/0.61          = ( im @ Y ) )
% 0.29/0.61       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
% 0.29/0.61          = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cmod_Re_le_iff
% 0.29/0.61  thf(fact_9516_csqrt__principal,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 0.29/0.61        | ( ( ( re @ ( csqrt @ Z ) )
% 0.29/0.61            = zero_zero_real )
% 0.29/0.61          & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % csqrt_principal
% 0.29/0.61  thf(fact_9517_cmod__le,axiom,
% 0.29/0.61      ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cmod_le
% 0.29/0.61  thf(fact_9518_sin__n__Im__cis__pow__n,axiom,
% 0.29/0.61      ! [N: nat,A: real] :
% 0.29/0.61        ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.29/0.61        = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % sin_n_Im_cis_pow_n
% 0.29/0.61  thf(fact_9519_cmod__power2,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.61        = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cmod_power2
% 0.29/0.61  thf(fact_9520_Im__power2,axiom,
% 0.29/0.61      ! [X: complex] :
% 0.29/0.61        ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.61        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_power2
% 0.29/0.61  thf(fact_9521_Re__power2,axiom,
% 0.29/0.61      ! [X: complex] :
% 0.29/0.61        ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.61        = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_power2
% 0.29/0.61  thf(fact_9522_complex__eq__0,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( Z = zero_zero_complex )
% 0.29/0.61        = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.61          = zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_eq_0
% 0.29/0.61  thf(fact_9523_norm__complex__def,axiom,
% 0.29/0.61      ( real_V1022390504157884413omplex
% 0.29/0.61      = ( ^ [Z4: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % norm_complex_def
% 0.29/0.61  thf(fact_9524_inverse__complex_Osimps_I1_J,axiom,
% 0.29/0.61      ! [X: complex] :
% 0.29/0.61        ( ( re @ ( invers8013647133539491842omplex @ X ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % inverse_complex.simps(1)
% 0.29/0.61  thf(fact_9525_complex__neq__0,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( Z != zero_zero_complex )
% 0.29/0.61        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_neq_0
% 0.29/0.61  thf(fact_9526_Re__divide,axiom,
% 0.29/0.61      ! [X: complex,Y: complex] :
% 0.29/0.61        ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_divide
% 0.29/0.61  thf(fact_9527_csqrt__square,axiom,
% 0.29/0.61      ! [B: complex] :
% 0.29/0.61        ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 0.29/0.61          | ( ( ( re @ B )
% 0.29/0.61              = zero_zero_real )
% 0.29/0.61            & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 0.29/0.61       => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.61          = B ) ) ).
% 0.29/0.61  
% 0.29/0.61  % csqrt_square
% 0.29/0.61  thf(fact_9528_csqrt__unique,axiom,
% 0.29/0.61      ! [W: complex,Z: complex] :
% 0.29/0.61        ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.61          = Z )
% 0.29/0.61       => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 0.29/0.61            | ( ( ( re @ W )
% 0.29/0.61                = zero_zero_real )
% 0.29/0.61              & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 0.29/0.61         => ( ( csqrt @ Z )
% 0.29/0.61            = W ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % csqrt_unique
% 0.29/0.61  thf(fact_9529_inverse__complex_Osimps_I2_J,axiom,
% 0.29/0.61      ! [X: complex] :
% 0.29/0.61        ( ( im @ ( invers8013647133539491842omplex @ X ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % inverse_complex.simps(2)
% 0.29/0.61  thf(fact_9530_Im__divide,axiom,
% 0.29/0.61      ! [X: complex,Y: complex] :
% 0.29/0.61        ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_divide
% 0.29/0.61  thf(fact_9531_complex__abs__le__norm,axiom,
% 0.29/0.61      ! [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 ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_abs_le_norm
% 0.29/0.61  thf(fact_9532_complex__unit__circle,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( Z != zero_zero_complex )
% 0.29/0.61       => ( ( 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 ) ) ) )
% 0.29/0.61          = one_one_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_unit_circle
% 0.29/0.61  thf(fact_9533_inverse__complex_Ocode,axiom,
% 0.29/0.61      ( invers8013647133539491842omplex
% 0.29/0.61      = ( ^ [X4: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X4 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X4 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % inverse_complex.code
% 0.29/0.61  thf(fact_9534_Complex__divide,axiom,
% 0.29/0.61      ( divide1717551699836669952omplex
% 0.29/0.61      = ( ^ [X4: complex,Y6: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Complex_divide
% 0.29/0.61  thf(fact_9535_Im__Reals__divide,axiom,
% 0.29/0.61      ! [R2: complex,Z: complex] :
% 0.29/0.61        ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.29/0.61       => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 0.29/0.61          = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_Reals_divide
% 0.29/0.61  thf(fact_9536_Re__Reals__divide,axiom,
% 0.29/0.61      ! [R2: complex,Z: complex] :
% 0.29/0.61        ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.29/0.61       => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 0.29/0.61          = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_Reals_divide
% 0.29/0.61  thf(fact_9537_complex__mult__cnj,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_mult_cnj
% 0.29/0.61  thf(fact_9538_complex__cnj__divide,axiom,
% 0.29/0.61      ! [X: complex,Y: complex] :
% 0.29/0.61        ( ( cnj @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.29/0.61        = ( divide1717551699836669952omplex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_cnj_divide
% 0.29/0.61  thf(fact_9539_complex__cnj__zero,axiom,
% 0.29/0.61      ( ( cnj @ zero_zero_complex )
% 0.29/0.61      = zero_zero_complex ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_cnj_zero
% 0.29/0.61  thf(fact_9540_complex__cnj__zero__iff,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( ( cnj @ Z )
% 0.29/0.61          = zero_zero_complex )
% 0.29/0.61        = ( Z = zero_zero_complex ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_cnj_zero_iff
% 0.29/0.61  thf(fact_9541_complex__cnj__one,axiom,
% 0.29/0.61      ( ( cnj @ one_one_complex )
% 0.29/0.61      = one_one_complex ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_cnj_one
% 0.29/0.61  thf(fact_9542_complex__cnj__one__iff,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( ( cnj @ Z )
% 0.29/0.61          = one_one_complex )
% 0.29/0.61        = ( Z = one_one_complex ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_cnj_one_iff
% 0.29/0.61  thf(fact_9543_complex__cnj__power,axiom,
% 0.29/0.61      ! [X: complex,N: nat] :
% 0.29/0.61        ( ( cnj @ ( power_power_complex @ X @ N ) )
% 0.29/0.61        = ( power_power_complex @ ( cnj @ X ) @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_cnj_power
% 0.29/0.61  thf(fact_9544_complex__cnj__numeral,axiom,
% 0.29/0.61      ! [W: num] :
% 0.29/0.61        ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
% 0.29/0.61        = ( numera6690914467698888265omplex @ W ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_cnj_numeral
% 0.29/0.61  thf(fact_9545_complex__cnj__of__int,axiom,
% 0.29/0.61      ! [Z: int] :
% 0.29/0.61        ( ( cnj @ ( ring_17405671764205052669omplex @ Z ) )
% 0.29/0.61        = ( ring_17405671764205052669omplex @ Z ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_cnj_of_int
% 0.29/0.61  thf(fact_9546_complex__cnj__neg__numeral,axiom,
% 0.29/0.61      ! [W: num] :
% 0.29/0.61        ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.29/0.61        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_cnj_neg_numeral
% 0.29/0.61  thf(fact_9547_complex__In__mult__cnj__zero,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.29/0.61        = zero_zero_real ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_In_mult_cnj_zero
% 0.29/0.61  thf(fact_9548_Re__divide__Reals,axiom,
% 0.29/0.61      ! [R2: complex,Z: complex] :
% 0.29/0.61        ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.29/0.61       => ( ( re @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 0.29/0.61          = ( divide_divide_real @ ( re @ Z ) @ ( re @ R2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_divide_Reals
% 0.29/0.61  thf(fact_9549_real__eq__imaginary__iff,axiom,
% 0.29/0.61      ! [Y: complex,X: complex] :
% 0.29/0.61        ( ( member_complex @ Y @ real_V2521375963428798218omplex )
% 0.29/0.61       => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 0.29/0.61         => ( ( X
% 0.29/0.61              = ( times_times_complex @ imaginary_unit @ Y ) )
% 0.29/0.61            = ( ( X = zero_zero_complex )
% 0.29/0.61              & ( Y = zero_zero_complex ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % real_eq_imaginary_iff
% 0.29/0.61  thf(fact_9550_imaginary__eq__real__iff,axiom,
% 0.29/0.61      ! [Y: complex,X: complex] :
% 0.29/0.61        ( ( member_complex @ Y @ real_V2521375963428798218omplex )
% 0.29/0.61       => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 0.29/0.61         => ( ( ( times_times_complex @ imaginary_unit @ Y )
% 0.29/0.61              = X )
% 0.29/0.61            = ( ( X = zero_zero_complex )
% 0.29/0.61              & ( Y = zero_zero_complex ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % imaginary_eq_real_iff
% 0.29/0.61  thf(fact_9551_Im__divide__Reals,axiom,
% 0.29/0.61      ! [R2: complex,Z: complex] :
% 0.29/0.61        ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.29/0.61       => ( ( im @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 0.29/0.61          = ( divide_divide_real @ ( im @ Z ) @ ( re @ R2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_divide_Reals
% 0.29/0.61  thf(fact_9552_complex__is__Real__iff,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( member_complex @ Z @ real_V2521375963428798218omplex )
% 0.29/0.61        = ( ( im @ Z )
% 0.29/0.61          = zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_is_Real_iff
% 0.29/0.61  thf(fact_9553_Complex__in__Reals,axiom,
% 0.29/0.61      ! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).
% 0.29/0.61  
% 0.29/0.61  % Complex_in_Reals
% 0.29/0.61  thf(fact_9554_Re__complex__div__eq__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61        = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.29/0.61          = zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_complex_div_eq_0
% 0.29/0.61  thf(fact_9555_Im__complex__div__eq__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.61          = zero_zero_real )
% 0.29/0.61        = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.29/0.61          = zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_complex_div_eq_0
% 0.29/0.61  thf(fact_9556_Re__complex__div__gt__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.29/0.61        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_complex_div_gt_0
% 0.29/0.61  thf(fact_9557_Re__complex__div__lt__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.29/0.61        = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_complex_div_lt_0
% 0.29/0.61  thf(fact_9558_Re__complex__div__le__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.29/0.61        = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_complex_div_le_0
% 0.29/0.61  thf(fact_9559_Re__complex__div__ge__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.29/0.61        = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Re_complex_div_ge_0
% 0.29/0.61  thf(fact_9560_Im__complex__div__gt__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.29/0.61        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_complex_div_gt_0
% 0.29/0.61  thf(fact_9561_Im__complex__div__lt__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.29/0.61        = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_complex_div_lt_0
% 0.29/0.61  thf(fact_9562_Im__complex__div__le__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.29/0.61        = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_complex_div_le_0
% 0.29/0.61  thf(fact_9563_Im__complex__div__ge__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.29/0.61        = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Im_complex_div_ge_0
% 0.29/0.61  thf(fact_9564_complex__mod__mult__cnj,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.29/0.61        = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_mod_mult_cnj
% 0.29/0.61  thf(fact_9565_complex__div__gt__0,axiom,
% 0.29/0.61      ! [A: complex,B: complex] :
% 0.29/0.61        ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.29/0.61          = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 0.29/0.61        & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.29/0.61          = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_div_gt_0
% 0.29/0.61  thf(fact_9566_complex__norm__square,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.61        = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_norm_square
% 0.29/0.61  thf(fact_9567_complex__add__cnj,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 0.29/0.61        = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_add_cnj
% 0.29/0.61  thf(fact_9568_complex__diff__cnj,axiom,
% 0.29/0.61      ! [Z: complex] :
% 0.29/0.61        ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 0.29/0.61        = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_diff_cnj
% 0.29/0.61  thf(fact_9569_complex__div__cnj,axiom,
% 0.29/0.61      ( divide1717551699836669952omplex
% 0.29/0.61      = ( ^ [A3: complex,B4: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B4 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % complex_div_cnj
% 0.29/0.61  thf(fact_9570_cnj__add__mult__eq__Re,axiom,
% 0.29/0.61      ! [Z: complex,W: complex] :
% 0.29/0.61        ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 0.29/0.61        = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % cnj_add_mult_eq_Re
% 0.29/0.61  thf(fact_9571_card__Collect__less__nat,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( finite_card_nat
% 0.29/0.61          @ ( collect_nat
% 0.29/0.61            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
% 0.29/0.61        = N ) ).
% 0.29/0.61  
% 0.29/0.61  % card_Collect_less_nat
% 0.29/0.61  thf(fact_9572_card__atMost,axiom,
% 0.29/0.61      ! [U: nat] :
% 0.29/0.61        ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 0.29/0.61        = ( suc @ U ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_atMost
% 0.29/0.61  thf(fact_9573_card__Collect__le__nat,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( finite_card_nat
% 0.29/0.61          @ ( collect_nat
% 0.29/0.61            @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N ) ) )
% 0.29/0.61        = ( suc @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_Collect_le_nat
% 0.29/0.61  thf(fact_9574_card__atLeastAtMost,axiom,
% 0.29/0.61      ! [L: nat,U: nat] :
% 0.29/0.61        ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.29/0.61        = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_atLeastAtMost
% 0.29/0.61  thf(fact_9575_card__atLeastAtMost__int,axiom,
% 0.29/0.61      ! [L: int,U: int] :
% 0.29/0.61        ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.29/0.61        = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_atLeastAtMost_int
% 0.29/0.61  thf(fact_9576_nat_Odisc__eq__case_I1_J,axiom,
% 0.29/0.61      ! [Nat: nat] :
% 0.29/0.61        ( ( Nat = zero_zero_nat )
% 0.29/0.61        = ( case_nat_o @ $true
% 0.29/0.61          @ ^ [Uu3: nat] : $false
% 0.29/0.61          @ Nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % nat.disc_eq_case(1)
% 0.29/0.61  thf(fact_9577_nat_Odisc__eq__case_I2_J,axiom,
% 0.29/0.61      ! [Nat: nat] :
% 0.29/0.61        ( ( Nat != zero_zero_nat )
% 0.29/0.61        = ( case_nat_o @ $false
% 0.29/0.61          @ ^ [Uu3: nat] : $true
% 0.29/0.61          @ Nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % nat.disc_eq_case(2)
% 0.29/0.61  thf(fact_9578_card__less,axiom,
% 0.29/0.61      ! [M2: set_nat,I: nat] :
% 0.29/0.61        ( ( member_nat @ zero_zero_nat @ M2 )
% 0.29/0.61       => ( ( finite_card_nat
% 0.29/0.61            @ ( collect_nat
% 0.29/0.61              @ ^ [K2: nat] :
% 0.29/0.61                  ( ( member_nat @ K2 @ M2 )
% 0.29/0.61                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
% 0.29/0.61         != zero_zero_nat ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_less
% 0.29/0.61  thf(fact_9579_card__less__Suc,axiom,
% 0.29/0.61      ! [M2: set_nat,I: nat] :
% 0.29/0.61        ( ( member_nat @ zero_zero_nat @ M2 )
% 0.29/0.61       => ( ( suc
% 0.29/0.61            @ ( finite_card_nat
% 0.29/0.61              @ ( collect_nat
% 0.29/0.61                @ ^ [K2: nat] :
% 0.29/0.61                    ( ( member_nat @ ( suc @ K2 ) @ M2 )
% 0.29/0.61                    & ( ord_less_nat @ K2 @ I ) ) ) ) )
% 0.29/0.61          = ( finite_card_nat
% 0.29/0.61            @ ( collect_nat
% 0.29/0.61              @ ^ [K2: nat] :
% 0.29/0.61                  ( ( member_nat @ K2 @ M2 )
% 0.29/0.61                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_less_Suc
% 0.29/0.61  thf(fact_9580_card__less__Suc2,axiom,
% 0.29/0.61      ! [M2: set_nat,I: nat] :
% 0.29/0.61        ( ~ ( member_nat @ zero_zero_nat @ M2 )
% 0.29/0.61       => ( ( finite_card_nat
% 0.29/0.61            @ ( collect_nat
% 0.29/0.61              @ ^ [K2: nat] :
% 0.29/0.61                  ( ( member_nat @ ( suc @ K2 ) @ M2 )
% 0.29/0.61                  & ( ord_less_nat @ K2 @ I ) ) ) )
% 0.29/0.61          = ( finite_card_nat
% 0.29/0.61            @ ( collect_nat
% 0.29/0.61              @ ^ [K2: nat] :
% 0.29/0.61                  ( ( member_nat @ K2 @ M2 )
% 0.29/0.61                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_less_Suc2
% 0.29/0.61  thf(fact_9581_card__atLeastZeroLessThan__int,axiom,
% 0.29/0.61      ! [U: int] :
% 0.29/0.61        ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
% 0.29/0.61        = ( nat2 @ U ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_atLeastZeroLessThan_int
% 0.29/0.61  thf(fact_9582_subset__card__intvl__is__intvl,axiom,
% 0.29/0.61      ! [A2: set_nat,K: nat] :
% 0.29/0.61        ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 0.29/0.61       => ( A2
% 0.29/0.61          = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % subset_card_intvl_is_intvl
% 0.29/0.61  thf(fact_9583_less__eq__nat_Osimps_I2_J,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.29/0.61        = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % less_eq_nat.simps(2)
% 0.29/0.61  thf(fact_9584_card__le__Suc__Max,axiom,
% 0.29/0.61      ! [S2: set_nat] :
% 0.29/0.61        ( ( finite_finite_nat @ S2 )
% 0.29/0.61       => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_le_Suc_Max
% 0.29/0.61  thf(fact_9585_subset__eq__atLeast0__lessThan__card,axiom,
% 0.29/0.61      ! [N4: set_nat,N: nat] :
% 0.29/0.61        ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.29/0.61       => ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % subset_eq_atLeast0_lessThan_card
% 0.29/0.61  thf(fact_9586_card__sum__le__nat__sum,axiom,
% 0.29/0.61      ! [S2: set_nat] :
% 0.29/0.61        ( ord_less_eq_nat
% 0.29/0.61        @ ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [X4: nat] : X4
% 0.29/0.61          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 0.29/0.61        @ ( groups3542108847815614940at_nat
% 0.29/0.61          @ ^ [X4: nat] : X4
% 0.29/0.61          @ S2 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_sum_le_nat_sum
% 0.29/0.61  thf(fact_9587_card__nth__roots,axiom,
% 0.29/0.61      ! [C: complex,N: nat] :
% 0.29/0.61        ( ( C != zero_zero_complex )
% 0.29/0.61       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61         => ( ( finite_card_complex
% 0.29/0.61              @ ( collect_complex
% 0.29/0.61                @ ^ [Z4: complex] :
% 0.29/0.61                    ( ( power_power_complex @ Z4 @ N )
% 0.29/0.61                    = C ) ) )
% 0.29/0.61            = N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_nth_roots
% 0.29/0.61  thf(fact_9588_card__roots__unity__eq,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.61       => ( ( finite_card_complex
% 0.29/0.61            @ ( collect_complex
% 0.29/0.61              @ ^ [Z4: complex] :
% 0.29/0.61                  ( ( power_power_complex @ Z4 @ N )
% 0.29/0.61                  = one_one_complex ) ) )
% 0.29/0.61          = N ) ) ).
% 0.29/0.61  
% 0.29/0.61  % card_roots_unity_eq
% 0.29/0.61  thf(fact_9589_diff__Suc,axiom,
% 0.29/0.61      ! [M: nat,N: nat] :
% 0.29/0.61        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.29/0.61        = ( case_nat_nat @ zero_zero_nat
% 0.29/0.61          @ ^ [K2: nat] : K2
% 0.29/0.61          @ ( minus_minus_nat @ M @ N ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % diff_Suc
% 0.29/0.61  thf(fact_9590_pred__def,axiom,
% 0.29/0.61      ( pred
% 0.29/0.61      = ( case_nat_nat @ zero_zero_nat
% 0.29/0.61        @ ^ [X24: nat] : X24 ) ) ).
% 0.29/0.61  
% 0.29/0.61  % pred_def
% 0.29/0.61  thf(fact_9591_or__int__rec,axiom,
% 0.29/0.61      ( bit_se1409905431419307370or_int
% 0.29/0.61      = ( ^ [K2: int,L3: int] :
% 0.29/0.61            ( plus_plus_int
% 0.29/0.61            @ ( zero_n2684676970156552555ol_int
% 0.29/0.61              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.29/0.61                | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 0.29/0.61            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_int_rec
% 0.29/0.61  thf(fact_9592_or__nonnegative__int__iff,axiom,
% 0.29/0.61      ! [K: int,L: int] :
% 0.29/0.61        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 0.29/0.61        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.61          & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_nonnegative_int_iff
% 0.29/0.61  thf(fact_9593_or__negative__int__iff,axiom,
% 0.29/0.61      ! [K: int,L: int] :
% 0.29/0.61        ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 0.29/0.61        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.29/0.61          | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_negative_int_iff
% 0.29/0.61  thf(fact_9594_or__minus__numerals_I6_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.29/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_minus_numerals(6)
% 0.29/0.61  thf(fact_9595_or__minus__numerals_I2_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_minus_numerals(2)
% 0.29/0.61  thf(fact_9596_or__minus__minus__numerals,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.61        = ( 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 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_minus_minus_numerals
% 0.29/0.61  thf(fact_9597_and__minus__minus__numerals,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.61        = ( 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 ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % and_minus_minus_numerals
% 0.29/0.61  thf(fact_9598_or__greater__eq,axiom,
% 0.29/0.61      ! [L: int,K: int] :
% 0.29/0.61        ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.29/0.61       => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_greater_eq
% 0.29/0.61  thf(fact_9599_OR__lower,axiom,
% 0.29/0.61      ! [X: int,Y: int] :
% 0.29/0.61        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.61       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.61         => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % OR_lower
% 0.29/0.61  thf(fact_9600_plus__and__or,axiom,
% 0.29/0.61      ! [X: int,Y: int] :
% 0.29/0.61        ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
% 0.29/0.61        = ( plus_plus_int @ X @ Y ) ) ).
% 0.29/0.61  
% 0.29/0.61  % plus_and_or
% 0.29/0.61  thf(fact_9601_or__not__numerals_I1_J,axiom,
% 0.29/0.61      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.29/0.61      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_numerals(1)
% 0.29/0.61  thf(fact_9602_set__bit__int__def,axiom,
% 0.29/0.61      ( bit_se7879613467334960850it_int
% 0.29/0.61      = ( ^ [N3: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % set_bit_int_def
% 0.29/0.61  thf(fact_9603_or__not__numerals_I2_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.29/0.61        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_numerals(2)
% 0.29/0.61  thf(fact_9604_or__not__numerals_I4_J,axiom,
% 0.29/0.61      ! [M: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.29/0.61        = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_numerals(4)
% 0.29/0.61  thf(fact_9605_or__not__numerals_I3_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.29/0.61        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_numerals(3)
% 0.29/0.61  thf(fact_9606_or__not__numerals_I7_J,axiom,
% 0.29/0.61      ! [M: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.29/0.61        = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_numerals(7)
% 0.29/0.61  thf(fact_9607_binomial__def,axiom,
% 0.29/0.61      ( binomial
% 0.29/0.61      = ( ^ [N3: nat,K2: nat] :
% 0.29/0.61            ( finite_card_set_nat
% 0.29/0.61            @ ( collect_set_nat
% 0.29/0.61              @ ^ [K7: set_nat] :
% 0.29/0.61                  ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
% 0.29/0.61                  & ( ( finite_card_nat @ K7 )
% 0.29/0.61                    = K2 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % binomial_def
% 0.29/0.61  thf(fact_9608_or__not__numerals_I6_J,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.29/0.61        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_numerals(6)
% 0.29/0.61  thf(fact_9609_OR__upper,axiom,
% 0.29/0.61      ! [X: int,N: nat,Y: int] :
% 0.29/0.61        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.61       => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.61         => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.61           => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % OR_upper
% 0.29/0.61  thf(fact_9610_or__not__numerals_I5_J,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_numerals(5)
% 0.29/0.61  thf(fact_9611_or__not__numerals_I8_J,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_numerals(8)
% 0.29/0.61  thf(fact_9612_or__not__numerals_I9_J,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.29/0.61        = ( 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 ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_numerals(9)
% 0.29/0.61  thf(fact_9613_or__minus__numerals_I1_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_minus_numerals(1)
% 0.29/0.61  thf(fact_9614_or__minus__numerals_I5_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.29/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_minus_numerals(5)
% 0.29/0.61  thf(fact_9615_or__nat__numerals_I4_J,axiom,
% 0.29/0.61      ! [X: num] :
% 0.29/0.61        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.29/0.61        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_nat_numerals(4)
% 0.29/0.61  thf(fact_9616_or__nat__numerals_I2_J,axiom,
% 0.29/0.61      ! [Y: num] :
% 0.29/0.61        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.29/0.61        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_nat_numerals(2)
% 0.29/0.61  thf(fact_9617_or__nat__numerals_I3_J,axiom,
% 0.29/0.61      ! [X: num] :
% 0.29/0.61        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.29/0.61        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_nat_numerals(3)
% 0.29/0.61  thf(fact_9618_or__nat__numerals_I1_J,axiom,
% 0.29/0.61      ! [Y: num] :
% 0.29/0.61        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.29/0.61        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_nat_numerals(1)
% 0.29/0.61  thf(fact_9619_or__minus__numerals_I8_J,axiom,
% 0.29/0.61      ! [N: num,M: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_minus_numerals(8)
% 0.29/0.61  thf(fact_9620_or__minus__numerals_I4_J,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_minus_numerals(4)
% 0.29/0.61  thf(fact_9621_or__minus__numerals_I3_J,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_minus_numerals(3)
% 0.29/0.61  thf(fact_9622_or__minus__numerals_I7_J,axiom,
% 0.29/0.61      ! [N: num,M: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_minus_numerals(7)
% 0.29/0.61  thf(fact_9623_or__not__num__neg_Osimps_I1_J,axiom,
% 0.29/0.61      ( ( bit_or_not_num_neg @ one @ one )
% 0.29/0.61      = one ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_num_neg.simps(1)
% 0.29/0.61  thf(fact_9624_set__bit__nat__def,axiom,
% 0.29/0.61      ( bit_se7882103937844011126it_nat
% 0.29/0.61      = ( ^ [M6: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % set_bit_nat_def
% 0.29/0.61  thf(fact_9625_or__not__num__neg_Osimps_I4_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 0.29/0.61        = ( bit0 @ one ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_num_neg.simps(4)
% 0.29/0.61  thf(fact_9626_or__not__num__neg_Osimps_I6_J,axiom,
% 0.29/0.61      ! [N: num,M: num] :
% 0.29/0.61        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 0.29/0.61        = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_num_neg.simps(6)
% 0.29/0.61  thf(fact_9627_or__not__num__neg_Osimps_I7_J,axiom,
% 0.29/0.61      ! [N: num] :
% 0.29/0.61        ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 0.29/0.61        = one ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_num_neg.simps(7)
% 0.29/0.61  thf(fact_9628_or__not__num__neg_Osimps_I3_J,axiom,
% 0.29/0.61      ! [M: num] :
% 0.29/0.61        ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 0.29/0.61        = ( bit1 @ M ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_num_neg.simps(3)
% 0.29/0.61  thf(fact_9629_or__not__num__neg_Osimps_I5_J,axiom,
% 0.29/0.61      ! [N: num,M: num] :
% 0.29/0.61        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 0.29/0.61        = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_num_neg.simps(5)
% 0.29/0.61  thf(fact_9630_or__nat__def,axiom,
% 0.29/0.61      ( bit_se1412395901928357646or_nat
% 0.29/0.61      = ( ^ [M6: nat,N3: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_nat_def
% 0.29/0.61  thf(fact_9631_or__not__num__neg_Osimps_I2_J,axiom,
% 0.29/0.61      ! [M: num] :
% 0.29/0.61        ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 0.29/0.61        = ( bit1 @ M ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_num_neg.simps(2)
% 0.29/0.61  thf(fact_9632_or__not__num__neg_Osimps_I8_J,axiom,
% 0.29/0.61      ! [N: num,M: num] :
% 0.29/0.61        ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 0.29/0.61        = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_num_neg.simps(8)
% 0.29/0.61  thf(fact_9633_or__not__num__neg_Oelims,axiom,
% 0.29/0.61      ! [X: num,Xa: num,Y: num] :
% 0.29/0.61        ( ( ( bit_or_not_num_neg @ X @ Xa )
% 0.29/0.61          = Y )
% 0.29/0.61       => ( ( ( X = one )
% 0.29/0.61           => ( ( Xa = one )
% 0.29/0.61             => ( Y != one ) ) )
% 0.29/0.61         => ( ( ( X = one )
% 0.29/0.61             => ! [M5: num] :
% 0.29/0.61                  ( ( Xa
% 0.29/0.61                    = ( bit0 @ M5 ) )
% 0.29/0.61                 => ( Y
% 0.29/0.61                   != ( bit1 @ M5 ) ) ) )
% 0.29/0.61           => ( ( ( X = one )
% 0.29/0.61               => ! [M5: num] :
% 0.29/0.61                    ( ( Xa
% 0.29/0.61                      = ( bit1 @ M5 ) )
% 0.29/0.61                   => ( Y
% 0.29/0.61                     != ( bit1 @ M5 ) ) ) )
% 0.29/0.61             => ( ( ? [N2: num] :
% 0.29/0.61                      ( X
% 0.29/0.61                      = ( bit0 @ N2 ) )
% 0.29/0.61                 => ( ( Xa = one )
% 0.29/0.61                   => ( Y
% 0.29/0.61                     != ( bit0 @ one ) ) ) )
% 0.29/0.61               => ( ! [N2: num] :
% 0.29/0.61                      ( ( X
% 0.29/0.61                        = ( bit0 @ N2 ) )
% 0.29/0.61                     => ! [M5: num] :
% 0.29/0.61                          ( ( Xa
% 0.29/0.61                            = ( bit0 @ M5 ) )
% 0.29/0.61                         => ( Y
% 0.29/0.61                           != ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) ) ) )
% 0.29/0.61                 => ( ! [N2: num] :
% 0.29/0.61                        ( ( X
% 0.29/0.61                          = ( bit0 @ N2 ) )
% 0.29/0.61                       => ! [M5: num] :
% 0.29/0.61                            ( ( Xa
% 0.29/0.61                              = ( bit1 @ M5 ) )
% 0.29/0.61                           => ( Y
% 0.29/0.61                             != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M5 ) ) ) ) )
% 0.29/0.61                   => ( ( ? [N2: num] :
% 0.29/0.61                            ( X
% 0.29/0.61                            = ( bit1 @ N2 ) )
% 0.29/0.61                       => ( ( Xa = one )
% 0.29/0.61                         => ( Y != one ) ) )
% 0.29/0.61                     => ( ! [N2: num] :
% 0.29/0.61                            ( ( X
% 0.29/0.61                              = ( bit1 @ N2 ) )
% 0.29/0.61                           => ! [M5: num] :
% 0.29/0.61                                ( ( Xa
% 0.29/0.61                                  = ( bit0 @ M5 ) )
% 0.29/0.61                               => ( Y
% 0.29/0.61                                 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) ) ) )
% 0.29/0.61                       => ~ ! [N2: num] :
% 0.29/0.61                              ( ( X
% 0.29/0.61                                = ( bit1 @ N2 ) )
% 0.29/0.61                             => ! [M5: num] :
% 0.29/0.61                                  ( ( Xa
% 0.29/0.61                                    = ( bit1 @ M5 ) )
% 0.29/0.61                                 => ( Y
% 0.29/0.61                                   != ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_not_num_neg.elims
% 0.29/0.61  thf(fact_9634_numeral__or__not__num__eq,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % numeral_or_not_num_eq
% 0.29/0.61  thf(fact_9635_int__numeral__not__or__num__neg,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % int_numeral_not_or_num_neg
% 0.29/0.61  thf(fact_9636_int__numeral__or__not__num__neg,axiom,
% 0.29/0.61      ! [M: num,N: num] :
% 0.29/0.61        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % int_numeral_or_not_num_neg
% 0.29/0.61  thf(fact_9637_Suc__0__or__eq,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.61        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % Suc_0_or_eq
% 0.29/0.61  thf(fact_9638_or__Suc__0__eq,axiom,
% 0.29/0.61      ! [N: nat] :
% 0.29/0.61        ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.61        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_Suc_0_eq
% 0.29/0.61  thf(fact_9639_or__nat__rec,axiom,
% 0.29/0.61      ( bit_se1412395901928357646or_nat
% 0.29/0.61      = ( ^ [M6: nat,N3: nat] :
% 0.29/0.61            ( plus_plus_nat
% 0.29/0.61            @ ( zero_n2687167440665602831ol_nat
% 0.29/0.61              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 0.29/0.61                | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.29/0.61            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.61  
% 0.29/0.61  % or_nat_rec
% 0.29/0.61  thf(fact_9640_or__not__num__neg_Opelims,axiom,
% 0.29/0.61      ! [X: num,Xa: num,Y: num] :
% 0.29/0.61        ( ( ( bit_or_not_num_neg @ X @ Xa )
% 0.29/0.61          = Y )
% 0.29/0.61       => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.29/0.61         => ( ( ( X = one )
% 0.29/0.61             => ( ( Xa = one )
% 0.29/0.61               => ( ( Y = one )
% 0.29/0.61                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.29/0.61           => ( ( ( X = one )
% 0.29/0.61               => ! [M5: num] :
% 0.29/0.61                    ( ( Xa
% 0.29/0.61                      = ( bit0 @ M5 ) )
% 0.29/0.61                   => ( ( Y
% 0.29/0.61                        = ( bit1 @ M5 ) )
% 0.29/0.61                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M5 ) ) ) ) ) )
% 0.29/0.61             => ( ( ( X = one )
% 0.29/0.61                 => ! [M5: num] :
% 0.29/0.61                      ( ( Xa
% 0.29/0.61                        = ( bit1 @ M5 ) )
% 0.29/0.61                     => ( ( Y
% 0.29/0.61                          = ( bit1 @ M5 ) )
% 0.29/0.61                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M5 ) ) ) ) ) )
% 0.29/0.61               => ( ! [N2: num] :
% 0.29/0.61                      ( ( X
% 0.29/0.61                        = ( bit0 @ N2 ) )
% 0.29/0.61                     => ( ( Xa = one )
% 0.29/0.61                       => ( ( Y
% 0.29/0.61                            = ( bit0 @ one ) )
% 0.29/0.61                         => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ one ) ) ) ) )
% 0.29/0.61                 => ( ! [N2: num] :
% 0.29/0.61                        ( ( X
% 0.29/0.61                          = ( bit0 @ N2 ) )
% 0.29/0.61                       => ! [M5: num] :
% 0.29/0.61                            ( ( Xa
% 0.29/0.61                              = ( bit0 @ M5 ) )
% 0.29/0.62                           => ( ( Y
% 0.29/0.62                                = ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) )
% 0.29/0.62                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit0 @ M5 ) ) ) ) ) )
% 0.29/0.62                   => ( ! [N2: num] :
% 0.29/0.62                          ( ( X
% 0.29/0.62                            = ( bit0 @ N2 ) )
% 0.29/0.62                         => ! [M5: num] :
% 0.29/0.62                              ( ( Xa
% 0.29/0.62                                = ( bit1 @ M5 ) )
% 0.29/0.62                             => ( ( Y
% 0.29/0.62                                  = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M5 ) ) )
% 0.29/0.62                               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit1 @ M5 ) ) ) ) ) )
% 0.29/0.62                     => ( ! [N2: num] :
% 0.29/0.62                            ( ( X
% 0.29/0.62                              = ( bit1 @ N2 ) )
% 0.29/0.62                           => ( ( Xa = one )
% 0.29/0.62                             => ( ( Y = one )
% 0.29/0.62                               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ one ) ) ) ) )
% 0.29/0.62                       => ( ! [N2: num] :
% 0.29/0.62                              ( ( X
% 0.29/0.62                                = ( bit1 @ N2 ) )
% 0.29/0.62                             => ! [M5: num] :
% 0.29/0.62                                  ( ( Xa
% 0.29/0.62                                    = ( bit0 @ M5 ) )
% 0.29/0.62                                 => ( ( Y
% 0.29/0.62                                      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) )
% 0.29/0.62                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit0 @ M5 ) ) ) ) ) )
% 0.29/0.62                         => ~ ! [N2: num] :
% 0.29/0.62                                ( ( X
% 0.29/0.62                                  = ( bit1 @ N2 ) )
% 0.29/0.62                               => ! [M5: num] :
% 0.29/0.62                                    ( ( Xa
% 0.29/0.62                                      = ( bit1 @ M5 ) )
% 0.29/0.62                                   => ( ( Y
% 0.29/0.62                                        = ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) )
% 0.29/0.62                                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit1 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % or_not_num_neg.pelims
% 0.29/0.62  thf(fact_9641_or__int__unfold,axiom,
% 0.29/0.62      ( bit_se1409905431419307370or_int
% 0.29/0.62      = ( ^ [K2: int,L3: int] :
% 0.29/0.62            ( if_int
% 0.29/0.62            @ ( ( K2
% 0.29/0.62                = ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.62              | ( L3
% 0.29/0.62                = ( uminus_uminus_int @ one_one_int ) ) )
% 0.29/0.62            @ ( uminus_uminus_int @ one_one_int )
% 0.29/0.62            @ ( if_int @ ( K2 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % or_int_unfold
% 0.29/0.62  thf(fact_9642_max__enat__simps_I2_J,axiom,
% 0.29/0.62      ! [Q3: extended_enat] :
% 0.29/0.62        ( ( ord_ma741700101516333627d_enat @ Q3 @ zero_z5237406670263579293d_enat )
% 0.29/0.62        = Q3 ) ).
% 0.29/0.62  
% 0.29/0.62  % max_enat_simps(2)
% 0.29/0.62  thf(fact_9643_max__enat__simps_I3_J,axiom,
% 0.29/0.62      ! [Q3: extended_enat] :
% 0.29/0.62        ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q3 )
% 0.29/0.62        = Q3 ) ).
% 0.29/0.62  
% 0.29/0.62  % max_enat_simps(3)
% 0.29/0.62  thf(fact_9644_max__Suc__Suc,axiom,
% 0.29/0.62      ! [M: nat,N: nat] :
% 0.29/0.62        ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.29/0.62        = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % max_Suc_Suc
% 0.29/0.62  thf(fact_9645_max__nat_Oeq__neutr__iff,axiom,
% 0.29/0.62      ! [A: nat,B: nat] :
% 0.29/0.62        ( ( ( ord_max_nat @ A @ B )
% 0.29/0.62          = zero_zero_nat )
% 0.29/0.62        = ( ( A = zero_zero_nat )
% 0.29/0.62          & ( B = zero_zero_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % max_nat.eq_neutr_iff
% 0.29/0.62  thf(fact_9646_max__nat_Oleft__neutral,axiom,
% 0.29/0.62      ! [A: nat] :
% 0.29/0.62        ( ( ord_max_nat @ zero_zero_nat @ A )
% 0.29/0.62        = A ) ).
% 0.29/0.62  
% 0.29/0.62  % max_nat.left_neutral
% 0.29/0.62  thf(fact_9647_max__nat_Oneutr__eq__iff,axiom,
% 0.29/0.62      ! [A: nat,B: nat] :
% 0.29/0.62        ( ( zero_zero_nat
% 0.29/0.62          = ( ord_max_nat @ A @ B ) )
% 0.29/0.62        = ( ( A = zero_zero_nat )
% 0.29/0.62          & ( B = zero_zero_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % max_nat.neutr_eq_iff
% 0.29/0.62  thf(fact_9648_max__nat_Oright__neutral,axiom,
% 0.29/0.62      ! [A: nat] :
% 0.29/0.62        ( ( ord_max_nat @ A @ zero_zero_nat )
% 0.29/0.62        = A ) ).
% 0.29/0.62  
% 0.29/0.62  % max_nat.right_neutral
% 0.29/0.62  thf(fact_9649_max__0L,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( ord_max_nat @ zero_zero_nat @ N )
% 0.29/0.62        = N ) ).
% 0.29/0.62  
% 0.29/0.62  % max_0L
% 0.29/0.62  thf(fact_9650_max__0R,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( ord_max_nat @ N @ zero_zero_nat )
% 0.29/0.62        = N ) ).
% 0.29/0.62  
% 0.29/0.62  % max_0R
% 0.29/0.62  thf(fact_9651_max__Suc__numeral,axiom,
% 0.29/0.62      ! [N: nat,K: num] :
% 0.29/0.62        ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.29/0.62        = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % max_Suc_numeral
% 0.29/0.62  thf(fact_9652_max__numeral__Suc,axiom,
% 0.29/0.62      ! [K: num,N: nat] :
% 0.29/0.62        ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.29/0.62        = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % max_numeral_Suc
% 0.29/0.62  thf(fact_9653_nat__add__max__right,axiom,
% 0.29/0.62      ! [M: nat,N: nat,Q3: nat] :
% 0.29/0.62        ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 0.29/0.62        = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_add_max_right
% 0.29/0.62  thf(fact_9654_nat__add__max__left,axiom,
% 0.29/0.62      ! [M: nat,N: nat,Q3: nat] :
% 0.29/0.62        ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 0.29/0.62        = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_add_max_left
% 0.29/0.62  thf(fact_9655_nat__mult__max__right,axiom,
% 0.29/0.62      ! [M: nat,N: nat,Q3: nat] :
% 0.29/0.62        ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 0.29/0.62        = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_mult_max_right
% 0.29/0.62  thf(fact_9656_nat__mult__max__left,axiom,
% 0.29/0.62      ! [M: nat,N: nat,Q3: nat] :
% 0.29/0.62        ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 0.29/0.62        = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_mult_max_left
% 0.29/0.62  thf(fact_9657_nat__minus__add__max,axiom,
% 0.29/0.62      ! [N: nat,M: nat] :
% 0.29/0.62        ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 0.29/0.62        = ( ord_max_nat @ N @ M ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_minus_add_max
% 0.29/0.62  thf(fact_9658_max__Suc2,axiom,
% 0.29/0.62      ! [M: nat,N: nat] :
% 0.29/0.62        ( ( ord_max_nat @ M @ ( suc @ N ) )
% 0.29/0.62        = ( case_nat_nat @ ( suc @ N )
% 0.29/0.62          @ ^ [M7: nat] : ( suc @ ( ord_max_nat @ M7 @ N ) )
% 0.29/0.62          @ M ) ) ).
% 0.29/0.62  
% 0.29/0.62  % max_Suc2
% 0.29/0.62  thf(fact_9659_max__Suc1,axiom,
% 0.29/0.62      ! [N: nat,M: nat] :
% 0.29/0.62        ( ( ord_max_nat @ ( suc @ N ) @ M )
% 0.29/0.62        = ( case_nat_nat @ ( suc @ N )
% 0.29/0.62          @ ^ [M7: nat] : ( suc @ ( ord_max_nat @ N @ M7 ) )
% 0.29/0.62          @ M ) ) ).
% 0.29/0.62  
% 0.29/0.62  % max_Suc1
% 0.29/0.62  thf(fact_9660_or__nat__unfold,axiom,
% 0.29/0.62      ( bit_se1412395901928357646or_nat
% 0.29/0.62      = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % or_nat_unfold
% 0.29/0.62  thf(fact_9661_prod__decode__aux_Osimps,axiom,
% 0.29/0.62      ( nat_prod_decode_aux
% 0.29/0.62      = ( ^ [K2: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K2 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K2 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M6 @ ( suc @ K2 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % prod_decode_aux.simps
% 0.29/0.62  thf(fact_9662_prod__decode__aux_Oelims,axiom,
% 0.29/0.62      ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
% 0.29/0.62        ( ( ( nat_prod_decode_aux @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( ( ord_less_eq_nat @ Xa @ X )
% 0.29/0.62           => ( Y
% 0.29/0.62              = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 0.29/0.62          & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 0.29/0.62           => ( Y
% 0.29/0.62              = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % prod_decode_aux.elims
% 0.29/0.62  thf(fact_9663_bezw__0,axiom,
% 0.29/0.62      ! [X: nat] :
% 0.29/0.62        ( ( bezw @ X @ zero_zero_nat )
% 0.29/0.62        = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % bezw_0
% 0.29/0.62  thf(fact_9664_prod__decode__aux_Opelims,axiom,
% 0.29/0.62      ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
% 0.29/0.62        ( ( ( nat_prod_decode_aux @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.29/0.62         => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
% 0.29/0.62                 => ( Y
% 0.29/0.62                    = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 0.29/0.62                & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 0.29/0.62                 => ( Y
% 0.29/0.62                    = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
% 0.29/0.62             => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % prod_decode_aux.pelims
% 0.29/0.62  thf(fact_9665_Suc__0__div__numeral,axiom,
% 0.29/0.62      ! [K: num] :
% 0.29/0.62        ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.29/0.62        = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Suc_0_div_numeral
% 0.29/0.62  thf(fact_9666_drop__bit__numeral__minus__bit1,axiom,
% 0.29/0.62      ! [L: num,K: num] :
% 0.29/0.62        ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.29/0.62        = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % drop_bit_numeral_minus_bit1
% 0.29/0.62  thf(fact_9667_drop__bit__nonnegative__int__iff,axiom,
% 0.29/0.62      ! [N: nat,K: int] :
% 0.29/0.62        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 0.29/0.62        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.29/0.62  
% 0.29/0.62  % drop_bit_nonnegative_int_iff
% 0.29/0.62  thf(fact_9668_drop__bit__negative__int__iff,axiom,
% 0.29/0.62      ! [N: nat,K: int] :
% 0.29/0.62        ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 0.29/0.62        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % drop_bit_negative_int_iff
% 0.29/0.62  thf(fact_9669_drop__bit__minus__one,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.29/0.62        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % drop_bit_minus_one
% 0.29/0.62  thf(fact_9670_fst__divmod__nat,axiom,
% 0.29/0.62      ! [M: nat,N: nat] :
% 0.29/0.62        ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.29/0.62        = ( divide_divide_nat @ M @ N ) ) ).
% 0.29/0.62  
% 0.29/0.62  % fst_divmod_nat
% 0.29/0.62  thf(fact_9671_drop__bit__Suc__minus__bit0,axiom,
% 0.29/0.62      ! [N: nat,K: num] :
% 0.29/0.62        ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.29/0.62        = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % drop_bit_Suc_minus_bit0
% 0.29/0.62  thf(fact_9672_drop__bit__numeral__minus__bit0,axiom,
% 0.29/0.62      ! [L: num,K: num] :
% 0.29/0.62        ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.29/0.62        = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % drop_bit_numeral_minus_bit0
% 0.29/0.62  thf(fact_9673_drop__bit__Suc__minus__bit1,axiom,
% 0.29/0.62      ! [N: nat,K: num] :
% 0.29/0.62        ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.29/0.62        = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % drop_bit_Suc_minus_bit1
% 0.29/0.62  thf(fact_9674_drop__bit__int__def,axiom,
% 0.29/0.62      ( bit_se8568078237143864401it_int
% 0.29/0.62      = ( ^ [N3: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % drop_bit_int_def
% 0.29/0.62  thf(fact_9675_Suc__0__mod__numeral,axiom,
% 0.29/0.62      ! [K: num] :
% 0.29/0.62        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.29/0.62        = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Suc_0_mod_numeral
% 0.29/0.62  thf(fact_9676_finite__enumerate,axiom,
% 0.29/0.62      ! [S2: set_nat] :
% 0.29/0.62        ( ( finite_finite_nat @ S2 )
% 0.29/0.62       => ? [R3: nat > nat] :
% 0.29/0.62            ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
% 0.29/0.62            & ! [N8: nat] :
% 0.29/0.62                ( ( ord_less_nat @ N8 @ ( finite_card_nat @ S2 ) )
% 0.29/0.62               => ( member_nat @ ( R3 @ N8 ) @ S2 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % finite_enumerate
% 0.29/0.62  thf(fact_9677_fst__divmod__integer,axiom,
% 0.29/0.62      ! [K: code_integer,L: code_integer] :
% 0.29/0.62        ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L ) )
% 0.29/0.62        = ( divide6298287555418463151nteger @ K @ L ) ) ).
% 0.29/0.62  
% 0.29/0.62  % fst_divmod_integer
% 0.29/0.62  thf(fact_9678_fst__divmod__abs,axiom,
% 0.29/0.62      ! [K: code_integer,L: code_integer] :
% 0.29/0.62        ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L ) )
% 0.29/0.62        = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % fst_divmod_abs
% 0.29/0.62  thf(fact_9679_drop__bit__of__Suc__0,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.62        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % drop_bit_of_Suc_0
% 0.29/0.62  thf(fact_9680_drop__bit__nat__def,axiom,
% 0.29/0.62      ( bit_se8570568707652914677it_nat
% 0.29/0.62      = ( ^ [N3: nat,M6: nat] : ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % drop_bit_nat_def
% 0.29/0.62  thf(fact_9681_rat__sgn__code,axiom,
% 0.29/0.62      ! [P5: rat] :
% 0.29/0.62        ( ( quotient_of @ ( sgn_sgn_rat @ P5 ) )
% 0.29/0.62        = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P5 ) ) ) @ one_one_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % rat_sgn_code
% 0.29/0.62  thf(fact_9682_Sup__nat__empty,axiom,
% 0.29/0.62      ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 0.29/0.62      = zero_zero_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % Sup_nat_empty
% 0.29/0.62  thf(fact_9683_Inf__real__def,axiom,
% 0.29/0.62      ( comple4887499456419720421f_real
% 0.29/0.62      = ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Inf_real_def
% 0.29/0.62  thf(fact_9684_Sup__nat__def,axiom,
% 0.29/0.62      ( complete_Sup_Sup_nat
% 0.29/0.62      = ( ^ [X6: set_nat] : ( if_nat @ ( X6 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X6 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Sup_nat_def
% 0.29/0.62  thf(fact_9685_quotient__of__denom__pos_H,axiom,
% 0.29/0.62      ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % quotient_of_denom_pos'
% 0.29/0.62  thf(fact_9686_bezw__non__0,axiom,
% 0.29/0.62      ! [Y: nat,X: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ Y )
% 0.29/0.62       => ( ( bezw @ X @ Y )
% 0.29/0.62          = ( 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 ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % bezw_non_0
% 0.29/0.62  thf(fact_9687_bezw_Osimps,axiom,
% 0.29/0.62      ( bezw
% 0.29/0.62      = ( ^ [X4: nat,Y6: nat] : ( if_Pro3027730157355071871nt_int @ ( Y6 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X4 @ Y6 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X4 @ Y6 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X4 @ Y6 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y6 ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % bezw.simps
% 0.29/0.62  thf(fact_9688_bezw_Oelims,axiom,
% 0.29/0.62      ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 0.29/0.62        ( ( ( bezw @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( ( Xa = zero_zero_nat )
% 0.29/0.62           => ( Y
% 0.29/0.62              = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.29/0.62          & ( ( Xa != zero_zero_nat )
% 0.29/0.62           => ( Y
% 0.29/0.62              = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % bezw.elims
% 0.29/0.62  thf(fact_9689_bezw_Opelims,axiom,
% 0.29/0.62      ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 0.29/0.62        ( ( ( bezw @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.29/0.62         => ~ ( ( ( ( Xa = zero_zero_nat )
% 0.29/0.62                 => ( Y
% 0.29/0.62                    = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.29/0.62                & ( ( Xa != zero_zero_nat )
% 0.29/0.62                 => ( Y
% 0.29/0.62                    = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
% 0.29/0.62             => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % bezw.pelims
% 0.29/0.62  thf(fact_9690_minus__one__mod__numeral,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62        = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % minus_one_mod_numeral
% 0.29/0.62  thf(fact_9691_numeral__mod__minus__numeral,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62        = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % numeral_mod_minus_numeral
% 0.29/0.62  thf(fact_9692_minus__numeral__mod__numeral,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62        = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % minus_numeral_mod_numeral
% 0.29/0.62  thf(fact_9693_one__mod__minus__numeral,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62        = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % one_mod_minus_numeral
% 0.29/0.62  thf(fact_9694_Divides_Oadjust__mod__def,axiom,
% 0.29/0.62      ( adjust_mod
% 0.29/0.62      = ( ^ [L3: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L3 @ R5 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Divides.adjust_mod_def
% 0.29/0.62  thf(fact_9695_normalize__def,axiom,
% 0.29/0.62      ( normalize
% 0.29/0.62      = ( ^ [P4: product_prod_int_int] :
% 0.29/0.62            ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P4 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) )
% 0.29/0.62            @ ( if_Pro3027730157355071871nt_int
% 0.29/0.62              @ ( ( product_snd_int_int @ P4 )
% 0.29/0.62                = zero_zero_int )
% 0.29/0.62              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.29/0.62              @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % normalize_def
% 0.29/0.62  thf(fact_9696_gcd__1__int,axiom,
% 0.29/0.62      ! [M: int] :
% 0.29/0.62        ( ( gcd_gcd_int @ M @ one_one_int )
% 0.29/0.62        = one_one_int ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_1_int
% 0.29/0.62  thf(fact_9697_gcd__pos__int,axiom,
% 0.29/0.62      ! [M: int,N: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 0.29/0.62        = ( ( M != zero_zero_int )
% 0.29/0.62          | ( N != zero_zero_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_pos_int
% 0.29/0.62  thf(fact_9698_gcd__neg__numeral__2__int,axiom,
% 0.29/0.62      ! [X: int,N: num] :
% 0.29/0.62        ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62        = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_neg_numeral_2_int
% 0.29/0.62  thf(fact_9699_gcd__neg__numeral__1__int,axiom,
% 0.29/0.62      ! [N: num,X: int] :
% 0.29/0.62        ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
% 0.29/0.62        = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_neg_numeral_1_int
% 0.29/0.62  thf(fact_9700_gcd__0__left__int,axiom,
% 0.29/0.62      ! [X: int] :
% 0.29/0.62        ( ( gcd_gcd_int @ zero_zero_int @ X )
% 0.29/0.62        = ( abs_abs_int @ X ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_0_left_int
% 0.29/0.62  thf(fact_9701_gcd__0__int,axiom,
% 0.29/0.62      ! [X: int] :
% 0.29/0.62        ( ( gcd_gcd_int @ X @ zero_zero_int )
% 0.29/0.62        = ( abs_abs_int @ X ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_0_int
% 0.29/0.62  thf(fact_9702_bezout__int,axiom,
% 0.29/0.62      ! [X: int,Y: int] :
% 0.29/0.62      ? [U3: int,V3: int] :
% 0.29/0.62        ( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V3 @ Y ) )
% 0.29/0.62        = ( gcd_gcd_int @ X @ Y ) ) ).
% 0.29/0.62  
% 0.29/0.62  % bezout_int
% 0.29/0.62  thf(fact_9703_gcd__ge__0__int,axiom,
% 0.29/0.62      ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_ge_0_int
% 0.29/0.62  thf(fact_9704_gcd__le1__int,axiom,
% 0.29/0.62      ! [A: int,B: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.62       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_le1_int
% 0.29/0.62  thf(fact_9705_gcd__le2__int,axiom,
% 0.29/0.62      ! [B: int,A: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.62       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_le2_int
% 0.29/0.62  thf(fact_9706_gcd__cases__int,axiom,
% 0.29/0.62      ! [X: int,Y: int,P: int > $o] :
% 0.29/0.62        ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.62         => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.62           => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
% 0.29/0.62       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.62           => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.29/0.62             => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
% 0.29/0.62         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.29/0.62             => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.62               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
% 0.29/0.62           => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.29/0.62               => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.29/0.62                 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
% 0.29/0.62             => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_cases_int
% 0.29/0.62  thf(fact_9707_gcd__unique__int,axiom,
% 0.29/0.62      ! [D: int,A: int,B: int] :
% 0.29/0.62        ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 0.29/0.62          & ( dvd_dvd_int @ D @ A )
% 0.29/0.62          & ( dvd_dvd_int @ D @ B )
% 0.29/0.62          & ! [E3: int] :
% 0.29/0.62              ( ( ( dvd_dvd_int @ E3 @ A )
% 0.29/0.62                & ( dvd_dvd_int @ E3 @ B ) )
% 0.29/0.62             => ( dvd_dvd_int @ E3 @ D ) ) )
% 0.29/0.62        = ( D
% 0.29/0.62          = ( gcd_gcd_int @ A @ B ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_unique_int
% 0.29/0.62  thf(fact_9708_gcd__non__0__int,axiom,
% 0.29/0.62      ! [Y: int,X: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ Y )
% 0.29/0.62       => ( ( gcd_gcd_int @ X @ Y )
% 0.29/0.62          = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_non_0_int
% 0.29/0.62  thf(fact_9709_gcd__code__int,axiom,
% 0.29/0.62      ( gcd_gcd_int
% 0.29/0.62      = ( ^ [K2: int,L3: int] : ( abs_abs_int @ ( if_int @ ( L3 = zero_zero_int ) @ K2 @ ( gcd_gcd_int @ L3 @ ( modulo_modulo_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_code_int
% 0.29/0.62  thf(fact_9710_gcd__is__Max__divisors__int,axiom,
% 0.29/0.62      ! [N: int,M: int] :
% 0.29/0.62        ( ( N != zero_zero_int )
% 0.29/0.62       => ( ( gcd_gcd_int @ M @ N )
% 0.29/0.62          = ( lattic8263393255366662781ax_int
% 0.29/0.62            @ ( collect_int
% 0.29/0.62              @ ^ [D3: int] :
% 0.29/0.62                  ( ( dvd_dvd_int @ D3 @ M )
% 0.29/0.62                  & ( dvd_dvd_int @ D3 @ N ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_is_Max_divisors_int
% 0.29/0.62  thf(fact_9711_suminf__eq__SUP__real,axiom,
% 0.29/0.62      ! [X8: nat > real] :
% 0.29/0.62        ( ( summable_real @ X8 )
% 0.29/0.62       => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I2 ) )
% 0.29/0.62         => ( ( suminf_real @ X8 )
% 0.29/0.62            = ( comple1385675409528146559p_real
% 0.29/0.62              @ ( image_nat_real
% 0.29/0.62                @ ^ [I3: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I3 ) )
% 0.29/0.62                @ top_top_set_nat ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % suminf_eq_SUP_real
% 0.29/0.62  thf(fact_9712_gcd__0__left__nat,axiom,
% 0.29/0.62      ! [X: nat] :
% 0.29/0.62        ( ( gcd_gcd_nat @ zero_zero_nat @ X )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_0_left_nat
% 0.29/0.62  thf(fact_9713_gcd__0__nat,axiom,
% 0.29/0.62      ! [X: nat] :
% 0.29/0.62        ( ( gcd_gcd_nat @ X @ zero_zero_nat )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_0_nat
% 0.29/0.62  thf(fact_9714_gcd__nat_Oright__neutral,axiom,
% 0.29/0.62      ! [A: nat] :
% 0.29/0.62        ( ( gcd_gcd_nat @ A @ zero_zero_nat )
% 0.29/0.62        = A ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_nat.right_neutral
% 0.29/0.62  thf(fact_9715_gcd__nat_Oneutr__eq__iff,axiom,
% 0.29/0.62      ! [A: nat,B: nat] :
% 0.29/0.62        ( ( zero_zero_nat
% 0.29/0.62          = ( gcd_gcd_nat @ A @ B ) )
% 0.29/0.62        = ( ( A = zero_zero_nat )
% 0.29/0.62          & ( B = zero_zero_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_nat.neutr_eq_iff
% 0.29/0.62  thf(fact_9716_gcd__nat_Oleft__neutral,axiom,
% 0.29/0.62      ! [A: nat] :
% 0.29/0.62        ( ( gcd_gcd_nat @ zero_zero_nat @ A )
% 0.29/0.62        = A ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_nat.left_neutral
% 0.29/0.62  thf(fact_9717_gcd__nat_Oeq__neutr__iff,axiom,
% 0.29/0.62      ! [A: nat,B: nat] :
% 0.29/0.62        ( ( ( gcd_gcd_nat @ A @ B )
% 0.29/0.62          = zero_zero_nat )
% 0.29/0.62        = ( ( A = zero_zero_nat )
% 0.29/0.62          & ( B = zero_zero_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_nat.eq_neutr_iff
% 0.29/0.62  thf(fact_9718_gcd__1__nat,axiom,
% 0.29/0.62      ! [M: nat] :
% 0.29/0.62        ( ( gcd_gcd_nat @ M @ one_one_nat )
% 0.29/0.62        = one_one_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_1_nat
% 0.29/0.62  thf(fact_9719_gcd__Suc__0,axiom,
% 0.29/0.62      ! [M: nat] :
% 0.29/0.62        ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.29/0.62        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_Suc_0
% 0.29/0.62  thf(fact_9720_gcd__pos__nat,axiom,
% 0.29/0.62      ! [M: nat,N: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 0.29/0.62        = ( ( M != zero_zero_nat )
% 0.29/0.62          | ( N != zero_zero_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_pos_nat
% 0.29/0.62  thf(fact_9721_gcd__int__int__eq,axiom,
% 0.29/0.62      ! [M: nat,N: nat] :
% 0.29/0.62        ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.29/0.62        = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_int_int_eq
% 0.29/0.62  thf(fact_9722_gcd__nat__abs__left__eq,axiom,
% 0.29/0.62      ! [K: int,N: nat] :
% 0.29/0.62        ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.29/0.62        = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_nat_abs_left_eq
% 0.29/0.62  thf(fact_9723_gcd__nat__abs__right__eq,axiom,
% 0.29/0.62      ! [N: nat,K: int] :
% 0.29/0.62        ( ( gcd_gcd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.29/0.62        = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_nat_abs_right_eq
% 0.29/0.62  thf(fact_9724_gcd__non__0__nat,axiom,
% 0.29/0.62      ! [Y: nat,X: nat] :
% 0.29/0.62        ( ( Y != zero_zero_nat )
% 0.29/0.62       => ( ( gcd_gcd_nat @ X @ Y )
% 0.29/0.62          = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_non_0_nat
% 0.29/0.62  thf(fact_9725_gcd__nat_Osimps,axiom,
% 0.29/0.62      ( gcd_gcd_nat
% 0.29/0.62      = ( ^ [X4: nat,Y6: nat] : ( if_nat @ ( Y6 = zero_zero_nat ) @ X4 @ ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X4 @ Y6 ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_nat.simps
% 0.29/0.62  thf(fact_9726_gcd__nat_Oelims,axiom,
% 0.29/0.62      ! [X: nat,Xa: nat,Y: nat] :
% 0.29/0.62        ( ( ( gcd_gcd_nat @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( ( Xa = zero_zero_nat )
% 0.29/0.62           => ( Y = X ) )
% 0.29/0.62          & ( ( Xa != zero_zero_nat )
% 0.29/0.62           => ( Y
% 0.29/0.62              = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_nat.elims
% 0.29/0.62  thf(fact_9727_gcd__le1__nat,axiom,
% 0.29/0.62      ! [A: nat,B: nat] :
% 0.29/0.62        ( ( A != zero_zero_nat )
% 0.29/0.62       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_le1_nat
% 0.29/0.62  thf(fact_9728_gcd__le2__nat,axiom,
% 0.29/0.62      ! [B: nat,A: nat] :
% 0.29/0.62        ( ( B != zero_zero_nat )
% 0.29/0.62       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_le2_nat
% 0.29/0.62  thf(fact_9729_gcd__diff1__nat,axiom,
% 0.29/0.62      ! [N: nat,M: nat] :
% 0.29/0.62        ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.62       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 0.29/0.62          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_diff1_nat
% 0.29/0.62  thf(fact_9730_gcd__diff2__nat,axiom,
% 0.29/0.62      ! [M: nat,N: nat] :
% 0.29/0.62        ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.62       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 0.29/0.62          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_diff2_nat
% 0.29/0.62  thf(fact_9731_bezout__nat,axiom,
% 0.29/0.62      ! [A: nat,B: nat] :
% 0.29/0.62        ( ( A != zero_zero_nat )
% 0.29/0.62       => ? [X3: nat,Y4: nat] :
% 0.29/0.62            ( ( times_times_nat @ A @ X3 )
% 0.29/0.62            = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % bezout_nat
% 0.29/0.62  thf(fact_9732_bezout__gcd__nat_H,axiom,
% 0.29/0.62      ! [B: nat,A: nat] :
% 0.29/0.62      ? [X3: nat,Y4: nat] :
% 0.29/0.62        ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X3 ) )
% 0.29/0.62          & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
% 0.29/0.62            = ( gcd_gcd_nat @ A @ B ) ) )
% 0.29/0.62        | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X3 ) )
% 0.29/0.62          & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.29/0.62            = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % bezout_gcd_nat'
% 0.29/0.62  thf(fact_9733_gcd__code__integer,axiom,
% 0.29/0.62      ( gcd_gcd_Code_integer
% 0.29/0.62      = ( ^ [K2: code_integer,L3: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L3 = zero_z3403309356797280102nteger ) @ K2 @ ( gcd_gcd_Code_integer @ L3 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L3 ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_code_integer
% 0.29/0.62  thf(fact_9734_gcd__int__def,axiom,
% 0.29/0.62      ( gcd_gcd_int
% 0.29/0.62      = ( ^ [X4: int,Y6: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X4 ) ) @ ( nat2 @ ( abs_abs_int @ Y6 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_int_def
% 0.29/0.62  thf(fact_9735_gcd__is__Max__divisors__nat,axiom,
% 0.29/0.62      ! [N: nat,M: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( gcd_gcd_nat @ M @ N )
% 0.29/0.62          = ( lattic8265883725875713057ax_nat
% 0.29/0.62            @ ( collect_nat
% 0.29/0.62              @ ^ [D3: nat] :
% 0.29/0.62                  ( ( dvd_dvd_nat @ D3 @ M )
% 0.29/0.62                  & ( dvd_dvd_nat @ D3 @ N ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_is_Max_divisors_nat
% 0.29/0.62  thf(fact_9736_UNIV__nat__eq,axiom,
% 0.29/0.62      ( top_top_set_nat
% 0.29/0.62      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % UNIV_nat_eq
% 0.29/0.62  thf(fact_9737_range__mod,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( image_nat_nat
% 0.29/0.62            @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N )
% 0.29/0.62            @ top_top_set_nat )
% 0.29/0.62          = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % range_mod
% 0.29/0.62  thf(fact_9738_bezw__aux,axiom,
% 0.29/0.62      ! [X: nat,Y: nat] :
% 0.29/0.62        ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
% 0.29/0.62        = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % bezw_aux
% 0.29/0.62  thf(fact_9739_gcd__nat_Opelims,axiom,
% 0.29/0.62      ! [X: nat,Xa: nat,Y: nat] :
% 0.29/0.62        ( ( ( gcd_gcd_nat @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.29/0.62         => ~ ( ( ( ( Xa = zero_zero_nat )
% 0.29/0.62                 => ( Y = X ) )
% 0.29/0.62                & ( ( Xa != zero_zero_nat )
% 0.29/0.62                 => ( Y
% 0.29/0.62                    = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
% 0.29/0.62             => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_nat.pelims
% 0.29/0.62  thf(fact_9740_card__UNIV__unit,axiom,
% 0.29/0.62      ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 0.29/0.62      = one_one_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % card_UNIV_unit
% 0.29/0.62  thf(fact_9741_card__UNIV__bool,axiom,
% 0.29/0.62      ( ( finite_card_o @ top_top_set_o )
% 0.29/0.62      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % card_UNIV_bool
% 0.29/0.62  thf(fact_9742_range__mult,axiom,
% 0.29/0.62      ! [A: real] :
% 0.29/0.62        ( ( ( A = zero_zero_real )
% 0.29/0.62         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.29/0.62            = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.29/0.62        & ( ( A != zero_zero_real )
% 0.29/0.62         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.29/0.62            = top_top_set_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % range_mult
% 0.29/0.62  thf(fact_9743_infinite__UNIV__int,axiom,
% 0.29/0.62      ~ ( finite_finite_int @ top_top_set_int ) ).
% 0.29/0.62  
% 0.29/0.62  % infinite_UNIV_int
% 0.29/0.62  thf(fact_9744_int__in__range__abs,axiom,
% 0.29/0.62      ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % int_in_range_abs
% 0.29/0.62  thf(fact_9745_nat__descend__induct,axiom,
% 0.29/0.62      ! [N: nat,P: nat > $o,M: nat] :
% 0.29/0.62        ( ! [K3: nat] :
% 0.29/0.62            ( ( ord_less_nat @ N @ K3 )
% 0.29/0.62           => ( P @ K3 ) )
% 0.29/0.62       => ( ! [K3: nat] :
% 0.29/0.62              ( ( ord_less_eq_nat @ K3 @ N )
% 0.29/0.62             => ( ! [I4: nat] :
% 0.29/0.62                    ( ( ord_less_nat @ K3 @ I4 )
% 0.29/0.62                   => ( P @ I4 ) )
% 0.29/0.62               => ( P @ K3 ) ) )
% 0.29/0.62         => ( P @ M ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_descend_induct
% 0.29/0.62  thf(fact_9746_root__def,axiom,
% 0.29/0.62      ( root
% 0.29/0.62      = ( ^ [N3: nat,X4: real] :
% 0.29/0.62            ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
% 0.29/0.62            @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.29/0.62              @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N3 ) )
% 0.29/0.62              @ X4 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % root_def
% 0.29/0.62  thf(fact_9747_card__UNIV__char,axiom,
% 0.29/0.62      ( ( finite_card_char @ top_top_set_char )
% 0.29/0.62      = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % card_UNIV_char
% 0.29/0.62  thf(fact_9748_UNIV__bool,axiom,
% 0.29/0.62      ( top_top_set_o
% 0.29/0.62      = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % UNIV_bool
% 0.29/0.62  thf(fact_9749_UNIV__char__of__nat,axiom,
% 0.29/0.62      ( top_top_set_char
% 0.29/0.62      = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % UNIV_char_of_nat
% 0.29/0.62  thf(fact_9750_char_Osize_I2_J,axiom,
% 0.29/0.62      ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 0.29/0.62        ( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 0.29/0.62        = zero_zero_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % char.size(2)
% 0.29/0.62  thf(fact_9751_nat__of__char__less__256,axiom,
% 0.29/0.62      ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_of_char_less_256
% 0.29/0.62  thf(fact_9752_range__nat__of__char,axiom,
% 0.29/0.62      ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 0.29/0.62      = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % range_nat_of_char
% 0.29/0.62  thf(fact_9753_integer__of__char__code,axiom,
% 0.29/0.62      ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 0.29/0.62        ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 0.29/0.62        = ( 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 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % integer_of_char_code
% 0.29/0.62  thf(fact_9754_char_Osize__gen,axiom,
% 0.29/0.62      ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 0.29/0.62        ( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 0.29/0.62        = zero_zero_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % char.size_gen
% 0.29/0.62  thf(fact_9755_String_Ochar__of__ascii__of,axiom,
% 0.29/0.62      ! [C: char] :
% 0.29/0.62        ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 0.29/0.62        = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % String.char_of_ascii_of
% 0.29/0.62  thf(fact_9756_DERIV__real__root__generic,axiom,
% 0.29/0.62      ! [N: nat,X: real,D2: real] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( X != zero_zero_real )
% 0.29/0.62         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.62             => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.62               => ( D2
% 0.29/0.62                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.29/0.62           => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.62               => ( ( ord_less_real @ X @ zero_zero_real )
% 0.29/0.62                 => ( D2
% 0.29/0.62                    = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 0.29/0.62             => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.62                 => ( D2
% 0.29/0.62                    = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.29/0.62               => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_real_root_generic
% 0.29/0.62  thf(fact_9757_DERIV__even__real__root,axiom,
% 0.29/0.62      ! [N: nat,X: real] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.62         => ( ( ord_less_real @ X @ zero_zero_real )
% 0.29/0.62           => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_even_real_root
% 0.29/0.62  thf(fact_9758_DERIV__arctan__series,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.62       => ( has_fi5821293074295781190e_real
% 0.29/0.62          @ ^ [X9: real] :
% 0.29/0.62              ( suminf_real
% 0.29/0.62              @ ^ [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 @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 0.29/0.62          @ ( suminf_real
% 0.29/0.62            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_arctan_series
% 0.29/0.62  thf(fact_9759_has__real__derivative__pos__inc__right,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.29/0.62         => ? [D5: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.29/0.62              & ! [H2: real] :
% 0.29/0.62                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.29/0.62                 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S2 )
% 0.29/0.62                   => ( ( ord_less_real @ H2 @ D5 )
% 0.29/0.62                     => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % has_real_derivative_pos_inc_right
% 0.29/0.62  thf(fact_9760_has__real__derivative__neg__dec__right,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.29/0.62       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.29/0.62         => ? [D5: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.29/0.62              & ! [H2: real] :
% 0.29/0.62                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.29/0.62                 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S2 )
% 0.29/0.62                   => ( ( ord_less_real @ H2 @ D5 )
% 0.29/0.62                     => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % has_real_derivative_neg_dec_right
% 0.29/0.62  thf(fact_9761_has__real__derivative__neg__dec__left,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.29/0.62       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.29/0.62         => ? [D5: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.29/0.62              & ! [H2: real] :
% 0.29/0.62                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.29/0.62                 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S2 )
% 0.29/0.62                   => ( ( ord_less_real @ H2 @ D5 )
% 0.29/0.62                     => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % has_real_derivative_neg_dec_left
% 0.29/0.62  thf(fact_9762_has__real__derivative__pos__inc__left,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.29/0.62         => ? [D5: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.29/0.62              & ! [H2: real] :
% 0.29/0.62                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.29/0.62                 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S2 )
% 0.29/0.62                   => ( ( ord_less_real @ H2 @ D5 )
% 0.29/0.62                     => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % has_real_derivative_pos_inc_left
% 0.29/0.62  thf(fact_9763_DERIV__const__ratio__const2,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real,K: real] :
% 0.29/0.62        ( ( A != B )
% 0.29/0.62       => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62         => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 0.29/0.62            = K ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_const_ratio_const2
% 0.29/0.62  thf(fact_9764_DERIV__pos__inc__left,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.29/0.62         => ? [D5: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.29/0.62              & ! [H2: real] :
% 0.29/0.62                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.29/0.62                 => ( ( ord_less_real @ H2 @ D5 )
% 0.29/0.62                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_pos_inc_left
% 0.29/0.62  thf(fact_9765_DERIV__neg__dec__left,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.29/0.62         => ? [D5: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.29/0.62              & ! [H2: real] :
% 0.29/0.62                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.29/0.62                 => ( ( ord_less_real @ H2 @ D5 )
% 0.29/0.62                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_neg_dec_left
% 0.29/0.62  thf(fact_9766_DERIV__isconst__all,axiom,
% 0.29/0.62      ! [F: real > real,X: real,Y: real] :
% 0.29/0.62        ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62       => ( ( F @ X )
% 0.29/0.62          = ( F @ Y ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_isconst_all
% 0.29/0.62  thf(fact_9767_DERIV__pos__inc__right,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.29/0.62         => ? [D5: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.29/0.62              & ! [H2: real] :
% 0.29/0.62                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.29/0.62                 => ( ( ord_less_real @ H2 @ D5 )
% 0.29/0.62                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_pos_inc_right
% 0.29/0.62  thf(fact_9768_DERIV__neg__dec__right,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.29/0.62         => ? [D5: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.29/0.62              & ! [H2: real] :
% 0.29/0.62                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.29/0.62                 => ( ( ord_less_real @ H2 @ D5 )
% 0.29/0.62                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_neg_dec_right
% 0.29/0.62  thf(fact_9769_DERIV__neg__imp__decreasing,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( ord_less_eq_real @ A @ X3 )
% 0.29/0.62             => ( ( ord_less_eq_real @ X3 @ B )
% 0.29/0.62               => ? [Y5: real] :
% 0.29/0.62                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62                    & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.29/0.62         => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_neg_imp_decreasing
% 0.29/0.62  thf(fact_9770_DERIV__pos__imp__increasing,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( ord_less_eq_real @ A @ X3 )
% 0.29/0.62             => ( ( ord_less_eq_real @ X3 @ B )
% 0.29/0.62               => ? [Y5: real] :
% 0.29/0.62                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62                    & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.29/0.62         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_pos_imp_increasing
% 0.29/0.62  thf(fact_9771_DERIV__nonneg__imp__nondecreasing,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( ord_less_eq_real @ A @ X3 )
% 0.29/0.62             => ( ( ord_less_eq_real @ X3 @ B )
% 0.29/0.62               => ? [Y5: real] :
% 0.29/0.62                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62                    & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 0.29/0.62         => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_nonneg_imp_nondecreasing
% 0.29/0.62  thf(fact_9772_DERIV__nonpos__imp__nonincreasing,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( ord_less_eq_real @ A @ X3 )
% 0.29/0.62             => ( ( ord_less_eq_real @ X3 @ B )
% 0.29/0.62               => ? [Y5: real] :
% 0.29/0.62                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62                    & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 0.29/0.62         => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_nonpos_imp_nonincreasing
% 0.29/0.62  thf(fact_9773_deriv__nonneg__imp__mono,axiom,
% 0.29/0.62      ! [A: real,B: real,G: real > real,G2: real > real] :
% 0.29/0.62        ( ! [X3: real] :
% 0.29/0.62            ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.29/0.62           => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.29/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
% 0.29/0.62         => ( ( ord_less_eq_real @ A @ B )
% 0.29/0.62           => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % deriv_nonneg_imp_mono
% 0.29/0.62  thf(fact_9774_MVT2,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real,F3: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( ord_less_eq_real @ A @ X3 )
% 0.29/0.62             => ( ( ord_less_eq_real @ X3 @ B )
% 0.29/0.62               => ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.29/0.62         => ? [Z3: real] :
% 0.29/0.62              ( ( ord_less_real @ A @ Z3 )
% 0.29/0.62              & ( ord_less_real @ Z3 @ B )
% 0.29/0.62              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.29/0.62                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F3 @ Z3 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % MVT2
% 0.29/0.62  thf(fact_9775_DERIV__local__const,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real,D: real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.29/0.62         => ( ! [Y4: real] :
% 0.29/0.62                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.29/0.62               => ( ( F @ X )
% 0.29/0.62                  = ( F @ Y4 ) ) )
% 0.29/0.62           => ( L = zero_zero_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_local_const
% 0.29/0.62  thf(fact_9776_DERIV__ln,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.62       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_ln
% 0.29/0.62  thf(fact_9777_DERIV__const__average,axiom,
% 0.29/0.62      ! [A: real,B: real,V: real > real,K: real] :
% 0.29/0.62        ( ( A != B )
% 0.29/0.62       => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62         => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.62            = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_const_average
% 0.29/0.62  thf(fact_9778_DERIV__local__min,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real,D: real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.29/0.62         => ( ! [Y4: real] :
% 0.29/0.62                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.29/0.62               => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.62           => ( L = zero_zero_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_local_min
% 0.29/0.62  thf(fact_9779_DERIV__local__max,axiom,
% 0.29/0.62      ! [F: real > real,L: real,X: real,D: real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.29/0.62         => ( ! [Y4: real] :
% 0.29/0.62                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.29/0.62               => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
% 0.29/0.62           => ( L = zero_zero_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_local_max
% 0.29/0.62  thf(fact_9780_DERIV__ln__divide,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.62       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_ln_divide
% 0.29/0.62  thf(fact_9781_DERIV__pow,axiom,
% 0.29/0.62      ! [N: nat,X: real,S: set_real] :
% 0.29/0.62        ( has_fi5821293074295781190e_real
% 0.29/0.62        @ ^ [X4: real] : ( power_power_real @ X4 @ N )
% 0.29/0.62        @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.29/0.62        @ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_pow
% 0.29/0.62  thf(fact_9782_DERIV__fun__pow,axiom,
% 0.29/0.62      ! [G: real > real,M: real,X: real,N: nat] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( has_fi5821293074295781190e_real
% 0.29/0.62          @ ^ [X4: real] : ( power_power_real @ ( G @ X4 ) @ N )
% 0.29/0.62          @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 0.29/0.62          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_fun_pow
% 0.29/0.62  thf(fact_9783_has__real__derivative__powr,axiom,
% 0.29/0.62      ! [Z: real,R2: real] :
% 0.29/0.62        ( ( ord_less_real @ zero_zero_real @ Z )
% 0.29/0.62       => ( has_fi5821293074295781190e_real
% 0.29/0.62          @ ^ [Z4: real] : ( powr_real @ Z4 @ R2 )
% 0.29/0.62          @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 0.29/0.62          @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % has_real_derivative_powr
% 0.29/0.62  thf(fact_9784_DERIV__log,axiom,
% 0.29/0.62      ! [X: real,B: real] :
% 0.29/0.62        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.62       => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_log
% 0.29/0.62  thf(fact_9785_DERIV__fun__powr,axiom,
% 0.29/0.62      ! [G: real > real,M: real,X: real,R2: real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.29/0.62         => ( has_fi5821293074295781190e_real
% 0.29/0.62            @ ^ [X4: real] : ( powr_real @ ( G @ X4 ) @ R2 )
% 0.29/0.62            @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.29/0.62            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_fun_powr
% 0.29/0.62  thf(fact_9786_DERIV__powr,axiom,
% 0.29/0.62      ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.29/0.62         => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62           => ( has_fi5821293074295781190e_real
% 0.29/0.62              @ ^ [X4: real] : ( powr_real @ ( G @ X4 ) @ ( F @ X4 ) )
% 0.29/0.62              @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_powr
% 0.29/0.62  thf(fact_9787_DERIV__real__sqrt,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.62       => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_real_sqrt
% 0.29/0.62  thf(fact_9788_DERIV__arctan,axiom,
% 0.29/0.62      ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_arctan
% 0.29/0.62  thf(fact_9789_arsinh__real__has__field__derivative,axiom,
% 0.29/0.62      ! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).
% 0.29/0.62  
% 0.29/0.62  % arsinh_real_has_field_derivative
% 0.29/0.62  thf(fact_9790_DERIV__real__sqrt__generic,axiom,
% 0.29/0.62      ! [X: real,D2: real] :
% 0.29/0.62        ( ( X != zero_zero_real )
% 0.29/0.62       => ( ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.62           => ( D2
% 0.29/0.62              = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62         => ( ( ( ord_less_real @ X @ zero_zero_real )
% 0.29/0.62             => ( D2
% 0.29/0.62                = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62           => ( has_fi5821293074295781190e_real @ sqrt @ D2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_real_sqrt_generic
% 0.29/0.62  thf(fact_9791_arcosh__real__has__field__derivative,axiom,
% 0.29/0.62      ! [X: real,A2: set_real] :
% 0.29/0.62        ( ( ord_less_real @ one_one_real @ X )
% 0.29/0.62       => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % arcosh_real_has_field_derivative
% 0.29/0.62  thf(fact_9792_artanh__real__has__field__derivative,axiom,
% 0.29/0.62      ! [X: real,A2: set_real] :
% 0.29/0.62        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.62       => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % artanh_real_has_field_derivative
% 0.29/0.62  thf(fact_9793_DERIV__real__root,axiom,
% 0.29/0.62      ! [N: nat,X: real] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.62         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_real_root
% 0.29/0.62  thf(fact_9794_DERIV__arccos,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.62       => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.62         => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_arccos
% 0.29/0.62  thf(fact_9795_DERIV__arcsin,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.62       => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.62         => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_arcsin
% 0.29/0.62  thf(fact_9796_Maclaurin__all__le,axiom,
% 0.29/0.62      ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 0.29/0.62        ( ( ( Diff @ zero_zero_nat )
% 0.29/0.62          = F )
% 0.29/0.62       => ( ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62         => ? [T3: real] :
% 0.29/0.62              ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.29/0.62              & ( ( F @ X )
% 0.29/0.62                = ( plus_plus_real
% 0.29/0.62                  @ ( groups6591440286371151544t_real
% 0.29/0.62                    @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.62                    @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.62                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Maclaurin_all_le
% 0.29/0.62  thf(fact_9797_Maclaurin__all__le__objl,axiom,
% 0.29/0.62      ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 0.29/0.62        ( ( ( ( Diff @ zero_zero_nat )
% 0.29/0.62            = F )
% 0.29/0.62          & ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.29/0.62       => ? [T3: real] :
% 0.29/0.62            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.29/0.62            & ( ( F @ X )
% 0.29/0.62              = ( plus_plus_real
% 0.29/0.62                @ ( groups6591440286371151544t_real
% 0.29/0.62                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.62                  @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.62                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Maclaurin_all_le_objl
% 0.29/0.62  thf(fact_9798_DERIV__odd__real__root,axiom,
% 0.29/0.62      ! [N: nat,X: real] :
% 0.29/0.62        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.62       => ( ( X != zero_zero_real )
% 0.29/0.62         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_odd_real_root
% 0.29/0.62  thf(fact_9799_Maclaurin__minus,axiom,
% 0.29/0.62      ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ H @ zero_zero_real )
% 0.29/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62         => ( ( ( Diff @ zero_zero_nat )
% 0.29/0.62              = F )
% 0.29/0.62           => ( ! [M5: nat,T3: real] :
% 0.29/0.62                  ( ( ( ord_less_nat @ M5 @ N )
% 0.29/0.62                    & ( ord_less_eq_real @ H @ T3 )
% 0.29/0.62                    & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
% 0.29/0.62                 => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.29/0.62             => ? [T3: real] :
% 0.29/0.62                  ( ( ord_less_real @ H @ T3 )
% 0.29/0.62                  & ( ord_less_real @ T3 @ zero_zero_real )
% 0.29/0.62                  & ( ( F @ H )
% 0.29/0.62                    = ( plus_plus_real
% 0.29/0.62                      @ ( groups6591440286371151544t_real
% 0.29/0.62                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
% 0.29/0.62                        @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.62                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Maclaurin_minus
% 0.29/0.62  thf(fact_9800_Maclaurin2,axiom,
% 0.29/0.62      ! [H: real,Diff: nat > real > real,F: real > real,N: nat] :
% 0.29/0.62        ( ( ord_less_real @ zero_zero_real @ H )
% 0.29/0.62       => ( ( ( Diff @ zero_zero_nat )
% 0.29/0.62            = F )
% 0.29/0.62         => ( ! [M5: nat,T3: real] :
% 0.29/0.62                ( ( ( ord_less_nat @ M5 @ N )
% 0.29/0.62                  & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.29/0.62                  & ( ord_less_eq_real @ T3 @ H ) )
% 0.29/0.62               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.29/0.62           => ? [T3: real] :
% 0.29/0.62                ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.29/0.62                & ( ord_less_eq_real @ T3 @ H )
% 0.29/0.62                & ( ( F @ H )
% 0.29/0.62                  = ( plus_plus_real
% 0.29/0.62                    @ ( groups6591440286371151544t_real
% 0.29/0.62                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
% 0.29/0.62                      @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.62                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Maclaurin2
% 0.29/0.62  thf(fact_9801_Maclaurin,axiom,
% 0.29/0.62      ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ zero_zero_real @ H )
% 0.29/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62         => ( ( ( Diff @ zero_zero_nat )
% 0.29/0.62              = F )
% 0.29/0.62           => ( ! [M5: nat,T3: real] :
% 0.29/0.62                  ( ( ( ord_less_nat @ M5 @ N )
% 0.29/0.62                    & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.29/0.62                    & ( ord_less_eq_real @ T3 @ H ) )
% 0.29/0.62                 => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.29/0.62             => ? [T3: real] :
% 0.29/0.62                  ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.29/0.62                  & ( ord_less_real @ T3 @ H )
% 0.29/0.62                  & ( ( F @ H )
% 0.29/0.62                    = ( plus_plus_real
% 0.29/0.62                      @ ( groups6591440286371151544t_real
% 0.29/0.62                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
% 0.29/0.62                        @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.62                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Maclaurin
% 0.29/0.62  thf(fact_9802_Maclaurin__all__lt,axiom,
% 0.29/0.62      ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 0.29/0.62        ( ( ( Diff @ zero_zero_nat )
% 0.29/0.62          = F )
% 0.29/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62         => ( ( X != zero_zero_real )
% 0.29/0.62           => ( ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62             => ? [T3: real] :
% 0.29/0.62                  ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 0.29/0.62                  & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.29/0.62                  & ( ( F @ X )
% 0.29/0.62                    = ( plus_plus_real
% 0.29/0.62                      @ ( groups6591440286371151544t_real
% 0.29/0.62                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.62                        @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.62                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Maclaurin_all_lt
% 0.29/0.62  thf(fact_9803_Maclaurin__bi__le,axiom,
% 0.29/0.62      ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 0.29/0.62        ( ( ( Diff @ zero_zero_nat )
% 0.29/0.62          = F )
% 0.29/0.62       => ( ! [M5: nat,T3: real] :
% 0.29/0.62              ( ( ( ord_less_nat @ M5 @ N )
% 0.29/0.62                & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
% 0.29/0.62             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.29/0.62         => ? [T3: real] :
% 0.29/0.62              ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.29/0.62              & ( ( F @ X )
% 0.29/0.62                = ( plus_plus_real
% 0.29/0.62                  @ ( groups6591440286371151544t_real
% 0.29/0.62                    @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.29/0.62                    @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.62                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Maclaurin_bi_le
% 0.29/0.62  thf(fact_9804_Taylor__down,axiom,
% 0.29/0.62      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( ( Diff @ zero_zero_nat )
% 0.29/0.62            = F )
% 0.29/0.62         => ( ! [M5: nat,T3: real] :
% 0.29/0.62                ( ( ( ord_less_nat @ M5 @ N )
% 0.29/0.62                  & ( ord_less_eq_real @ A @ T3 )
% 0.29/0.62                  & ( ord_less_eq_real @ T3 @ B ) )
% 0.29/0.62               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.29/0.62           => ( ( ord_less_real @ A @ C )
% 0.29/0.62             => ( ( ord_less_eq_real @ C @ B )
% 0.29/0.62               => ? [T3: real] :
% 0.29/0.62                    ( ( ord_less_real @ A @ T3 )
% 0.29/0.62                    & ( ord_less_real @ T3 @ C )
% 0.29/0.62                    & ( ( F @ A )
% 0.29/0.62                      = ( plus_plus_real
% 0.29/0.62                        @ ( groups6591440286371151544t_real
% 0.29/0.62                          @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
% 0.29/0.62                          @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.62                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Taylor_down
% 0.29/0.62  thf(fact_9805_Taylor__up,axiom,
% 0.29/0.62      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( ( Diff @ zero_zero_nat )
% 0.29/0.62            = F )
% 0.29/0.62         => ( ! [M5: nat,T3: real] :
% 0.29/0.62                ( ( ( ord_less_nat @ M5 @ N )
% 0.29/0.62                  & ( ord_less_eq_real @ A @ T3 )
% 0.29/0.62                  & ( ord_less_eq_real @ T3 @ B ) )
% 0.29/0.62               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.29/0.62           => ( ( ord_less_eq_real @ A @ C )
% 0.29/0.62             => ( ( ord_less_real @ C @ B )
% 0.29/0.62               => ? [T3: real] :
% 0.29/0.62                    ( ( ord_less_real @ C @ T3 )
% 0.29/0.62                    & ( ord_less_real @ T3 @ B )
% 0.29/0.62                    & ( ( F @ B )
% 0.29/0.62                      = ( plus_plus_real
% 0.29/0.62                        @ ( groups6591440286371151544t_real
% 0.29/0.62                          @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M6 ) )
% 0.29/0.62                          @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.62                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Taylor_up
% 0.29/0.62  thf(fact_9806_Taylor,axiom,
% 0.29/0.62      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( ( Diff @ zero_zero_nat )
% 0.29/0.62            = F )
% 0.29/0.62         => ( ! [M5: nat,T3: real] :
% 0.29/0.62                ( ( ( ord_less_nat @ M5 @ N )
% 0.29/0.62                  & ( ord_less_eq_real @ A @ T3 )
% 0.29/0.62                  & ( ord_less_eq_real @ T3 @ B ) )
% 0.29/0.62               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.29/0.62           => ( ( ord_less_eq_real @ A @ C )
% 0.29/0.62             => ( ( ord_less_eq_real @ C @ B )
% 0.29/0.62               => ( ( ord_less_eq_real @ A @ X )
% 0.29/0.62                 => ( ( ord_less_eq_real @ X @ B )
% 0.29/0.62                   => ( ( X != C )
% 0.29/0.62                     => ? [T3: real] :
% 0.29/0.62                          ( ( ( ord_less_real @ X @ C )
% 0.29/0.62                           => ( ( ord_less_real @ X @ T3 )
% 0.29/0.62                              & ( ord_less_real @ T3 @ C ) ) )
% 0.29/0.62                          & ( ~ ( ord_less_real @ X @ C )
% 0.29/0.62                           => ( ( ord_less_real @ C @ T3 )
% 0.29/0.62                              & ( ord_less_real @ T3 @ X ) ) )
% 0.29/0.62                          & ( ( F @ X )
% 0.29/0.62                            = ( plus_plus_real
% 0.29/0.62                              @ ( groups6591440286371151544t_real
% 0.29/0.62                                @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M6 ) )
% 0.29/0.62                                @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.62                              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Taylor
% 0.29/0.62  thf(fact_9807_Maclaurin__lemma2,axiom,
% 0.29/0.62      ! [N: nat,H: real,Diff: nat > real > real,K: nat,B2: real] :
% 0.29/0.62        ( ! [M5: nat,T3: real] :
% 0.29/0.62            ( ( ( ord_less_nat @ M5 @ N )
% 0.29/0.62              & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.29/0.62              & ( ord_less_eq_real @ T3 @ H ) )
% 0.29/0.62           => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.29/0.62       => ( ( N
% 0.29/0.62            = ( suc @ K ) )
% 0.29/0.62         => ! [M3: nat,T4: real] :
% 0.29/0.62              ( ( ( ord_less_nat @ M3 @ N )
% 0.29/0.62                & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.29/0.62                & ( ord_less_eq_real @ T4 @ H ) )
% 0.29/0.62             => ( has_fi5821293074295781190e_real
% 0.29/0.62                @ ^ [U2: real] :
% 0.29/0.62                    ( minus_minus_real @ ( Diff @ M3 @ U2 )
% 0.29/0.62                    @ ( plus_plus_real
% 0.29/0.62                      @ ( groups6591440286371151544t_real
% 0.29/0.62                        @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M3 @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ U2 @ P4 ) )
% 0.29/0.62                        @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M3 ) ) )
% 0.29/0.62                      @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M3 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M3 ) ) ) ) ) )
% 0.29/0.62                @ ( minus_minus_real @ ( Diff @ ( suc @ M3 ) @ T4 )
% 0.29/0.62                  @ ( plus_plus_real
% 0.29/0.62                    @ ( groups6591440286371151544t_real
% 0.29/0.62                      @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M3 ) @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ T4 @ P4 ) )
% 0.29/0.62                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) )
% 0.29/0.62                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Maclaurin_lemma2
% 0.29/0.62  thf(fact_9808_DERIV__power__series_H,axiom,
% 0.29/0.62      ! [R: real,F: nat > real,X0: real] :
% 0.29/0.62        ( ! [X3: real] :
% 0.29/0.62            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.29/0.62           => ( summable_real
% 0.29/0.62              @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X3 @ N3 ) ) ) )
% 0.29/0.62       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.29/0.62         => ( ( ord_less_real @ zero_zero_real @ R )
% 0.29/0.62           => ( has_fi5821293074295781190e_real
% 0.29/0.62              @ ^ [X4: real] :
% 0.29/0.62                  ( suminf_real
% 0.29/0.62                  @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X4 @ ( suc @ N3 ) ) ) )
% 0.29/0.62              @ ( suminf_real
% 0.29/0.62                @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_power_series'
% 0.29/0.62  thf(fact_9809_tanh__real__bounds,axiom,
% 0.29/0.62      ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % tanh_real_bounds
% 0.29/0.62  thf(fact_9810_DERIV__isconst3,axiom,
% 0.29/0.62      ! [A: real,B: real,X: real,Y: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.29/0.62         => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.29/0.62           => ( ! [X3: real] :
% 0.29/0.62                  ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.29/0.62                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.29/0.62             => ( ( F @ X )
% 0.29/0.62                = ( F @ Y ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_isconst3
% 0.29/0.62  thf(fact_9811_DERIV__series_H,axiom,
% 0.29/0.62      ! [F: real > nat > real,F3: real > nat > real,X0: real,A: real,B: real,L4: nat > real] :
% 0.29/0.62        ( ! [N2: nat] :
% 0.29/0.62            ( has_fi5821293074295781190e_real
% 0.29/0.62            @ ^ [X4: real] : ( F @ X4 @ N2 )
% 0.29/0.62            @ ( F3 @ X0 @ N2 )
% 0.29/0.62            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.29/0.62             => ( summable_real @ ( F @ X3 ) ) )
% 0.29/0.62         => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.29/0.62           => ( ( summable_real @ ( F3 @ X0 ) )
% 0.29/0.62             => ( ( summable_real @ L4 )
% 0.29/0.62               => ( ! [N2: nat,X3: real,Y4: real] :
% 0.29/0.62                      ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.29/0.62                     => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.29/0.62                       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N2 ) @ ( F @ Y4 @ N2 ) ) ) @ ( times_times_real @ ( L4 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) ) ) ) )
% 0.29/0.62                 => ( has_fi5821293074295781190e_real
% 0.29/0.62                    @ ^ [X4: real] : ( suminf_real @ ( F @ X4 ) )
% 0.29/0.62                    @ ( suminf_real @ ( F3 @ X0 ) )
% 0.29/0.62                    @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_series'
% 0.29/0.62  thf(fact_9812_divmod__integer__eq__cases,axiom,
% 0.29/0.62      ( code_divmod_integer
% 0.29/0.62      = ( ^ [K2: code_integer,L3: code_integer] :
% 0.29/0.62            ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.29/0.62            @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 0.29/0.62              @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L3
% 0.29/0.62                @ ( if_Pro6119634080678213985nteger
% 0.29/0.62                  @ ( ( sgn_sgn_Code_integer @ K2 )
% 0.29/0.62                    = ( sgn_sgn_Code_integer @ L3 ) )
% 0.29/0.62                  @ ( code_divmod_abs @ K2 @ L3 )
% 0.29/0.62                  @ ( produc6916734918728496179nteger
% 0.29/0.62                    @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L3 ) @ S5 ) ) )
% 0.29/0.62                    @ ( code_divmod_abs @ K2 @ L3 ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % divmod_integer_eq_cases
% 0.29/0.62  thf(fact_9813_card__greaterThanLessThan,axiom,
% 0.29/0.62      ! [L: nat,U: nat] :
% 0.29/0.62        ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 0.29/0.62        = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % card_greaterThanLessThan
% 0.29/0.62  thf(fact_9814_card__greaterThanLessThan__int,axiom,
% 0.29/0.62      ! [L: int,U: int] :
% 0.29/0.62        ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
% 0.29/0.62        = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % card_greaterThanLessThan_int
% 0.29/0.62  thf(fact_9815_atLeastSucLessThan__greaterThanLessThan,axiom,
% 0.29/0.62      ! [L: nat,U: nat] :
% 0.29/0.62        ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 0.29/0.62        = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 0.29/0.62  
% 0.29/0.62  % atLeastSucLessThan_greaterThanLessThan
% 0.29/0.62  thf(fact_9816_isCont__Lb__Ub,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( ( ord_less_eq_real @ A @ X3 )
% 0.29/0.62                & ( ord_less_eq_real @ X3 @ B ) )
% 0.29/0.62             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 0.29/0.62         => ? [L5: real,M9: real] :
% 0.29/0.62              ( ! [X2: real] :
% 0.29/0.62                  ( ( ( ord_less_eq_real @ A @ X2 )
% 0.29/0.62                    & ( ord_less_eq_real @ X2 @ B ) )
% 0.29/0.62                 => ( ( ord_less_eq_real @ L5 @ ( F @ X2 ) )
% 0.29/0.62                    & ( ord_less_eq_real @ ( F @ X2 ) @ M9 ) ) )
% 0.29/0.62              & ! [Y5: real] :
% 0.29/0.62                  ( ( ( ord_less_eq_real @ L5 @ Y5 )
% 0.29/0.62                    & ( ord_less_eq_real @ Y5 @ M9 ) )
% 0.29/0.62                 => ? [X3: real] :
% 0.29/0.62                      ( ( ord_less_eq_real @ A @ X3 )
% 0.29/0.62                      & ( ord_less_eq_real @ X3 @ B )
% 0.29/0.62                      & ( ( F @ X3 )
% 0.29/0.62                        = Y5 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % isCont_Lb_Ub
% 0.29/0.62  thf(fact_9817_LIM__fun__gt__zero,axiom,
% 0.29/0.62      ! [F: real > real,L: real,C: real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.29/0.62         => ? [R3: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.29/0.62              & ! [X2: real] :
% 0.29/0.62                  ( ( ( X2 != C )
% 0.29/0.62                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 0.29/0.62                 => ( ord_less_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIM_fun_gt_zero
% 0.29/0.62  thf(fact_9818_LIM__fun__not__zero,axiom,
% 0.29/0.62      ! [F: real > real,L: real,C: real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.29/0.62       => ( ( L != zero_zero_real )
% 0.29/0.62         => ? [R3: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.29/0.62              & ! [X2: real] :
% 0.29/0.62                  ( ( ( X2 != C )
% 0.29/0.62                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 0.29/0.62                 => ( ( F @ X2 )
% 0.29/0.62                   != zero_zero_real ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIM_fun_not_zero
% 0.29/0.62  thf(fact_9819_LIM__fun__less__zero,axiom,
% 0.29/0.62      ! [F: real > real,L: real,C: real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.29/0.62       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.29/0.62         => ? [R3: real] :
% 0.29/0.62              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.29/0.62              & ! [X2: real] :
% 0.29/0.62                  ( ( ( X2 != C )
% 0.29/0.62                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 0.29/0.62                 => ( ord_less_real @ ( F @ X2 ) @ zero_zero_real ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIM_fun_less_zero
% 0.29/0.62  thf(fact_9820_isCont__arctan,axiom,
% 0.29/0.62      ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arctan ) ).
% 0.29/0.62  
% 0.29/0.62  % isCont_arctan
% 0.29/0.62  thf(fact_9821_isCont__arsinh,axiom,
% 0.29/0.62      ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arsinh_real ) ).
% 0.29/0.62  
% 0.29/0.62  % isCont_arsinh
% 0.29/0.62  thf(fact_9822_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 0.29/0.62      ! [L: int,U: int] :
% 0.29/0.62        ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 0.29/0.62        = ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 0.29/0.62  
% 0.29/0.62  % atLeastPlusOneLessThan_greaterThanLessThan_int
% 0.29/0.62  thf(fact_9823_isCont__inverse__function2,axiom,
% 0.29/0.62      ! [A: real,X: real,B: real,G: real > real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ X )
% 0.29/0.62       => ( ( ord_less_real @ X @ B )
% 0.29/0.62         => ( ! [Z3: real] :
% 0.29/0.62                ( ( ord_less_eq_real @ A @ Z3 )
% 0.29/0.62               => ( ( ord_less_eq_real @ Z3 @ B )
% 0.29/0.62                 => ( ( G @ ( F @ Z3 ) )
% 0.29/0.62                    = Z3 ) ) )
% 0.29/0.62           => ( ! [Z3: real] :
% 0.29/0.62                  ( ( ord_less_eq_real @ A @ Z3 )
% 0.29/0.62                 => ( ( ord_less_eq_real @ Z3 @ B )
% 0.29/0.62                   => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
% 0.29/0.62             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % isCont_inverse_function2
% 0.29/0.62  thf(fact_9824_isCont__ln,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( X != zero_zero_real )
% 0.29/0.62       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ln_ln_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % isCont_ln
% 0.29/0.62  thf(fact_9825_isCont__arcosh,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ one_one_real @ X )
% 0.29/0.62       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % isCont_arcosh
% 0.29/0.62  thf(fact_9826_LIM__cos__div__sin,axiom,
% 0.29/0.62      ( filterlim_real_real
% 0.29/0.62      @ ^ [X4: real] : ( divide_divide_real @ ( cos_real @ X4 ) @ ( sin_real @ X4 ) )
% 0.29/0.62      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62      @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIM_cos_div_sin
% 0.29/0.62  thf(fact_9827_DERIV__inverse__function,axiom,
% 0.29/0.62      ! [F: real > real,D2: real,G: real > real,X: real,A: real,B: real] :
% 0.29/0.62        ( ( has_fi5821293074295781190e_real @ F @ D2 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
% 0.29/0.62       => ( ( D2 != zero_zero_real )
% 0.29/0.62         => ( ( ord_less_real @ A @ X )
% 0.29/0.62           => ( ( ord_less_real @ X @ B )
% 0.29/0.62             => ( ! [Y4: real] :
% 0.29/0.62                    ( ( ord_less_real @ A @ Y4 )
% 0.29/0.62                   => ( ( ord_less_real @ Y4 @ B )
% 0.29/0.62                     => ( ( F @ ( G @ Y4 ) )
% 0.29/0.62                        = Y4 ) ) )
% 0.29/0.62               => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
% 0.29/0.62                 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D2 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_inverse_function
% 0.29/0.62  thf(fact_9828_isCont__arccos,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.62       => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.62         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % isCont_arccos
% 0.29/0.62  thf(fact_9829_isCont__arcsin,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.62       => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.62         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % isCont_arcsin
% 0.29/0.62  thf(fact_9830_LIM__less__bound,axiom,
% 0.29/0.62      ! [B: real,X: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ B @ X )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ X ) )
% 0.29/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.29/0.62         => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
% 0.29/0.62           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIM_less_bound
% 0.29/0.62  thf(fact_9831_isCont__artanh,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.29/0.62       => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.62         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % isCont_artanh
% 0.29/0.62  thf(fact_9832_isCont__inverse__function,axiom,
% 0.29/0.62      ! [D: real,X: real,G: real > real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ zero_zero_real @ D )
% 0.29/0.62       => ( ! [Z3: real] :
% 0.29/0.62              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
% 0.29/0.62             => ( ( G @ ( F @ Z3 ) )
% 0.29/0.62                = Z3 ) )
% 0.29/0.62         => ( ! [Z3: real] :
% 0.29/0.62                ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
% 0.29/0.62               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
% 0.29/0.62           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % isCont_inverse_function
% 0.29/0.62  thf(fact_9833_GMVT_H,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F3: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ! [Z3: real] :
% 0.29/0.62              ( ( ord_less_eq_real @ A @ Z3 )
% 0.29/0.62             => ( ( ord_less_eq_real @ Z3 @ B )
% 0.29/0.62               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
% 0.29/0.62         => ( ! [Z3: real] :
% 0.29/0.62                ( ( ord_less_eq_real @ A @ Z3 )
% 0.29/0.62               => ( ( ord_less_eq_real @ Z3 @ B )
% 0.29/0.62                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
% 0.29/0.62           => ( ! [Z3: real] :
% 0.29/0.62                  ( ( ord_less_real @ A @ Z3 )
% 0.29/0.62                 => ( ( ord_less_real @ Z3 @ B )
% 0.29/0.62                   => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
% 0.29/0.62             => ( ! [Z3: real] :
% 0.29/0.62                    ( ( ord_less_real @ A @ Z3 )
% 0.29/0.62                   => ( ( ord_less_real @ Z3 @ B )
% 0.29/0.62                     => ( has_fi5821293074295781190e_real @ F @ ( F3 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
% 0.29/0.62               => ? [C4: real] :
% 0.29/0.62                    ( ( ord_less_real @ A @ C4 )
% 0.29/0.62                    & ( ord_less_real @ C4 @ B )
% 0.29/0.62                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C4 ) )
% 0.29/0.62                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F3 @ C4 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % GMVT'
% 0.29/0.62  thf(fact_9834_summable__Leibniz_I3_J,axiom,
% 0.29/0.62      ! [A: nat > real] :
% 0.29/0.62        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62       => ( ( topolo6980174941875973593q_real @ A )
% 0.29/0.62         => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 0.29/0.62           => ! [N8: nat] :
% 0.29/0.62                ( member_real
% 0.29/0.62                @ ( suminf_real
% 0.29/0.62                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 0.29/0.62                @ ( set_or1222579329274155063t_real
% 0.29/0.62                  @ ( groups6591440286371151544t_real
% 0.29/0.62                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) )
% 0.29/0.62                  @ ( groups6591440286371151544t_real
% 0.29/0.62                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable_Leibniz(3)
% 0.29/0.62  thf(fact_9835_summable__Leibniz_I2_J,axiom,
% 0.29/0.62      ! [A: nat > real] :
% 0.29/0.62        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62       => ( ( topolo6980174941875973593q_real @ A )
% 0.29/0.62         => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 0.29/0.62           => ! [N8: nat] :
% 0.29/0.62                ( member_real
% 0.29/0.62                @ ( suminf_real
% 0.29/0.62                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 0.29/0.62                @ ( set_or1222579329274155063t_real
% 0.29/0.62                  @ ( groups6591440286371151544t_real
% 0.29/0.62                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) )
% 0.29/0.62                  @ ( groups6591440286371151544t_real
% 0.29/0.62                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable_Leibniz(2)
% 0.29/0.62  thf(fact_9836_summable__Leibniz_H_I4_J,axiom,
% 0.29/0.62      ! [A: nat > real,N: nat] :
% 0.29/0.62        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.29/0.62         => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.29/0.62           => ( ord_less_eq_real
% 0.29/0.62              @ ( suminf_real
% 0.29/0.62                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 0.29/0.62              @ ( groups6591440286371151544t_real
% 0.29/0.62                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable_Leibniz'(4)
% 0.29/0.62  thf(fact_9837_filterlim__Suc,axiom,
% 0.29/0.62      filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_Suc
% 0.29/0.62  thf(fact_9838_card_Ocomp__fun__commute__on,axiom,
% 0.29/0.62      ( ( comp_nat_nat_nat @ suc @ suc )
% 0.29/0.62      = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 0.29/0.62  
% 0.29/0.62  % card.comp_fun_commute_on
% 0.29/0.62  thf(fact_9839_mult__nat__right__at__top,axiom,
% 0.29/0.62      ! [C: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.62       => ( filterlim_nat_nat
% 0.29/0.62          @ ^ [X4: nat] : ( times_times_nat @ X4 @ C )
% 0.29/0.62          @ at_top_nat
% 0.29/0.62          @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % mult_nat_right_at_top
% 0.29/0.62  thf(fact_9840_mult__nat__left__at__top,axiom,
% 0.29/0.62      ! [C: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.62       => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % mult_nat_left_at_top
% 0.29/0.62  thf(fact_9841_monoseq__convergent,axiom,
% 0.29/0.62      ! [X8: nat > real,B2: real] :
% 0.29/0.62        ( ( topolo6980174941875973593q_real @ X8 )
% 0.29/0.62       => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I2 ) ) @ B2 )
% 0.29/0.62         => ~ ! [L5: real] :
% 0.29/0.62                ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % monoseq_convergent
% 0.29/0.62  thf(fact_9842_LIMSEQ__root,axiom,
% 0.29/0.62      ( filterlim_nat_real
% 0.29/0.62      @ ^ [N3: nat] : ( root @ N3 @ ( semiri5074537144036343181t_real @ N3 ) )
% 0.29/0.62      @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.29/0.62      @ at_top_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_root
% 0.29/0.62  thf(fact_9843_nested__sequence__unique,axiom,
% 0.29/0.62      ! [F: nat > real,G: nat > real] :
% 0.29/0.62        ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.62       => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
% 0.29/0.62         => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.29/0.62           => ( ( filterlim_nat_real
% 0.29/0.62                @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.29/0.62                @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62                @ at_top_nat )
% 0.29/0.62             => ? [L2: real] :
% 0.29/0.62                  ( ! [N8: nat] : ( ord_less_eq_real @ ( F @ N8 ) @ L2 )
% 0.29/0.62                  & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat )
% 0.29/0.62                  & ! [N8: nat] : ( ord_less_eq_real @ L2 @ ( G @ N8 ) )
% 0.29/0.62                  & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nested_sequence_unique
% 0.29/0.62  thf(fact_9844_LIMSEQ__inverse__zero,axiom,
% 0.29/0.62      ! [X8: nat > real] :
% 0.29/0.62        ( ! [R3: real] :
% 0.29/0.62          ? [N6: nat] :
% 0.29/0.62          ! [N2: nat] :
% 0.29/0.62            ( ( ord_less_eq_nat @ N6 @ N2 )
% 0.29/0.62           => ( ord_less_real @ R3 @ ( X8 @ N2 ) ) )
% 0.29/0.62       => ( filterlim_nat_real
% 0.29/0.62          @ ^ [N3: nat] : ( inverse_inverse_real @ ( X8 @ N3 ) )
% 0.29/0.62          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62          @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_inverse_zero
% 0.29/0.62  thf(fact_9845_lim__inverse__n_H,axiom,
% 0.29/0.62      ( filterlim_nat_real
% 0.29/0.62      @ ^ [N3: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
% 0.29/0.62      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62      @ at_top_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % lim_inverse_n'
% 0.29/0.62  thf(fact_9846_LIMSEQ__root__const,axiom,
% 0.29/0.62      ! [C: real] :
% 0.29/0.62        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.62       => ( filterlim_nat_real
% 0.29/0.62          @ ^ [N3: nat] : ( root @ N3 @ C )
% 0.29/0.62          @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.29/0.62          @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_root_const
% 0.29/0.62  thf(fact_9847_LIMSEQ__inverse__real__of__nat,axiom,
% 0.29/0.62      ( filterlim_nat_real
% 0.29/0.62      @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
% 0.29/0.62      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62      @ at_top_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_inverse_real_of_nat
% 0.29/0.62  thf(fact_9848_LIMSEQ__inverse__real__of__nat__add,axiom,
% 0.29/0.62      ! [R2: real] :
% 0.29/0.62        ( filterlim_nat_real
% 0.29/0.62        @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 0.29/0.62        @ ( topolo2815343760600316023s_real @ R2 )
% 0.29/0.62        @ at_top_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_inverse_real_of_nat_add
% 0.29/0.62  thf(fact_9849_increasing__LIMSEQ,axiom,
% 0.29/0.62      ! [F: nat > real,L: real] :
% 0.29/0.62        ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.62       => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
% 0.29/0.62         => ( ! [E: real] :
% 0.29/0.62                ( ( ord_less_real @ zero_zero_real @ E )
% 0.29/0.62               => ? [N8: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N8 ) @ E ) ) )
% 0.29/0.62           => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % increasing_LIMSEQ
% 0.29/0.62  thf(fact_9850_continuous__floor,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ~ ( member_real @ X @ ring_1_Ints_real )
% 0.29/0.62       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( comp_int_real_real @ ring_1_of_int_real @ archim6058952711729229775r_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % continuous_floor
% 0.29/0.62  thf(fact_9851_LIMSEQ__realpow__zero,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.62       => ( ( ord_less_real @ X @ one_one_real )
% 0.29/0.62         => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_realpow_zero
% 0.29/0.62  thf(fact_9852_LIMSEQ__divide__realpow__zero,axiom,
% 0.29/0.62      ! [X: real,A: real] :
% 0.29/0.62        ( ( ord_less_real @ one_one_real @ X )
% 0.29/0.62       => ( filterlim_nat_real
% 0.29/0.62          @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N3 ) )
% 0.29/0.62          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62          @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_divide_realpow_zero
% 0.29/0.62  thf(fact_9853_LIMSEQ__abs__realpow__zero2,axiom,
% 0.29/0.62      ! [C: real] :
% 0.29/0.62        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.29/0.62       => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_abs_realpow_zero2
% 0.29/0.62  thf(fact_9854_LIMSEQ__abs__realpow__zero,axiom,
% 0.29/0.62      ! [C: real] :
% 0.29/0.62        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.29/0.62       => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_abs_realpow_zero
% 0.29/0.62  thf(fact_9855_LIMSEQ__inverse__realpow__zero,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_real @ one_one_real @ X )
% 0.29/0.62       => ( filterlim_nat_real
% 0.29/0.62          @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N3 ) )
% 0.29/0.62          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62          @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_inverse_realpow_zero
% 0.29/0.62  thf(fact_9856_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 0.29/0.62      ! [R2: real] :
% 0.29/0.62        ( filterlim_nat_real
% 0.29/0.62        @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
% 0.29/0.62        @ ( topolo2815343760600316023s_real @ R2 )
% 0.29/0.62        @ at_top_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_inverse_real_of_nat_add_minus
% 0.29/0.62  thf(fact_9857_tendsto__exp__limit__sequentially,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( filterlim_nat_real
% 0.29/0.62        @ ^ [N3: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 )
% 0.29/0.62        @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.29/0.62        @ at_top_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % tendsto_exp_limit_sequentially
% 0.29/0.62  thf(fact_9858_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 0.29/0.62      ! [R2: real] :
% 0.29/0.62        ( filterlim_nat_real
% 0.29/0.62        @ ^ [N3: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
% 0.29/0.62        @ ( topolo2815343760600316023s_real @ R2 )
% 0.29/0.62        @ at_top_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % LIMSEQ_inverse_real_of_nat_add_minus_mult
% 0.29/0.62  thf(fact_9859_summable__Leibniz_I1_J,axiom,
% 0.29/0.62      ! [A: nat > real] :
% 0.29/0.62        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62       => ( ( topolo6980174941875973593q_real @ A )
% 0.29/0.62         => ( summable_real
% 0.29/0.62            @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable_Leibniz(1)
% 0.29/0.62  thf(fact_9860_summable,axiom,
% 0.29/0.62      ! [A: nat > real] :
% 0.29/0.62        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.29/0.62         => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.29/0.62           => ( summable_real
% 0.29/0.62              @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable
% 0.29/0.62  thf(fact_9861_cos__diff__limit__1,axiom,
% 0.29/0.62      ! [Theta: nat > real,Theta2: real] :
% 0.29/0.62        ( ( filterlim_nat_real
% 0.29/0.62          @ ^ [J2: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J2 ) @ Theta2 ) )
% 0.29/0.62          @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.29/0.62          @ at_top_nat )
% 0.29/0.62       => ~ ! [K3: nat > int] :
% 0.29/0.62              ~ ( filterlim_nat_real
% 0.29/0.62                @ ^ [J2: nat] : ( minus_minus_real @ ( Theta @ J2 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.29/0.62                @ ( topolo2815343760600316023s_real @ Theta2 )
% 0.29/0.62                @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % cos_diff_limit_1
% 0.29/0.62  thf(fact_9862_cos__limit__1,axiom,
% 0.29/0.62      ! [Theta: nat > real] :
% 0.29/0.62        ( ( filterlim_nat_real
% 0.29/0.62          @ ^ [J2: nat] : ( cos_real @ ( Theta @ J2 ) )
% 0.29/0.62          @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.29/0.62          @ at_top_nat )
% 0.29/0.62       => ? [K3: nat > int] :
% 0.29/0.62            ( filterlim_nat_real
% 0.29/0.62            @ ^ [J2: nat] : ( minus_minus_real @ ( Theta @ J2 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.29/0.62            @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62            @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % cos_limit_1
% 0.29/0.62  thf(fact_9863_summable__Leibniz_I4_J,axiom,
% 0.29/0.62      ! [A: nat > real] :
% 0.29/0.62        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62       => ( ( topolo6980174941875973593q_real @ A )
% 0.29/0.62         => ( filterlim_nat_real
% 0.29/0.62            @ ^ [N3: nat] :
% 0.29/0.62                ( groups6591440286371151544t_real
% 0.29/0.62                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.29/0.62            @ ( topolo2815343760600316023s_real
% 0.29/0.62              @ ( suminf_real
% 0.29/0.62                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 0.29/0.62            @ at_top_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable_Leibniz(4)
% 0.29/0.62  thf(fact_9864_zeroseq__arctan__series,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.29/0.62       => ( filterlim_nat_real
% 0.29/0.62          @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 0.29/0.62          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62          @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % zeroseq_arctan_series
% 0.29/0.62  thf(fact_9865_summable__Leibniz_H_I3_J,axiom,
% 0.29/0.62      ! [A: nat > real] :
% 0.29/0.62        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.29/0.62         => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.29/0.62           => ( filterlim_nat_real
% 0.29/0.62              @ ^ [N3: nat] :
% 0.29/0.62                  ( groups6591440286371151544t_real
% 0.29/0.62                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.29/0.62              @ ( topolo2815343760600316023s_real
% 0.29/0.62                @ ( suminf_real
% 0.29/0.62                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 0.29/0.62              @ at_top_nat ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable_Leibniz'(3)
% 0.29/0.62  thf(fact_9866_summable__Leibniz_H_I2_J,axiom,
% 0.29/0.62      ! [A: nat > real,N: nat] :
% 0.29/0.62        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.29/0.62         => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.29/0.62           => ( ord_less_eq_real
% 0.29/0.62              @ ( groups6591440286371151544t_real
% 0.29/0.62                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.29/0.62              @ ( suminf_real
% 0.29/0.62                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable_Leibniz'(2)
% 0.29/0.62  thf(fact_9867_sums__alternating__upper__lower,axiom,
% 0.29/0.62      ! [A: nat > real] :
% 0.29/0.62        ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.29/0.62       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.29/0.62         => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62           => ? [L2: real] :
% 0.29/0.62                ( ! [N8: nat] :
% 0.29/0.62                    ( ord_less_eq_real
% 0.29/0.62                    @ ( groups6591440286371151544t_real
% 0.29/0.62                      @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) )
% 0.29/0.62                    @ L2 )
% 0.29/0.62                & ( filterlim_nat_real
% 0.29/0.62                  @ ^ [N3: nat] :
% 0.29/0.62                      ( groups6591440286371151544t_real
% 0.29/0.62                      @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ L2 )
% 0.29/0.62                  @ at_top_nat )
% 0.29/0.62                & ! [N8: nat] :
% 0.29/0.62                    ( ord_less_eq_real @ L2
% 0.29/0.62                    @ ( groups6591440286371151544t_real
% 0.29/0.62                      @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) ) )
% 0.29/0.62                & ( filterlim_nat_real
% 0.29/0.62                  @ ^ [N3: nat] :
% 0.29/0.62                      ( groups6591440286371151544t_real
% 0.29/0.62                      @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ L2 )
% 0.29/0.62                  @ at_top_nat ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sums_alternating_upper_lower
% 0.29/0.62  thf(fact_9868_summable__Leibniz_I5_J,axiom,
% 0.29/0.62      ! [A: nat > real] :
% 0.29/0.62        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62       => ( ( topolo6980174941875973593q_real @ A )
% 0.29/0.62         => ( filterlim_nat_real
% 0.29/0.62            @ ^ [N3: nat] :
% 0.29/0.62                ( groups6591440286371151544t_real
% 0.29/0.62                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.29/0.62            @ ( topolo2815343760600316023s_real
% 0.29/0.62              @ ( suminf_real
% 0.29/0.62                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 0.29/0.62            @ at_top_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable_Leibniz(5)
% 0.29/0.62  thf(fact_9869_summable__Leibniz_H_I5_J,axiom,
% 0.29/0.62      ! [A: nat > real] :
% 0.29/0.62        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.29/0.62       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.29/0.62         => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.29/0.62           => ( filterlim_nat_real
% 0.29/0.62              @ ^ [N3: nat] :
% 0.29/0.62                  ( groups6591440286371151544t_real
% 0.29/0.62                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.29/0.62                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.29/0.62              @ ( topolo2815343760600316023s_real
% 0.29/0.62                @ ( suminf_real
% 0.29/0.62                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 0.29/0.62              @ at_top_nat ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable_Leibniz'(5)
% 0.29/0.62  thf(fact_9870_Code__Numeral_Onegative__def,axiom,
% 0.29/0.62      ( code_negative
% 0.29/0.62      = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Code_Numeral.negative_def
% 0.29/0.62  thf(fact_9871_Code__Target__Int_Onegative__def,axiom,
% 0.29/0.62      ( code_Target_negative
% 0.29/0.62      = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Code_Target_Int.negative_def
% 0.29/0.62  thf(fact_9872_tendsto__exp__limit__at__right,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( filterlim_real_real
% 0.29/0.62        @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y6 ) ) @ ( divide_divide_real @ one_one_real @ Y6 ) )
% 0.29/0.62        @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.29/0.62        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % tendsto_exp_limit_at_right
% 0.29/0.62  thf(fact_9873_tendsto__arcosh__at__left__1,axiom,
% 0.29/0.62      filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % tendsto_arcosh_at_left_1
% 0.29/0.62  thf(fact_9874_filterlim__tan__at__right,axiom,
% 0.29/0.62      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 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_tan_at_right
% 0.29/0.62  thf(fact_9875_sinh__real__at__bot,axiom,
% 0.29/0.62      filterlim_real_real @ sinh_real @ at_bot_real @ at_bot_real ).
% 0.29/0.62  
% 0.29/0.62  % sinh_real_at_bot
% 0.29/0.62  thf(fact_9876_arsinh__real__at__bot,axiom,
% 0.29/0.62      filterlim_real_real @ arsinh_real @ at_bot_real @ at_bot_real ).
% 0.29/0.62  
% 0.29/0.62  % arsinh_real_at_bot
% 0.29/0.62  thf(fact_9877_greaterThan__0,axiom,
% 0.29/0.62      ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 0.29/0.62      = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % greaterThan_0
% 0.29/0.62  thf(fact_9878_exp__at__bot,axiom,
% 0.29/0.62      filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).
% 0.29/0.62  
% 0.29/0.62  % exp_at_bot
% 0.29/0.62  thf(fact_9879_greaterThan__Suc,axiom,
% 0.29/0.62      ! [K: nat] :
% 0.29/0.62        ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 0.29/0.62        = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % greaterThan_Suc
% 0.29/0.62  thf(fact_9880_filterlim__inverse__at__bot__neg,axiom,
% 0.29/0.62      filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_inverse_at_bot_neg
% 0.29/0.62  thf(fact_9881_tanh__real__at__bot,axiom,
% 0.29/0.62      filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 0.29/0.62  
% 0.29/0.62  % tanh_real_at_bot
% 0.29/0.62  thf(fact_9882_ln__at__0,axiom,
% 0.29/0.62      filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % ln_at_0
% 0.29/0.62  thf(fact_9883_artanh__real__at__right__1,axiom,
% 0.29/0.62      filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % artanh_real_at_right_1
% 0.29/0.62  thf(fact_9884_DERIV__pos__imp__increasing__at__bot,axiom,
% 0.29/0.62      ! [B: real,F: real > real,Flim: real] :
% 0.29/0.62        ( ! [X3: real] :
% 0.29/0.62            ( ( ord_less_eq_real @ X3 @ B )
% 0.29/0.62           => ? [Y5: real] :
% 0.29/0.62                ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62                & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 0.29/0.62       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 0.29/0.62         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_pos_imp_increasing_at_bot
% 0.29/0.62  thf(fact_9885_filterlim__pow__at__bot__odd,axiom,
% 0.29/0.62      ! [N: nat,F: real > real,F4: filter_real] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 0.29/0.62         => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.62           => ( filterlim_real_real
% 0.29/0.62              @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
% 0.29/0.62              @ at_bot_real
% 0.29/0.62              @ F4 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_pow_at_bot_odd
% 0.29/0.62  thf(fact_9886_tendsto__arctan__at__bot,axiom,
% 0.29/0.62      filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 0.29/0.62  
% 0.29/0.62  % tendsto_arctan_at_bot
% 0.29/0.62  thf(fact_9887_filterlim__pow__at__bot__even,axiom,
% 0.29/0.62      ! [N: nat,F: real > real,F4: filter_real] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 0.29/0.62         => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.29/0.62           => ( filterlim_real_real
% 0.29/0.62              @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
% 0.29/0.62              @ at_top_real
% 0.29/0.62              @ F4 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_pow_at_bot_even
% 0.29/0.62  thf(fact_9888_at__bot__le__at__infinity,axiom,
% 0.29/0.62      ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 0.29/0.62  
% 0.29/0.62  % at_bot_le_at_infinity
% 0.29/0.62  thf(fact_9889_cosh__real__at__top,axiom,
% 0.29/0.62      filterlim_real_real @ cosh_real @ at_top_real @ at_top_real ).
% 0.29/0.62  
% 0.29/0.62  % cosh_real_at_top
% 0.29/0.62  thf(fact_9890_sinh__real__at__top,axiom,
% 0.29/0.62      filterlim_real_real @ sinh_real @ at_top_real @ at_top_real ).
% 0.29/0.62  
% 0.29/0.62  % sinh_real_at_top
% 0.29/0.62  thf(fact_9891_at__top__le__at__infinity,axiom,
% 0.29/0.62      ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 0.29/0.62  
% 0.29/0.62  % at_top_le_at_infinity
% 0.29/0.62  thf(fact_9892_arcosh__real__at__top,axiom,
% 0.29/0.62      filterlim_real_real @ arcosh_real @ at_top_real @ at_top_real ).
% 0.29/0.62  
% 0.29/0.62  % arcosh_real_at_top
% 0.29/0.62  thf(fact_9893_arsinh__real__at__top,axiom,
% 0.29/0.62      filterlim_real_real @ arsinh_real @ at_top_real @ at_top_real ).
% 0.29/0.62  
% 0.29/0.62  % arsinh_real_at_top
% 0.29/0.62  thf(fact_9894_ln__at__top,axiom,
% 0.29/0.62      filterlim_real_real @ ln_ln_real @ at_top_real @ at_top_real ).
% 0.29/0.62  
% 0.29/0.62  % ln_at_top
% 0.29/0.62  thf(fact_9895_exp__at__top,axiom,
% 0.29/0.62      filterlim_real_real @ exp_real @ at_top_real @ at_top_real ).
% 0.29/0.62  
% 0.29/0.62  % exp_at_top
% 0.29/0.62  thf(fact_9896_cosh__real__at__bot,axiom,
% 0.29/0.62      filterlim_real_real @ cosh_real @ at_top_real @ at_bot_real ).
% 0.29/0.62  
% 0.29/0.62  % cosh_real_at_bot
% 0.29/0.62  thf(fact_9897_filterlim__real__sequentially,axiom,
% 0.29/0.62      filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_real_sequentially
% 0.29/0.62  thf(fact_9898_filterlim__real__at__infinity__sequentially,axiom,
% 0.29/0.62      filterlim_nat_real @ semiri5074537144036343181t_real @ at_infinity_real @ at_top_nat ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_real_at_infinity_sequentially
% 0.29/0.62  thf(fact_9899_tanh__real__at__top,axiom,
% 0.29/0.62      filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 0.29/0.62  
% 0.29/0.62  % tanh_real_at_top
% 0.29/0.62  thf(fact_9900_artanh__real__at__left__1,axiom,
% 0.29/0.62      filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % artanh_real_at_left_1
% 0.29/0.62  thf(fact_9901_ln__x__over__x__tendsto__0,axiom,
% 0.29/0.62      ( filterlim_real_real
% 0.29/0.62      @ ^ [X4: real] : ( divide_divide_real @ ( ln_ln_real @ X4 ) @ X4 )
% 0.29/0.62      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62      @ at_top_real ) ).
% 0.29/0.62  
% 0.29/0.62  % ln_x_over_x_tendsto_0
% 0.29/0.62  thf(fact_9902_filterlim__inverse__at__right__top,axiom,
% 0.29/0.62      filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_inverse_at_right_top
% 0.29/0.62  thf(fact_9903_filterlim__inverse__at__top__right,axiom,
% 0.29/0.62      filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_inverse_at_top_right
% 0.29/0.62  thf(fact_9904_tendsto__power__div__exp__0,axiom,
% 0.29/0.62      ! [K: nat] :
% 0.29/0.62        ( filterlim_real_real
% 0.29/0.62        @ ^ [X4: real] : ( divide_divide_real @ ( power_power_real @ X4 @ K ) @ ( exp_real @ X4 ) )
% 0.29/0.62        @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.29/0.62        @ at_top_real ) ).
% 0.29/0.62  
% 0.29/0.62  % tendsto_power_div_exp_0
% 0.29/0.62  thf(fact_9905_tendsto__exp__limit__at__top,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( filterlim_real_real
% 0.29/0.62        @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y6 ) ) @ Y6 )
% 0.29/0.62        @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.29/0.62        @ at_top_real ) ).
% 0.29/0.62  
% 0.29/0.62  % tendsto_exp_limit_at_top
% 0.29/0.62  thf(fact_9906_filterlim__tan__at__left,axiom,
% 0.29/0.62      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 ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_tan_at_left
% 0.29/0.62  thf(fact_9907_DERIV__neg__imp__decreasing__at__top,axiom,
% 0.29/0.62      ! [B: real,F: real > real,Flim: real] :
% 0.29/0.62        ( ! [X3: real] :
% 0.29/0.62            ( ( ord_less_eq_real @ B @ X3 )
% 0.29/0.62           => ? [Y5: real] :
% 0.29/0.62                ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62                & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 0.29/0.62       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 0.29/0.62         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_neg_imp_decreasing_at_top
% 0.29/0.62  thf(fact_9908_tendsto__arctan__at__top,axiom,
% 0.29/0.62      filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 0.29/0.62  
% 0.29/0.62  % tendsto_arctan_at_top
% 0.29/0.62  thf(fact_9909_lhopital__left__at__top,axiom,
% 0.29/0.62      ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 0.29/0.62        ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62       => ( ( eventually_real
% 0.29/0.62            @ ^ [X4: real] :
% 0.29/0.62                ( ( G2 @ X4 )
% 0.29/0.62               != zero_zero_real )
% 0.29/0.62            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ Y )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ Y )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_left_at_top
% 0.29/0.62  thf(fact_9910_eventually__sequentially__Suc,axiom,
% 0.29/0.62      ! [P: nat > $o] :
% 0.29/0.62        ( ( eventually_nat
% 0.29/0.62          @ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
% 0.29/0.62          @ at_top_nat )
% 0.29/0.62        = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eventually_sequentially_Suc
% 0.29/0.62  thf(fact_9911_eventually__sequentially__seg,axiom,
% 0.29/0.62      ! [P: nat > $o,K: nat] :
% 0.29/0.62        ( ( eventually_nat
% 0.29/0.62          @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K ) )
% 0.29/0.62          @ at_top_nat )
% 0.29/0.62        = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eventually_sequentially_seg
% 0.29/0.62  thf(fact_9912_eventually__sequentially,axiom,
% 0.29/0.62      ! [P: nat > $o] :
% 0.29/0.62        ( ( eventually_nat @ P @ at_top_nat )
% 0.29/0.62        = ( ? [N7: nat] :
% 0.29/0.62            ! [N3: nat] :
% 0.29/0.62              ( ( ord_less_eq_nat @ N7 @ N3 )
% 0.29/0.62             => ( P @ N3 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eventually_sequentially
% 0.29/0.62  thf(fact_9913_eventually__sequentiallyI,axiom,
% 0.29/0.62      ! [C: nat,P: nat > $o] :
% 0.29/0.62        ( ! [X3: nat] :
% 0.29/0.62            ( ( ord_less_eq_nat @ C @ X3 )
% 0.29/0.62           => ( P @ X3 ) )
% 0.29/0.62       => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eventually_sequentiallyI
% 0.29/0.62  thf(fact_9914_sequentially__offset,axiom,
% 0.29/0.62      ! [P: nat > $o,K: nat] :
% 0.29/0.62        ( ( eventually_nat @ P @ at_top_nat )
% 0.29/0.62       => ( eventually_nat
% 0.29/0.62          @ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
% 0.29/0.62          @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sequentially_offset
% 0.29/0.62  thf(fact_9915_le__sequentially,axiom,
% 0.29/0.62      ! [F4: filter_nat] :
% 0.29/0.62        ( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
% 0.29/0.62        = ( ! [N7: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N7 ) @ F4 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % le_sequentially
% 0.29/0.62  thf(fact_9916_eventually__at__right__to__0,axiom,
% 0.29/0.62      ! [P: real > $o,A: real] :
% 0.29/0.62        ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62        = ( eventually_real
% 0.29/0.62          @ ^ [X4: real] : ( P @ ( plus_plus_real @ X4 @ A ) )
% 0.29/0.62          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eventually_at_right_to_0
% 0.29/0.62  thf(fact_9917_eventually__at__right__real,axiom,
% 0.29/0.62      ! [A: real,B: real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( eventually_real
% 0.29/0.62          @ ^ [X4: real] : ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.29/0.62          @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eventually_at_right_real
% 0.29/0.62  thf(fact_9918_eventually__at__left__real,axiom,
% 0.29/0.62      ! [B: real,A: real] :
% 0.29/0.62        ( ( ord_less_real @ B @ A )
% 0.29/0.62       => ( eventually_real
% 0.29/0.62          @ ^ [X4: real] : ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ A ) )
% 0.29/0.62          @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eventually_at_left_real
% 0.29/0.62  thf(fact_9919_eventually__at__top__to__right,axiom,
% 0.29/0.62      ! [P: real > $o] :
% 0.29/0.62        ( ( eventually_real @ P @ at_top_real )
% 0.29/0.62        = ( eventually_real
% 0.29/0.62          @ ^ [X4: real] : ( P @ ( inverse_inverse_real @ X4 ) )
% 0.29/0.62          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eventually_at_top_to_right
% 0.29/0.62  thf(fact_9920_eventually__at__right__to__top,axiom,
% 0.29/0.62      ! [P: real > $o] :
% 0.29/0.62        ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62        = ( eventually_real
% 0.29/0.62          @ ^ [X4: real] : ( P @ ( inverse_inverse_real @ X4 ) )
% 0.29/0.62          @ at_top_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eventually_at_right_to_top
% 0.29/0.62  thf(fact_9921_lhopital__at__top__at__top,axiom,
% 0.29/0.62      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.29/0.62       => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ at_top_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ at_top_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_at_top_at_top
% 0.29/0.62  thf(fact_9922_lhopital,axiom,
% 0.29/0.62      ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] :
% 0.29/0.62                  ( ( G @ X4 )
% 0.29/0.62                 != zero_zero_real )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] :
% 0.29/0.62                    ( ( G2 @ X4 )
% 0.29/0.62                   != zero_zero_real )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62             => ( ( eventually_real
% 0.29/0.62                  @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62               => ( ( eventually_real
% 0.29/0.62                    @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62                 => ( ( filterlim_real_real
% 0.29/0.62                      @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                      @ F4
% 0.29/0.62                      @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62                   => ( filterlim_real_real
% 0.29/0.62                      @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                      @ F4
% 0.29/0.62                      @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital
% 0.29/0.62  thf(fact_9923_lhopital__right__at__top__at__top,axiom,
% 0.29/0.62      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62       => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ at_top_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ at_top_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_right_at_top_at_top
% 0.29/0.62  thf(fact_9924_lhopital__at__top__at__bot,axiom,
% 0.29/0.62      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.29/0.62       => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ at_bot_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ at_bot_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_at_top_at_bot
% 0.29/0.62  thf(fact_9925_lhopital__left__at__top__at__top,axiom,
% 0.29/0.62      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.29/0.62       => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ at_top_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ at_top_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_left_at_top_at_top
% 0.29/0.62  thf(fact_9926_lhospital__at__top__at__top,axiom,
% 0.29/0.62      ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 0.29/0.62        ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 0.29/0.62       => ( ( eventually_real
% 0.29/0.62            @ ^ [X4: real] :
% 0.29/0.62                ( ( G2 @ X4 )
% 0.29/0.62               != zero_zero_real )
% 0.29/0.62            @ at_top_real )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ at_top_real )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ at_top_real )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ X )
% 0.29/0.62                  @ at_top_real )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ X )
% 0.29/0.62                  @ at_top_real ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhospital_at_top_at_top
% 0.29/0.62  thf(fact_9927_lhopital__at__top,axiom,
% 0.29/0.62      ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 0.29/0.62        ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62       => ( ( eventually_real
% 0.29/0.62            @ ^ [X4: real] :
% 0.29/0.62                ( ( G2 @ X4 )
% 0.29/0.62               != zero_zero_real )
% 0.29/0.62            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ Y )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ Y )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_at_top
% 0.29/0.62  thf(fact_9928_lhopital__right__0,axiom,
% 0.29/0.62      ! [F0: real > real,G0: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.29/0.62        ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62       => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] :
% 0.29/0.62                  ( ( G0 @ X4 )
% 0.29/0.62                 != zero_zero_real )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] :
% 0.29/0.62                    ( ( G2 @ X4 )
% 0.29/0.62                   != zero_zero_real )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62             => ( ( eventually_real
% 0.29/0.62                  @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62               => ( ( eventually_real
% 0.29/0.62                    @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62                 => ( ( filterlim_real_real
% 0.29/0.62                      @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                      @ F4
% 0.29/0.62                      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62                   => ( filterlim_real_real
% 0.29/0.62                      @ ^ [X4: real] : ( divide_divide_real @ ( F0 @ X4 ) @ ( G0 @ X4 ) )
% 0.29/0.62                      @ F4
% 0.29/0.62                      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_right_0
% 0.29/0.62  thf(fact_9929_lhopital__right,axiom,
% 0.29/0.62      ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62       => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] :
% 0.29/0.62                  ( ( G @ X4 )
% 0.29/0.62                 != zero_zero_real )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] :
% 0.29/0.62                    ( ( G2 @ X4 )
% 0.29/0.62                   != zero_zero_real )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62             => ( ( eventually_real
% 0.29/0.62                  @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62               => ( ( eventually_real
% 0.29/0.62                    @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62                 => ( ( filterlim_real_real
% 0.29/0.62                      @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                      @ F4
% 0.29/0.62                      @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62                   => ( filterlim_real_real
% 0.29/0.62                      @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                      @ F4
% 0.29/0.62                      @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_right
% 0.29/0.62  thf(fact_9930_lhopital__left,axiom,
% 0.29/0.62      ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62       => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] :
% 0.29/0.62                  ( ( G @ X4 )
% 0.29/0.62                 != zero_zero_real )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] :
% 0.29/0.62                    ( ( G2 @ X4 )
% 0.29/0.62                   != zero_zero_real )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62             => ( ( eventually_real
% 0.29/0.62                  @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62               => ( ( eventually_real
% 0.29/0.62                    @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62                 => ( ( filterlim_real_real
% 0.29/0.62                      @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                      @ F4
% 0.29/0.62                      @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.29/0.62                   => ( filterlim_real_real
% 0.29/0.62                      @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                      @ F4
% 0.29/0.62                      @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_left
% 0.29/0.62  thf(fact_9931_lhopital__right__at__top__at__bot,axiom,
% 0.29/0.62      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62       => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ at_bot_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ at_bot_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_right_at_top_at_bot
% 0.29/0.62  thf(fact_9932_lhopital__left__at__top__at__bot,axiom,
% 0.29/0.62      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.29/0.62        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.29/0.62       => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ at_bot_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ at_bot_real
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_left_at_top_at_bot
% 0.29/0.62  thf(fact_9933_lhopital__right__at__top,axiom,
% 0.29/0.62      ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 0.29/0.62        ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62       => ( ( eventually_real
% 0.29/0.62            @ ^ [X4: real] :
% 0.29/0.62                ( ( G2 @ X4 )
% 0.29/0.62               != zero_zero_real )
% 0.29/0.62            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ Y )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ Y )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_right_at_top
% 0.29/0.62  thf(fact_9934_lhopital__right__0__at__top,axiom,
% 0.29/0.62      ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 0.29/0.62        ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62       => ( ( eventually_real
% 0.29/0.62            @ ^ [X4: real] :
% 0.29/0.62                ( ( G2 @ X4 )
% 0.29/0.62               != zero_zero_real )
% 0.29/0.62            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62         => ( ( eventually_real
% 0.29/0.62              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62           => ( ( eventually_real
% 0.29/0.62                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.29/0.62                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62             => ( ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ X )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.29/0.62               => ( filterlim_real_real
% 0.29/0.62                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.62                  @ ( topolo2815343760600316023s_real @ X )
% 0.29/0.62                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % lhopital_right_0_at_top
% 0.29/0.62  thf(fact_9935_filterlim__int__sequentially,axiom,
% 0.29/0.62      filterlim_nat_int @ semiri1314217659103216013at_int @ at_top_int @ at_top_nat ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_int_sequentially
% 0.29/0.62  thf(fact_9936_filterlim__real__of__int__at__top,axiom,
% 0.29/0.62      filterlim_int_real @ ring_1_of_int_real @ at_top_real @ at_top_int ).
% 0.29/0.62  
% 0.29/0.62  % filterlim_real_of_int_at_top
% 0.29/0.62  thf(fact_9937_Bseq__eq__bounded,axiom,
% 0.29/0.62      ! [F: nat > real,A: real,B: real] :
% 0.29/0.62        ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.29/0.62       => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Bseq_eq_bounded
% 0.29/0.62  thf(fact_9938_Bseq__realpow,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.62       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.29/0.62         => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Bseq_realpow
% 0.29/0.62  thf(fact_9939_atLeastSucAtMost__greaterThanAtMost,axiom,
% 0.29/0.62      ! [L: nat,U: nat] :
% 0.29/0.62        ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 0.29/0.62        = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 0.29/0.62  
% 0.29/0.62  % atLeastSucAtMost_greaterThanAtMost
% 0.29/0.62  thf(fact_9940_GreatestI__nat,axiom,
% 0.29/0.62      ! [P: nat > $o,K: nat,B: nat] :
% 0.29/0.62        ( ( P @ K )
% 0.29/0.62       => ( ! [Y4: nat] :
% 0.29/0.62              ( ( P @ Y4 )
% 0.29/0.62             => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.29/0.62         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % GreatestI_nat
% 0.29/0.62  thf(fact_9941_Greatest__le__nat,axiom,
% 0.29/0.62      ! [P: nat > $o,K: nat,B: nat] :
% 0.29/0.62        ( ( P @ K )
% 0.29/0.62       => ( ! [Y4: nat] :
% 0.29/0.62              ( ( P @ Y4 )
% 0.29/0.62             => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.29/0.62         => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Greatest_le_nat
% 0.29/0.62  thf(fact_9942_GreatestI__ex__nat,axiom,
% 0.29/0.62      ! [P: nat > $o,B: nat] :
% 0.29/0.62        ( ? [X_1: nat] : ( P @ X_1 )
% 0.29/0.62       => ( ! [Y4: nat] :
% 0.29/0.62              ( ( P @ Y4 )
% 0.29/0.62             => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.29/0.62         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % GreatestI_ex_nat
% 0.29/0.62  thf(fact_9943_atLeast__0,axiom,
% 0.29/0.62      ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 0.29/0.62      = top_top_set_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % atLeast_0
% 0.29/0.62  thf(fact_9944_atLeast__Suc__greaterThan,axiom,
% 0.29/0.62      ! [K: nat] :
% 0.29/0.62        ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.29/0.62        = ( set_or1210151606488870762an_nat @ K ) ) ).
% 0.29/0.62  
% 0.29/0.62  % atLeast_Suc_greaterThan
% 0.29/0.62  thf(fact_9945_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 0.29/0.62      ! [L: int,U: int] :
% 0.29/0.62        ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 0.29/0.62        = ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 0.29/0.62  
% 0.29/0.62  % atLeastPlusOneAtMost_greaterThanAtMost_int
% 0.29/0.62  thf(fact_9946_decseq__bounded,axiom,
% 0.29/0.62      ! [X8: nat > real,B2: real] :
% 0.29/0.62        ( ( order_9091379641038594480t_real @ X8 )
% 0.29/0.62       => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X8 @ I2 ) )
% 0.29/0.62         => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % decseq_bounded
% 0.29/0.62  thf(fact_9947_decseq__convergent,axiom,
% 0.29/0.62      ! [X8: nat > real,B2: real] :
% 0.29/0.62        ( ( order_9091379641038594480t_real @ X8 )
% 0.29/0.62       => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X8 @ I2 ) )
% 0.29/0.62         => ~ ! [L5: real] :
% 0.29/0.62                ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat )
% 0.29/0.62               => ~ ! [I4: nat] : ( ord_less_eq_real @ L5 @ ( X8 @ I4 ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % decseq_convergent
% 0.29/0.62  thf(fact_9948_atLeast__Suc,axiom,
% 0.29/0.62      ! [K: nat] :
% 0.29/0.62        ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.29/0.62        = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % atLeast_Suc
% 0.29/0.62  thf(fact_9949_GMVT,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real,G: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( ( ord_less_eq_real @ A @ X3 )
% 0.29/0.62                & ( ord_less_eq_real @ X3 @ B ) )
% 0.29/0.62             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 0.29/0.62         => ( ! [X3: real] :
% 0.29/0.62                ( ( ( ord_less_real @ A @ X3 )
% 0.29/0.62                  & ( ord_less_real @ X3 @ B ) )
% 0.29/0.62               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.29/0.62           => ( ! [X3: real] :
% 0.29/0.62                  ( ( ( ord_less_eq_real @ A @ X3 )
% 0.29/0.62                    & ( ord_less_eq_real @ X3 @ B ) )
% 0.29/0.62                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
% 0.29/0.62             => ( ! [X3: real] :
% 0.29/0.62                    ( ( ( ord_less_real @ A @ X3 )
% 0.29/0.62                      & ( ord_less_real @ X3 @ B ) )
% 0.29/0.62                   => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.29/0.62               => ? [G_c: real,F_c: real,C4: real] :
% 0.29/0.62                    ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C4 @ top_top_set_real ) )
% 0.29/0.62                    & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C4 @ top_top_set_real ) )
% 0.29/0.62                    & ( ord_less_real @ A @ C4 )
% 0.29/0.62                    & ( ord_less_real @ C4 @ B )
% 0.29/0.62                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 0.29/0.62                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % GMVT
% 0.29/0.62  thf(fact_9950_MVT,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.29/0.62         => ( ! [X3: real] :
% 0.29/0.62                ( ( ord_less_real @ A @ X3 )
% 0.29/0.62               => ( ( ord_less_real @ X3 @ B )
% 0.29/0.62                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.29/0.62           => ? [L2: real,Z3: real] :
% 0.29/0.62                ( ( ord_less_real @ A @ Z3 )
% 0.29/0.62                & ( ord_less_real @ Z3 @ B )
% 0.29/0.62                & ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
% 0.29/0.62                & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.29/0.62                  = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L2 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % MVT
% 0.29/0.62  thf(fact_9951_continuous__on__arsinh,axiom,
% 0.29/0.62      ! [A2: set_real] : ( topolo5044208981011980120l_real @ A2 @ arsinh_real ) ).
% 0.29/0.62  
% 0.29/0.62  % continuous_on_arsinh
% 0.29/0.62  thf(fact_9952_continuous__on__arsinh_H,axiom,
% 0.29/0.62      ! [A2: set_real,F: real > real] :
% 0.29/0.62        ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.29/0.62       => ( topolo5044208981011980120l_real @ A2
% 0.29/0.62          @ ^ [X4: real] : ( arsinh_real @ ( F @ X4 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % continuous_on_arsinh'
% 0.29/0.62  thf(fact_9953_continuous__on__cos__real,axiom,
% 0.29/0.62      ! [A: real,B: real] : ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ cos_real ) ).
% 0.29/0.62  
% 0.29/0.62  % continuous_on_cos_real
% 0.29/0.62  thf(fact_9954_continuous__on__sin__real,axiom,
% 0.29/0.62      ! [A: real,B: real] : ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ sin_real ) ).
% 0.29/0.62  
% 0.29/0.62  % continuous_on_sin_real
% 0.29/0.62  thf(fact_9955_continuous__on__arcosh_H,axiom,
% 0.29/0.62      ! [A2: set_real,F: real > real] :
% 0.29/0.62        ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( member_real @ X3 @ A2 )
% 0.29/0.62             => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.29/0.62         => ( topolo5044208981011980120l_real @ A2
% 0.29/0.62            @ ^ [X4: real] : ( arcosh_real @ ( F @ X4 ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % continuous_on_arcosh'
% 0.29/0.62  thf(fact_9956_continuous__image__closed__interval,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.62       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.29/0.62         => ? [C4: real,D5: real] :
% 0.29/0.62              ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.29/0.62                = ( set_or1222579329274155063t_real @ C4 @ D5 ) )
% 0.29/0.62              & ( ord_less_eq_real @ C4 @ D5 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % continuous_image_closed_interval
% 0.29/0.62  thf(fact_9957_continuous__on__arcosh,axiom,
% 0.29/0.62      ! [A2: set_real] :
% 0.29/0.62        ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
% 0.29/0.62       => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % continuous_on_arcosh
% 0.29/0.62  thf(fact_9958_continuous__on__arccos_H,axiom,
% 0.29/0.62      topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 0.29/0.62  
% 0.29/0.62  % continuous_on_arccos'
% 0.29/0.62  thf(fact_9959_continuous__on__arcsin_H,axiom,
% 0.29/0.62      topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 0.29/0.62  
% 0.29/0.62  % continuous_on_arcsin'
% 0.29/0.62  thf(fact_9960_continuous__on__artanh_H,axiom,
% 0.29/0.62      ! [A2: set_real,F: real > real] :
% 0.29/0.62        ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( member_real @ X3 @ A2 )
% 0.29/0.62             => ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 0.29/0.62         => ( topolo5044208981011980120l_real @ A2
% 0.29/0.62            @ ^ [X4: real] : ( artanh_real @ ( F @ X4 ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % continuous_on_artanh'
% 0.29/0.62  thf(fact_9961_Rolle__deriv,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real,F3: real > real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ( ( F @ A )
% 0.29/0.62            = ( F @ B ) )
% 0.29/0.62         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.29/0.62           => ( ! [X3: real] :
% 0.29/0.62                  ( ( ord_less_real @ A @ X3 )
% 0.29/0.62                 => ( ( ord_less_real @ X3 @ B )
% 0.29/0.62                   => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.29/0.62             => ? [Z3: real] :
% 0.29/0.62                  ( ( ord_less_real @ A @ Z3 )
% 0.29/0.62                  & ( ord_less_real @ Z3 @ B )
% 0.29/0.62                  & ( ( F3 @ Z3 )
% 0.29/0.62                    = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Rolle_deriv
% 0.29/0.62  thf(fact_9962_mvt,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real,F3: real > real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.29/0.62         => ( ! [X3: real] :
% 0.29/0.62                ( ( ord_less_real @ A @ X3 )
% 0.29/0.62               => ( ( ord_less_real @ X3 @ B )
% 0.29/0.62                 => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.29/0.62           => ~ ! [Xi: real] :
% 0.29/0.62                  ( ( ord_less_real @ A @ Xi )
% 0.29/0.62                 => ( ( ord_less_real @ Xi @ B )
% 0.29/0.62                   => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.29/0.62                     != ( F3 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % mvt
% 0.29/0.62  thf(fact_9963_DERIV__isconst__end,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.29/0.62         => ( ! [X3: real] :
% 0.29/0.62                ( ( ord_less_real @ A @ X3 )
% 0.29/0.62               => ( ( ord_less_real @ X3 @ B )
% 0.29/0.62                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.29/0.62           => ( ( F @ B )
% 0.29/0.62              = ( F @ A ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_isconst_end
% 0.29/0.62  thf(fact_9964_DERIV__neg__imp__decreasing__open,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( ord_less_real @ A @ X3 )
% 0.29/0.62             => ( ( ord_less_real @ X3 @ B )
% 0.29/0.62               => ? [Y5: real] :
% 0.29/0.62                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62                    & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.29/0.62         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.29/0.62           => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_neg_imp_decreasing_open
% 0.29/0.62  thf(fact_9965_DERIV__pos__imp__increasing__open,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ! [X3: real] :
% 0.29/0.62              ( ( ord_less_real @ A @ X3 )
% 0.29/0.62             => ( ( ord_less_real @ X3 @ B )
% 0.29/0.62               => ? [Y5: real] :
% 0.29/0.62                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62                    & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.29/0.62         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.29/0.62           => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_pos_imp_increasing_open
% 0.29/0.62  thf(fact_9966_continuous__on__artanh,axiom,
% 0.29/0.62      ! [A2: set_real] :
% 0.29/0.62        ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 0.29/0.62       => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % continuous_on_artanh
% 0.29/0.62  thf(fact_9967_DERIV__isconst2,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real,X: real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.29/0.62         => ( ! [X3: real] :
% 0.29/0.62                ( ( ord_less_real @ A @ X3 )
% 0.29/0.62               => ( ( ord_less_real @ X3 @ B )
% 0.29/0.62                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.29/0.62           => ( ( ord_less_eq_real @ A @ X )
% 0.29/0.62             => ( ( ord_less_eq_real @ X @ B )
% 0.29/0.62               => ( ( F @ X )
% 0.29/0.62                  = ( F @ A ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % DERIV_isconst2
% 0.29/0.62  thf(fact_9968_Rolle,axiom,
% 0.29/0.62      ! [A: real,B: real,F: real > real] :
% 0.29/0.62        ( ( ord_less_real @ A @ B )
% 0.29/0.62       => ( ( ( F @ A )
% 0.29/0.62            = ( F @ B ) )
% 0.29/0.62         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.29/0.62           => ( ! [X3: real] :
% 0.29/0.62                  ( ( ord_less_real @ A @ X3 )
% 0.29/0.62                 => ( ( ord_less_real @ X3 @ B )
% 0.29/0.62                   => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.29/0.62             => ? [Z3: real] :
% 0.29/0.62                  ( ( ord_less_real @ A @ Z3 )
% 0.29/0.62                  & ( ord_less_real @ Z3 @ B )
% 0.29/0.62                  & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Rolle
% 0.29/0.62  thf(fact_9969_mono__Suc,axiom,
% 0.29/0.62      order_mono_nat_nat @ suc ).
% 0.29/0.62  
% 0.29/0.62  % mono_Suc
% 0.29/0.62  thf(fact_9970_mono__times__nat,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % mono_times_nat
% 0.29/0.62  thf(fact_9971_incseq__bounded,axiom,
% 0.29/0.62      ! [X8: nat > real,B2: real] :
% 0.29/0.62        ( ( order_mono_nat_real @ X8 )
% 0.29/0.62       => ( ! [I2: nat] : ( ord_less_eq_real @ ( X8 @ I2 ) @ B2 )
% 0.29/0.62         => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % incseq_bounded
% 0.29/0.62  thf(fact_9972_incseq__convergent,axiom,
% 0.29/0.62      ! [X8: nat > real,B2: real] :
% 0.29/0.62        ( ( order_mono_nat_real @ X8 )
% 0.29/0.62       => ( ! [I2: nat] : ( ord_less_eq_real @ ( X8 @ I2 ) @ B2 )
% 0.29/0.62         => ~ ! [L5: real] :
% 0.29/0.62                ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat )
% 0.29/0.62               => ~ ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ L5 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % incseq_convergent
% 0.29/0.62  thf(fact_9973_mono__ge2__power__minus__self,axiom,
% 0.29/0.62      ! [K: nat] :
% 0.29/0.62        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.29/0.62       => ( order_mono_nat_nat
% 0.29/0.62          @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % mono_ge2_power_minus_self
% 0.29/0.62  thf(fact_9974_xor__minus__numerals_I2_J,axiom,
% 0.29/0.62      ! [K: int,N: num] :
% 0.29/0.62        ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62        = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_minus_numerals(2)
% 0.29/0.62  thf(fact_9975_xor__minus__numerals_I1_J,axiom,
% 0.29/0.62      ! [N: num,K: int] :
% 0.29/0.62        ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 0.29/0.62        = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_minus_numerals(1)
% 0.29/0.62  thf(fact_9976_sub__BitM__One__eq,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 0.29/0.62        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sub_BitM_One_eq
% 0.29/0.62  thf(fact_9977_tendsto__at__topI__sequentially__real,axiom,
% 0.29/0.62      ! [F: real > real,Y: real] :
% 0.29/0.62        ( ( order_mono_real_real @ F )
% 0.29/0.62       => ( ( filterlim_nat_real
% 0.29/0.62            @ ^ [N3: nat] : ( F @ ( semiri5074537144036343181t_real @ N3 ) )
% 0.29/0.62            @ ( topolo2815343760600316023s_real @ Y )
% 0.29/0.62            @ at_top_nat )
% 0.29/0.62         => ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Y ) @ at_top_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % tendsto_at_topI_sequentially_real
% 0.29/0.62  thf(fact_9978_nonneg__incseq__Bseq__subseq__iff,axiom,
% 0.29/0.62      ! [F: nat > real,G: nat > nat] :
% 0.29/0.62        ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.29/0.62       => ( ( order_mono_nat_real @ F )
% 0.29/0.62         => ( ( order_5726023648592871131at_nat @ G )
% 0.29/0.62           => ( ( bfun_nat_real
% 0.29/0.62                @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
% 0.29/0.62                @ at_top_nat )
% 0.29/0.62              = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nonneg_incseq_Bseq_subseq_iff
% 0.29/0.62  thf(fact_9979_range__abs__Nats,axiom,
% 0.29/0.62      ( ( image_int_int @ abs_abs_int @ top_top_set_int )
% 0.29/0.62      = semiring_1_Nats_int ) ).
% 0.29/0.62  
% 0.29/0.62  % range_abs_Nats
% 0.29/0.62  thf(fact_9980_strict__mono__imp__increasing,axiom,
% 0.29/0.62      ! [F: nat > nat,N: nat] :
% 0.29/0.62        ( ( order_5726023648592871131at_nat @ F )
% 0.29/0.62       => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % strict_mono_imp_increasing
% 0.29/0.62  thf(fact_9981_tanh__real__strict__mono,axiom,
% 0.29/0.62      order_7092887310737990675l_real @ tanh_real ).
% 0.29/0.62  
% 0.29/0.62  % tanh_real_strict_mono
% 0.29/0.62  thf(fact_9982_sinh__real__strict__mono,axiom,
% 0.29/0.62      order_7092887310737990675l_real @ sinh_real ).
% 0.29/0.62  
% 0.29/0.62  % sinh_real_strict_mono
% 0.29/0.62  thf(fact_9983_complex__is__Nat__iff,axiom,
% 0.29/0.62      ! [Z: complex] :
% 0.29/0.62        ( ( member_complex @ Z @ semiri3842193898606819883omplex )
% 0.29/0.62        = ( ( ( im @ Z )
% 0.29/0.62            = zero_zero_real )
% 0.29/0.62          & ? [I3: nat] :
% 0.29/0.62              ( ( re @ Z )
% 0.29/0.62              = ( semiri5074537144036343181t_real @ I3 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % complex_is_Nat_iff
% 0.29/0.62  thf(fact_9984_pos__deriv__imp__strict__mono,axiom,
% 0.29/0.62      ! [F: real > real,F3: real > real] :
% 0.29/0.62        ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.29/0.62       => ( ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( F3 @ X3 ) )
% 0.29/0.62         => ( order_7092887310737990675l_real @ F ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % pos_deriv_imp_strict_mono
% 0.29/0.62  thf(fact_9985_nat__of__integer__non__positive,axiom,
% 0.29/0.62      ! [K: code_integer] :
% 0.29/0.62        ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 0.29/0.62       => ( ( code_nat_of_integer @ K )
% 0.29/0.62          = zero_zero_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_of_integer_non_positive
% 0.29/0.62  thf(fact_9986_Suc__funpow,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( compow_nat_nat @ N @ suc )
% 0.29/0.62        = ( plus_plus_nat @ N ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Suc_funpow
% 0.29/0.62  thf(fact_9987_of__nat__of__integer,axiom,
% 0.29/0.62      ! [K: code_integer] :
% 0.29/0.62        ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
% 0.29/0.62        = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).
% 0.29/0.62  
% 0.29/0.62  % of_nat_of_integer
% 0.29/0.62  thf(fact_9988_nat__of__integer__code__post_I1_J,axiom,
% 0.29/0.62      ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 0.29/0.62      = zero_zero_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_of_integer_code_post(1)
% 0.29/0.62  thf(fact_9989_nat__of__integer__code__post_I3_J,axiom,
% 0.29/0.62      ! [K: num] :
% 0.29/0.62        ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.29/0.62        = ( numeral_numeral_nat @ K ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_of_integer_code_post(3)
% 0.29/0.62  thf(fact_9990_nat__of__integer__code__post_I2_J,axiom,
% 0.29/0.62      ( ( code_nat_of_integer @ one_one_Code_integer )
% 0.29/0.62      = one_one_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_of_integer_code_post(2)
% 0.29/0.62  thf(fact_9991_nat__of__integer__code,axiom,
% 0.29/0.62      ( code_nat_of_integer
% 0.29/0.62      = ( ^ [K2: code_integer] :
% 0.29/0.62            ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.29/0.62            @ ( produc1555791787009142072er_nat
% 0.29/0.62              @ ^ [L3: code_integer,J2: code_integer] : ( if_nat @ ( J2 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ one_one_nat ) )
% 0.29/0.62              @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_of_integer_code
% 0.29/0.62  thf(fact_9992_inj__sgn__power,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( inj_on_real_real
% 0.29/0.62          @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 0.29/0.62          @ top_top_set_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % inj_sgn_power
% 0.29/0.62  thf(fact_9993_log__inj,axiom,
% 0.29/0.62      ! [B: real] :
% 0.29/0.62        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.62       => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % log_inj
% 0.29/0.62  thf(fact_9994_inj__on__diff__nat,axiom,
% 0.29/0.62      ! [N4: set_nat,K: nat] :
% 0.29/0.62        ( ! [N2: nat] :
% 0.29/0.62            ( ( member_nat @ N2 @ N4 )
% 0.29/0.62           => ( ord_less_eq_nat @ K @ N2 ) )
% 0.29/0.62       => ( inj_on_nat_nat
% 0.29/0.62          @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
% 0.29/0.62          @ N4 ) ) ).
% 0.29/0.62  
% 0.29/0.62  % inj_on_diff_nat
% 0.29/0.62  thf(fact_9995_inj__Suc,axiom,
% 0.29/0.62      ! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% 0.29/0.62  
% 0.29/0.62  % inj_Suc
% 0.29/0.62  thf(fact_9996_summable__reindex,axiom,
% 0.29/0.62      ! [F: nat > real,G: nat > nat] :
% 0.29/0.62        ( ( summable_real @ F )
% 0.29/0.62       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.29/0.62         => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.29/0.62           => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % summable_reindex
% 0.29/0.62  thf(fact_9997_suminf__reindex__mono,axiom,
% 0.29/0.62      ! [F: nat > real,G: nat > nat] :
% 0.29/0.62        ( ( summable_real @ F )
% 0.29/0.62       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.29/0.62         => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.29/0.62           => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % suminf_reindex_mono
% 0.29/0.62  thf(fact_9998_inj__on__char__of__nat,axiom,
% 0.29/0.62      inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % inj_on_char_of_nat
% 0.29/0.62  thf(fact_9999_suminf__reindex,axiom,
% 0.29/0.62      ! [F: nat > real,G: nat > nat] :
% 0.29/0.62        ( ( summable_real @ F )
% 0.29/0.62       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.29/0.62         => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.29/0.62           => ( ! [X3: nat] :
% 0.29/0.62                  ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 0.29/0.62                 => ( ( F @ X3 )
% 0.29/0.62                    = zero_zero_real ) )
% 0.29/0.62             => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 0.29/0.62                = ( suminf_real @ F ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % suminf_reindex
% 0.29/0.62  thf(fact_10000_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.29/0.62      ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.29/0.62      @ ^ [X4: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X4 )
% 0.29/0.62      @ ^ [X4: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X4 ) ) ).
% 0.29/0.62  
% 0.29/0.62  % max_nat.semilattice_neutr_order_axioms
% 0.29/0.62  thf(fact_10001_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.29/0.62      ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 0.29/0.62      @ ^ [M6: nat,N3: nat] :
% 0.29/0.62          ( ( dvd_dvd_nat @ M6 @ N3 )
% 0.29/0.62          & ( M6 != N3 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % gcd_nat.semilattice_neutr_order_axioms
% 0.29/0.62  thf(fact_10002_int__of__integer__code,axiom,
% 0.29/0.62      ( code_int_of_integer
% 0.29/0.62      = ( ^ [K2: code_integer] :
% 0.29/0.62            ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
% 0.29/0.62            @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.29/0.62              @ ( produc1553301316500091796er_int
% 0.29/0.62                @ ^ [L3: code_integer,J2: code_integer] : ( if_int @ ( J2 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ one_one_int ) )
% 0.29/0.62                @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % int_of_integer_code
% 0.29/0.62  thf(fact_10003_of__int__integer__of,axiom,
% 0.29/0.62      ! [K: code_integer] :
% 0.29/0.62        ( ( ring_18347121197199848620nteger @ ( code_int_of_integer @ K ) )
% 0.29/0.62        = K ) ).
% 0.29/0.62  
% 0.29/0.62  % of_int_integer_of
% 0.29/0.62  thf(fact_10004_int__of__integer__of__int,axiom,
% 0.29/0.62      ! [K: int] :
% 0.29/0.62        ( ( code_int_of_integer @ ( ring_18347121197199848620nteger @ K ) )
% 0.29/0.62        = K ) ).
% 0.29/0.62  
% 0.29/0.62  % int_of_integer_of_int
% 0.29/0.62  thf(fact_10005_zero__integer_Orep__eq,axiom,
% 0.29/0.62      ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
% 0.29/0.62      = zero_zero_int ) ).
% 0.29/0.62  
% 0.29/0.62  % zero_integer.rep_eq
% 0.29/0.62  thf(fact_10006_int__of__integer__numeral,axiom,
% 0.29/0.62      ! [K: num] :
% 0.29/0.62        ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.29/0.62        = ( numeral_numeral_int @ K ) ) ).
% 0.29/0.62  
% 0.29/0.62  % int_of_integer_numeral
% 0.29/0.62  thf(fact_10007_plus__integer_Orep__eq,axiom,
% 0.29/0.62      ! [X: code_integer,Xa: code_integer] :
% 0.29/0.62        ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa ) )
% 0.29/0.62        = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % plus_integer.rep_eq
% 0.29/0.62  thf(fact_10008_one__integer_Orep__eq,axiom,
% 0.29/0.62      ( ( code_int_of_integer @ one_one_Code_integer )
% 0.29/0.62      = one_one_int ) ).
% 0.29/0.62  
% 0.29/0.62  % one_integer.rep_eq
% 0.29/0.62  thf(fact_10009_divide__integer_Orep__eq,axiom,
% 0.29/0.62      ! [X: code_integer,Xa: code_integer] :
% 0.29/0.62        ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
% 0.29/0.62        = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % divide_integer.rep_eq
% 0.29/0.62  thf(fact_10010_int__of__integer__of__nat,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.29/0.62        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.29/0.62  
% 0.29/0.62  % int_of_integer_of_nat
% 0.29/0.62  thf(fact_10011_less__eq__integer_Orep__eq,axiom,
% 0.29/0.62      ( ord_le3102999989581377725nteger
% 0.29/0.62      = ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % less_eq_integer.rep_eq
% 0.29/0.62  thf(fact_10012_integer__less__eq__iff,axiom,
% 0.29/0.62      ( ord_le3102999989581377725nteger
% 0.29/0.62      = ( ^ [K2: code_integer,L3: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % integer_less_eq_iff
% 0.29/0.62  thf(fact_10013_integer__less__iff,axiom,
% 0.29/0.62      ( ord_le6747313008572928689nteger
% 0.29/0.62      = ( ^ [K2: code_integer,L3: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % integer_less_iff
% 0.29/0.62  thf(fact_10014_less__integer_Orep__eq,axiom,
% 0.29/0.62      ( ord_le6747313008572928689nteger
% 0.29/0.62      = ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % less_integer.rep_eq
% 0.29/0.62  thf(fact_10015_times__int_Oabs__eq,axiom,
% 0.29/0.62      ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.29/0.62        ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.29/0.62        = ( abs_Integ
% 0.29/0.62          @ ( produc27273713700761075at_nat
% 0.29/0.62            @ ^ [X4: nat,Y6: nat] :
% 0.29/0.62                ( produc2626176000494625587at_nat
% 0.29/0.62                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y6 @ U2 ) ) ) )
% 0.29/0.62            @ Xa
% 0.29/0.62            @ X ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % times_int.abs_eq
% 0.29/0.62  thf(fact_10016_nat_Oabs__eq,axiom,
% 0.29/0.62      ! [X: product_prod_nat_nat] :
% 0.29/0.62        ( ( nat2 @ ( abs_Integ @ X ) )
% 0.29/0.62        = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat.abs_eq
% 0.29/0.62  thf(fact_10017_int_Oabs__induct,axiom,
% 0.29/0.62      ! [P: int > $o,X: int] :
% 0.29/0.62        ( ! [Y4: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y4 ) )
% 0.29/0.62       => ( P @ X ) ) ).
% 0.29/0.62  
% 0.29/0.62  % int.abs_induct
% 0.29/0.62  thf(fact_10018_eq__Abs__Integ,axiom,
% 0.29/0.62      ! [Z: int] :
% 0.29/0.62        ~ ! [X3: nat,Y4: nat] :
% 0.29/0.62            ( Z
% 0.29/0.62           != ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y4 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eq_Abs_Integ
% 0.29/0.62  thf(fact_10019_zero__int__def,axiom,
% 0.29/0.62      ( zero_zero_int
% 0.29/0.62      = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % zero_int_def
% 0.29/0.62  thf(fact_10020_int__def,axiom,
% 0.29/0.62      ( semiri1314217659103216013at_int
% 0.29/0.62      = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % int_def
% 0.29/0.62  thf(fact_10021_uminus__int_Oabs__eq,axiom,
% 0.29/0.62      ! [X: product_prod_nat_nat] :
% 0.29/0.62        ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
% 0.29/0.62        = ( abs_Integ
% 0.29/0.62          @ ( produc2626176000494625587at_nat
% 0.29/0.62            @ ^ [X4: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X4 )
% 0.29/0.62            @ X ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % uminus_int.abs_eq
% 0.29/0.62  thf(fact_10022_one__int__def,axiom,
% 0.29/0.62      ( one_one_int
% 0.29/0.62      = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % one_int_def
% 0.29/0.62  thf(fact_10023_less__int_Oabs__eq,axiom,
% 0.29/0.62      ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.29/0.62        ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.29/0.62        = ( produc8739625826339149834_nat_o
% 0.29/0.62          @ ^ [X4: nat,Y6: nat] :
% 0.29/0.62              ( produc6081775807080527818_nat_o
% 0.29/0.62              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y6 ) ) )
% 0.29/0.62          @ Xa
% 0.29/0.62          @ X ) ) ).
% 0.29/0.62  
% 0.29/0.62  % less_int.abs_eq
% 0.29/0.62  thf(fact_10024_less__eq__int_Oabs__eq,axiom,
% 0.29/0.62      ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.29/0.62        ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.29/0.62        = ( produc8739625826339149834_nat_o
% 0.29/0.62          @ ^ [X4: nat,Y6: nat] :
% 0.29/0.62              ( produc6081775807080527818_nat_o
% 0.29/0.62              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y6 ) ) )
% 0.29/0.62          @ Xa
% 0.29/0.62          @ X ) ) ).
% 0.29/0.62  
% 0.29/0.62  % less_eq_int.abs_eq
% 0.29/0.62  thf(fact_10025_plus__int_Oabs__eq,axiom,
% 0.29/0.62      ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.29/0.62        ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.29/0.62        = ( abs_Integ
% 0.29/0.62          @ ( produc27273713700761075at_nat
% 0.29/0.62            @ ^ [X4: nat,Y6: nat] :
% 0.29/0.62                ( produc2626176000494625587at_nat
% 0.29/0.62                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) )
% 0.29/0.62            @ Xa
% 0.29/0.62            @ X ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % plus_int.abs_eq
% 0.29/0.62  thf(fact_10026_minus__int_Oabs__eq,axiom,
% 0.29/0.62      ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.29/0.62        ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.29/0.62        = ( abs_Integ
% 0.29/0.62          @ ( produc27273713700761075at_nat
% 0.29/0.62            @ ^ [X4: nat,Y6: nat] :
% 0.29/0.62                ( produc2626176000494625587at_nat
% 0.29/0.62                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y6 @ U2 ) ) )
% 0.29/0.62            @ Xa
% 0.29/0.62            @ X ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % minus_int.abs_eq
% 0.29/0.62  thf(fact_10027_Gcd__remove0__nat,axiom,
% 0.29/0.62      ! [M2: set_nat] :
% 0.29/0.62        ( ( finite_finite_nat @ M2 )
% 0.29/0.62       => ( ( gcd_Gcd_nat @ M2 )
% 0.29/0.62          = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Gcd_remove0_nat
% 0.29/0.62  thf(fact_10028_Gcd__nat__eq__one,axiom,
% 0.29/0.62      ! [N4: set_nat] :
% 0.29/0.62        ( ( member_nat @ one_one_nat @ N4 )
% 0.29/0.62       => ( ( gcd_Gcd_nat @ N4 )
% 0.29/0.62          = one_one_nat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Gcd_nat_eq_one
% 0.29/0.62  thf(fact_10029_Gcd__eq__Max,axiom,
% 0.29/0.62      ! [M2: set_nat] :
% 0.29/0.62        ( ( finite_finite_nat @ M2 )
% 0.29/0.62       => ( ( M2 != bot_bot_set_nat )
% 0.29/0.62         => ( ~ ( member_nat @ zero_zero_nat @ M2 )
% 0.29/0.62           => ( ( gcd_Gcd_nat @ M2 )
% 0.29/0.62              = ( lattic8265883725875713057ax_nat
% 0.29/0.62                @ ( comple7806235888213564991et_nat
% 0.29/0.62                  @ ( image_nat_set_nat
% 0.29/0.62                    @ ^ [M6: nat] :
% 0.29/0.62                        ( collect_nat
% 0.29/0.62                        @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M6 ) )
% 0.29/0.62                    @ M2 ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Gcd_eq_Max
% 0.29/0.62  thf(fact_10030_less__eq__int_Orep__eq,axiom,
% 0.29/0.62      ( ord_less_eq_int
% 0.29/0.62      = ( ^ [X4: int,Xa4: int] :
% 0.29/0.62            ( produc8739625826339149834_nat_o
% 0.29/0.62            @ ^ [Y6: nat,Z4: nat] :
% 0.29/0.62                ( produc6081775807080527818_nat_o
% 0.29/0.62                @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U2 @ Z4 ) ) )
% 0.29/0.62            @ ( rep_Integ @ X4 )
% 0.29/0.62            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % less_eq_int.rep_eq
% 0.29/0.62  thf(fact_10031_less__int_Orep__eq,axiom,
% 0.29/0.62      ( ord_less_int
% 0.29/0.62      = ( ^ [X4: int,Xa4: int] :
% 0.29/0.62            ( produc8739625826339149834_nat_o
% 0.29/0.62            @ ^ [Y6: nat,Z4: nat] :
% 0.29/0.62                ( produc6081775807080527818_nat_o
% 0.29/0.62                @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U2 @ Z4 ) ) )
% 0.29/0.62            @ ( rep_Integ @ X4 )
% 0.29/0.62            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % less_int.rep_eq
% 0.29/0.62  thf(fact_10032_Gcd__int__eq,axiom,
% 0.29/0.62      ! [N4: set_nat] :
% 0.29/0.62        ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N4 ) )
% 0.29/0.62        = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N4 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Gcd_int_eq
% 0.29/0.62  thf(fact_10033_Gcd__int__greater__eq__0,axiom,
% 0.29/0.62      ! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K4 ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Gcd_int_greater_eq_0
% 0.29/0.62  thf(fact_10034_nat_Orep__eq,axiom,
% 0.29/0.62      ( nat2
% 0.29/0.62      = ( ^ [X4: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X4 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat.rep_eq
% 0.29/0.62  thf(fact_10035_Gcd__int__def,axiom,
% 0.29/0.62      ( gcd_Gcd_int
% 0.29/0.62      = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Gcd_int_def
% 0.29/0.62  thf(fact_10036_uminus__int__def,axiom,
% 0.29/0.62      ( uminus_uminus_int
% 0.29/0.62      = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 0.29/0.62        @ ( produc2626176000494625587at_nat
% 0.29/0.62          @ ^ [X4: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X4 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % uminus_int_def
% 0.29/0.62  thf(fact_10037_times__int__def,axiom,
% 0.29/0.62      ( times_times_int
% 0.29/0.62      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.29/0.62        @ ( produc27273713700761075at_nat
% 0.29/0.62          @ ^ [X4: nat,Y6: nat] :
% 0.29/0.62              ( produc2626176000494625587at_nat
% 0.29/0.62              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y6 @ U2 ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % times_int_def
% 0.29/0.62  thf(fact_10038_minus__int__def,axiom,
% 0.29/0.62      ( minus_minus_int
% 0.29/0.62      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.29/0.62        @ ( produc27273713700761075at_nat
% 0.29/0.62          @ ^ [X4: nat,Y6: nat] :
% 0.29/0.62              ( produc2626176000494625587at_nat
% 0.29/0.62              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y6 @ U2 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % minus_int_def
% 0.29/0.62  thf(fact_10039_plus__int__def,axiom,
% 0.29/0.62      ( plus_plus_int
% 0.29/0.62      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.29/0.62        @ ( produc27273713700761075at_nat
% 0.29/0.62          @ ^ [X4: nat,Y6: nat] :
% 0.29/0.62              ( produc2626176000494625587at_nat
% 0.29/0.62              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % plus_int_def
% 0.29/0.62  thf(fact_10040_measure__function__int,axiom,
% 0.29/0.62      fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% 0.29/0.62  
% 0.29/0.62  % measure_function_int
% 0.29/0.62  thf(fact_10041_prod__encode__def,axiom,
% 0.29/0.62      ( nat_prod_encode
% 0.29/0.62      = ( produc6842872674320459806at_nat
% 0.29/0.62        @ ^ [M6: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N3 ) ) @ M6 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % prod_encode_def
% 0.29/0.62  thf(fact_10042_le__prod__encode__2,axiom,
% 0.29/0.62      ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % le_prod_encode_2
% 0.29/0.62  thf(fact_10043_le__prod__encode__1,axiom,
% 0.29/0.62      ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % le_prod_encode_1
% 0.29/0.62  thf(fact_10044_prod__encode__prod__decode__aux,axiom,
% 0.29/0.62      ! [K: nat,M: nat] :
% 0.29/0.62        ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 0.29/0.62        = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 0.29/0.62  
% 0.29/0.62  % prod_encode_prod_decode_aux
% 0.29/0.62  thf(fact_10045_powr__real__of__int_H,axiom,
% 0.29/0.62      ! [X: real,N: int] :
% 0.29/0.62        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.62       => ( ( ( X != zero_zero_real )
% 0.29/0.62            | ( ord_less_int @ zero_zero_int @ N ) )
% 0.29/0.62         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.29/0.62            = ( power_int_real @ X @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % powr_real_of_int'
% 0.29/0.62  thf(fact_10046_uniformity__complex__def,axiom,
% 0.29/0.62      ( topolo896644834953643431omplex
% 0.29/0.62      = ( comple8358262395181532106omplex
% 0.29/0.62        @ ( image_5971271580939081552omplex
% 0.29/0.62          @ ^ [E3: real] :
% 0.29/0.62              ( princi3496590319149328850omplex
% 0.29/0.62              @ ( collec8663557070575231912omplex
% 0.29/0.62                @ ( produc6771430404735790350plex_o
% 0.29/0.62                  @ ^ [X4: complex,Y6: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X4 @ Y6 ) @ E3 ) ) ) )
% 0.29/0.62          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % uniformity_complex_def
% 0.29/0.62  thf(fact_10047_uniformity__real__def,axiom,
% 0.29/0.62      ( topolo1511823702728130853y_real
% 0.29/0.62      = ( comple2936214249959783750l_real
% 0.29/0.62        @ ( image_2178119161166701260l_real
% 0.29/0.62          @ ^ [E3: real] :
% 0.29/0.62              ( princi6114159922880469582l_real
% 0.29/0.62              @ ( collec3799799289383736868l_real
% 0.29/0.62                @ ( produc5414030515140494994real_o
% 0.29/0.62                  @ ^ [X4: real,Y6: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X4 @ Y6 ) @ E3 ) ) ) )
% 0.29/0.62          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % uniformity_real_def
% 0.29/0.62  thf(fact_10048_num__of__nat_Osimps_I2_J,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62         => ( ( num_of_nat @ ( suc @ N ) )
% 0.29/0.62            = ( inc @ ( num_of_nat @ N ) ) ) )
% 0.29/0.62        & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62         => ( ( num_of_nat @ ( suc @ N ) )
% 0.29/0.62            = one ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % num_of_nat.simps(2)
% 0.29/0.62  thf(fact_10049_num__of__nat__numeral__eq,axiom,
% 0.29/0.62      ! [Q3: num] :
% 0.29/0.62        ( ( num_of_nat @ ( numeral_numeral_nat @ Q3 ) )
% 0.29/0.62        = Q3 ) ).
% 0.29/0.62  
% 0.29/0.62  % num_of_nat_numeral_eq
% 0.29/0.62  thf(fact_10050_num__of__nat_Osimps_I1_J,axiom,
% 0.29/0.62      ( ( num_of_nat @ zero_zero_nat )
% 0.29/0.62      = one ) ).
% 0.29/0.62  
% 0.29/0.62  % num_of_nat.simps(1)
% 0.29/0.62  thf(fact_10051_numeral__num__of__nat,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 0.29/0.62          = N ) ) ).
% 0.29/0.62  
% 0.29/0.62  % numeral_num_of_nat
% 0.29/0.62  thf(fact_10052_num__of__nat__One,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( ord_less_eq_nat @ N @ one_one_nat )
% 0.29/0.62       => ( ( num_of_nat @ N )
% 0.29/0.62          = one ) ) ).
% 0.29/0.62  
% 0.29/0.62  % num_of_nat_One
% 0.29/0.62  thf(fact_10053_num__of__nat__double,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62       => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 0.29/0.62          = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % num_of_nat_double
% 0.29/0.62  thf(fact_10054_num__of__nat__plus__distrib,axiom,
% 0.29/0.62      ! [M: nat,N: nat] :
% 0.29/0.62        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.62         => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 0.29/0.62            = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % num_of_nat_plus_distrib
% 0.29/0.62  thf(fact_10055_upto__aux__rec,axiom,
% 0.29/0.62      ( upto_aux
% 0.29/0.62      = ( ^ [I3: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_aux_rec
% 0.29/0.62  thf(fact_10056_upto_Opsimps,axiom,
% 0.29/0.62      ! [I: int,J: int] :
% 0.29/0.62        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 0.29/0.62       => ( ( ( ord_less_eq_int @ I @ J )
% 0.29/0.62           => ( ( upto @ I @ J )
% 0.29/0.62              = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 0.29/0.62          & ( ~ ( ord_less_eq_int @ I @ J )
% 0.29/0.62           => ( ( upto @ I @ J )
% 0.29/0.62              = nil_int ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto.psimps
% 0.29/0.62  thf(fact_10057_upto_Opelims,axiom,
% 0.29/0.62      ! [X: int,Xa: int,Y: list_int] :
% 0.29/0.62        ( ( ( upto @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 0.29/0.62         => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
% 0.29/0.62                 => ( Y
% 0.29/0.62                    = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 0.29/0.62                & ( ~ ( ord_less_eq_int @ X @ Xa )
% 0.29/0.62                 => ( Y = nil_int ) ) )
% 0.29/0.62             => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto.pelims
% 0.29/0.62  thf(fact_10058_upto__empty,axiom,
% 0.29/0.62      ! [J: int,I: int] :
% 0.29/0.62        ( ( ord_less_int @ J @ I )
% 0.29/0.62       => ( ( upto @ I @ J )
% 0.29/0.62          = nil_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_empty
% 0.29/0.62  thf(fact_10059_upto__Nil2,axiom,
% 0.29/0.62      ! [I: int,J: int] :
% 0.29/0.62        ( ( nil_int
% 0.29/0.62          = ( upto @ I @ J ) )
% 0.29/0.62        = ( ord_less_int @ J @ I ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_Nil2
% 0.29/0.62  thf(fact_10060_upto__Nil,axiom,
% 0.29/0.62      ! [I: int,J: int] :
% 0.29/0.62        ( ( ( upto @ I @ J )
% 0.29/0.62          = nil_int )
% 0.29/0.62        = ( ord_less_int @ J @ I ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_Nil
% 0.29/0.62  thf(fact_10061_upto__single,axiom,
% 0.29/0.62      ! [I: int] :
% 0.29/0.62        ( ( upto @ I @ I )
% 0.29/0.62        = ( cons_int @ I @ nil_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_single
% 0.29/0.62  thf(fact_10062_nth__upto,axiom,
% 0.29/0.62      ! [I: int,K: nat,J: int] :
% 0.29/0.62        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.29/0.62       => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 0.29/0.62          = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nth_upto
% 0.29/0.62  thf(fact_10063_length__upto,axiom,
% 0.29/0.62      ! [I: int,J: int] :
% 0.29/0.62        ( ( size_size_list_int @ ( upto @ I @ J ) )
% 0.29/0.62        = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % length_upto
% 0.29/0.62  thf(fact_10064_upto__rec__numeral_I1_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.29/0.62        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62            = nil_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_rec_numeral(1)
% 0.29/0.62  thf(fact_10065_upto__rec__numeral_I4_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62            = ( 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 ) ) ) ) ) )
% 0.29/0.62        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62            = nil_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_rec_numeral(4)
% 0.29/0.62  thf(fact_10066_upto__rec__numeral_I3_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62            = ( 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 ) ) ) ) )
% 0.29/0.62        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62            = nil_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_rec_numeral(3)
% 0.29/0.62  thf(fact_10067_upto__rec__numeral_I2_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.29/0.62        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62            = nil_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_rec_numeral(2)
% 0.29/0.62  thf(fact_10068_upto__aux__def,axiom,
% 0.29/0.62      ( upto_aux
% 0.29/0.62      = ( ^ [I3: int,J2: int] : ( append_int @ ( upto @ I3 @ J2 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_aux_def
% 0.29/0.62  thf(fact_10069_upto__code,axiom,
% 0.29/0.62      ( upto
% 0.29/0.62      = ( ^ [I3: int,J2: int] : ( upto_aux @ I3 @ J2 @ nil_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_code
% 0.29/0.62  thf(fact_10070_atLeastAtMost__upto,axiom,
% 0.29/0.62      ( set_or1266510415728281911st_int
% 0.29/0.62      = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ I3 @ J2 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % atLeastAtMost_upto
% 0.29/0.62  thf(fact_10071_distinct__upto,axiom,
% 0.29/0.62      ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).
% 0.29/0.62  
% 0.29/0.62  % distinct_upto
% 0.29/0.62  thf(fact_10072_upto__split2,axiom,
% 0.29/0.62      ! [I: int,J: int,K: int] :
% 0.29/0.62        ( ( ord_less_eq_int @ I @ J )
% 0.29/0.62       => ( ( ord_less_eq_int @ J @ K )
% 0.29/0.62         => ( ( upto @ I @ K )
% 0.29/0.62            = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_split2
% 0.29/0.62  thf(fact_10073_upto__split1,axiom,
% 0.29/0.62      ! [I: int,J: int,K: int] :
% 0.29/0.62        ( ( ord_less_eq_int @ I @ J )
% 0.29/0.62       => ( ( ord_less_eq_int @ J @ K )
% 0.29/0.62         => ( ( upto @ I @ K )
% 0.29/0.62            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_split1
% 0.29/0.62  thf(fact_10074_atLeastLessThan__upto,axiom,
% 0.29/0.62      ( set_or4662586982721622107an_int
% 0.29/0.62      = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % atLeastLessThan_upto
% 0.29/0.62  thf(fact_10075_greaterThanAtMost__upto,axiom,
% 0.29/0.62      ( set_or6656581121297822940st_int
% 0.29/0.62      = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % greaterThanAtMost_upto
% 0.29/0.62  thf(fact_10076_upto__rec1,axiom,
% 0.29/0.62      ! [I: int,J: int] :
% 0.29/0.62        ( ( ord_less_eq_int @ I @ J )
% 0.29/0.62       => ( ( upto @ I @ J )
% 0.29/0.62          = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_rec1
% 0.29/0.62  thf(fact_10077_upto_Oelims,axiom,
% 0.29/0.62      ! [X: int,Xa: int,Y: list_int] :
% 0.29/0.62        ( ( ( upto @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( ( ord_less_eq_int @ X @ Xa )
% 0.29/0.62           => ( Y
% 0.29/0.62              = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 0.29/0.62          & ( ~ ( ord_less_eq_int @ X @ Xa )
% 0.29/0.62           => ( Y = nil_int ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto.elims
% 0.29/0.62  thf(fact_10078_upto_Osimps,axiom,
% 0.29/0.62      ( upto
% 0.29/0.62      = ( ^ [I3: int,J2: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J2 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) @ nil_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto.simps
% 0.29/0.62  thf(fact_10079_upto__rec2,axiom,
% 0.29/0.62      ! [I: int,J: int] :
% 0.29/0.62        ( ( ord_less_eq_int @ I @ J )
% 0.29/0.62       => ( ( upto @ I @ J )
% 0.29/0.62          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_rec2
% 0.29/0.62  thf(fact_10080_greaterThanLessThan__upto,axiom,
% 0.29/0.62      ( set_or5832277885323065728an_int
% 0.29/0.62      = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % greaterThanLessThan_upto
% 0.29/0.62  thf(fact_10081_upto__split3,axiom,
% 0.29/0.62      ! [I: int,J: int,K: int] :
% 0.29/0.62        ( ( ord_less_eq_int @ I @ J )
% 0.29/0.62       => ( ( ord_less_eq_int @ J @ K )
% 0.29/0.62         => ( ( upto @ I @ K )
% 0.29/0.62            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % upto_split3
% 0.29/0.62  thf(fact_10082_sup__enat__def,axiom,
% 0.29/0.62      sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 0.29/0.62  
% 0.29/0.62  % sup_enat_def
% 0.29/0.62  thf(fact_10083_sup__nat__def,axiom,
% 0.29/0.62      sup_sup_nat = ord_max_nat ).
% 0.29/0.62  
% 0.29/0.62  % sup_nat_def
% 0.29/0.62  thf(fact_10084_sup__int__def,axiom,
% 0.29/0.62      sup_sup_int = ord_max_int ).
% 0.29/0.62  
% 0.29/0.62  % sup_int_def
% 0.29/0.62  thf(fact_10085_atLeastLessThan__add__Un,axiom,
% 0.29/0.62      ! [I: nat,J: nat,K: nat] :
% 0.29/0.62        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.62       => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 0.29/0.62          = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % atLeastLessThan_add_Un
% 0.29/0.62  thf(fact_10086_list__encode_Oelims,axiom,
% 0.29/0.62      ! [X: list_nat,Y: nat] :
% 0.29/0.62        ( ( ( nat_list_encode @ X )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( ( X = nil_nat )
% 0.29/0.62           => ( Y != zero_zero_nat ) )
% 0.29/0.62         => ~ ! [X3: nat,Xs2: list_nat] :
% 0.29/0.62                ( ( X
% 0.29/0.62                  = ( cons_nat @ X3 @ Xs2 ) )
% 0.29/0.62               => ( Y
% 0.29/0.62                 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % list_encode.elims
% 0.29/0.62  thf(fact_10087_list__encode_Osimps_I1_J,axiom,
% 0.29/0.62      ( ( nat_list_encode @ nil_nat )
% 0.29/0.62      = zero_zero_nat ) ).
% 0.29/0.62  
% 0.29/0.62  % list_encode.simps(1)
% 0.29/0.62  thf(fact_10088_list__encode_Osimps_I2_J,axiom,
% 0.29/0.62      ! [X: nat,Xs: list_nat] :
% 0.29/0.62        ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
% 0.29/0.62        = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % list_encode.simps(2)
% 0.29/0.62  thf(fact_10089_list__encode_Opelims,axiom,
% 0.29/0.62      ! [X: list_nat,Y: nat] :
% 0.29/0.62        ( ( ( nat_list_encode @ X )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( accp_list_nat @ nat_list_encode_rel @ X )
% 0.29/0.62         => ( ( ( X = nil_nat )
% 0.29/0.62             => ( ( Y = zero_zero_nat )
% 0.29/0.62               => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 0.29/0.62           => ~ ! [X3: nat,Xs2: list_nat] :
% 0.29/0.62                  ( ( X
% 0.29/0.62                    = ( cons_nat @ X3 @ Xs2 ) )
% 0.29/0.62                 => ( ( Y
% 0.29/0.62                      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) )
% 0.29/0.62                   => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % list_encode.pelims
% 0.29/0.62  thf(fact_10090_pred__nat__def,axiom,
% 0.29/0.62      ( pred_nat
% 0.29/0.62      = ( collec3392354462482085612at_nat
% 0.29/0.62        @ ( produc6081775807080527818_nat_o
% 0.29/0.62          @ ^ [M6: nat,N3: nat] :
% 0.29/0.62              ( N3
% 0.29/0.62              = ( suc @ M6 ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % pred_nat_def
% 0.29/0.62  thf(fact_10091_sorted__list__of__set__lessThan__Suc,axiom,
% 0.29/0.62      ! [K: nat] :
% 0.29/0.62        ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 0.29/0.62        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sorted_list_of_set_lessThan_Suc
% 0.29/0.62  thf(fact_10092_sorted__list__of__set__atMost__Suc,axiom,
% 0.29/0.62      ! [K: nat] :
% 0.29/0.62        ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 0.29/0.62        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sorted_list_of_set_atMost_Suc
% 0.29/0.62  thf(fact_10093_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.29/0.62      ! [I: nat,J: nat] :
% 0.29/0.62        ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 0.29/0.62       => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 0.29/0.62          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sorted_list_of_set_greaterThanAtMost
% 0.29/0.62  thf(fact_10094_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.29/0.62      ! [I: nat,J: nat] :
% 0.29/0.62        ( ( ord_less_nat @ ( suc @ I ) @ J )
% 0.29/0.62       => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 0.29/0.62          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sorted_list_of_set_greaterThanLessThan
% 0.29/0.62  thf(fact_10095_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.29/0.62      ! [N: nat,J: nat,I: nat] :
% 0.29/0.62        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 0.29/0.62       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 0.29/0.62          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nth_sorted_list_of_set_greaterThanAtMost
% 0.29/0.62  thf(fact_10096_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.29/0.62      ! [N: nat,J: nat,I: nat] :
% 0.29/0.62        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 0.29/0.62       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 0.29/0.62          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nth_sorted_list_of_set_greaterThanLessThan
% 0.29/0.62  thf(fact_10097_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.29/0.62      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.62        ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 0.29/0.62       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.29/0.62                ( X
% 0.29/0.62                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.29/0.62           => ( Xa = one_one_nat ) )
% 0.29/0.62         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.29/0.62                ( ( X
% 0.29/0.62                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.62               => ( ( Deg2 = Xa )
% 0.29/0.62                  & ! [X3: vEBT_VEBT] :
% 0.29/0.62                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                     => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.62                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                  & ( case_o184042715313410164at_nat
% 0.29/0.62                    @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.29/0.62                      & ! [X4: vEBT_VEBT] :
% 0.29/0.62                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                         => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                    @ ( produc6081775807080527818_nat_o
% 0.29/0.62                      @ ^ [Mi3: nat,Ma3: nat] :
% 0.29/0.62                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.62                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                          & ! [I3: nat] :
% 0.29/0.62                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                             => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 0.29/0.62                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.29/0.62                          & ( ( Mi3 = Ma3 )
% 0.29/0.62                           => ! [X4: vEBT_VEBT] :
% 0.29/0.62                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                          & ( ( Mi3 != Ma3 )
% 0.29/0.62                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.29/0.62                              & ! [X4: nat] :
% 0.29/0.62                                  ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 0.29/0.62                                   => ( ( ord_less_nat @ Mi3 @ X4 )
% 0.29/0.62                                      & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 0.29/0.62                    @ Mima ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % VEBT_internal.valid'.elims(3)
% 0.29/0.62  thf(fact_10098_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.29/0.62      ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 0.29/0.62        ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
% 0.29/0.62        = ( ( Deg = Deg4 )
% 0.29/0.62          & ! [X4: vEBT_VEBT] :
% 0.29/0.62              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.62             => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62          & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62          & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.29/0.62            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62          & ( case_o184042715313410164at_nat
% 0.29/0.62            @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
% 0.29/0.62              & ! [X4: vEBT_VEBT] :
% 0.29/0.62                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.62                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62            @ ( produc6081775807080527818_nat_o
% 0.29/0.62              @ ^ [Mi3: nat,Ma3: nat] :
% 0.29/0.62                  ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.62                  & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.29/0.62                  & ! [I3: nat] :
% 0.29/0.62                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
% 0.29/0.62                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.29/0.62                  & ( ( Mi3 = Ma3 )
% 0.29/0.62                   => ! [X4: vEBT_VEBT] :
% 0.29/0.62                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.62                       => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                  & ( ( Mi3 != Ma3 )
% 0.29/0.62                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 0.29/0.62                      & ! [X4: nat] :
% 0.29/0.62                          ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.29/0.62                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
% 0.29/0.62                           => ( ( ord_less_nat @ Mi3 @ X4 )
% 0.29/0.62                              & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 0.29/0.62            @ Mima2 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % VEBT_internal.valid'.simps(2)
% 0.29/0.62  thf(fact_10099_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.29/0.62      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.29/0.62        ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.29/0.62                ( X
% 0.29/0.62                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.29/0.62           => ( Y
% 0.29/0.62              = ( Xa != one_one_nat ) ) )
% 0.29/0.62         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.29/0.62                ( ( X
% 0.29/0.62                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.62               => ( Y
% 0.29/0.62                  = ( ~ ( ( Deg2 = Xa )
% 0.29/0.62                        & ! [X4: vEBT_VEBT] :
% 0.29/0.62                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                           => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.62                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                        & ( case_o184042715313410164at_nat
% 0.29/0.62                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.29/0.62                            & ! [X4: vEBT_VEBT] :
% 0.29/0.62                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                          @ ( produc6081775807080527818_nat_o
% 0.29/0.62                            @ ^ [Mi3: nat,Ma3: nat] :
% 0.29/0.62                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.62                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                                & ! [I3: nat] :
% 0.29/0.62                                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 0.29/0.62                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.29/0.62                                & ( ( Mi3 = Ma3 )
% 0.29/0.62                                 => ! [X4: vEBT_VEBT] :
% 0.29/0.62                                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                                & ( ( Mi3 != Ma3 )
% 0.29/0.62                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.29/0.62                                    & ! [X4: nat] :
% 0.29/0.62                                        ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 0.29/0.62                                         => ( ( ord_less_nat @ Mi3 @ X4 )
% 0.29/0.62                                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 0.29/0.62                          @ Mima ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % VEBT_internal.valid'.elims(1)
% 0.29/0.62  thf(fact_10100_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.29/0.62      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.62        ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.29/0.62       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.29/0.62                ( X
% 0.29/0.62                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.29/0.62           => ( Xa != one_one_nat ) )
% 0.29/0.62         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.29/0.62                ( ( X
% 0.29/0.62                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.62               => ~ ( ( Deg2 = Xa )
% 0.29/0.62                    & ! [X2: vEBT_VEBT] :
% 0.29/0.62                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                       => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.62                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                    & ( case_o184042715313410164at_nat
% 0.29/0.62                      @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.29/0.62                        & ! [X4: vEBT_VEBT] :
% 0.29/0.62                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                           => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                      @ ( produc6081775807080527818_nat_o
% 0.29/0.62                        @ ^ [Mi3: nat,Ma3: nat] :
% 0.29/0.62                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.62                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                            & ! [I3: nat] :
% 0.29/0.62                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                               => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 0.29/0.62                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.29/0.62                            & ( ( Mi3 = Ma3 )
% 0.29/0.62                             => ! [X4: vEBT_VEBT] :
% 0.29/0.62                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                            & ( ( Mi3 != Ma3 )
% 0.29/0.62                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.29/0.62                                & ! [X4: nat] :
% 0.29/0.62                                    ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 0.29/0.62                                     => ( ( ord_less_nat @ Mi3 @ X4 )
% 0.29/0.62                                        & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 0.29/0.62                      @ Mima ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % VEBT_internal.valid'.elims(2)
% 0.29/0.62  thf(fact_10101_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.29/0.62      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.29/0.62        ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.29/0.62         => ( ! [Uu2: $o,Uv2: $o] :
% 0.29/0.62                ( ( X
% 0.29/0.62                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.29/0.62               => ( ( Y
% 0.29/0.62                    = ( Xa = one_one_nat ) )
% 0.29/0.62                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 0.29/0.62           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.29/0.62                  ( ( X
% 0.29/0.62                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.62                 => ( ( Y
% 0.29/0.62                      = ( ( Deg2 = Xa )
% 0.29/0.62                        & ! [X4: vEBT_VEBT] :
% 0.29/0.62                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                           => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.62                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                        & ( case_o184042715313410164at_nat
% 0.29/0.62                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.29/0.62                            & ! [X4: vEBT_VEBT] :
% 0.29/0.62                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                          @ ( produc6081775807080527818_nat_o
% 0.29/0.62                            @ ^ [Mi3: nat,Ma3: nat] :
% 0.29/0.62                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.62                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                                & ! [I3: nat] :
% 0.29/0.62                                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 0.29/0.62                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.29/0.62                                & ( ( Mi3 = Ma3 )
% 0.29/0.62                                 => ! [X4: vEBT_VEBT] :
% 0.29/0.62                                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                                & ( ( Mi3 != Ma3 )
% 0.29/0.62                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.29/0.62                                    & ! [X4: nat] :
% 0.29/0.62                                        ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 0.29/0.62                                         => ( ( ord_less_nat @ Mi3 @ X4 )
% 0.29/0.62                                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 0.29/0.62                          @ Mima ) ) )
% 0.29/0.62                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % VEBT_internal.valid'.pelims(1)
% 0.29/0.62  thf(fact_10102_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.29/0.62      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.62        ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.29/0.62       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.29/0.62         => ( ! [Uu2: $o,Uv2: $o] :
% 0.29/0.62                ( ( X
% 0.29/0.62                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.29/0.62               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 0.29/0.62                 => ( Xa != one_one_nat ) ) )
% 0.29/0.62           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.29/0.62                  ( ( X
% 0.29/0.62                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.62                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 0.29/0.62                   => ~ ( ( Deg2 = Xa )
% 0.29/0.62                        & ! [X2: vEBT_VEBT] :
% 0.29/0.62                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                           => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.62                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                        & ( case_o184042715313410164at_nat
% 0.29/0.62                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.29/0.62                            & ! [X4: vEBT_VEBT] :
% 0.29/0.62                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                          @ ( produc6081775807080527818_nat_o
% 0.29/0.62                            @ ^ [Mi3: nat,Ma3: nat] :
% 0.29/0.62                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.62                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                                & ! [I3: nat] :
% 0.29/0.62                                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 0.29/0.62                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.29/0.62                                & ( ( Mi3 = Ma3 )
% 0.29/0.62                                 => ! [X4: vEBT_VEBT] :
% 0.29/0.62                                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                                & ( ( Mi3 != Ma3 )
% 0.29/0.62                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.29/0.62                                    & ! [X4: nat] :
% 0.29/0.62                                        ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 0.29/0.62                                         => ( ( ord_less_nat @ Mi3 @ X4 )
% 0.29/0.62                                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 0.29/0.62                          @ Mima ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % VEBT_internal.valid'.pelims(2)
% 0.29/0.62  thf(fact_10103_Sup__int__def,axiom,
% 0.29/0.62      ( complete_Sup_Sup_int
% 0.29/0.62      = ( ^ [X6: set_int] :
% 0.29/0.62            ( the_int
% 0.29/0.62            @ ^ [X4: int] :
% 0.29/0.62                ( ( member_int @ X4 @ X6 )
% 0.29/0.62                & ! [Y6: int] :
% 0.29/0.62                    ( ( member_int @ Y6 @ X6 )
% 0.29/0.62                   => ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Sup_int_def
% 0.29/0.62  thf(fact_10104_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.29/0.62      ! [X: vEBT_VEBT,Xa: nat] :
% 0.29/0.62        ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 0.29/0.62       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.29/0.62         => ( ! [Uu2: $o,Uv2: $o] :
% 0.29/0.62                ( ( X
% 0.29/0.62                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.29/0.62               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 0.29/0.62                 => ( Xa = one_one_nat ) ) )
% 0.29/0.62           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.29/0.62                  ( ( X
% 0.29/0.62                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.62                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 0.29/0.62                   => ( ( Deg2 = Xa )
% 0.29/0.62                      & ! [X3: vEBT_VEBT] :
% 0.29/0.62                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                         => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.62                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.62                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                      & ( case_o184042715313410164at_nat
% 0.29/0.62                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.29/0.62                          & ! [X4: vEBT_VEBT] :
% 0.29/0.62                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                        @ ( produc6081775807080527818_nat_o
% 0.29/0.62                          @ ^ [Mi3: nat,Ma3: nat] :
% 0.29/0.62                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.62                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                              & ! [I3: nat] :
% 0.29/0.62                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.29/0.62                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 0.29/0.62                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.29/0.62                              & ( ( Mi3 = Ma3 )
% 0.29/0.62                               => ! [X4: vEBT_VEBT] :
% 0.29/0.62                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.62                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 0.29/0.62                              & ( ( Mi3 != Ma3 )
% 0.29/0.62                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.29/0.62                                  & ! [X4: nat] :
% 0.29/0.62                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.62                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 0.29/0.62                                       => ( ( ord_less_nat @ Mi3 @ X4 )
% 0.29/0.62                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 0.29/0.62                        @ Mima ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % VEBT_internal.valid'.pelims(3)
% 0.29/0.62  thf(fact_10105_take__bit__numeral__minus__numeral__int,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62        = ( case_option_int_num @ zero_zero_int
% 0.29/0.62          @ ^ [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 ) ) )
% 0.29/0.62          @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % take_bit_numeral_minus_numeral_int
% 0.29/0.62  thf(fact_10106_take__bit__num__simps_I1_J,axiom,
% 0.29/0.62      ! [M: num] :
% 0.29/0.62        ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.29/0.62        = none_num ) ).
% 0.29/0.62  
% 0.29/0.62  % take_bit_num_simps(1)
% 0.29/0.62  thf(fact_10107_take__bit__num__simps_I2_J,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 0.29/0.62        = ( some_num @ one ) ) ).
% 0.29/0.62  
% 0.29/0.62  % take_bit_num_simps(2)
% 0.29/0.62  thf(fact_10108_take__bit__num__simps_I5_J,axiom,
% 0.29/0.62      ! [R2: num] :
% 0.29/0.62        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
% 0.29/0.62        = ( some_num @ one ) ) ).
% 0.29/0.62  
% 0.29/0.62  % take_bit_num_simps(5)
% 0.29/0.62  thf(fact_10109_take__bit__num__simps_I3_J,axiom,
% 0.29/0.62      ! [N: nat,M: num] :
% 0.29/0.62        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 0.29/0.62        = ( case_o6005452278849405969um_num @ none_num
% 0.29/0.62          @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.29/0.62          @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % take_bit_num_simps(3)
% 0.29/0.62  thf(fact_10110_take__bit__num__simps_I4_J,axiom,
% 0.29/0.62      ! [N: nat,M: num] :
% 0.29/0.62        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 0.29/0.62        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % take_bit_num_simps(4)
% 0.29/0.62  thf(fact_10111_take__bit__num__simps_I6_J,axiom,
% 0.29/0.62      ! [R2: num,M: num] :
% 0.29/0.62        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 0.29/0.62        = ( case_o6005452278849405969um_num @ none_num
% 0.29/0.62          @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.29/0.62          @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % take_bit_num_simps(6)
% 0.29/0.62  thf(fact_10112_take__bit__num__simps_I7_J,axiom,
% 0.29/0.62      ! [R2: num,M: num] :
% 0.29/0.62        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
% 0.29/0.62        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % take_bit_num_simps(7)
% 0.29/0.62  thf(fact_10113_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 0.29/0.62      ! [N: nat,M: num] :
% 0.29/0.62        ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 0.29/0.62        = ( case_nat_option_num @ none_num
% 0.29/0.62          @ ^ [N3: nat] :
% 0.29/0.62              ( case_o6005452278849405969um_num @ none_num
% 0.29/0.62              @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.29/0.62              @ ( bit_take_bit_num @ N3 @ M ) )
% 0.29/0.62          @ N ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Code_Abstract_Nat.take_bit_num_code(2)
% 0.29/0.62  thf(fact_10114_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 0.29/0.62      ! [N: nat,M: num] :
% 0.29/0.62        ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 0.29/0.62        = ( case_nat_option_num @ none_num
% 0.29/0.62          @ ^ [N3: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N3 @ M ) ) )
% 0.29/0.62          @ N ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Code_Abstract_Nat.take_bit_num_code(3)
% 0.29/0.62  thf(fact_10115_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 0.29/0.62      ! [N: nat] :
% 0.29/0.62        ( ( bit_take_bit_num @ N @ one )
% 0.29/0.62        = ( case_nat_option_num @ none_num
% 0.29/0.62          @ ^ [N3: nat] : ( some_num @ one )
% 0.29/0.62          @ N ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Code_Abstract_Nat.take_bit_num_code(1)
% 0.29/0.62  thf(fact_10116_take__bit__num__def,axiom,
% 0.29/0.62      ( bit_take_bit_num
% 0.29/0.62      = ( ^ [N3: nat,M6: num] :
% 0.29/0.62            ( if_option_num
% 0.29/0.62            @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M6 ) )
% 0.29/0.62              = zero_zero_nat )
% 0.29/0.62            @ none_num
% 0.29/0.62            @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % take_bit_num_def
% 0.29/0.62  thf(fact_10117_and__minus__numerals_I7_J,axiom,
% 0.29/0.62      ! [N: num,M: num] :
% 0.29/0.62        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.29/0.62        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_minus_numerals(7)
% 0.29/0.62  thf(fact_10118_and__minus__numerals_I3_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.29/0.62        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_minus_numerals(3)
% 0.29/0.62  thf(fact_10119_and__minus__numerals_I8_J,axiom,
% 0.29/0.62      ! [N: num,M: num] :
% 0.29/0.62        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.29/0.62        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_minus_numerals(8)
% 0.29/0.62  thf(fact_10120_and__minus__numerals_I4_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.29/0.62        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_minus_numerals(4)
% 0.29/0.62  thf(fact_10121_and__not__num_Osimps_I2_J,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 0.29/0.62        = ( some_num @ one ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.simps(2)
% 0.29/0.62  thf(fact_10122_and__not__num_Osimps_I4_J,axiom,
% 0.29/0.62      ! [M: num] :
% 0.29/0.62        ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 0.29/0.62        = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.simps(4)
% 0.29/0.62  thf(fact_10123_and__not__num_Osimps_I3_J,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 0.29/0.62        = none_num ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.simps(3)
% 0.29/0.62  thf(fact_10124_and__not__num_Osimps_I1_J,axiom,
% 0.29/0.62      ( ( bit_and_not_num @ one @ one )
% 0.29/0.62      = none_num ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.simps(1)
% 0.29/0.62  thf(fact_10125_and__not__num_Osimps_I7_J,axiom,
% 0.29/0.62      ! [M: num] :
% 0.29/0.62        ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 0.29/0.62        = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.simps(7)
% 0.29/0.62  thf(fact_10126_and__not__num__eq__Some__iff,axiom,
% 0.29/0.62      ! [M: num,N: num,Q3: num] :
% 0.29/0.62        ( ( ( bit_and_not_num @ M @ N )
% 0.29/0.62          = ( some_num @ Q3 ) )
% 0.29/0.62        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62          = ( numeral_numeral_int @ Q3 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num_eq_Some_iff
% 0.29/0.62  thf(fact_10127_and__not__num_Osimps_I8_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.29/0.62        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.29/0.62          @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.29/0.62          @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.simps(8)
% 0.29/0.62  thf(fact_10128_and__not__num__eq__None__iff,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( ( bit_and_not_num @ M @ N )
% 0.29/0.62          = none_num )
% 0.29/0.62        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62          = zero_zero_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num_eq_None_iff
% 0.29/0.62  thf(fact_10129_int__numeral__and__not__num,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.29/0.62        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % int_numeral_and_not_num
% 0.29/0.62  thf(fact_10130_int__numeral__not__and__num,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.29/0.62        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % int_numeral_not_and_num
% 0.29/0.62  thf(fact_10131_Rats__eq__int__div__nat,axiom,
% 0.29/0.62      ( field_5140801741446780682s_real
% 0.29/0.62      = ( collect_real
% 0.29/0.62        @ ^ [Uu3: real] :
% 0.29/0.62          ? [I3: int,N3: nat] :
% 0.29/0.62            ( ( Uu3
% 0.29/0.62              = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 0.29/0.62            & ( N3 != zero_zero_nat ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Rats_eq_int_div_nat
% 0.29/0.62  thf(fact_10132_Rats__abs__iff,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62        ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
% 0.29/0.62        = ( member_real @ X @ field_5140801741446780682s_real ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Rats_abs_iff
% 0.29/0.62  thf(fact_10133_Rats__no__top__le,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62      ? [X3: real] :
% 0.29/0.62        ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 0.29/0.62        & ( ord_less_eq_real @ X @ X3 ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Rats_no_top_le
% 0.29/0.62  thf(fact_10134_Rats__dense__in__real,axiom,
% 0.29/0.62      ! [X: real,Y: real] :
% 0.29/0.62        ( ( ord_less_real @ X @ Y )
% 0.29/0.62       => ? [X3: real] :
% 0.29/0.62            ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 0.29/0.62            & ( ord_less_real @ X @ X3 )
% 0.29/0.62            & ( ord_less_real @ X3 @ Y ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Rats_dense_in_real
% 0.29/0.62  thf(fact_10135_Rats__no__bot__less,axiom,
% 0.29/0.62      ! [X: real] :
% 0.29/0.62      ? [X3: real] :
% 0.29/0.62        ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 0.29/0.62        & ( ord_less_real @ X3 @ X ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Rats_no_bot_less
% 0.29/0.62  thf(fact_10136_Rats__eq__int__div__int,axiom,
% 0.29/0.62      ( field_5140801741446780682s_real
% 0.29/0.62      = ( collect_real
% 0.29/0.62        @ ^ [Uu3: real] :
% 0.29/0.62          ? [I3: int,J2: int] :
% 0.29/0.62            ( ( Uu3
% 0.29/0.62              = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( ring_1_of_int_real @ J2 ) ) )
% 0.29/0.62            & ( J2 != zero_zero_int ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Rats_eq_int_div_int
% 0.29/0.62  thf(fact_10137_and__not__num_Oelims,axiom,
% 0.29/0.62      ! [X: num,Xa: num,Y: option_num] :
% 0.29/0.62        ( ( ( bit_and_not_num @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( ( X = one )
% 0.29/0.62           => ( ( Xa = one )
% 0.29/0.62             => ( Y != none_num ) ) )
% 0.29/0.62         => ( ( ( X = one )
% 0.29/0.62             => ( ? [N2: num] :
% 0.29/0.62                    ( Xa
% 0.29/0.62                    = ( bit0 @ N2 ) )
% 0.29/0.62               => ( Y
% 0.29/0.62                 != ( some_num @ one ) ) ) )
% 0.29/0.62           => ( ( ( X = one )
% 0.29/0.62               => ( ? [N2: num] :
% 0.29/0.62                      ( Xa
% 0.29/0.62                      = ( bit1 @ N2 ) )
% 0.29/0.62                 => ( Y != none_num ) ) )
% 0.29/0.62             => ( ! [M5: num] :
% 0.29/0.62                    ( ( X
% 0.29/0.62                      = ( bit0 @ M5 ) )
% 0.29/0.62                   => ( ( Xa = one )
% 0.29/0.62                     => ( Y
% 0.29/0.62                       != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 0.29/0.62               => ( ! [M5: num] :
% 0.29/0.62                      ( ( X
% 0.29/0.62                        = ( bit0 @ M5 ) )
% 0.29/0.62                     => ! [N2: num] :
% 0.29/0.62                          ( ( Xa
% 0.29/0.62                            = ( bit0 @ N2 ) )
% 0.29/0.62                         => ( Y
% 0.29/0.62                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) ) ) )
% 0.29/0.62                 => ( ! [M5: num] :
% 0.29/0.62                        ( ( X
% 0.29/0.62                          = ( bit0 @ M5 ) )
% 0.29/0.62                       => ! [N2: num] :
% 0.29/0.62                            ( ( Xa
% 0.29/0.62                              = ( bit1 @ N2 ) )
% 0.29/0.62                           => ( Y
% 0.29/0.62                             != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) ) ) )
% 0.29/0.62                   => ( ! [M5: num] :
% 0.29/0.62                          ( ( X
% 0.29/0.62                            = ( bit1 @ M5 ) )
% 0.29/0.62                         => ( ( Xa = one )
% 0.29/0.62                           => ( Y
% 0.29/0.62                             != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 0.29/0.62                     => ( ! [M5: num] :
% 0.29/0.62                            ( ( X
% 0.29/0.62                              = ( bit1 @ M5 ) )
% 0.29/0.62                           => ! [N2: num] :
% 0.29/0.62                                ( ( Xa
% 0.29/0.62                                  = ( bit0 @ N2 ) )
% 0.29/0.62                               => ( Y
% 0.29/0.62                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.29/0.62                                    @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.29/0.62                                    @ ( bit_and_not_num @ M5 @ N2 ) ) ) ) )
% 0.29/0.62                       => ~ ! [M5: num] :
% 0.29/0.62                              ( ( X
% 0.29/0.62                                = ( bit1 @ M5 ) )
% 0.29/0.62                             => ! [N2: num] :
% 0.29/0.62                                  ( ( Xa
% 0.29/0.62                                    = ( bit1 @ N2 ) )
% 0.29/0.62                                 => ( Y
% 0.29/0.62                                   != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.elims
% 0.29/0.62  thf(fact_10138_and__not__num_Osimps_I5_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.29/0.62        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.simps(5)
% 0.29/0.62  thf(fact_10139_and__not__num_Osimps_I6_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.62        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.simps(6)
% 0.29/0.62  thf(fact_10140_and__not__num_Osimps_I9_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.62        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.simps(9)
% 0.29/0.62  thf(fact_10141_and__not__num_Opelims,axiom,
% 0.29/0.62      ! [X: num,Xa: num,Y: option_num] :
% 0.29/0.62        ( ( ( bit_and_not_num @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.29/0.62         => ( ( ( X = one )
% 0.29/0.62             => ( ( Xa = one )
% 0.29/0.62               => ( ( Y = none_num )
% 0.29/0.62                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.29/0.62           => ( ( ( X = one )
% 0.29/0.62               => ! [N2: num] :
% 0.29/0.62                    ( ( Xa
% 0.29/0.62                      = ( bit0 @ N2 ) )
% 0.29/0.62                   => ( ( Y
% 0.29/0.62                        = ( some_num @ one ) )
% 0.29/0.62                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.29/0.62             => ( ( ( X = one )
% 0.29/0.62                 => ! [N2: num] :
% 0.29/0.62                      ( ( Xa
% 0.29/0.62                        = ( bit1 @ N2 ) )
% 0.29/0.62                     => ( ( Y = none_num )
% 0.29/0.62                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.29/0.62               => ( ! [M5: num] :
% 0.29/0.62                      ( ( X
% 0.29/0.62                        = ( bit0 @ M5 ) )
% 0.29/0.62                     => ( ( Xa = one )
% 0.29/0.62                       => ( ( Y
% 0.29/0.62                            = ( some_num @ ( bit0 @ M5 ) ) )
% 0.29/0.62                         => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 0.29/0.62                 => ( ! [M5: num] :
% 0.29/0.62                        ( ( X
% 0.29/0.62                          = ( bit0 @ M5 ) )
% 0.29/0.62                       => ! [N2: num] :
% 0.29/0.62                            ( ( Xa
% 0.29/0.62                              = ( bit0 @ N2 ) )
% 0.29/0.62                           => ( ( Y
% 0.29/0.62                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) )
% 0.29/0.62                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.29/0.62                   => ( ! [M5: num] :
% 0.29/0.62                          ( ( X
% 0.29/0.62                            = ( bit0 @ M5 ) )
% 0.29/0.62                         => ! [N2: num] :
% 0.29/0.62                              ( ( Xa
% 0.29/0.62                                = ( bit1 @ N2 ) )
% 0.29/0.62                             => ( ( Y
% 0.29/0.62                                  = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) )
% 0.29/0.62                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.29/0.62                     => ( ! [M5: num] :
% 0.29/0.62                            ( ( X
% 0.29/0.62                              = ( bit1 @ M5 ) )
% 0.29/0.62                           => ( ( Xa = one )
% 0.29/0.62                             => ( ( Y
% 0.29/0.62                                  = ( some_num @ ( bit0 @ M5 ) ) )
% 0.29/0.62                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 0.29/0.62                       => ( ! [M5: num] :
% 0.29/0.62                              ( ( X
% 0.29/0.62                                = ( bit1 @ M5 ) )
% 0.29/0.62                             => ! [N2: num] :
% 0.29/0.62                                  ( ( Xa
% 0.29/0.62                                    = ( bit0 @ N2 ) )
% 0.29/0.62                                 => ( ( Y
% 0.29/0.62                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.29/0.62                                        @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.29/0.62                                        @ ( bit_and_not_num @ M5 @ N2 ) ) )
% 0.29/0.62                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.29/0.62                         => ~ ! [M5: num] :
% 0.29/0.62                                ( ( X
% 0.29/0.62                                  = ( bit1 @ M5 ) )
% 0.29/0.62                               => ! [N2: num] :
% 0.29/0.62                                    ( ( Xa
% 0.29/0.62                                      = ( bit1 @ N2 ) )
% 0.29/0.62                                   => ( ( Y
% 0.29/0.62                                        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) )
% 0.29/0.62                                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_not_num.pelims
% 0.29/0.62  thf(fact_10142_and__num_Oelims,axiom,
% 0.29/0.62      ! [X: num,Xa: num,Y: option_num] :
% 0.29/0.62        ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( ( X = one )
% 0.29/0.62           => ( ( Xa = one )
% 0.29/0.62             => ( Y
% 0.29/0.62               != ( some_num @ one ) ) ) )
% 0.29/0.62         => ( ( ( X = one )
% 0.29/0.62             => ( ? [N2: num] :
% 0.29/0.62                    ( Xa
% 0.29/0.62                    = ( bit0 @ N2 ) )
% 0.29/0.62               => ( Y != none_num ) ) )
% 0.29/0.62           => ( ( ( X = one )
% 0.29/0.62               => ( ? [N2: num] :
% 0.29/0.62                      ( Xa
% 0.29/0.62                      = ( bit1 @ N2 ) )
% 0.29/0.62                 => ( Y
% 0.29/0.62                   != ( some_num @ one ) ) ) )
% 0.29/0.62             => ( ( ? [M5: num] :
% 0.29/0.62                      ( X
% 0.29/0.62                      = ( bit0 @ M5 ) )
% 0.29/0.62                 => ( ( Xa = one )
% 0.29/0.62                   => ( Y != none_num ) ) )
% 0.29/0.62               => ( ! [M5: num] :
% 0.29/0.62                      ( ( X
% 0.29/0.62                        = ( bit0 @ M5 ) )
% 0.29/0.62                     => ! [N2: num] :
% 0.29/0.62                          ( ( Xa
% 0.29/0.62                            = ( bit0 @ N2 ) )
% 0.29/0.62                         => ( Y
% 0.29/0.62                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) ) ) )
% 0.29/0.62                 => ( ! [M5: num] :
% 0.29/0.62                        ( ( X
% 0.29/0.62                          = ( bit0 @ M5 ) )
% 0.29/0.62                       => ! [N2: num] :
% 0.29/0.62                            ( ( Xa
% 0.29/0.62                              = ( bit1 @ N2 ) )
% 0.29/0.62                           => ( Y
% 0.29/0.62                             != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) ) ) )
% 0.29/0.62                   => ( ( ? [M5: num] :
% 0.29/0.62                            ( X
% 0.29/0.62                            = ( bit1 @ M5 ) )
% 0.29/0.62                       => ( ( Xa = one )
% 0.29/0.62                         => ( Y
% 0.29/0.62                           != ( some_num @ one ) ) ) )
% 0.29/0.62                     => ( ! [M5: num] :
% 0.29/0.62                            ( ( X
% 0.29/0.62                              = ( bit1 @ M5 ) )
% 0.29/0.62                           => ! [N2: num] :
% 0.29/0.62                                ( ( Xa
% 0.29/0.62                                  = ( bit0 @ N2 ) )
% 0.29/0.62                               => ( Y
% 0.29/0.62                                 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) ) ) )
% 0.29/0.62                       => ~ ! [M5: num] :
% 0.29/0.62                              ( ( X
% 0.29/0.62                                = ( bit1 @ M5 ) )
% 0.29/0.62                             => ! [N2: num] :
% 0.29/0.62                                  ( ( Xa
% 0.29/0.62                                    = ( bit1 @ N2 ) )
% 0.29/0.62                                 => ( Y
% 0.29/0.62                                   != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.29/0.62                                      @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.29/0.62                                      @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.elims
% 0.29/0.62  thf(fact_10143_xor__num_Oelims,axiom,
% 0.29/0.62      ! [X: num,Xa: num,Y: option_num] :
% 0.29/0.62        ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( ( X = one )
% 0.29/0.62           => ( ( Xa = one )
% 0.29/0.62             => ( Y != none_num ) ) )
% 0.29/0.62         => ( ( ( X = one )
% 0.29/0.62             => ! [N2: num] :
% 0.29/0.62                  ( ( Xa
% 0.29/0.62                    = ( bit0 @ N2 ) )
% 0.29/0.62                 => ( Y
% 0.29/0.62                   != ( some_num @ ( bit1 @ N2 ) ) ) ) )
% 0.29/0.62           => ( ( ( X = one )
% 0.29/0.62               => ! [N2: num] :
% 0.29/0.62                    ( ( Xa
% 0.29/0.62                      = ( bit1 @ N2 ) )
% 0.29/0.62                   => ( Y
% 0.29/0.62                     != ( some_num @ ( bit0 @ N2 ) ) ) ) )
% 0.29/0.62             => ( ! [M5: num] :
% 0.29/0.62                    ( ( X
% 0.29/0.62                      = ( bit0 @ M5 ) )
% 0.29/0.62                   => ( ( Xa = one )
% 0.29/0.62                     => ( Y
% 0.29/0.62                       != ( some_num @ ( bit1 @ M5 ) ) ) ) )
% 0.29/0.62               => ( ! [M5: num] :
% 0.29/0.62                      ( ( X
% 0.29/0.62                        = ( bit0 @ M5 ) )
% 0.29/0.62                     => ! [N2: num] :
% 0.29/0.62                          ( ( Xa
% 0.29/0.62                            = ( bit0 @ N2 ) )
% 0.29/0.62                         => ( Y
% 0.29/0.62                           != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) ) )
% 0.29/0.62                 => ( ! [M5: num] :
% 0.29/0.62                        ( ( X
% 0.29/0.62                          = ( bit0 @ M5 ) )
% 0.29/0.62                       => ! [N2: num] :
% 0.29/0.62                            ( ( Xa
% 0.29/0.62                              = ( bit1 @ N2 ) )
% 0.29/0.62                           => ( Y
% 0.29/0.62                             != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) ) ) )
% 0.29/0.62                   => ( ! [M5: num] :
% 0.29/0.62                          ( ( X
% 0.29/0.62                            = ( bit1 @ M5 ) )
% 0.29/0.62                         => ( ( Xa = one )
% 0.29/0.62                           => ( Y
% 0.29/0.62                             != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 0.29/0.62                     => ( ! [M5: num] :
% 0.29/0.62                            ( ( X
% 0.29/0.62                              = ( bit1 @ M5 ) )
% 0.29/0.62                           => ! [N2: num] :
% 0.29/0.62                                ( ( Xa
% 0.29/0.62                                  = ( bit0 @ N2 ) )
% 0.29/0.62                               => ( Y
% 0.29/0.62                                 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) ) ) )
% 0.29/0.62                       => ~ ! [M5: num] :
% 0.29/0.62                              ( ( X
% 0.29/0.62                                = ( bit1 @ M5 ) )
% 0.29/0.62                             => ! [N2: num] :
% 0.29/0.62                                  ( ( Xa
% 0.29/0.62                                    = ( bit1 @ N2 ) )
% 0.29/0.62                                 => ( Y
% 0.29/0.62                                   != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.elims
% 0.29/0.62  thf(fact_10144_xor__num_Osimps_I5_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.29/0.62        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.simps(5)
% 0.29/0.62  thf(fact_10145_and__num_Osimps_I5_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.29/0.62        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.simps(5)
% 0.29/0.62  thf(fact_10146_xor__num_Osimps_I1_J,axiom,
% 0.29/0.62      ( ( bit_un2480387367778600638or_num @ one @ one )
% 0.29/0.62      = none_num ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.simps(1)
% 0.29/0.62  thf(fact_10147_and__num_Osimps_I1_J,axiom,
% 0.29/0.62      ( ( bit_un7362597486090784418nd_num @ one @ one )
% 0.29/0.62      = ( some_num @ one ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.simps(1)
% 0.29/0.62  thf(fact_10148_and__num_Osimps_I7_J,axiom,
% 0.29/0.62      ! [M: num] :
% 0.29/0.62        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 0.29/0.62        = ( some_num @ one ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.simps(7)
% 0.29/0.62  thf(fact_10149_and__num_Osimps_I3_J,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 0.29/0.62        = ( some_num @ one ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.simps(3)
% 0.29/0.62  thf(fact_10150_and__num_Osimps_I2_J,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 0.29/0.62        = none_num ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.simps(2)
% 0.29/0.62  thf(fact_10151_and__num_Osimps_I4_J,axiom,
% 0.29/0.62      ! [M: num] :
% 0.29/0.62        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 0.29/0.62        = none_num ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.simps(4)
% 0.29/0.62  thf(fact_10152_and__num_Osimps_I8_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.29/0.62        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.simps(8)
% 0.29/0.62  thf(fact_10153_and__num_Osimps_I6_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.62        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.simps(6)
% 0.29/0.62  thf(fact_10154_xor__num_Osimps_I9_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.62        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.simps(9)
% 0.29/0.62  thf(fact_10155_xor__num_Osimps_I7_J,axiom,
% 0.29/0.62      ! [M: num] :
% 0.29/0.62        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 0.29/0.62        = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.simps(7)
% 0.29/0.62  thf(fact_10156_xor__num_Osimps_I4_J,axiom,
% 0.29/0.62      ! [M: num] :
% 0.29/0.62        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 0.29/0.62        = ( some_num @ ( bit1 @ M ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.simps(4)
% 0.29/0.62  thf(fact_10157_xor__num_Osimps_I3_J,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 0.29/0.62        = ( some_num @ ( bit0 @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.simps(3)
% 0.29/0.62  thf(fact_10158_xor__num_Osimps_I2_J,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 0.29/0.62        = ( some_num @ ( bit1 @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.simps(2)
% 0.29/0.62  thf(fact_10159_and__num_Osimps_I9_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.29/0.62        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.29/0.62          @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.29/0.62          @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.simps(9)
% 0.29/0.62  thf(fact_10160_xor__num_Osimps_I8_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.29/0.62        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.simps(8)
% 0.29/0.62  thf(fact_10161_xor__num_Osimps_I6_J,axiom,
% 0.29/0.62      ! [M: num,N: num] :
% 0.29/0.62        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.29/0.62        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.simps(6)
% 0.29/0.62  thf(fact_10162_and__num_Opelims,axiom,
% 0.29/0.62      ! [X: num,Xa: num,Y: option_num] :
% 0.29/0.62        ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.29/0.62         => ( ( ( X = one )
% 0.29/0.62             => ( ( Xa = one )
% 0.29/0.62               => ( ( Y
% 0.29/0.62                    = ( some_num @ one ) )
% 0.29/0.62                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.29/0.62           => ( ( ( X = one )
% 0.29/0.62               => ! [N2: num] :
% 0.29/0.62                    ( ( Xa
% 0.29/0.62                      = ( bit0 @ N2 ) )
% 0.29/0.62                   => ( ( Y = none_num )
% 0.29/0.62                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.29/0.62             => ( ( ( X = one )
% 0.29/0.62                 => ! [N2: num] :
% 0.29/0.62                      ( ( Xa
% 0.29/0.62                        = ( bit1 @ N2 ) )
% 0.29/0.62                     => ( ( Y
% 0.29/0.62                          = ( some_num @ one ) )
% 0.29/0.62                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.29/0.62               => ( ! [M5: num] :
% 0.29/0.62                      ( ( X
% 0.29/0.62                        = ( bit0 @ M5 ) )
% 0.29/0.62                     => ( ( Xa = one )
% 0.29/0.62                       => ( ( Y = none_num )
% 0.29/0.62                         => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 0.29/0.62                 => ( ! [M5: num] :
% 0.29/0.62                        ( ( X
% 0.29/0.62                          = ( bit0 @ M5 ) )
% 0.29/0.62                       => ! [N2: num] :
% 0.29/0.62                            ( ( Xa
% 0.29/0.62                              = ( bit0 @ N2 ) )
% 0.29/0.62                           => ( ( Y
% 0.29/0.62                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) )
% 0.29/0.62                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.29/0.62                   => ( ! [M5: num] :
% 0.29/0.62                          ( ( X
% 0.29/0.62                            = ( bit0 @ M5 ) )
% 0.29/0.62                         => ! [N2: num] :
% 0.29/0.62                              ( ( Xa
% 0.29/0.62                                = ( bit1 @ N2 ) )
% 0.29/0.62                             => ( ( Y
% 0.29/0.62                                  = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) )
% 0.29/0.62                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.29/0.62                     => ( ! [M5: num] :
% 0.29/0.62                            ( ( X
% 0.29/0.62                              = ( bit1 @ M5 ) )
% 0.29/0.62                           => ( ( Xa = one )
% 0.29/0.62                             => ( ( Y
% 0.29/0.62                                  = ( some_num @ one ) )
% 0.29/0.62                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 0.29/0.62                       => ( ! [M5: num] :
% 0.29/0.62                              ( ( X
% 0.29/0.62                                = ( bit1 @ M5 ) )
% 0.29/0.62                             => ! [N2: num] :
% 0.29/0.62                                  ( ( Xa
% 0.29/0.62                                    = ( bit0 @ N2 ) )
% 0.29/0.62                                 => ( ( Y
% 0.29/0.62                                      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) )
% 0.29/0.62                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.29/0.62                         => ~ ! [M5: num] :
% 0.29/0.62                                ( ( X
% 0.29/0.62                                  = ( bit1 @ M5 ) )
% 0.29/0.62                               => ! [N2: num] :
% 0.29/0.62                                    ( ( Xa
% 0.29/0.62                                      = ( bit1 @ N2 ) )
% 0.29/0.62                                   => ( ( Y
% 0.29/0.62                                        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.29/0.62                                          @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.29/0.62                                          @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) )
% 0.29/0.62                                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % and_num.pelims
% 0.29/0.62  thf(fact_10163_xor__num_Opelims,axiom,
% 0.29/0.62      ! [X: num,Xa: num,Y: option_num] :
% 0.29/0.62        ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 0.29/0.62          = Y )
% 0.29/0.62       => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.29/0.62         => ( ( ( X = one )
% 0.29/0.62             => ( ( Xa = one )
% 0.29/0.62               => ( ( Y = none_num )
% 0.29/0.62                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.29/0.62           => ( ( ( X = one )
% 0.29/0.62               => ! [N2: num] :
% 0.29/0.62                    ( ( Xa
% 0.29/0.62                      = ( bit0 @ N2 ) )
% 0.29/0.62                   => ( ( Y
% 0.29/0.62                        = ( some_num @ ( bit1 @ N2 ) ) )
% 0.29/0.62                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.29/0.62             => ( ( ( X = one )
% 0.29/0.62                 => ! [N2: num] :
% 0.29/0.62                      ( ( Xa
% 0.29/0.62                        = ( bit1 @ N2 ) )
% 0.29/0.62                     => ( ( Y
% 0.29/0.62                          = ( some_num @ ( bit0 @ N2 ) ) )
% 0.29/0.62                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.29/0.62               => ( ! [M5: num] :
% 0.29/0.62                      ( ( X
% 0.29/0.62                        = ( bit0 @ M5 ) )
% 0.29/0.62                     => ( ( Xa = one )
% 0.29/0.62                       => ( ( Y
% 0.29/0.62                            = ( some_num @ ( bit1 @ M5 ) ) )
% 0.29/0.62                         => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 0.29/0.62                 => ( ! [M5: num] :
% 0.29/0.62                        ( ( X
% 0.29/0.62                          = ( bit0 @ M5 ) )
% 0.29/0.62                       => ! [N2: num] :
% 0.29/0.62                            ( ( Xa
% 0.29/0.62                              = ( bit0 @ N2 ) )
% 0.29/0.62                           => ( ( Y
% 0.29/0.62                                = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) )
% 0.29/0.62                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.29/0.62                   => ( ! [M5: num] :
% 0.29/0.62                          ( ( X
% 0.29/0.62                            = ( bit0 @ M5 ) )
% 0.29/0.62                         => ! [N2: num] :
% 0.29/0.62                              ( ( Xa
% 0.29/0.62                                = ( bit1 @ N2 ) )
% 0.29/0.62                             => ( ( Y
% 0.29/0.62                                  = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) )
% 0.29/0.62                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.29/0.62                     => ( ! [M5: num] :
% 0.29/0.62                            ( ( X
% 0.29/0.62                              = ( bit1 @ M5 ) )
% 0.29/0.62                           => ( ( Xa = one )
% 0.29/0.62                             => ( ( Y
% 0.29/0.62                                  = ( some_num @ ( bit0 @ M5 ) ) )
% 0.29/0.62                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 0.29/0.62                       => ( ! [M5: num] :
% 0.29/0.62                              ( ( X
% 0.29/0.62                                = ( bit1 @ M5 ) )
% 0.29/0.62                             => ! [N2: num] :
% 0.29/0.62                                  ( ( Xa
% 0.29/0.62                                    = ( bit0 @ N2 ) )
% 0.29/0.62                                 => ( ( Y
% 0.29/0.62                                      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) )
% 0.29/0.62                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.29/0.62                         => ~ ! [M5: num] :
% 0.29/0.62                                ( ( X
% 0.29/0.62                                  = ( bit1 @ M5 ) )
% 0.29/0.62                               => ! [N2: num] :
% 0.29/0.62                                    ( ( Xa
% 0.29/0.62                                      = ( bit1 @ N2 ) )
% 0.29/0.62                                   => ( ( Y
% 0.29/0.62                                        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) )
% 0.29/0.62                                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % xor_num.pelims
% 0.29/0.62  thf(fact_10164_Bit__Operations_Otake__bit__num__code,axiom,
% 0.29/0.62      ( bit_take_bit_num
% 0.29/0.62      = ( ^ [N3: nat,M6: num] :
% 0.29/0.62            ( produc478579273971653890on_num
% 0.29/0.62            @ ^ [A3: nat,X4: num] :
% 0.29/0.62                ( case_nat_option_num @ none_num
% 0.29/0.62                @ ^ [O: nat] :
% 0.29/0.62                    ( case_num_option_num @ ( some_num @ one )
% 0.29/0.62                    @ ^ [P4: num] :
% 0.29/0.62                        ( case_o6005452278849405969um_num @ none_num
% 0.29/0.62                        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.29/0.62                        @ ( bit_take_bit_num @ O @ P4 ) )
% 0.29/0.62                    @ ^ [P4: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P4 ) ) )
% 0.29/0.62                    @ X4 )
% 0.29/0.62                @ A3 )
% 0.29/0.62            @ ( product_Pair_nat_num @ N3 @ M6 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Bit_Operations.take_bit_num_code
% 0.29/0.62  thf(fact_10165_less__eq,axiom,
% 0.29/0.62      ! [M: nat,N: nat] :
% 0.29/0.62        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 0.29/0.62        = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.62  
% 0.29/0.62  % less_eq
% 0.29/0.62  thf(fact_10166_pow_Osimps_I3_J,axiom,
% 0.29/0.62      ! [X: num,Y: num] :
% 0.29/0.62        ( ( pow @ X @ ( bit1 @ Y ) )
% 0.29/0.62        = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% 0.29/0.62  
% 0.29/0.62  % pow.simps(3)
% 0.29/0.62  thf(fact_10167_sqr__conv__mult,axiom,
% 0.29/0.62      ( sqr
% 0.29/0.62      = ( ^ [X4: num] : ( times_times_num @ X4 @ X4 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sqr_conv_mult
% 0.29/0.62  thf(fact_10168_sqr_Osimps_I2_J,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( sqr @ ( bit0 @ N ) )
% 0.29/0.62        = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sqr.simps(2)
% 0.29/0.62  thf(fact_10169_sqr_Osimps_I1_J,axiom,
% 0.29/0.62      ( ( sqr @ one )
% 0.29/0.62      = one ) ).
% 0.29/0.62  
% 0.29/0.62  % sqr.simps(1)
% 0.29/0.62  thf(fact_10170_pred__nat__trancl__eq__le,axiom,
% 0.29/0.62      ! [M: nat,N: nat] :
% 0.29/0.62        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 0.29/0.62        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.62  
% 0.29/0.62  % pred_nat_trancl_eq_le
% 0.29/0.62  thf(fact_10171_pow_Osimps_I2_J,axiom,
% 0.29/0.62      ! [X: num,Y: num] :
% 0.29/0.62        ( ( pow @ X @ ( bit0 @ Y ) )
% 0.29/0.62        = ( sqr @ ( pow @ X @ Y ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % pow.simps(2)
% 0.29/0.62  thf(fact_10172_sqr_Osimps_I3_J,axiom,
% 0.29/0.62      ! [N: num] :
% 0.29/0.62        ( ( sqr @ ( bit1 @ N ) )
% 0.29/0.62        = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sqr.simps(3)
% 0.29/0.62  thf(fact_10173_rat__floor__lemma,axiom,
% 0.29/0.62      ! [A: int,B: int] :
% 0.29/0.62        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 0.29/0.62        & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % rat_floor_lemma
% 0.29/0.62  thf(fact_10174_divide__rat,axiom,
% 0.29/0.62      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.62        ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.29/0.62        = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % divide_rat
% 0.29/0.62  thf(fact_10175_floor__Fract,axiom,
% 0.29/0.62      ! [A: int,B: int] :
% 0.29/0.62        ( ( archim3151403230148437115or_rat @ ( fract @ A @ B ) )
% 0.29/0.62        = ( divide_divide_int @ A @ B ) ) ).
% 0.29/0.62  
% 0.29/0.62  % floor_Fract
% 0.29/0.62  thf(fact_10176_diff__rat,axiom,
% 0.29/0.62      ! [B: int,D: int,A: int,C: int] :
% 0.29/0.62        ( ( B != zero_zero_int )
% 0.29/0.62       => ( ( D != zero_zero_int )
% 0.29/0.62         => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.29/0.62            = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % diff_rat
% 0.29/0.62  thf(fact_10177_less__rat,axiom,
% 0.29/0.62      ! [B: int,D: int,A: int,C: int] :
% 0.29/0.62        ( ( B != zero_zero_int )
% 0.29/0.62       => ( ( D != zero_zero_int )
% 0.29/0.62         => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.29/0.62            = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % less_rat
% 0.29/0.62  thf(fact_10178_add__rat,axiom,
% 0.29/0.62      ! [B: int,D: int,A: int,C: int] :
% 0.29/0.62        ( ( B != zero_zero_int )
% 0.29/0.62       => ( ( D != zero_zero_int )
% 0.29/0.62         => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.29/0.62            = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % add_rat
% 0.29/0.62  thf(fact_10179_le__rat,axiom,
% 0.29/0.62      ! [B: int,D: int,A: int,C: int] :
% 0.29/0.62        ( ( B != zero_zero_int )
% 0.29/0.62       => ( ( D != zero_zero_int )
% 0.29/0.62         => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.29/0.62            = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % le_rat
% 0.29/0.62  thf(fact_10180_sgn__rat,axiom,
% 0.29/0.62      ! [A: int,B: int] :
% 0.29/0.62        ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
% 0.29/0.62        = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % sgn_rat
% 0.29/0.62  thf(fact_10181_rat__number__collapse_I3_J,axiom,
% 0.29/0.62      ! [W: num] :
% 0.29/0.62        ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 0.29/0.62        = ( numeral_numeral_rat @ W ) ) ).
% 0.29/0.62  
% 0.29/0.62  % rat_number_collapse(3)
% 0.29/0.62  thf(fact_10182_rat__number__expand_I3_J,axiom,
% 0.29/0.62      ( numeral_numeral_rat
% 0.29/0.62      = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % rat_number_expand(3)
% 0.29/0.62  thf(fact_10183_Fract__of__int__eq,axiom,
% 0.29/0.62      ! [K: int] :
% 0.29/0.62        ( ( fract @ K @ one_one_int )
% 0.29/0.62        = ( ring_1_of_int_rat @ K ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Fract_of_int_eq
% 0.29/0.62  thf(fact_10184_Rat__induct__pos,axiom,
% 0.29/0.62      ! [P: rat > $o,Q3: rat] :
% 0.29/0.62        ( ! [A5: int,B6: int] :
% 0.29/0.62            ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.29/0.62           => ( P @ ( fract @ A5 @ B6 ) ) )
% 0.29/0.62       => ( P @ Q3 ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Rat_induct_pos
% 0.29/0.62  thf(fact_10185_Fract__coprime,axiom,
% 0.29/0.62      ! [A: int,B: int] :
% 0.29/0.62        ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( divide_divide_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
% 0.29/0.62        = ( fract @ A @ B ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Fract_coprime
% 0.29/0.62  thf(fact_10186_rat__number__collapse_I6_J,axiom,
% 0.29/0.62      ! [K: int] :
% 0.29/0.62        ( ( fract @ K @ zero_zero_int )
% 0.29/0.62        = zero_zero_rat ) ).
% 0.29/0.62  
% 0.29/0.62  % rat_number_collapse(6)
% 0.29/0.62  thf(fact_10187_rat__number__collapse_I1_J,axiom,
% 0.29/0.62      ! [K: int] :
% 0.29/0.62        ( ( fract @ zero_zero_int @ K )
% 0.29/0.62        = zero_zero_rat ) ).
% 0.29/0.62  
% 0.29/0.62  % rat_number_collapse(1)
% 0.29/0.62  thf(fact_10188_eq__rat_I3_J,axiom,
% 0.29/0.62      ! [A: int,C: int] :
% 0.29/0.62        ( ( fract @ zero_zero_int @ A )
% 0.29/0.62        = ( fract @ zero_zero_int @ C ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eq_rat(3)
% 0.29/0.62  thf(fact_10189_mult__rat__cancel,axiom,
% 0.29/0.62      ! [C: int,A: int,B: int] :
% 0.29/0.62        ( ( C != zero_zero_int )
% 0.29/0.62       => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.62          = ( fract @ A @ B ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % mult_rat_cancel
% 0.29/0.62  thf(fact_10190_eq__rat_I1_J,axiom,
% 0.29/0.62      ! [B: int,D: int,A: int,C: int] :
% 0.29/0.62        ( ( B != zero_zero_int )
% 0.29/0.62       => ( ( D != zero_zero_int )
% 0.29/0.62         => ( ( ( fract @ A @ B )
% 0.29/0.62              = ( fract @ C @ D ) )
% 0.29/0.62            = ( ( times_times_int @ A @ D )
% 0.29/0.62              = ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eq_rat(1)
% 0.29/0.62  thf(fact_10191_eq__rat_I2_J,axiom,
% 0.29/0.62      ! [A: int] :
% 0.29/0.62        ( ( fract @ A @ zero_zero_int )
% 0.29/0.62        = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % eq_rat(2)
% 0.29/0.62  thf(fact_10192_One__rat__def,axiom,
% 0.29/0.62      ( one_one_rat
% 0.29/0.62      = ( fract @ one_one_int @ one_one_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % One_rat_def
% 0.29/0.62  thf(fact_10193_Fract__of__int__quotient,axiom,
% 0.29/0.62      ( fract
% 0.29/0.62      = ( ^ [K2: int,L3: int] : ( divide_divide_rat @ ( ring_1_of_int_rat @ K2 ) @ ( ring_1_of_int_rat @ L3 ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Fract_of_int_quotient
% 0.29/0.62  thf(fact_10194_Fract__of__nat__eq,axiom,
% 0.29/0.62      ! [K: nat] :
% 0.29/0.62        ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
% 0.29/0.62        = ( semiri681578069525770553at_rat @ K ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Fract_of_nat_eq
% 0.29/0.62  thf(fact_10195_Zero__rat__def,axiom,
% 0.29/0.62      ( zero_zero_rat
% 0.29/0.62      = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Zero_rat_def
% 0.29/0.62  thf(fact_10196_Fract__less__zero__iff,axiom,
% 0.29/0.62      ! [B: int,A: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.62       => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.29/0.62          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Fract_less_zero_iff
% 0.29/0.62  thf(fact_10197_zero__less__Fract__iff,axiom,
% 0.29/0.62      ! [B: int,A: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.62       => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.29/0.62          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % zero_less_Fract_iff
% 0.29/0.62  thf(fact_10198_one__less__Fract__iff,axiom,
% 0.29/0.62      ! [B: int,A: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.62       => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.29/0.62          = ( ord_less_int @ B @ A ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % one_less_Fract_iff
% 0.29/0.62  thf(fact_10199_Fract__less__one__iff,axiom,
% 0.29/0.62      ! [B: int,A: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.62       => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.29/0.62          = ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Fract_less_one_iff
% 0.29/0.62  thf(fact_10200_rat__number__collapse_I5_J,axiom,
% 0.29/0.62      ( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.29/0.62      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.29/0.62  
% 0.29/0.62  % rat_number_collapse(5)
% 0.29/0.62  thf(fact_10201_Fract__add__one,axiom,
% 0.29/0.62      ! [N: int,M: int] :
% 0.29/0.62        ( ( N != zero_zero_int )
% 0.29/0.62       => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 0.29/0.62          = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Fract_add_one
% 0.29/0.62  thf(fact_10202_Fract__le__zero__iff,axiom,
% 0.29/0.62      ! [B: int,A: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.62       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.29/0.62          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Fract_le_zero_iff
% 0.29/0.62  thf(fact_10203_zero__le__Fract__iff,axiom,
% 0.29/0.62      ! [B: int,A: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.62       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.29/0.62          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % zero_le_Fract_iff
% 0.29/0.62  thf(fact_10204_one__le__Fract__iff,axiom,
% 0.29/0.62      ! [B: int,A: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.62       => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.29/0.62          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % one_le_Fract_iff
% 0.29/0.62  thf(fact_10205_Fract__le__one__iff,axiom,
% 0.29/0.62      ! [B: int,A: int] :
% 0.29/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.62       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.29/0.62          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % Fract_le_one_iff
% 0.29/0.62  thf(fact_10206_rat__number__expand_I5_J,axiom,
% 0.29/0.62      ! [K: num] :
% 0.29/0.62        ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 0.29/0.62        = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.29/0.62  
% 0.29/0.62  % rat_number_expand(5)
% 0.29/0.62  thf(fact_10207_rat__number__collapse_I4_J,axiom,
% 0.29/0.62      ! [W: num] :
% 0.29/0.62        ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 0.29/0.62        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % rat_number_collapse(4)
% 0.29/0.62  thf(fact_10208_of__nat__eq__id,axiom,
% 0.29/0.62      semiri1316708129612266289at_nat = id_nat ).
% 0.29/0.62  
% 0.29/0.62  % of_nat_eq_id
% 0.29/0.62  thf(fact_10209_less__eq__int__def,axiom,
% 0.29/0.62      ( ord_less_eq_int
% 0.29/0.62      = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.29/0.62        @ ( produc8739625826339149834_nat_o
% 0.29/0.62          @ ^ [X4: nat,Y6: nat] :
% 0.29/0.62              ( produc6081775807080527818_nat_o
% 0.29/0.62              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y6 ) ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % less_eq_int_def
% 0.29/0.62  thf(fact_10210_nat__def,axiom,
% 0.29/0.62      ( nat2
% 0.29/0.62      = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  % nat_def
% 0.29/0.62  
% 0.29/0.62  % Helper facts (30)
% 0.29/0.62  thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.29/0.62      ! [X: int,Y: int] :
% 0.29/0.62        ( ( if_int @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.29/0.62      ! [X: int,Y: int] :
% 0.29/0.62        ( ( if_int @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.29/0.62      ! [X: nat,Y: nat] :
% 0.29/0.62        ( ( if_nat @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.29/0.62      ! [X: nat,Y: nat] :
% 0.29/0.62        ( ( if_nat @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.29/0.62      ! [X: rat,Y: rat] :
% 0.29/0.62        ( ( if_rat @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.29/0.62      ! [X: rat,Y: rat] :
% 0.29/0.62        ( ( if_rat @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.29/0.62      ! [X: real,Y: real] :
% 0.29/0.62        ( ( if_real @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.29/0.62      ! [X: real,Y: real] :
% 0.29/0.62        ( ( if_real @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 0.29/0.62      ! [P: real > $o] :
% 0.29/0.62        ( ( P @ ( fChoice_real @ P ) )
% 0.29/0.62        = ( ? [X6: real] : ( P @ X6 ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.29/0.62      ! [X: complex,Y: complex] :
% 0.29/0.62        ( ( if_complex @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.29/0.62      ! [X: complex,Y: complex] :
% 0.29/0.62        ( ( if_complex @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.29/0.62      ! [X: code_integer,Y: code_integer] :
% 0.29/0.62        ( ( if_Code_integer @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.29/0.62      ! [X: code_integer,Y: code_integer] :
% 0.29/0.62        ( ( if_Code_integer @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.29/0.62      ! [X: set_int,Y: set_int] :
% 0.29/0.62        ( ( if_set_int @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.29/0.62      ! [X: set_int,Y: set_int] :
% 0.29/0.62        ( ( if_set_int @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.29/0.62      ! [X: list_int,Y: list_int] :
% 0.29/0.62        ( ( if_list_int @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.29/0.62      ! [X: list_int,Y: list_int] :
% 0.29/0.62        ( ( if_list_int @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.29/0.62      ! [X: int > int,Y: int > int] :
% 0.29/0.62        ( ( if_int_int @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.29/0.62      ! [X: int > int,Y: int > int] :
% 0.29/0.62        ( ( if_int_int @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.29/0.62      ! [X: option_num,Y: option_num] :
% 0.29/0.62        ( ( if_option_num @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.29/0.62      ! [X: option_num,Y: option_num] :
% 0.29/0.62        ( ( if_option_num @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.29/0.62      ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.29/0.62        ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.29/0.62      ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.29/0.62        ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.29/0.62      ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.29/0.62        ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.29/0.62      ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.29/0.62        ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.29/0.62      ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 0.29/0.62        ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.29/0.62      ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 0.29/0.62        ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.29/0.62      ! [P: $o] :
% 0.29/0.62        ( ( P = $true )
% 0.29/0.62        | ( P = $false ) ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.29/0.62      ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 0.29/0.62        ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
% 0.29/0.62        = Y ) ).
% 0.29/0.62  
% 0.29/0.62  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.29/0.62      ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 0.29/0.62        ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
% 0.29/0.62        = X ) ).
% 0.29/0.62  
% 0.29/0.62  % Conjectures (1)
% 0.29/0.62  thf(conj_0,conjecture,
% 0.29/0.62      ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ ( vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary ) ) )
% 0.29/0.62      = ( plus_plus_int @ one_one_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ) ) ).
% 0.29/0.62  
% 0.29/0.62  %------------------------------------------------------------------------------
% 0.29/0.62  ------------------------
% 6.48/2.43  ---- Embedded in THF ---
% 6.48/2.43  %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 6.48/2.43  %%% Generated on Thu Feb 26 08:42:53 EST 2026
% 6.48/2.43  %%% using '$$dhol' embedding, version 1.3.0.
% 6.48/2.43  %%% Logic specification used:
% 6.48/2.43  %%% thf(spec, logic, $$dhol).
% 6.48/2.43  
% 6.48/2.43  % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2DTF32338.p
% See solution above
% 6.48/2.47  ------------------------
% 9.03/4.90  ---- Cleaned THF ---
% 9.03/4.90  thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_J,type,
% 9.03/4.90      set_fi4554929511873752355omplex: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.03/4.90      list_P7413028617227757229T_VEBT: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
% 9.03/4.90      set_fi7789364187291644575l_real: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.03/4.90      set_op4508134149509766951at_nat: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 9.03/4.90      filter6041513312241820739omplex: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 9.03/4.90      list_P7037539587688870467BT_nat: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 9.03/4.90      list_P4547456442757143711BT_int: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.03/4.90      list_P5647936690300460905T_VEBT: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.03/4.90      list_P7524865323317820941T_VEBT: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.03/4.90      produc8243902056947475879T_VEBT: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 9.03/4.90      set_Pr5085853215250843933omplex: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.03/4.90      produc8923325533196201883nteger: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 9.03/4.90      list_P3126845725202233233VEBT_o: $tType ).
% 9.03/4.90  
% 9.03/4.90  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.03/4.91      list_P7495141550334521929T_VEBT: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 9.03/4.91      filter2146258269922977983l_real: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
% 9.03/4.91      list_P8526636022914148096eger_o: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.03/4.91      option4927543243414619207at_nat: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 9.03/4.91      set_Pr6218003697084177305l_real: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
% 9.03/4.91      list_P3744719386663036955um_num: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
% 9.03/4.91      list_P1726324292696863441at_num: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.03/4.91      list_P6011104703257516679at_nat: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
% 9.03/4.91      list_P3521021558325789923at_int: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.03/4.91      list_P5707943133018811711nt_int: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 9.03/4.91      produc9072475918466114483BT_nat: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.03/4.91      produc8025551001238799321T_VEBT: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.03/4.91      set_Pr1261947904930325089at_nat: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.03/4.91      set_Pr958786334691620121nt_int: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 9.03/4.91      produc4411394909380815293omplex: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 9.03/4.91      list_P7333126701944960589_nat_o: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_M_Eo_J_J,type,
% 9.03/4.91      list_P5087981734274514673_int_o: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 9.03/4.91      list_P3795440434834930179_o_int: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 9.03/4.91      produc334124729049499915VEBT_o: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 9.03/4.91      produc6271795597528267376eger_o: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Set__Oset_It__Set__Oset_It__VEBT____Definitions__OVEBT_J_J,type,
% 9.03/4.91      set_set_VEBT_VEBT: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
% 9.03/4.91      set_set_set_nat: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 9.03/4.91      produc2422161461964618553l_real: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 9.03/4.91      product_prod_num_num: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 9.03/4.91      product_prod_nat_num: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.03/4.91      product_prod_nat_nat: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
% 9.03/4.91      product_prod_nat_int: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.03/4.91      product_prod_int_int: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 9.03/4.91      list_P4002435161011370285od_o_o: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
% 9.03/4.91      set_set_complex: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Num__Onum_J_J,type,
% 9.03/4.91      set_option_num: $tType ).
% 9.03/4.91  
% 9.03/4.91  thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.03/4.91      list_VEBT_VEBT: $tType ).
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% 9.03/4.91  thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 9.03/4.91      code_nat_of_integer: code_integer > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Code__Numeral_Onegative,type,
% 9.03/4.91      code_negative: num > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Code__Numeral_Opositive,type,
% 9.03/4.91      code_positive: num > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Code__Target__Int_Onegative,type,
% 9.03/4.91      code_Target_negative: num > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Code__Target__Int_Opositive,type,
% 9.03/4.91      code_Target_positive: num > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Code__Target__Nat_Oint__of__nat,type,
% 9.03/4.91      code_T6385005292777649522of_nat: nat > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 9.03/4.91      comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 9.03/4.91      comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
% 9.03/4.91      comple4887499456419720421f_real: set_real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.03/4.91      comple7806235888213564991et_nat: set_set_nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
% 9.03/4.91      complete_Sup_Sup_int: set_int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
% 9.03/4.91      complete_Sup_Sup_nat: set_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
% 9.03/4.91      comple1385675409528146559p_real: set_real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complex_OArg,type,
% 9.03/4.91      arg: complex > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complex_Ocis,type,
% 9.03/4.91      cis: real > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complex_Ocnj,type,
% 9.03/4.91      cnj: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complex_Ocomplex_OComplex,type,
% 9.03/4.91      complex2: real > real > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complex_Ocomplex_OIm,type,
% 9.03/4.91      im: complex > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complex_Ocomplex_ORe,type,
% 9.03/4.91      re: complex > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complex_Ocsqrt,type,
% 9.03/4.91      csqrt: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Complex_Oimaginary__unit,type,
% 9.03/4.91      imaginary_unit: complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.03/4.91      differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.03/4.91      has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
% 9.03/4.91      has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Oadjust__div,type,
% 9.03/4.91      adjust_div: product_prod_int_int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Oadjust__mod,type,
% 9.03/4.91      adjust_mod: int > int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Odivmod__nat,type,
% 9.03/4.91      divmod_nat: nat > nat > product_prod_nat_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Oeucl__rel__int,type,
% 9.03/4.91      eucl_rel_int: int > int > product_prod_int_int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
% 9.03/4.91      unique6319869463603278526ux_int: product_prod_int_int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
% 9.03/4.91      unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
% 9.03/4.91      unique5052692396658037445od_int: num > num > product_prod_int_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
% 9.03/4.91      unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
% 9.03/4.91      unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
% 9.03/4.91      unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
% 9.03/4.91      comm_s2602460028002588243omplex: complex > nat > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
% 9.03/4.91      comm_s4660882817536571857er_int: int > nat > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
% 9.03/4.91      comm_s4663373288045622133er_nat: nat > nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
% 9.03/4.91      comm_s4028243227959126397er_rat: rat > nat > rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
% 9.03/4.91      comm_s7457072308508201937r_real: real > nat > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      semiri3624122377584611663nteger: nat > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
% 9.03/4.91      semiri5044797733671781792omplex: nat > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
% 9.03/4.91      semiri1406184849735516958ct_int: nat > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
% 9.03/4.91      semiri1408675320244567234ct_nat: nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
% 9.03/4.91      semiri773545260158071498ct_rat: nat > rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 9.03/4.91      semiri2265585572941072030t_real: nat > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
% 9.03/4.91      invers8013647133539491842omplex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 9.03/4.91      inverse_inverse_rat: rat > rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 9.03/4.91      inverse_inverse_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 9.03/4.91      at_bot_real: filter_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Oat__top_001t__Int__Oint,type,
% 9.03/4.91      at_top_int: filter_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 9.03/4.91      at_top_nat: filter_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 9.03/4.91      at_top_real: filter_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 9.03/4.91      eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 9.03/4.91      eventually_real: ( real > $o ) > filter_real > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Ofilterlim_001t__Int__Oint_001t__Real__Oreal,type,
% 9.03/4.91      filterlim_int_real: ( int > real ) > filter_real > filter_int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Int__Oint,type,
% 9.03/4.91      filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.03/4.91      filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.03/4.91      filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.03/4.91      filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 9.03/4.91      princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 9.03/4.91      princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Finite__Set_Ocard_001_Eo,type,
% 9.03/4.91      finite_card_o: set_o > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 9.03/4.91      finite_card_complex: set_complex > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
% 9.03/4.91      finite_card_int: set_int > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 9.03/4.91      finite_card_nat: set_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
% 9.03/4.91      finite410649719033368117t_unit: set_Product_unit > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.03/4.91      finite_card_set_nat: set_set_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
% 9.03/4.91      finite_card_char: set_char > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 9.03/4.91      finite3207457112153483333omplex: set_complex > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
% 9.03/4.91      finite_finite_int: set_int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 9.03/4.91      finite_finite_nat: set_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 9.03/4.91      bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 9.03/4.91      bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.03/4.91      bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  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,
% 9.03/4.91      comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.03/4.91  
% 9.03/4.91  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,
% 9.03/4.91      comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 9.03/4.91      comp_C3531382070062128313er_num: ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 9.03/4.91      comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.03/4.91      comp_int_real_real: ( int > real ) > ( real > int ) > real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.03/4.91      comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.03/4.91      comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Oid_001_Eo,type,
% 9.03/4.91      id_o: $o > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
% 9.03/4.91      id_nat: nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.03/4.91      inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
% 9.03/4.91      inj_on_nat_char: ( nat > char ) > set_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.03/4.91      inj_on_real_real: ( real > real ) > set_real > $o ).
% 9.03/4.91  
% 9.03/4.91  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,
% 9.03/4.91      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 ).
% 9.03/4.91  
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% 9.03/4.91  thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
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% 9.03/4.91      bezw: nat > nat > product_prod_int_int ).
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% 9.03/4.91      the_int: ( int > $o ) > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
% 9.03/4.91      the_real: ( real > $o ) > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.03/4.91      if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      if_Code_integer: $o > code_integer > code_integer > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__Complex__Ocomplex,type,
% 9.03/4.91      if_complex: $o > complex > complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__Int__Oint,type,
% 9.03/4.91      if_int: $o > int > int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
% 9.03/4.91      if_list_int: $o > list_int > list_int > list_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__Nat__Onat,type,
% 9.03/4.91      if_nat: $o > nat > nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.03/4.91      if_option_num: $o > option_num > option_num > option_num ).
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% 9.03/4.91  thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.03/4.91      if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__Rat__Orat,type,
% 9.03/4.91      if_rat: $o > rat > rat > rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__Real__Oreal,type,
% 9.03/4.91      if_real: $o > real > real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
% 9.03/4.91      if_set_int: $o > set_int > set_int > set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_OAbs__Integ,type,
% 9.03/4.91      abs_Integ: product_prod_nat_nat > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_ORep__Integ,type,
% 9.03/4.91      rep_Integ: int > product_prod_nat_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oint__ge__less__than,type,
% 9.03/4.91      int_ge_less_than: int > set_Pr958786334691620121nt_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oint__ge__less__than2,type,
% 9.03/4.91      int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Onat,type,
% 9.03/4.91      nat2: int > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
% 9.03/4.91      power_int_real: real > int > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      ring_11222124179247155820nteger: set_Code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
% 9.03/4.91      ring_1_Ints_complex: set_complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
% 9.03/4.91      ring_1_Ints_int: set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
% 9.03/4.91      ring_1_Ints_rat: set_rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 9.03/4.91      ring_1_Ints_real: set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      ring_18347121197199848620nteger: int > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
% 9.03/4.91      ring_17405671764205052669omplex: int > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 9.03/4.91      ring_1_of_int_int: int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
% 9.03/4.91      ring_1_of_int_rat: int > rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 9.03/4.91      ring_1_of_int_real: int > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 9.03/4.91      semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
% 9.03/4.91      sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
% 9.03/4.91      sup_sup_int: int > int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 9.03/4.91      sup_sup_nat: nat > nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.03/4.91      sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001_Eo,type,
% 9.03/4.91      lattic1921953407002678535_Max_o: set_o > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
% 9.03/4.91      lattic8263393255366662781ax_int: set_int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 9.03/4.91      lattic8265883725875713057ax_nat: set_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Real__Oreal,type,
% 9.03/4.91      lattic4275903605611617917x_real: set_real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.03/4.91      bfun_nat_real: ( nat > real ) > filter_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
% 9.03/4.91      at_infinity_real: filter_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 9.03/4.91      append_int: list_int > list_int > list_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 9.03/4.91      append_nat: list_nat > list_nat > list_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Odistinct_001t__Int__Oint,type,
% 9.03/4.91      distinct_int: list_int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 9.03/4.91      linord2614967742042102400et_nat: set_nat > list_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 9.03/4.91      cons_int: int > list_int > list_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 9.03/4.91      cons_nat: nat > list_nat > list_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 9.03/4.91      nil_int: list_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 9.03/4.91      nil_nat: list_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_Oset_001_Eo,type,
% 9.03/4.91      set_o2: list_o > set_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 9.03/4.91      set_complex2: list_complex > set_complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 9.03/4.91      set_int2: list_int > set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 9.03/4.91      set_nat2: list_nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 9.03/4.91      set_real2: list_real > set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.03/4.91      set_set_nat2: list_set_nat > set_set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001_Eo,type,
% 9.03/4.91      nth_o: list_o > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      nth_Code_integer: list_Code_integer > nat > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 9.03/4.91      nth_complex: list_complex > nat > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Int__Oint,type,
% 9.03/4.91      nth_int: list_int > nat > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 9.03/4.91      nth_nat: list_nat > nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Num__Onum,type,
% 9.03/4.91      nth_num: list_num > nat > num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 9.03/4.91      nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
% 9.03/4.91      nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
% 9.03/4.91      nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.03/4.91      nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 9.03/4.91      nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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% 9.03/4.91  
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% 9.03/4.91      nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.03/4.91      nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 9.03/4.91      nth_real: list_real > nat > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.03/4.91      nth_set_nat: list_set_nat > nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 9.03/4.91      product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 9.03/4.91      product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
% 9.03/4.91      produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
% 9.03/4.91      product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
% 9.03/4.91      product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 9.03/4.91      product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
% 9.03/4.91      product_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.03/4.91      product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
% 9.03/4.91      product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
% 9.03/4.91      product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
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% 9.03/4.91  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 9.03/4.91      produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oupto,type,
% 9.03/4.91      upto: int > int > list_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oupto__aux,type,
% 9.03/4.91      upto_aux: int > int > list_int > list_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_List_Oupto__rel,type,
% 9.03/4.91      upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_OSuc,type,
% 9.03/4.91      suc: nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.03/4.91      compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 9.03/4.91      case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 9.03/4.91      case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.03/4.91      case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Onat_Opred,type,
% 9.03/4.91      pred: nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Complex__Ocomplex,type,
% 9.03/4.91      semiri3842193898606819883omplex: set_complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
% 9.03/4.91      semiring_1_Nats_int: set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      semiri4939895301339042750nteger: nat > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 9.03/4.91      semiri8010041392384452111omplex: nat > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 9.03/4.91      semiri1314217659103216013at_int: nat > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 9.03/4.91      semiri1316708129612266289at_nat: nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 9.03/4.91      semiri681578069525770553at_rat: nat > rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 9.03/4.91      semiri5074537144036343181t_real: nat > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 9.03/4.91      size_size_list_o: list_o > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
% 9.03/4.91      size_s3445333598471063425nteger: list_Code_integer > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 9.03/4.91      size_s3451745648224563538omplex: list_complex > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 9.03/4.91      size_size_list_int: list_int > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 9.03/4.91      size_size_list_nat: list_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 9.03/4.91      size_size_list_num: list_num > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 9.03/4.91      size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 9.03/4.91      size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.03/4.91      size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_M_Eo_J_J,type,
% 9.03/4.91      size_s4246224855604898693_int_o: list_P5087981734274514673_int_o > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.03/4.91      size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.03/4.91      size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 9.03/4.91      size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 9.03/4.91      size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
% 9.03/4.91  
% 9.03/4.91  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,
% 9.03/4.91      size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 9.03/4.91      size_size_list_real: list_real > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.03/4.91      size_s3254054031482475050et_nat: list_set_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.03/4.91      size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 9.03/4.91      size_size_num: num > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.03/4.91      size_size_option_num: option_num > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.03/4.91      size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
% 9.03/4.91      size_size_char: char > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat__Bijection_Olist__encode,type,
% 9.03/4.91      nat_list_encode: list_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 9.03/4.91      nat_list_encode_rel: list_nat > list_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 9.03/4.91      nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 9.03/4.91      nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 9.03/4.91      nat_prod_encode: product_prod_nat_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat__Bijection_Oset__decode,type,
% 9.03/4.91      nat_set_decode: nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat__Bijection_Oset__encode,type,
% 9.03/4.91      nat_set_encode: set_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Nat__Bijection_Otriangle,type,
% 9.03/4.91      nat_triangle: nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_NthRoot_Oroot,type,
% 9.03/4.91      root: nat > real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_NthRoot_Osqrt,type,
% 9.03/4.91      sqrt: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_OBitM,type,
% 9.03/4.91      bitM: num > num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oinc,type,
% 9.03/4.91      inc: num > num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 9.03/4.91      neg_nu7009210354673126013omplex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 9.03/4.91      neg_numeral_dbl_int: int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 9.03/4.91      neg_numeral_dbl_rat: rat > rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 9.03/4.91      neg_numeral_dbl_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 9.03/4.91      neg_nu6511756317524482435omplex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 9.03/4.91      neg_nu3811975205180677377ec_int: int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 9.03/4.91      neg_nu3179335615603231917ec_rat: rat > rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 9.03/4.91      neg_nu6075765906172075777c_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      neg_nu5831290666863070958nteger: code_integer > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 9.03/4.91      neg_nu8557863876264182079omplex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 9.03/4.91      neg_nu5851722552734809277nc_int: int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 9.03/4.91      neg_nu5219082963157363817nc_rat: rat > rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 9.03/4.91      neg_nu8295874005876285629c_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 9.03/4.91      neg_numeral_sub_int: num > num > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onum_OBit0,type,
% 9.03/4.91      bit0: num > num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onum_OBit1,type,
% 9.03/4.91      bit1: num > num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onum_OOne,type,
% 9.03/4.91      one: num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.03/4.91      case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onum_Osize__num,type,
% 9.03/4.91      size_num: num > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onum__of__nat,type,
% 9.03/4.91      num_of_nat: nat > num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      numera6620942414471956472nteger: num > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 9.03/4.91      numera6690914467698888265omplex: num > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 9.03/4.91      numera1916890842035813515d_enat: num > extended_enat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 9.03/4.91      numeral_numeral_int: num > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 9.03/4.91      numeral_numeral_nat: num > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 9.03/4.91      numeral_numeral_rat: num > rat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 9.03/4.91      numeral_numeral_real: num > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Opow,type,
% 9.03/4.91      pow: num > num > num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Opred__numeral,type,
% 9.03/4.91      pred_numeral: num > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Num_Osqr,type,
% 9.03/4.91      sqr: num > num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 9.03/4.91      none_num: option_num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.03/4.91      none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 9.03/4.91      some_num: num > option_num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.03/4.91      some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.03/4.91      case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
% 9.03/4.91      case_option_int_num: int > ( num > int ) > option_num > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 9.03/4.91      case_option_num_num: num > ( num > num ) > option_num > num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 9.03/4.91      case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 9.03/4.91      map_option_num_num: ( num > num ) > option_num > option_num ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
% 9.03/4.91      size_option_num: ( num > nat ) > option_num > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.03/4.91      size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
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% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 9.03/4.91      bot_bot_set_nat_o: set_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
% 9.03/4.91      bot_bo4199563552545308370d_enat: extended_enat ).
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% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 9.03/4.91      bot_bot_nat: nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
% 9.03/4.91      bot_bot_set_o: set_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.03/4.91      bot_bot_set_complex: set_complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
% 9.03/4.91      bot_bot_set_int: set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.03/4.91      bot_bot_set_nat: set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.03/4.91      bot_bot_set_real: set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.03/4.91      bot_bot_set_set_nat: set_set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 9.03/4.91      bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 9.03/4.91      ord_less_int: int > int > $o ).
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 9.03/4.91      ord_less_nat: nat > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
% 9.03/4.91      ord_less_num: num > num > $o ).
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
% 9.03/4.91      ord_less_rat: rat > rat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 9.03/4.91      ord_less_real: real > real > $o ).
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 9.03/4.91      ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 9.03/4.91      ord_less_eq_int: int > int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 9.03/4.91      ord_less_eq_nat: nat > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
% 9.03/4.91      ord_less_eq_rat: rat > rat > $o ).
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 9.03/4.91      ord_less_eq_real: real > real > $o ).
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.03/4.91      ord_le211207098394363844omplex: set_complex > set_complex > $o ).
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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% 9.03/4.91  thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
% 9.03/4.91      ord_max_nat: nat > nat > nat ).
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% 9.03/4.91  thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
% 9.03/4.91      order_Greatest_nat: ( nat > $o ) > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
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% 9.03/4.91  thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oorder__class_Omono_001t__Real__Oreal_001t__Real__Oreal,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.03/4.91      order_5726023648592871131at_nat: ( nat > nat ) > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.03/4.91      order_7092887310737990675l_real: ( real > real ) > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
% 9.03/4.91      top_top_set_o: set_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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% 9.03/4.91  thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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% 9.03/4.91  thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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% 9.03/4.91  thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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% 9.03/4.91  thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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% 9.03/4.91  thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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% 9.03/4.91  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
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% 9.03/4.91  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
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% 9.03/4.91  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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% 9.03/4.91  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
% 9.03/4.91      product_Pair_nat_num: nat > num > product_prod_nat_num ).
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% 9.03/4.91  thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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% 9.03/4.91  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 9.03/4.91      produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
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% 9.03/4.91  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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% 9.03/4.91  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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% 9.03/4.91  
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% 9.03/4.91  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
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% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.03/4.91      image_real_nat: ( real > nat ) > set_real > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.03/4.91      image_real_real: ( real > real ) > set_real > set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      image_real_VEBT_VEBT: ( real > vEBT_VEBT ) > set_real > set_VEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
% 9.03/4.91      image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.03/4.91      image_7998606247489673935omplex: ( set_complex > set_complex ) > set_set_complex > set_set_complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
% 9.03/4.91      image_524474410958335435et_int: ( set_int > set_int ) > set_set_int > set_set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.03/4.91      image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.03/4.91      image_2436557299294012491t_real: ( set_real > set_real ) > set_set_real > set_set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.03/4.91      image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 9.03/4.91      image_1661326939266726661T_VEBT: ( set_VEBT_VEBT > set_VEBT_VEBT ) > set_set_VEBT_VEBT > set_set_VEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
% 9.03/4.91      image_char_nat: ( char > nat ) > set_char > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 9.03/4.91      image_VEBT_VEBT_o: ( vEBT_VEBT > $o ) > set_VEBT_VEBT > set_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 9.03/4.91      image_VEBT_VEBT_int: ( vEBT_VEBT > int ) > set_VEBT_VEBT > set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 9.03/4.91      image_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
% 9.03/4.91      image_VEBT_VEBT_real: ( vEBT_VEBT > real ) > set_VEBT_VEBT > set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      image_3375948659692109573T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > set_VEBT_VEBT > set_VEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oinsert_001_Eo,type,
% 9.03/4.91      insert_o: $o > set_o > set_o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
% 9.03/4.91      insert_complex: complex > set_complex > set_complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
% 9.03/4.91      insert_int: int > set_int > set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 9.03/4.91      insert_nat: nat > set_nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
% 9.03/4.91      insert_real: real > set_real > set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.03/4.91      insert_set_nat: set_nat > set_set_nat > set_set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Ois__singleton_001_Eo,type,
% 9.03/4.91      is_singleton_o: set_o > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
% 9.03/4.91      is_singleton_int: set_int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
% 9.03/4.91      is_singleton_nat: set_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
% 9.03/4.91      is_singleton_real: set_real > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Ois__singleton_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      is_sin24926331636114728T_VEBT: set_VEBT_VEBT > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Othe__elem_001_Eo,type,
% 9.03/4.91      the_elem_o: set_o > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
% 9.03/4.91      the_elem_int: set_int > int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
% 9.03/4.91      the_elem_nat: set_nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
% 9.03/4.91      the_elem_real: set_real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.03/4.91      the_elem_set_nat: set_set_nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set_Othe__elem_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      the_elem_VEBT_VEBT: set_VEBT_VEBT > vEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
% 9.03/4.91      set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
% 9.03/4.91      set_or1266510415728281911st_int: int > int > set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
% 9.03/4.91      set_or1269000886237332187st_nat: nat > nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 9.03/4.91      set_or1222579329274155063t_real: real > real > set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 9.03/4.91      set_or4662586982721622107an_int: int > int > set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 9.03/4.91      set_or4665077453230672383an_nat: nat > nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
% 9.03/4.91      set_ord_atLeast_nat: nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
% 9.03/4.91      set_ord_atLeast_real: real > set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
% 9.03/4.91      set_ord_atMost_int: int > set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 9.03/4.91      set_ord_atMost_nat: nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 9.03/4.91      set_or6656581121297822940st_int: int > int > set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 9.03/4.91      set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 9.03/4.91      set_or5832277885323065728an_int: int > int > set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 9.03/4.91      set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 9.03/4.91      set_or1633881224788618240n_real: real > real > set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 9.03/4.91      set_or1210151606488870762an_nat: nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 9.03/4.91      set_or5849166863359141190n_real: real > set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 9.03/4.91      set_ord_lessThan_int: int > set_int ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 9.03/4.91      set_ord_lessThan_nat: nat > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 9.03/4.91      set_or5984915006950818249n_real: real > set_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_String_Oascii__of,type,
% 9.03/4.91      ascii_of: char > char ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_String_Ochar_OChar,type,
% 9.03/4.91      char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_String_Ochar_Osize__char,type,
% 9.03/4.91      size_char: char > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 9.03/4.91      comm_s629917340098488124ar_nat: char > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_String_Ointeger__of__char,type,
% 9.03/4.91      integer_of_char: char > code_integer ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 9.03/4.91      unique3096191561947761185of_nat: nat > char ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.03/4.91      topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.03/4.91      topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 9.03/4.91      topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 9.03/4.91      topolo2177554685111907308n_real: real > set_real > filter_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 9.03/4.91      topolo2815343760600316023s_real: real > filter_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 9.03/4.91      topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 9.03/4.91      topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 9.03/4.91      topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Oarccos,type,
% 9.03/4.91      arccos: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 9.03/4.91      arcosh_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Oarcsin,type,
% 9.03/4.91      arcsin: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Oarctan,type,
% 9.03/4.91      arctan: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 9.03/4.91      arsinh_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 9.03/4.91      artanh_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 9.03/4.91      cos_complex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 9.03/4.91      cos_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Ocos__coeff,type,
% 9.03/4.91      cos_coeff: nat > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
% 9.03/4.91      cosh_complex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 9.03/4.91      cosh_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
% 9.03/4.91      cot_complex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 9.03/4.91      cot_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 9.03/4.91      diffs_real: ( nat > real ) > nat > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 9.03/4.91      exp_complex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 9.03/4.91      exp_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 9.03/4.91      ln_ln_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Olog,type,
% 9.03/4.91      log: real > real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Opi,type,
% 9.03/4.91      pi: real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 9.03/4.91      powr_real: real > real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 9.03/4.91      sin_complex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 9.03/4.91      sin_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Osin__coeff,type,
% 9.03/4.91      sin_coeff: nat > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
% 9.03/4.91      sinh_complex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 9.03/4.91      sinh_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 9.03/4.91      tan_complex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 9.03/4.91      tan_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 9.03/4.91      tanh_complex: complex > complex ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 9.03/4.91      tanh_real: real > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 9.03/4.91      transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
% 9.03/4.91      transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 9.03/4.91      vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 9.03/4.91      vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 9.03/4.91      vEBT_size_VEBT: vEBT_VEBT > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 9.03/4.91      vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 9.03/4.91      vEBT_VEBT_high: nat > nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 9.03/4.91      vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 9.03/4.91      vEBT_VEBT_low: nat > nat > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 9.03/4.91      vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 9.03/4.91      vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 9.03/4.91      vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 9.03/4.91      vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 9.03/4.91      vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 9.03/4.91      vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 9.03/4.91      vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 9.03/4.91      vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 9.03/4.91      vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 9.03/4.91      vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
% 9.03/4.91      vEBT_VEBT_height: vEBT_VEBT > nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
% 9.03/4.91      vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 9.03/4.91      accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 9.03/4.91      accp_nat: ( nat > nat > $o ) > nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.03/4.91      accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.03/4.91      accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 9.03/4.91      accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 9.03/4.91      accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_Wellfounded_Opred__nat,type,
% 9.03/4.91      pred_nat: set_Pr1261947904930325089at_nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_fChoice_001t__Real__Oreal,type,
% 9.03/4.91      fChoice_real: ( real > $o ) > real ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001_Eo,type,
% 9.03/4.91      member_o: $o > set_o > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
% 9.03/4.91      member_Code_integer: code_integer > set_Code_integer > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Complex__Ocomplex,type,
% 9.03/4.91      member_complex: complex > set_complex > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Int__Oint,type,
% 9.03/4.91      member_int: int > set_int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Nat__Onat,type,
% 9.03/4.91      member_nat: nat > set_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.03/4.91      member_option_num: option_num > set_option_num > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.03/4.91      member3954567711264315760at_nat: option4927543243414619207at_nat > set_op4508134149509766951at_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.03/4.91      member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Rat__Orat,type,
% 9.03/4.91      member_rat: rat > set_rat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Real__Oreal,type,
% 9.03/4.91      member_real: real > set_real > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
% 9.03/4.91      member_set_o: set_o > set_set_o > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.03/4.91      member_set_complex: set_complex > set_set_complex > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 9.03/4.91      member_set_int: set_int > set_set_int > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.03/4.91      member_set_nat: set_nat > set_set_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.03/4.91      member_set_real: set_real > set_set_real > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.03/4.91      member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 9.03/4.91      member_set_VEBT_VEBT: set_VEBT_VEBT > set_set_VEBT_VEBT > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 9.03/4.91      member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_v_deg____,type,
% 9.03/4.91      deg: nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_v_m____,type,
% 9.03/4.91      m: nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_v_na____,type,
% 9.03/4.91      na: nat ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_v_summary____,type,
% 9.03/4.91      summary: vEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(sy_v_treeList____,type,
% 9.03/4.91      treeList: list_VEBT_VEBT ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_0__C3_OIH_C_I2_J,axiom,
% 9.03/4.91      ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ summary ) )
% 9.03/4.91      = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_1__C3_Ohyps_C_I1_J,axiom,
% 9.03/4.91      vEBT_invar_vebt @ summary @ m ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_2__C3_C,axiom,
% 9.03/4.91      ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ summary @ ( set_VEBT_VEBT2 @ treeList ) ) ) ) )
% 9.03/4.91      = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_3__C3_Ohyps_C_I5_J,axiom,
% 9.03/4.91      ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ summary @ X_1 ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_4_ceiling__add__one,axiom,
% 9.03/4.91      ! [X: rat] :
% 9.03/4.91        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
% 9.03/4.91        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_5_ceiling__add__one,axiom,
% 9.03/4.91      ! [X: real] :
% 9.03/4.91        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
% 9.03/4.91        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_6_one__add__one,axiom,
% 9.03/4.91      ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 9.03/4.91      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_7_one__add__one,axiom,
% 9.03/4.91      ( ( plus_plus_real @ one_one_real @ one_one_real )
% 9.03/4.91      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_8_one__add__one,axiom,
% 9.03/4.91      ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 9.03/4.91      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_9_one__add__one,axiom,
% 9.03/4.91      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 9.03/4.91      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_10_one__add__one,axiom,
% 9.03/4.91      ( ( plus_plus_int @ one_one_int @ one_one_int )
% 9.03/4.91      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_11_ceiling__add__numeral,axiom,
% 9.03/4.91      ! [X: real,V: num] :
% 9.03/4.91        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 9.03/4.91        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_12_ceiling__add__numeral,axiom,
% 9.03/4.91      ! [X: rat,V: num] :
% 9.03/4.91        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 9.03/4.91        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_13_numeral__plus__one,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 9.03/4.91        = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_14_numeral__plus__one,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 9.03/4.91        = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_15_numeral__plus__one,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 9.03/4.91        = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_16_numeral__plus__one,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 9.03/4.91        = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_17_numeral__plus__one,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 9.03/4.91        = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_18_one__plus__numeral,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 9.03/4.91        = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_19_one__plus__numeral,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 9.03/4.91        = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_20_one__plus__numeral,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 9.03/4.91        = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_21_one__plus__numeral,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 9.03/4.91        = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_22_one__plus__numeral,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 9.03/4.91        = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_23_ceiling__of__nat,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.91        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_24_ceiling__of__nat,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( archim2889992004027027881ng_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.03/4.91        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_25_ceiling__one,axiom,
% 9.03/4.91      ( ( archim7802044766580827645g_real @ one_one_real )
% 9.03/4.91      = one_one_int ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_26_ceiling__one,axiom,
% 9.03/4.91      ( ( archim2889992004027027881ng_rat @ one_one_rat )
% 9.03/4.91      = one_one_int ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_27_numeral__eq__one__iff,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( ( numera6690914467698888265omplex @ N )
% 9.03/4.91          = one_one_complex )
% 9.03/4.91        = ( N = one ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_28_numeral__eq__one__iff,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( ( numeral_numeral_real @ N )
% 9.03/4.91          = one_one_real )
% 9.03/4.91        = ( N = one ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_29_numeral__eq__one__iff,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( ( numeral_numeral_rat @ N )
% 9.03/4.91          = one_one_rat )
% 9.03/4.91        = ( N = one ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_30_numeral__eq__one__iff,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( ( numeral_numeral_nat @ N )
% 9.03/4.91          = one_one_nat )
% 9.03/4.91        = ( N = one ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_31_numeral__eq__one__iff,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( ( numeral_numeral_int @ N )
% 9.03/4.91          = one_one_int )
% 9.03/4.91        = ( N = one ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_32_one__eq__numeral__iff,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( one_one_complex
% 9.03/4.91          = ( numera6690914467698888265omplex @ N ) )
% 9.03/4.91        = ( one = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_33_one__eq__numeral__iff,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( one_one_real
% 9.03/4.91          = ( numeral_numeral_real @ N ) )
% 9.03/4.91        = ( one = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_34_one__eq__numeral__iff,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( one_one_rat
% 9.03/4.91          = ( numeral_numeral_rat @ N ) )
% 9.03/4.91        = ( one = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_35_one__eq__numeral__iff,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( one_one_nat
% 9.03/4.91          = ( numeral_numeral_nat @ N ) )
% 9.03/4.91        = ( one = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_36_one__eq__numeral__iff,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( one_one_int
% 9.03/4.91          = ( numeral_numeral_int @ N ) )
% 9.03/4.91        = ( one = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_37_ceiling__numeral,axiom,
% 9.03/4.91      ! [V: num] :
% 9.03/4.91        ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
% 9.03/4.91        = ( numeral_numeral_int @ V ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_38_ceiling__numeral,axiom,
% 9.03/4.91      ! [V: num] :
% 9.03/4.91        ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
% 9.03/4.91        = ( numeral_numeral_int @ V ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_39_of__nat__add,axiom,
% 9.03/4.91      ! [M: nat,N: nat] :
% 9.03/4.91        ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.91        = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_40_of__nat__add,axiom,
% 9.03/4.91      ! [M: nat,N: nat] :
% 9.03/4.91        ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.91        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_41_of__nat__add,axiom,
% 9.03/4.91      ! [M: nat,N: nat] :
% 9.03/4.91        ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.91        = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_42_of__nat__add,axiom,
% 9.03/4.91      ! [M: nat,N: nat] :
% 9.03/4.91        ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.91        = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_43_of__nat__add,axiom,
% 9.03/4.91      ! [M: nat,N: nat] :
% 9.03/4.91        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.91        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_44_of__nat__1,axiom,
% 9.03/4.91      ( ( semiri8010041392384452111omplex @ one_one_nat )
% 9.03/4.91      = one_one_complex ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_45_of__nat__1,axiom,
% 9.03/4.91      ( ( semiri5074537144036343181t_real @ one_one_nat )
% 9.03/4.91      = one_one_real ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_46_of__nat__1,axiom,
% 9.03/4.91      ( ( semiri681578069525770553at_rat @ one_one_nat )
% 9.03/4.91      = one_one_rat ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_47_of__nat__1,axiom,
% 9.03/4.91      ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 9.03/4.91      = one_one_nat ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_48_of__nat__1,axiom,
% 9.03/4.91      ( ( semiri1314217659103216013at_int @ one_one_nat )
% 9.03/4.91      = one_one_int ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_49__C3_Ohyps_C_I6_J,axiom,
% 9.03/4.91      ! [X2: vEBT_VEBT] :
% 9.03/4.91        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 9.03/4.91       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_50_numeral__eq__iff,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( ( numera6690914467698888265omplex @ M )
% 9.03/4.91          = ( numera6690914467698888265omplex @ N ) )
% 9.03/4.91        = ( M = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_51_numeral__eq__iff,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( ( numeral_numeral_real @ M )
% 9.03/4.91          = ( numeral_numeral_real @ N ) )
% 9.03/4.91        = ( M = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_52_numeral__eq__iff,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( ( numeral_numeral_rat @ M )
% 9.03/4.91          = ( numeral_numeral_rat @ N ) )
% 9.03/4.91        = ( M = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_53_numeral__eq__iff,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( ( numeral_numeral_nat @ M )
% 9.03/4.91          = ( numeral_numeral_nat @ N ) )
% 9.03/4.91        = ( M = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_54_numeral__eq__iff,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( ( numeral_numeral_int @ M )
% 9.03/4.91          = ( numeral_numeral_int @ N ) )
% 9.03/4.91        = ( M = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_55_of__nat__eq__iff,axiom,
% 9.03/4.91      ! [M: nat,N: nat] :
% 9.03/4.91        ( ( ( semiri8010041392384452111omplex @ M )
% 9.03/4.91          = ( semiri8010041392384452111omplex @ N ) )
% 9.03/4.91        = ( M = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_56_of__nat__eq__iff,axiom,
% 9.03/4.91      ! [M: nat,N: nat] :
% 9.03/4.91        ( ( ( semiri5074537144036343181t_real @ M )
% 9.03/4.91          = ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.91        = ( M = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_57_of__nat__eq__iff,axiom,
% 9.03/4.91      ! [M: nat,N: nat] :
% 9.03/4.91        ( ( ( semiri681578069525770553at_rat @ M )
% 9.03/4.91          = ( semiri681578069525770553at_rat @ N ) )
% 9.03/4.91        = ( M = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_58_of__nat__eq__iff,axiom,
% 9.03/4.91      ! [M: nat,N: nat] :
% 9.03/4.91        ( ( ( semiri1316708129612266289at_nat @ M )
% 9.03/4.91          = ( semiri1316708129612266289at_nat @ N ) )
% 9.03/4.91        = ( M = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_59_of__nat__eq__iff,axiom,
% 9.03/4.91      ! [M: nat,N: nat] :
% 9.03/4.91        ( ( ( semiri1314217659103216013at_int @ M )
% 9.03/4.91          = ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.91        = ( M = N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_60_of__nat__numeral,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 9.03/4.91        = ( numera6690914467698888265omplex @ N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_61_of__nat__numeral,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 9.03/4.91        = ( numeral_numeral_real @ N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_62_of__nat__numeral,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 9.03/4.91        = ( numeral_numeral_rat @ N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_63_of__nat__numeral,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 9.03/4.91        = ( numeral_numeral_nat @ N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_64_of__nat__numeral,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 9.03/4.91        = ( numeral_numeral_int @ N ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_65_of__nat__eq__1__iff,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( ( semiri8010041392384452111omplex @ N )
% 9.03/4.91          = one_one_complex )
% 9.03/4.91        = ( N = one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_66_of__nat__eq__1__iff,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( ( semiri5074537144036343181t_real @ N )
% 9.03/4.91          = one_one_real )
% 9.03/4.91        = ( N = one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_67_of__nat__eq__1__iff,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( ( semiri681578069525770553at_rat @ N )
% 9.03/4.91          = one_one_rat )
% 9.03/4.91        = ( N = one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_68_of__nat__eq__1__iff,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( ( semiri1316708129612266289at_nat @ N )
% 9.03/4.91          = one_one_nat )
% 9.03/4.91        = ( N = one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_69_of__nat__eq__1__iff,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( ( semiri1314217659103216013at_int @ N )
% 9.03/4.91          = one_one_int )
% 9.03/4.91        = ( N = one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_70_of__nat__1__eq__iff,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( one_one_complex
% 9.03/4.91          = ( semiri8010041392384452111omplex @ N ) )
% 9.03/4.91        = ( N = one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_71_of__nat__1__eq__iff,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( one_one_real
% 9.03/4.91          = ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.91        = ( N = one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_72_of__nat__1__eq__iff,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( one_one_rat
% 9.03/4.91          = ( semiri681578069525770553at_rat @ N ) )
% 9.03/4.91        = ( N = one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_73_of__nat__1__eq__iff,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( one_one_nat
% 9.03/4.91          = ( semiri1316708129612266289at_nat @ N ) )
% 9.03/4.91        = ( N = one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_74_of__nat__1__eq__iff,axiom,
% 9.03/4.91      ! [N: nat] :
% 9.03/4.91        ( ( one_one_int
% 9.03/4.91          = ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.91        = ( N = one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_75_add__numeral__left,axiom,
% 9.03/4.91      ! [V: num,W: num,Z: complex] :
% 9.03/4.91        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 9.03/4.91        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_76_add__numeral__left,axiom,
% 9.03/4.91      ! [V: num,W: num,Z: real] :
% 9.03/4.91        ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 9.03/4.91        = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_77_add__numeral__left,axiom,
% 9.03/4.91      ! [V: num,W: num,Z: rat] :
% 9.03/4.91        ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 9.03/4.91        = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_78_add__numeral__left,axiom,
% 9.03/4.91      ! [V: num,W: num,Z: nat] :
% 9.03/4.91        ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 9.03/4.91        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_79_add__numeral__left,axiom,
% 9.03/4.91      ! [V: num,W: num,Z: int] :
% 9.03/4.91        ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 9.03/4.91        = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_80_numeral__plus__numeral,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 9.03/4.91        = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_81_numeral__plus__numeral,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.03/4.91        = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_82_numeral__plus__numeral,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.03/4.91        = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_83_numeral__plus__numeral,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.03/4.91        = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_84_numeral__plus__numeral,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.03/4.91        = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_85__C3_Ohyps_C_I4_J,axiom,
% 9.03/4.91      ( deg
% 9.03/4.91      = ( plus_plus_nat @ na @ m ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_86__C3_OIH_C_I1_J,axiom,
% 9.03/4.91      ! [X2: vEBT_VEBT] :
% 9.03/4.91        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 9.03/4.91       => ( ( vEBT_invar_vebt @ X2 @ na )
% 9.03/4.91          & ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X2 ) )
% 9.03/4.91            = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ na ) ) ) ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_87_numerals_I1_J,axiom,
% 9.03/4.91      ( ( numeral_numeral_nat @ one )
% 9.03/4.91      = one_one_nat ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_88_nat__1__add__1,axiom,
% 9.03/4.91      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 9.03/4.91      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_89_add__One__commute,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( plus_plus_num @ one @ N )
% 9.03/4.91        = ( plus_plus_num @ N @ one ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_90_VEBT__internal_Oheight_Osimps_I2_J,axiom,
% 9.03/4.91      ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.03/4.91        ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ Deg @ TreeList @ Summary ) )
% 9.03/4.91        = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_91_is__num__normalize_I1_J,axiom,
% 9.03/4.91      ! [A: real,B: real,C: real] :
% 9.03/4.91        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.03/4.91        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_92_is__num__normalize_I1_J,axiom,
% 9.03/4.91      ! [A: rat,B: rat,C: rat] :
% 9.03/4.91        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.03/4.91        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_93_is__num__normalize_I1_J,axiom,
% 9.03/4.91      ! [A: int,B: int,C: int] :
% 9.03/4.91        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.03/4.91        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_94_is__num__normalize_I1_J,axiom,
% 9.03/4.91      ! [A: complex,B: complex,C: complex] :
% 9.03/4.91        ( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 9.03/4.91        = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_95_one__plus__numeral__commute,axiom,
% 9.03/4.91      ! [X: num] :
% 9.03/4.91        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
% 9.03/4.91        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_96_one__plus__numeral__commute,axiom,
% 9.03/4.91      ! [X: num] :
% 9.03/4.91        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 9.03/4.91        = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_97_one__plus__numeral__commute,axiom,
% 9.03/4.91      ! [X: num] :
% 9.03/4.91        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 9.03/4.91        = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_98_one__plus__numeral__commute,axiom,
% 9.03/4.91      ! [X: num] :
% 9.03/4.91        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 9.03/4.91        = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_99_one__plus__numeral__commute,axiom,
% 9.03/4.91      ! [X: num] :
% 9.03/4.91        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 9.03/4.91        = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_100_numeral__Bit0,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 9.03/4.91        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_101_numeral__Bit0,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 9.03/4.91        = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_102_numeral__Bit0,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 9.03/4.91        = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_103_numeral__Bit0,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 9.03/4.91        = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_104_numeral__Bit0,axiom,
% 9.03/4.91      ! [N: num] :
% 9.03/4.91        ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 9.03/4.91        = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_105_numeral__One,axiom,
% 9.03/4.91      ( ( numera6690914467698888265omplex @ one )
% 9.03/4.91      = one_one_complex ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_106_numeral__One,axiom,
% 9.03/4.91      ( ( numeral_numeral_real @ one )
% 9.03/4.91      = one_one_real ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_107_numeral__One,axiom,
% 9.03/4.91      ( ( numeral_numeral_rat @ one )
% 9.03/4.91      = one_one_rat ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_108_numeral__One,axiom,
% 9.03/4.91      ( ( numeral_numeral_nat @ one )
% 9.03/4.91      = one_one_nat ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_109_numeral__One,axiom,
% 9.03/4.91      ( ( numeral_numeral_int @ one )
% 9.03/4.91      = one_one_int ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_110_setceilmax,axiom,
% 9.03/4.91      ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
% 9.03/4.91        ( ( vEBT_invar_vebt @ S @ M )
% 9.03/4.91       => ( ! [X3: vEBT_VEBT] :
% 9.03/4.91              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
% 9.03/4.91             => ( vEBT_invar_vebt @ X3 @ N ) )
% 9.03/4.91         => ( ( M
% 9.03/4.91              = ( suc @ N ) )
% 9.03/4.91           => ( ! [X3: vEBT_VEBT] :
% 9.03/4.91                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
% 9.03/4.91                 => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X3 ) )
% 9.03/4.91                    = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 9.03/4.91             => ( ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ S ) )
% 9.03/4.91                  = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) )
% 9.03/4.91               => ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ S @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
% 9.03/4.91                  = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_111_semiring__norm_I2_J,axiom,
% 9.03/4.91      ( ( plus_plus_num @ one @ one )
% 9.03/4.91      = ( bit0 @ one ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_112_semiring__norm_I6_J,axiom,
% 9.03/4.91      ! [M: num,N: num] :
% 9.03/4.91        ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.03/4.91        = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/4.91  
% 9.03/4.91  thf(fact_113_int__eq__iff__numeral,axiom,
% 9.03/4.92      ! [M: nat,V: num] :
% 9.03/4.92        ( ( ( semiri1314217659103216013at_int @ M )
% 9.03/4.92          = ( numeral_numeral_int @ V ) )
% 9.03/4.92        = ( M
% 9.03/4.92          = ( numeral_numeral_nat @ V ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_114_image__insert,axiom,
% 9.03/4.92      ! [F: vEBT_VEBT > vEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( image_3375948659692109573T_VEBT @ F @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 9.03/4.92        = ( insert_VEBT_VEBT @ ( F @ A ) @ ( image_3375948659692109573T_VEBT @ F @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_115_image__insert,axiom,
% 9.03/4.92      ! [F: vEBT_VEBT > nat,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( image_VEBT_VEBT_nat @ F @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 9.03/4.92        = ( insert_nat @ ( F @ A ) @ ( image_VEBT_VEBT_nat @ F @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_116_image__insert,axiom,
% 9.03/4.92      ! [F: vEBT_VEBT > int,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( image_VEBT_VEBT_int @ F @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 9.03/4.92        = ( insert_int @ ( F @ A ) @ ( image_VEBT_VEBT_int @ F @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_117_image__insert,axiom,
% 9.03/4.92      ! [F: vEBT_VEBT > real,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( image_VEBT_VEBT_real @ F @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 9.03/4.92        = ( insert_real @ ( F @ A ) @ ( image_VEBT_VEBT_real @ F @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_118_image__insert,axiom,
% 9.03/4.92      ! [F: vEBT_VEBT > $o,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( image_VEBT_VEBT_o @ F @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 9.03/4.92        = ( insert_o @ ( F @ A ) @ ( image_VEBT_VEBT_o @ F @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_119_image__insert,axiom,
% 9.03/4.92      ! [F: nat > vEBT_VEBT,A: nat,B2: set_nat] :
% 9.03/4.92        ( ( image_nat_VEBT_VEBT @ F @ ( insert_nat @ A @ B2 ) )
% 9.03/4.92        = ( insert_VEBT_VEBT @ ( F @ A ) @ ( image_nat_VEBT_VEBT @ F @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_120_image__insert,axiom,
% 9.03/4.92      ! [F: nat > nat,A: nat,B2: set_nat] :
% 9.03/4.92        ( ( image_nat_nat @ F @ ( insert_nat @ A @ B2 ) )
% 9.03/4.92        = ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_121_image__insert,axiom,
% 9.03/4.92      ! [F: nat > int,A: nat,B2: set_nat] :
% 9.03/4.92        ( ( image_nat_int @ F @ ( insert_nat @ A @ B2 ) )
% 9.03/4.92        = ( insert_int @ ( F @ A ) @ ( image_nat_int @ F @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_122_image__insert,axiom,
% 9.03/4.92      ! [F: nat > real,A: nat,B2: set_nat] :
% 9.03/4.92        ( ( image_nat_real @ F @ ( insert_nat @ A @ B2 ) )
% 9.03/4.92        = ( insert_real @ ( F @ A ) @ ( image_nat_real @ F @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_123_image__insert,axiom,
% 9.03/4.92      ! [F: nat > $o,A: nat,B2: set_nat] :
% 9.03/4.92        ( ( image_nat_o @ F @ ( insert_nat @ A @ B2 ) )
% 9.03/4.92        = ( insert_o @ ( F @ A ) @ ( image_nat_o @ F @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_124_insert__image,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,F: complex > vEBT_VEBT] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( ( insert_VEBT_VEBT @ ( F @ X ) @ ( image_932796090930683071T_VEBT @ F @ A2 ) )
% 9.03/4.92          = ( image_932796090930683071T_VEBT @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_125_insert__image,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,F: complex > nat] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( ( insert_nat @ ( F @ X ) @ ( image_complex_nat @ F @ A2 ) )
% 9.03/4.92          = ( image_complex_nat @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_126_insert__image,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,F: complex > int] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( ( insert_int @ ( F @ X ) @ ( image_complex_int @ F @ A2 ) )
% 9.03/4.92          = ( image_complex_int @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_127_insert__image,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,F: complex > real] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( ( insert_real @ ( F @ X ) @ ( image_complex_real @ F @ A2 ) )
% 9.03/4.92          = ( image_complex_real @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_128_insert__image,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,F: complex > $o] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( ( insert_o @ ( F @ X ) @ ( image_complex_o @ F @ A2 ) )
% 9.03/4.92          = ( image_complex_o @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_129_insert__image,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,F: real > vEBT_VEBT] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( ( insert_VEBT_VEBT @ ( F @ X ) @ ( image_real_VEBT_VEBT @ F @ A2 ) )
% 9.03/4.92          = ( image_real_VEBT_VEBT @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_130_insert__image,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,F: real > nat] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( ( insert_nat @ ( F @ X ) @ ( image_real_nat @ F @ A2 ) )
% 9.03/4.92          = ( image_real_nat @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_131_insert__image,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,F: real > int] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( ( insert_int @ ( F @ X ) @ ( image_real_int @ F @ A2 ) )
% 9.03/4.92          = ( image_real_int @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_132_insert__image,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,F: real > real] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( ( insert_real @ ( F @ X ) @ ( image_real_real @ F @ A2 ) )
% 9.03/4.92          = ( image_real_real @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_133_insert__image,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,F: real > $o] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( ( insert_o @ ( F @ X ) @ ( image_real_o @ F @ A2 ) )
% 9.03/4.92          = ( image_real_o @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_134_semiring__norm_I83_J,axiom,
% 9.03/4.92      ! [N: num] :
% 9.03/4.92        ( one
% 9.03/4.92       != ( bit0 @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_135_semiring__norm_I85_J,axiom,
% 9.03/4.92      ! [M: num] :
% 9.03/4.92        ( ( bit0 @ M )
% 9.03/4.92       != one ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_136_int__ops_I2_J,axiom,
% 9.03/4.92      ( ( semiri1314217659103216013at_int @ one_one_nat )
% 9.03/4.92      = one_one_int ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_137_zadd__int__left,axiom,
% 9.03/4.92      ! [M: nat,N: nat,Z: int] :
% 9.03/4.92        ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 9.03/4.92        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_138_mem__Collect__eq,axiom,
% 9.03/4.92      ! [A: complex,P: complex > $o] :
% 9.03/4.92        ( ( member_complex @ A @ ( collect_complex @ P ) )
% 9.03/4.92        = ( P @ A ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_139_mem__Collect__eq,axiom,
% 9.03/4.92      ! [A: real,P: real > $o] :
% 9.03/4.92        ( ( member_real @ A @ ( collect_real @ P ) )
% 9.03/4.92        = ( P @ A ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_140_mem__Collect__eq,axiom,
% 9.03/4.92      ! [A: set_nat,P: set_nat > $o] :
% 9.03/4.92        ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 9.03/4.92        = ( P @ A ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_141_mem__Collect__eq,axiom,
% 9.03/4.92      ! [A: nat,P: nat > $o] :
% 9.03/4.92        ( ( member_nat @ A @ ( collect_nat @ P ) )
% 9.03/4.92        = ( P @ A ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_142_mem__Collect__eq,axiom,
% 9.03/4.92      ! [A: int,P: int > $o] :
% 9.03/4.92        ( ( member_int @ A @ ( collect_int @ P ) )
% 9.03/4.92        = ( P @ A ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_143_Collect__mem__eq,axiom,
% 9.03/4.92      ! [A2: set_complex] :
% 9.03/4.92        ( ( collect_complex
% 9.03/4.92          @ ^ [X4: complex] : ( member_complex @ X4 @ A2 ) )
% 9.03/4.92        = A2 ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_144_Collect__mem__eq,axiom,
% 9.03/4.92      ! [A2: set_real] :
% 9.03/4.92        ( ( collect_real
% 9.03/4.92          @ ^ [X4: real] : ( member_real @ X4 @ A2 ) )
% 9.03/4.92        = A2 ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_145_Collect__mem__eq,axiom,
% 9.03/4.92      ! [A2: set_set_nat] :
% 9.03/4.92        ( ( collect_set_nat
% 9.03/4.92          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A2 ) )
% 9.03/4.92        = A2 ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_146_Collect__mem__eq,axiom,
% 9.03/4.92      ! [A2: set_nat] :
% 9.03/4.92        ( ( collect_nat
% 9.03/4.92          @ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
% 9.03/4.92        = A2 ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_147_Collect__mem__eq,axiom,
% 9.03/4.92      ! [A2: set_int] :
% 9.03/4.92        ( ( collect_int
% 9.03/4.92          @ ^ [X4: int] : ( member_int @ X4 @ A2 ) )
% 9.03/4.92        = A2 ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_148_Collect__cong,axiom,
% 9.03/4.92      ! [P: complex > $o,Q: complex > $o] :
% 9.03/4.92        ( ! [X3: complex] :
% 9.03/4.92            ( ( P @ X3 )
% 9.03/4.92            = ( Q @ X3 ) )
% 9.03/4.92       => ( ( collect_complex @ P )
% 9.03/4.92          = ( collect_complex @ Q ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_149_Collect__cong,axiom,
% 9.03/4.92      ! [P: real > $o,Q: real > $o] :
% 9.03/4.92        ( ! [X3: real] :
% 9.03/4.92            ( ( P @ X3 )
% 9.03/4.92            = ( Q @ X3 ) )
% 9.03/4.92       => ( ( collect_real @ P )
% 9.03/4.92          = ( collect_real @ Q ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_150_Collect__cong,axiom,
% 9.03/4.92      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.03/4.92        ( ! [X3: set_nat] :
% 9.03/4.92            ( ( P @ X3 )
% 9.03/4.92            = ( Q @ X3 ) )
% 9.03/4.92       => ( ( collect_set_nat @ P )
% 9.03/4.92          = ( collect_set_nat @ Q ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_151_Collect__cong,axiom,
% 9.03/4.92      ! [P: nat > $o,Q: nat > $o] :
% 9.03/4.92        ( ! [X3: nat] :
% 9.03/4.92            ( ( P @ X3 )
% 9.03/4.92            = ( Q @ X3 ) )
% 9.03/4.92       => ( ( collect_nat @ P )
% 9.03/4.92          = ( collect_nat @ Q ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_152_Collect__cong,axiom,
% 9.03/4.92      ! [P: int > $o,Q: int > $o] :
% 9.03/4.92        ( ! [X3: int] :
% 9.03/4.92            ( ( P @ X3 )
% 9.03/4.92            = ( Q @ X3 ) )
% 9.03/4.92       => ( ( collect_int @ P )
% 9.03/4.92          = ( collect_int @ Q ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_153_int__plus,axiom,
% 9.03/4.92      ! [N: nat,M: nat] :
% 9.03/4.92        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 9.03/4.92        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_154_int__ops_I5_J,axiom,
% 9.03/4.92      ! [A: nat,B: nat] :
% 9.03/4.92        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 9.03/4.92        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_155_even__odd__cases,axiom,
% 9.03/4.92      ! [X: nat] :
% 9.03/4.92        ( ! [N2: nat] :
% 9.03/4.92            ( X
% 9.03/4.92           != ( plus_plus_nat @ N2 @ N2 ) )
% 9.03/4.92       => ~ ! [N2: nat] :
% 9.03/4.92              ( X
% 9.03/4.92             != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_156__C3_Ohyps_C_I3_J,axiom,
% 9.03/4.92      ( m
% 9.03/4.92      = ( suc @ na ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_157_verit__eq__simplify_I8_J,axiom,
% 9.03/4.92      ! [X22: num,Y2: num] :
% 9.03/4.92        ( ( ( bit0 @ X22 )
% 9.03/4.92          = ( bit0 @ Y2 ) )
% 9.03/4.92        = ( X22 = Y2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_158_semiring__norm_I87_J,axiom,
% 9.03/4.92      ! [M: num,N: num] :
% 9.03/4.92        ( ( ( bit0 @ M )
% 9.03/4.92          = ( bit0 @ N ) )
% 9.03/4.92        = ( M = N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_159_image__eqI,axiom,
% 9.03/4.92      ! [B: complex,F: complex > complex,X: complex,A2: set_complex] :
% 9.03/4.92        ( ( B
% 9.03/4.92          = ( F @ X ) )
% 9.03/4.92       => ( ( member_complex @ X @ A2 )
% 9.03/4.92         => ( member_complex @ B @ ( image_1468599708987790691omplex @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_160_image__eqI,axiom,
% 9.03/4.92      ! [B: real,F: complex > real,X: complex,A2: set_complex] :
% 9.03/4.92        ( ( B
% 9.03/4.92          = ( F @ X ) )
% 9.03/4.92       => ( ( member_complex @ X @ A2 )
% 9.03/4.92         => ( member_real @ B @ ( image_complex_real @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_161_image__eqI,axiom,
% 9.03/4.92      ! [B: nat,F: complex > nat,X: complex,A2: set_complex] :
% 9.03/4.92        ( ( B
% 9.03/4.92          = ( F @ X ) )
% 9.03/4.92       => ( ( member_complex @ X @ A2 )
% 9.03/4.92         => ( member_nat @ B @ ( image_complex_nat @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_162_image__eqI,axiom,
% 9.03/4.92      ! [B: int,F: complex > int,X: complex,A2: set_complex] :
% 9.03/4.92        ( ( B
% 9.03/4.92          = ( F @ X ) )
% 9.03/4.92       => ( ( member_complex @ X @ A2 )
% 9.03/4.92         => ( member_int @ B @ ( image_complex_int @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_163_image__eqI,axiom,
% 9.03/4.92      ! [B: complex,F: real > complex,X: real,A2: set_real] :
% 9.03/4.92        ( ( B
% 9.03/4.92          = ( F @ X ) )
% 9.03/4.92       => ( ( member_real @ X @ A2 )
% 9.03/4.92         => ( member_complex @ B @ ( image_real_complex @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_164_image__eqI,axiom,
% 9.03/4.92      ! [B: real,F: real > real,X: real,A2: set_real] :
% 9.03/4.92        ( ( B
% 9.03/4.92          = ( F @ X ) )
% 9.03/4.92       => ( ( member_real @ X @ A2 )
% 9.03/4.92         => ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_165_image__eqI,axiom,
% 9.03/4.92      ! [B: nat,F: real > nat,X: real,A2: set_real] :
% 9.03/4.92        ( ( B
% 9.03/4.92          = ( F @ X ) )
% 9.03/4.92       => ( ( member_real @ X @ A2 )
% 9.03/4.92         => ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_166_image__eqI,axiom,
% 9.03/4.92      ! [B: int,F: real > int,X: real,A2: set_real] :
% 9.03/4.92        ( ( B
% 9.03/4.92          = ( F @ X ) )
% 9.03/4.92       => ( ( member_real @ X @ A2 )
% 9.03/4.92         => ( member_int @ B @ ( image_real_int @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_167_image__eqI,axiom,
% 9.03/4.92      ! [B: complex,F: nat > complex,X: nat,A2: set_nat] :
% 9.03/4.92        ( ( B
% 9.03/4.92          = ( F @ X ) )
% 9.03/4.92       => ( ( member_nat @ X @ A2 )
% 9.03/4.92         => ( member_complex @ B @ ( image_nat_complex @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_168_image__eqI,axiom,
% 9.03/4.92      ! [B: real,F: nat > real,X: nat,A2: set_nat] :
% 9.03/4.92        ( ( B
% 9.03/4.92          = ( F @ X ) )
% 9.03/4.92       => ( ( member_nat @ X @ A2 )
% 9.03/4.92         => ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_169_nat_Oinject,axiom,
% 9.03/4.92      ! [X22: nat,Y2: nat] :
% 9.03/4.92        ( ( ( suc @ X22 )
% 9.03/4.92          = ( suc @ Y2 ) )
% 9.03/4.92        = ( X22 = Y2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_170_old_Onat_Oinject,axiom,
% 9.03/4.92      ! [Nat: nat,Nat2: nat] :
% 9.03/4.92        ( ( ( suc @ Nat )
% 9.03/4.92          = ( suc @ Nat2 ) )
% 9.03/4.92        = ( Nat = Nat2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_171_insert__absorb2,axiom,
% 9.03/4.92      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( insert_VEBT_VEBT @ X @ ( insert_VEBT_VEBT @ X @ A2 ) )
% 9.03/4.92        = ( insert_VEBT_VEBT @ X @ A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_172_insert__absorb2,axiom,
% 9.03/4.92      ! [X: nat,A2: set_nat] :
% 9.03/4.92        ( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
% 9.03/4.92        = ( insert_nat @ X @ A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_173_insert__absorb2,axiom,
% 9.03/4.92      ! [X: int,A2: set_int] :
% 9.03/4.92        ( ( insert_int @ X @ ( insert_int @ X @ A2 ) )
% 9.03/4.92        = ( insert_int @ X @ A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_174_insert__absorb2,axiom,
% 9.03/4.92      ! [X: real,A2: set_real] :
% 9.03/4.92        ( ( insert_real @ X @ ( insert_real @ X @ A2 ) )
% 9.03/4.92        = ( insert_real @ X @ A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_175_insert__absorb2,axiom,
% 9.03/4.92      ! [X: $o,A2: set_o] :
% 9.03/4.92        ( ( insert_o @ X @ ( insert_o @ X @ A2 ) )
% 9.03/4.92        = ( insert_o @ X @ A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_176_insert__iff,axiom,
% 9.03/4.92      ! [A: vEBT_VEBT,B: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ A2 ) )
% 9.03/4.92        = ( ( A = B )
% 9.03/4.92          | ( member_VEBT_VEBT @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_177_insert__iff,axiom,
% 9.03/4.92      ! [A: $o,B: $o,A2: set_o] :
% 9.03/4.92        ( ( member_o @ A @ ( insert_o @ B @ A2 ) )
% 9.03/4.92        = ( ( A = B )
% 9.03/4.92          | ( member_o @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_178_insert__iff,axiom,
% 9.03/4.92      ! [A: complex,B: complex,A2: set_complex] :
% 9.03/4.92        ( ( member_complex @ A @ ( insert_complex @ B @ A2 ) )
% 9.03/4.92        = ( ( A = B )
% 9.03/4.92          | ( member_complex @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_179_insert__iff,axiom,
% 9.03/4.92      ! [A: real,B: real,A2: set_real] :
% 9.03/4.92        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 9.03/4.92        = ( ( A = B )
% 9.03/4.92          | ( member_real @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_180_insert__iff,axiom,
% 9.03/4.92      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 9.03/4.92        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 9.03/4.92        = ( ( A = B )
% 9.03/4.92          | ( member_set_nat @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_181_insert__iff,axiom,
% 9.03/4.92      ! [A: nat,B: nat,A2: set_nat] :
% 9.03/4.92        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 9.03/4.92        = ( ( A = B )
% 9.03/4.92          | ( member_nat @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_182_insert__iff,axiom,
% 9.03/4.92      ! [A: int,B: int,A2: set_int] :
% 9.03/4.92        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 9.03/4.92        = ( ( A = B )
% 9.03/4.92          | ( member_int @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_183_insertCI,axiom,
% 9.03/4.92      ! [A: vEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
% 9.03/4.92        ( ( ~ ( member_VEBT_VEBT @ A @ B2 )
% 9.03/4.92         => ( A = B ) )
% 9.03/4.92       => ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_184_insertCI,axiom,
% 9.03/4.92      ! [A: $o,B2: set_o,B: $o] :
% 9.03/4.92        ( ( ~ ( member_o @ A @ B2 )
% 9.03/4.92         => ( A = B ) )
% 9.03/4.92       => ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_185_insertCI,axiom,
% 9.03/4.92      ! [A: complex,B2: set_complex,B: complex] :
% 9.03/4.92        ( ( ~ ( member_complex @ A @ B2 )
% 9.03/4.92         => ( A = B ) )
% 9.03/4.92       => ( member_complex @ A @ ( insert_complex @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_186_insertCI,axiom,
% 9.03/4.92      ! [A: real,B2: set_real,B: real] :
% 9.03/4.92        ( ( ~ ( member_real @ A @ B2 )
% 9.03/4.92         => ( A = B ) )
% 9.03/4.92       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_187_insertCI,axiom,
% 9.03/4.92      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 9.03/4.92        ( ( ~ ( member_set_nat @ A @ B2 )
% 9.03/4.92         => ( A = B ) )
% 9.03/4.92       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_188_insertCI,axiom,
% 9.03/4.92      ! [A: nat,B2: set_nat,B: nat] :
% 9.03/4.92        ( ( ~ ( member_nat @ A @ B2 )
% 9.03/4.92         => ( A = B ) )
% 9.03/4.92       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_189_insertCI,axiom,
% 9.03/4.92      ! [A: int,B2: set_int,B: int] :
% 9.03/4.92        ( ( ~ ( member_int @ A @ B2 )
% 9.03/4.92         => ( A = B ) )
% 9.03/4.92       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_190_add__Suc__right,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 9.03/4.92        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_191_of__nat__Suc,axiom,
% 9.03/4.92      ! [M: nat] :
% 9.03/4.92        ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 9.03/4.92        = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_192_of__nat__Suc,axiom,
% 9.03/4.92      ! [M: nat] :
% 9.03/4.92        ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 9.03/4.92        = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_193_of__nat__Suc,axiom,
% 9.03/4.92      ! [M: nat] :
% 9.03/4.92        ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 9.03/4.92        = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_194_of__nat__Suc,axiom,
% 9.03/4.92      ! [M: nat] :
% 9.03/4.92        ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 9.03/4.92        = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_195_of__nat__Suc,axiom,
% 9.03/4.92      ! [M: nat] :
% 9.03/4.92        ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 9.03/4.92        = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_196_Suc__numeral,axiom,
% 9.03/4.92      ! [N: num] :
% 9.03/4.92        ( ( suc @ ( numeral_numeral_nat @ N ) )
% 9.03/4.92        = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_197_add__2__eq__Suc,axiom,
% 9.03/4.92      ! [N: nat] :
% 9.03/4.92        ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.03/4.92        = ( suc @ ( suc @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_198_add__2__eq__Suc_H,axiom,
% 9.03/4.92      ! [N: nat] :
% 9.03/4.92        ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.92        = ( suc @ ( suc @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_199_Suc__1,axiom,
% 9.03/4.92      ( ( suc @ one_one_nat )
% 9.03/4.92      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_200_Suc__inject,axiom,
% 9.03/4.92      ! [X: nat,Y: nat] :
% 9.03/4.92        ( ( ( suc @ X )
% 9.03/4.92          = ( suc @ Y ) )
% 9.03/4.92       => ( X = Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_201_n__not__Suc__n,axiom,
% 9.03/4.92      ! [N: nat] :
% 9.03/4.92        ( N
% 9.03/4.92       != ( suc @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_202_add__Suc__shift,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 9.03/4.92        = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_203_add__Suc,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 9.03/4.92        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_204_nat__arith_Osuc1,axiom,
% 9.03/4.92      ! [A2: nat,K: nat,A: nat] :
% 9.03/4.92        ( ( A2
% 9.03/4.92          = ( plus_plus_nat @ K @ A ) )
% 9.03/4.92       => ( ( suc @ A2 )
% 9.03/4.92          = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_205_Suc__eq__plus1__left,axiom,
% 9.03/4.92      ( suc
% 9.03/4.92      = ( plus_plus_nat @ one_one_nat ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_206_plus__1__eq__Suc,axiom,
% 9.03/4.92      ( ( plus_plus_nat @ one_one_nat )
% 9.03/4.92      = suc ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_207_Suc__eq__plus1,axiom,
% 9.03/4.92      ( suc
% 9.03/4.92      = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_208_int__ops_I4_J,axiom,
% 9.03/4.92      ! [A: nat] :
% 9.03/4.92        ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 9.03/4.92        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_209_int__Suc,axiom,
% 9.03/4.92      ! [N: nat] :
% 9.03/4.92        ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 9.03/4.92        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_210_rev__image__eqI,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,B: complex,F: complex > complex] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( ( B
% 9.03/4.92            = ( F @ X ) )
% 9.03/4.92         => ( member_complex @ B @ ( image_1468599708987790691omplex @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_211_rev__image__eqI,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,B: real,F: complex > real] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( ( B
% 9.03/4.92            = ( F @ X ) )
% 9.03/4.92         => ( member_real @ B @ ( image_complex_real @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_212_rev__image__eqI,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,B: nat,F: complex > nat] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( ( B
% 9.03/4.92            = ( F @ X ) )
% 9.03/4.92         => ( member_nat @ B @ ( image_complex_nat @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_213_rev__image__eqI,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,B: int,F: complex > int] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( ( B
% 9.03/4.92            = ( F @ X ) )
% 9.03/4.92         => ( member_int @ B @ ( image_complex_int @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_214_rev__image__eqI,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,B: complex,F: real > complex] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( ( B
% 9.03/4.92            = ( F @ X ) )
% 9.03/4.92         => ( member_complex @ B @ ( image_real_complex @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_215_rev__image__eqI,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,B: real,F: real > real] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( ( B
% 9.03/4.92            = ( F @ X ) )
% 9.03/4.92         => ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_216_rev__image__eqI,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,B: nat,F: real > nat] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( ( B
% 9.03/4.92            = ( F @ X ) )
% 9.03/4.92         => ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_217_rev__image__eqI,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,B: int,F: real > int] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( ( B
% 9.03/4.92            = ( F @ X ) )
% 9.03/4.92         => ( member_int @ B @ ( image_real_int @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_218_rev__image__eqI,axiom,
% 9.03/4.92      ! [X: nat,A2: set_nat,B: complex,F: nat > complex] :
% 9.03/4.92        ( ( member_nat @ X @ A2 )
% 9.03/4.92       => ( ( B
% 9.03/4.92            = ( F @ X ) )
% 9.03/4.92         => ( member_complex @ B @ ( image_nat_complex @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_219_rev__image__eqI,axiom,
% 9.03/4.92      ! [X: nat,A2: set_nat,B: real,F: nat > real] :
% 9.03/4.92        ( ( member_nat @ X @ A2 )
% 9.03/4.92       => ( ( B
% 9.03/4.92            = ( F @ X ) )
% 9.03/4.92         => ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_220_ball__imageD,axiom,
% 9.03/4.92      ! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
% 9.03/4.92        ( ! [X3: set_nat] :
% 9.03/4.92            ( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A2 ) )
% 9.03/4.92           => ( P @ X3 ) )
% 9.03/4.92       => ! [X2: nat] :
% 9.03/4.92            ( ( member_nat @ X2 @ A2 )
% 9.03/4.92           => ( P @ ( F @ X2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_221_ball__imageD,axiom,
% 9.03/4.92      ! [F: nat > nat,A2: set_nat,P: nat > $o] :
% 9.03/4.92        ( ! [X3: nat] :
% 9.03/4.92            ( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
% 9.03/4.92           => ( P @ X3 ) )
% 9.03/4.92       => ! [X2: nat] :
% 9.03/4.92            ( ( member_nat @ X2 @ A2 )
% 9.03/4.92           => ( P @ ( F @ X2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_222_ball__imageD,axiom,
% 9.03/4.92      ! [F: nat > int,A2: set_nat,P: int > $o] :
% 9.03/4.92        ( ! [X3: int] :
% 9.03/4.92            ( ( member_int @ X3 @ ( image_nat_int @ F @ A2 ) )
% 9.03/4.92           => ( P @ X3 ) )
% 9.03/4.92       => ! [X2: nat] :
% 9.03/4.92            ( ( member_nat @ X2 @ A2 )
% 9.03/4.92           => ( P @ ( F @ X2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_223_ball__imageD,axiom,
% 9.03/4.92      ! [F: int > nat,A2: set_int,P: nat > $o] :
% 9.03/4.92        ( ! [X3: nat] :
% 9.03/4.92            ( ( member_nat @ X3 @ ( image_int_nat @ F @ A2 ) )
% 9.03/4.92           => ( P @ X3 ) )
% 9.03/4.92       => ! [X2: int] :
% 9.03/4.92            ( ( member_int @ X2 @ A2 )
% 9.03/4.92           => ( P @ ( F @ X2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_224_ball__imageD,axiom,
% 9.03/4.92      ! [F: int > int,A2: set_int,P: int > $o] :
% 9.03/4.92        ( ! [X3: int] :
% 9.03/4.92            ( ( member_int @ X3 @ ( image_int_int @ F @ A2 ) )
% 9.03/4.92           => ( P @ X3 ) )
% 9.03/4.92       => ! [X2: int] :
% 9.03/4.92            ( ( member_int @ X2 @ A2 )
% 9.03/4.92           => ( P @ ( F @ X2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_225_image__cong,axiom,
% 9.03/4.92      ! [M2: set_nat,N4: set_nat,F: nat > set_nat,G: nat > set_nat] :
% 9.03/4.92        ( ( M2 = N4 )
% 9.03/4.92       => ( ! [X3: nat] :
% 9.03/4.92              ( ( member_nat @ X3 @ N4 )
% 9.03/4.92             => ( ( F @ X3 )
% 9.03/4.92                = ( G @ X3 ) ) )
% 9.03/4.92         => ( ( image_nat_set_nat @ F @ M2 )
% 9.03/4.92            = ( image_nat_set_nat @ G @ N4 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_226_image__cong,axiom,
% 9.03/4.92      ! [M2: set_nat,N4: set_nat,F: nat > nat,G: nat > nat] :
% 9.03/4.92        ( ( M2 = N4 )
% 9.03/4.92       => ( ! [X3: nat] :
% 9.03/4.92              ( ( member_nat @ X3 @ N4 )
% 9.03/4.92             => ( ( F @ X3 )
% 9.03/4.92                = ( G @ X3 ) ) )
% 9.03/4.92         => ( ( image_nat_nat @ F @ M2 )
% 9.03/4.92            = ( image_nat_nat @ G @ N4 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_227_image__cong,axiom,
% 9.03/4.92      ! [M2: set_nat,N4: set_nat,F: nat > int,G: nat > int] :
% 9.03/4.92        ( ( M2 = N4 )
% 9.03/4.92       => ( ! [X3: nat] :
% 9.03/4.92              ( ( member_nat @ X3 @ N4 )
% 9.03/4.92             => ( ( F @ X3 )
% 9.03/4.92                = ( G @ X3 ) ) )
% 9.03/4.92         => ( ( image_nat_int @ F @ M2 )
% 9.03/4.92            = ( image_nat_int @ G @ N4 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_228_image__cong,axiom,
% 9.03/4.92      ! [M2: set_int,N4: set_int,F: int > nat,G: int > nat] :
% 9.03/4.92        ( ( M2 = N4 )
% 9.03/4.92       => ( ! [X3: int] :
% 9.03/4.92              ( ( member_int @ X3 @ N4 )
% 9.03/4.92             => ( ( F @ X3 )
% 9.03/4.92                = ( G @ X3 ) ) )
% 9.03/4.92         => ( ( image_int_nat @ F @ M2 )
% 9.03/4.92            = ( image_int_nat @ G @ N4 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_229_image__cong,axiom,
% 9.03/4.92      ! [M2: set_int,N4: set_int,F: int > int,G: int > int] :
% 9.03/4.92        ( ( M2 = N4 )
% 9.03/4.92       => ( ! [X3: int] :
% 9.03/4.92              ( ( member_int @ X3 @ N4 )
% 9.03/4.92             => ( ( F @ X3 )
% 9.03/4.92                = ( G @ X3 ) ) )
% 9.03/4.92         => ( ( image_int_int @ F @ M2 )
% 9.03/4.92            = ( image_int_int @ G @ N4 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_230_bex__imageD,axiom,
% 9.03/4.92      ! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
% 9.03/4.92        ( ? [X2: set_nat] :
% 9.03/4.92            ( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A2 ) )
% 9.03/4.92            & ( P @ X2 ) )
% 9.03/4.92       => ? [X3: nat] :
% 9.03/4.92            ( ( member_nat @ X3 @ A2 )
% 9.03/4.92            & ( P @ ( F @ X3 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_231_bex__imageD,axiom,
% 9.03/4.92      ! [F: nat > nat,A2: set_nat,P: nat > $o] :
% 9.03/4.92        ( ? [X2: nat] :
% 9.03/4.92            ( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
% 9.03/4.92            & ( P @ X2 ) )
% 9.03/4.92       => ? [X3: nat] :
% 9.03/4.92            ( ( member_nat @ X3 @ A2 )
% 9.03/4.92            & ( P @ ( F @ X3 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_232_bex__imageD,axiom,
% 9.03/4.92      ! [F: nat > int,A2: set_nat,P: int > $o] :
% 9.03/4.92        ( ? [X2: int] :
% 9.03/4.92            ( ( member_int @ X2 @ ( image_nat_int @ F @ A2 ) )
% 9.03/4.92            & ( P @ X2 ) )
% 9.03/4.92       => ? [X3: nat] :
% 9.03/4.92            ( ( member_nat @ X3 @ A2 )
% 9.03/4.92            & ( P @ ( F @ X3 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_233_bex__imageD,axiom,
% 9.03/4.92      ! [F: int > nat,A2: set_int,P: nat > $o] :
% 9.03/4.92        ( ? [X2: nat] :
% 9.03/4.92            ( ( member_nat @ X2 @ ( image_int_nat @ F @ A2 ) )
% 9.03/4.92            & ( P @ X2 ) )
% 9.03/4.92       => ? [X3: int] :
% 9.03/4.92            ( ( member_int @ X3 @ A2 )
% 9.03/4.92            & ( P @ ( F @ X3 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_234_bex__imageD,axiom,
% 9.03/4.92      ! [F: int > int,A2: set_int,P: int > $o] :
% 9.03/4.92        ( ? [X2: int] :
% 9.03/4.92            ( ( member_int @ X2 @ ( image_int_int @ F @ A2 ) )
% 9.03/4.92            & ( P @ X2 ) )
% 9.03/4.92       => ? [X3: int] :
% 9.03/4.92            ( ( member_int @ X3 @ A2 )
% 9.03/4.92            & ( P @ ( F @ X3 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_235_image__iff,axiom,
% 9.03/4.92      ! [Z: set_nat,F: nat > set_nat,A2: set_nat] :
% 9.03/4.92        ( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A2 ) )
% 9.03/4.92        = ( ? [X4: nat] :
% 9.03/4.92              ( ( member_nat @ X4 @ A2 )
% 9.03/4.92              & ( Z
% 9.03/4.92                = ( F @ X4 ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_236_image__iff,axiom,
% 9.03/4.92      ! [Z: nat,F: nat > nat,A2: set_nat] :
% 9.03/4.92        ( ( member_nat @ Z @ ( image_nat_nat @ F @ A2 ) )
% 9.03/4.92        = ( ? [X4: nat] :
% 9.03/4.92              ( ( member_nat @ X4 @ A2 )
% 9.03/4.92              & ( Z
% 9.03/4.92                = ( F @ X4 ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_237_image__iff,axiom,
% 9.03/4.92      ! [Z: nat,F: int > nat,A2: set_int] :
% 9.03/4.92        ( ( member_nat @ Z @ ( image_int_nat @ F @ A2 ) )
% 9.03/4.92        = ( ? [X4: int] :
% 9.03/4.92              ( ( member_int @ X4 @ A2 )
% 9.03/4.92              & ( Z
% 9.03/4.92                = ( F @ X4 ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_238_image__iff,axiom,
% 9.03/4.92      ! [Z: int,F: nat > int,A2: set_nat] :
% 9.03/4.92        ( ( member_int @ Z @ ( image_nat_int @ F @ A2 ) )
% 9.03/4.92        = ( ? [X4: nat] :
% 9.03/4.92              ( ( member_nat @ X4 @ A2 )
% 9.03/4.92              & ( Z
% 9.03/4.92                = ( F @ X4 ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_239_image__iff,axiom,
% 9.03/4.92      ! [Z: int,F: int > int,A2: set_int] :
% 9.03/4.92        ( ( member_int @ Z @ ( image_int_int @ F @ A2 ) )
% 9.03/4.92        = ( ? [X4: int] :
% 9.03/4.92              ( ( member_int @ X4 @ A2 )
% 9.03/4.92              & ( Z
% 9.03/4.92                = ( F @ X4 ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_240_imageI,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,F: complex > complex] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( member_complex @ ( F @ X ) @ ( image_1468599708987790691omplex @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_241_imageI,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,F: complex > real] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( member_real @ ( F @ X ) @ ( image_complex_real @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_242_imageI,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,F: complex > nat] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( member_nat @ ( F @ X ) @ ( image_complex_nat @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_243_imageI,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,F: complex > int] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ( member_int @ ( F @ X ) @ ( image_complex_int @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_244_imageI,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,F: real > complex] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( member_complex @ ( F @ X ) @ ( image_real_complex @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_245_imageI,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,F: real > real] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_246_imageI,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,F: real > nat] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_247_imageI,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,F: real > int] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ( member_int @ ( F @ X ) @ ( image_real_int @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_248_imageI,axiom,
% 9.03/4.92      ! [X: nat,A2: set_nat,F: nat > complex] :
% 9.03/4.92        ( ( member_nat @ X @ A2 )
% 9.03/4.92       => ( member_complex @ ( F @ X ) @ ( image_nat_complex @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_249_imageI,axiom,
% 9.03/4.92      ! [X: nat,A2: set_nat,F: nat > real] :
% 9.03/4.92        ( ( member_nat @ X @ A2 )
% 9.03/4.92       => ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_250_Suc__nat__number__of__add,axiom,
% 9.03/4.92      ! [V: num,N: nat] :
% 9.03/4.92        ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 9.03/4.92        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_251_mk__disjoint__insert,axiom,
% 9.03/4.92      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( member_VEBT_VEBT @ A @ A2 )
% 9.03/4.92       => ? [B3: set_VEBT_VEBT] :
% 9.03/4.92            ( ( A2
% 9.03/4.92              = ( insert_VEBT_VEBT @ A @ B3 ) )
% 9.03/4.92            & ~ ( member_VEBT_VEBT @ A @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_252_mk__disjoint__insert,axiom,
% 9.03/4.92      ! [A: $o,A2: set_o] :
% 9.03/4.92        ( ( member_o @ A @ A2 )
% 9.03/4.92       => ? [B3: set_o] :
% 9.03/4.92            ( ( A2
% 9.03/4.92              = ( insert_o @ A @ B3 ) )
% 9.03/4.92            & ~ ( member_o @ A @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_253_mk__disjoint__insert,axiom,
% 9.03/4.92      ! [A: complex,A2: set_complex] :
% 9.03/4.92        ( ( member_complex @ A @ A2 )
% 9.03/4.92       => ? [B3: set_complex] :
% 9.03/4.92            ( ( A2
% 9.03/4.92              = ( insert_complex @ A @ B3 ) )
% 9.03/4.92            & ~ ( member_complex @ A @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_254_mk__disjoint__insert,axiom,
% 9.03/4.92      ! [A: real,A2: set_real] :
% 9.03/4.92        ( ( member_real @ A @ A2 )
% 9.03/4.92       => ? [B3: set_real] :
% 9.03/4.92            ( ( A2
% 9.03/4.92              = ( insert_real @ A @ B3 ) )
% 9.03/4.92            & ~ ( member_real @ A @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_255_mk__disjoint__insert,axiom,
% 9.03/4.92      ! [A: set_nat,A2: set_set_nat] :
% 9.03/4.92        ( ( member_set_nat @ A @ A2 )
% 9.03/4.92       => ? [B3: set_set_nat] :
% 9.03/4.92            ( ( A2
% 9.03/4.92              = ( insert_set_nat @ A @ B3 ) )
% 9.03/4.92            & ~ ( member_set_nat @ A @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_256_mk__disjoint__insert,axiom,
% 9.03/4.92      ! [A: nat,A2: set_nat] :
% 9.03/4.92        ( ( member_nat @ A @ A2 )
% 9.03/4.92       => ? [B3: set_nat] :
% 9.03/4.92            ( ( A2
% 9.03/4.92              = ( insert_nat @ A @ B3 ) )
% 9.03/4.92            & ~ ( member_nat @ A @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_257_mk__disjoint__insert,axiom,
% 9.03/4.92      ! [A: int,A2: set_int] :
% 9.03/4.92        ( ( member_int @ A @ A2 )
% 9.03/4.92       => ? [B3: set_int] :
% 9.03/4.92            ( ( A2
% 9.03/4.92              = ( insert_int @ A @ B3 ) )
% 9.03/4.92            & ~ ( member_int @ A @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_258_insert__commute,axiom,
% 9.03/4.92      ! [X: vEBT_VEBT,Y: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( insert_VEBT_VEBT @ X @ ( insert_VEBT_VEBT @ Y @ A2 ) )
% 9.03/4.92        = ( insert_VEBT_VEBT @ Y @ ( insert_VEBT_VEBT @ X @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_259_insert__commute,axiom,
% 9.03/4.92      ! [X: nat,Y: nat,A2: set_nat] :
% 9.03/4.92        ( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
% 9.03/4.92        = ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_260_insert__commute,axiom,
% 9.03/4.92      ! [X: int,Y: int,A2: set_int] :
% 9.03/4.92        ( ( insert_int @ X @ ( insert_int @ Y @ A2 ) )
% 9.03/4.92        = ( insert_int @ Y @ ( insert_int @ X @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_261_insert__commute,axiom,
% 9.03/4.92      ! [X: real,Y: real,A2: set_real] :
% 9.03/4.92        ( ( insert_real @ X @ ( insert_real @ Y @ A2 ) )
% 9.03/4.92        = ( insert_real @ Y @ ( insert_real @ X @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_262_insert__commute,axiom,
% 9.03/4.92      ! [X: $o,Y: $o,A2: set_o] :
% 9.03/4.92        ( ( insert_o @ X @ ( insert_o @ Y @ A2 ) )
% 9.03/4.92        = ( insert_o @ Y @ ( insert_o @ X @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_263_insert__eq__iff,axiom,
% 9.03/4.92      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.92        ( ~ ( member_VEBT_VEBT @ A @ A2 )
% 9.03/4.92       => ( ~ ( member_VEBT_VEBT @ B @ B2 )
% 9.03/4.92         => ( ( ( insert_VEBT_VEBT @ A @ A2 )
% 9.03/4.92              = ( insert_VEBT_VEBT @ B @ B2 ) )
% 9.03/4.92            = ( ( ( A = B )
% 9.03/4.92               => ( A2 = B2 ) )
% 9.03/4.92              & ( ( A != B )
% 9.03/4.92               => ? [C2: set_VEBT_VEBT] :
% 9.03/4.92                    ( ( A2
% 9.03/4.92                      = ( insert_VEBT_VEBT @ B @ C2 ) )
% 9.03/4.92                    & ~ ( member_VEBT_VEBT @ B @ C2 )
% 9.03/4.92                    & ( B2
% 9.03/4.92                      = ( insert_VEBT_VEBT @ A @ C2 ) )
% 9.03/4.92                    & ~ ( member_VEBT_VEBT @ A @ C2 ) ) ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_264_insert__eq__iff,axiom,
% 9.03/4.92      ! [A: $o,A2: set_o,B: $o,B2: set_o] :
% 9.03/4.92        ( ~ ( member_o @ A @ A2 )
% 9.03/4.92       => ( ~ ( member_o @ B @ B2 )
% 9.03/4.92         => ( ( ( insert_o @ A @ A2 )
% 9.03/4.92              = ( insert_o @ B @ B2 ) )
% 9.03/4.92            = ( ( ( A = B )
% 9.03/4.92               => ( A2 = B2 ) )
% 9.03/4.92              & ( ( A = ~ B )
% 9.03/4.92               => ? [C2: set_o] :
% 9.03/4.92                    ( ( A2
% 9.03/4.92                      = ( insert_o @ B @ C2 ) )
% 9.03/4.92                    & ~ ( member_o @ B @ C2 )
% 9.03/4.92                    & ( B2
% 9.03/4.92                      = ( insert_o @ A @ C2 ) )
% 9.03/4.92                    & ~ ( member_o @ A @ C2 ) ) ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_265_insert__eq__iff,axiom,
% 9.03/4.92      ! [A: complex,A2: set_complex,B: complex,B2: set_complex] :
% 9.03/4.92        ( ~ ( member_complex @ A @ A2 )
% 9.03/4.92       => ( ~ ( member_complex @ B @ B2 )
% 9.03/4.92         => ( ( ( insert_complex @ A @ A2 )
% 9.03/4.92              = ( insert_complex @ B @ B2 ) )
% 9.03/4.92            = ( ( ( A = B )
% 9.03/4.92               => ( A2 = B2 ) )
% 9.03/4.92              & ( ( A != B )
% 9.03/4.92               => ? [C2: set_complex] :
% 9.03/4.92                    ( ( A2
% 9.03/4.92                      = ( insert_complex @ B @ C2 ) )
% 9.03/4.92                    & ~ ( member_complex @ B @ C2 )
% 9.03/4.92                    & ( B2
% 9.03/4.92                      = ( insert_complex @ A @ C2 ) )
% 9.03/4.92                    & ~ ( member_complex @ A @ C2 ) ) ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_266_insert__eq__iff,axiom,
% 9.03/4.92      ! [A: real,A2: set_real,B: real,B2: set_real] :
% 9.03/4.92        ( ~ ( member_real @ A @ A2 )
% 9.03/4.92       => ( ~ ( member_real @ B @ B2 )
% 9.03/4.92         => ( ( ( insert_real @ A @ A2 )
% 9.03/4.92              = ( insert_real @ B @ B2 ) )
% 9.03/4.92            = ( ( ( A = B )
% 9.03/4.92               => ( A2 = B2 ) )
% 9.03/4.92              & ( ( A != B )
% 9.03/4.92               => ? [C2: set_real] :
% 9.03/4.92                    ( ( A2
% 9.03/4.92                      = ( insert_real @ B @ C2 ) )
% 9.03/4.92                    & ~ ( member_real @ B @ C2 )
% 9.03/4.92                    & ( B2
% 9.03/4.92                      = ( insert_real @ A @ C2 ) )
% 9.03/4.92                    & ~ ( member_real @ A @ C2 ) ) ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_267_insert__eq__iff,axiom,
% 9.03/4.92      ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
% 9.03/4.92        ( ~ ( member_set_nat @ A @ A2 )
% 9.03/4.92       => ( ~ ( member_set_nat @ B @ B2 )
% 9.03/4.92         => ( ( ( insert_set_nat @ A @ A2 )
% 9.03/4.92              = ( insert_set_nat @ B @ B2 ) )
% 9.03/4.92            = ( ( ( A = B )
% 9.03/4.92               => ( A2 = B2 ) )
% 9.03/4.92              & ( ( A != B )
% 9.03/4.92               => ? [C2: set_set_nat] :
% 9.03/4.92                    ( ( A2
% 9.03/4.92                      = ( insert_set_nat @ B @ C2 ) )
% 9.03/4.92                    & ~ ( member_set_nat @ B @ C2 )
% 9.03/4.92                    & ( B2
% 9.03/4.92                      = ( insert_set_nat @ A @ C2 ) )
% 9.03/4.92                    & ~ ( member_set_nat @ A @ C2 ) ) ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_268_insert__eq__iff,axiom,
% 9.03/4.92      ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
% 9.03/4.92        ( ~ ( member_nat @ A @ A2 )
% 9.03/4.92       => ( ~ ( member_nat @ B @ B2 )
% 9.03/4.92         => ( ( ( insert_nat @ A @ A2 )
% 9.03/4.92              = ( insert_nat @ B @ B2 ) )
% 9.03/4.92            = ( ( ( A = B )
% 9.03/4.92               => ( A2 = B2 ) )
% 9.03/4.92              & ( ( A != B )
% 9.03/4.92               => ? [C2: set_nat] :
% 9.03/4.92                    ( ( A2
% 9.03/4.92                      = ( insert_nat @ B @ C2 ) )
% 9.03/4.92                    & ~ ( member_nat @ B @ C2 )
% 9.03/4.92                    & ( B2
% 9.03/4.92                      = ( insert_nat @ A @ C2 ) )
% 9.03/4.92                    & ~ ( member_nat @ A @ C2 ) ) ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_269_insert__eq__iff,axiom,
% 9.03/4.92      ! [A: int,A2: set_int,B: int,B2: set_int] :
% 9.03/4.92        ( ~ ( member_int @ A @ A2 )
% 9.03/4.92       => ( ~ ( member_int @ B @ B2 )
% 9.03/4.92         => ( ( ( insert_int @ A @ A2 )
% 9.03/4.92              = ( insert_int @ B @ B2 ) )
% 9.03/4.92            = ( ( ( A = B )
% 9.03/4.92               => ( A2 = B2 ) )
% 9.03/4.92              & ( ( A != B )
% 9.03/4.92               => ? [C2: set_int] :
% 9.03/4.92                    ( ( A2
% 9.03/4.92                      = ( insert_int @ B @ C2 ) )
% 9.03/4.92                    & ~ ( member_int @ B @ C2 )
% 9.03/4.92                    & ( B2
% 9.03/4.92                      = ( insert_int @ A @ C2 ) )
% 9.03/4.92                    & ~ ( member_int @ A @ C2 ) ) ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_270_insert__absorb,axiom,
% 9.03/4.92      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( member_VEBT_VEBT @ A @ A2 )
% 9.03/4.92       => ( ( insert_VEBT_VEBT @ A @ A2 )
% 9.03/4.92          = A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_271_insert__absorb,axiom,
% 9.03/4.92      ! [A: $o,A2: set_o] :
% 9.03/4.92        ( ( member_o @ A @ A2 )
% 9.03/4.92       => ( ( insert_o @ A @ A2 )
% 9.03/4.92          = A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_272_insert__absorb,axiom,
% 9.03/4.92      ! [A: complex,A2: set_complex] :
% 9.03/4.92        ( ( member_complex @ A @ A2 )
% 9.03/4.92       => ( ( insert_complex @ A @ A2 )
% 9.03/4.92          = A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_273_insert__absorb,axiom,
% 9.03/4.92      ! [A: real,A2: set_real] :
% 9.03/4.92        ( ( member_real @ A @ A2 )
% 9.03/4.92       => ( ( insert_real @ A @ A2 )
% 9.03/4.92          = A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_274_insert__absorb,axiom,
% 9.03/4.92      ! [A: set_nat,A2: set_set_nat] :
% 9.03/4.92        ( ( member_set_nat @ A @ A2 )
% 9.03/4.92       => ( ( insert_set_nat @ A @ A2 )
% 9.03/4.92          = A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_275_insert__absorb,axiom,
% 9.03/4.92      ! [A: nat,A2: set_nat] :
% 9.03/4.92        ( ( member_nat @ A @ A2 )
% 9.03/4.92       => ( ( insert_nat @ A @ A2 )
% 9.03/4.92          = A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_276_insert__absorb,axiom,
% 9.03/4.92      ! [A: int,A2: set_int] :
% 9.03/4.92        ( ( member_int @ A @ A2 )
% 9.03/4.92       => ( ( insert_int @ A @ A2 )
% 9.03/4.92          = A2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_277_insert__ident,axiom,
% 9.03/4.92      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.92        ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 9.03/4.92       => ( ~ ( member_VEBT_VEBT @ X @ B2 )
% 9.03/4.92         => ( ( ( insert_VEBT_VEBT @ X @ A2 )
% 9.03/4.92              = ( insert_VEBT_VEBT @ X @ B2 ) )
% 9.03/4.92            = ( A2 = B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_278_insert__ident,axiom,
% 9.03/4.92      ! [X: $o,A2: set_o,B2: set_o] :
% 9.03/4.92        ( ~ ( member_o @ X @ A2 )
% 9.03/4.92       => ( ~ ( member_o @ X @ B2 )
% 9.03/4.92         => ( ( ( insert_o @ X @ A2 )
% 9.03/4.92              = ( insert_o @ X @ B2 ) )
% 9.03/4.92            = ( A2 = B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_279_insert__ident,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,B2: set_complex] :
% 9.03/4.92        ( ~ ( member_complex @ X @ A2 )
% 9.03/4.92       => ( ~ ( member_complex @ X @ B2 )
% 9.03/4.92         => ( ( ( insert_complex @ X @ A2 )
% 9.03/4.92              = ( insert_complex @ X @ B2 ) )
% 9.03/4.92            = ( A2 = B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_280_insert__ident,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,B2: set_real] :
% 9.03/4.92        ( ~ ( member_real @ X @ A2 )
% 9.03/4.92       => ( ~ ( member_real @ X @ B2 )
% 9.03/4.92         => ( ( ( insert_real @ X @ A2 )
% 9.03/4.92              = ( insert_real @ X @ B2 ) )
% 9.03/4.92            = ( A2 = B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_281_insert__ident,axiom,
% 9.03/4.92      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.92        ( ~ ( member_set_nat @ X @ A2 )
% 9.03/4.92       => ( ~ ( member_set_nat @ X @ B2 )
% 9.03/4.92         => ( ( ( insert_set_nat @ X @ A2 )
% 9.03/4.92              = ( insert_set_nat @ X @ B2 ) )
% 9.03/4.92            = ( A2 = B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_282_insert__ident,axiom,
% 9.03/4.92      ! [X: nat,A2: set_nat,B2: set_nat] :
% 9.03/4.92        ( ~ ( member_nat @ X @ A2 )
% 9.03/4.92       => ( ~ ( member_nat @ X @ B2 )
% 9.03/4.92         => ( ( ( insert_nat @ X @ A2 )
% 9.03/4.92              = ( insert_nat @ X @ B2 ) )
% 9.03/4.92            = ( A2 = B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_283_insert__ident,axiom,
% 9.03/4.92      ! [X: int,A2: set_int,B2: set_int] :
% 9.03/4.92        ( ~ ( member_int @ X @ A2 )
% 9.03/4.92       => ( ~ ( member_int @ X @ B2 )
% 9.03/4.92         => ( ( ( insert_int @ X @ A2 )
% 9.03/4.92              = ( insert_int @ X @ B2 ) )
% 9.03/4.92            = ( A2 = B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_284_Set_Oset__insert,axiom,
% 9.03/4.92      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( member_VEBT_VEBT @ X @ A2 )
% 9.03/4.92       => ~ ! [B3: set_VEBT_VEBT] :
% 9.03/4.92              ( ( A2
% 9.03/4.92                = ( insert_VEBT_VEBT @ X @ B3 ) )
% 9.03/4.92             => ( member_VEBT_VEBT @ X @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_285_Set_Oset__insert,axiom,
% 9.03/4.92      ! [X: $o,A2: set_o] :
% 9.03/4.92        ( ( member_o @ X @ A2 )
% 9.03/4.92       => ~ ! [B3: set_o] :
% 9.03/4.92              ( ( A2
% 9.03/4.92                = ( insert_o @ X @ B3 ) )
% 9.03/4.92             => ( member_o @ X @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_286_Set_Oset__insert,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex] :
% 9.03/4.92        ( ( member_complex @ X @ A2 )
% 9.03/4.92       => ~ ! [B3: set_complex] :
% 9.03/4.92              ( ( A2
% 9.03/4.92                = ( insert_complex @ X @ B3 ) )
% 9.03/4.92             => ( member_complex @ X @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_287_Set_Oset__insert,axiom,
% 9.03/4.92      ! [X: real,A2: set_real] :
% 9.03/4.92        ( ( member_real @ X @ A2 )
% 9.03/4.92       => ~ ! [B3: set_real] :
% 9.03/4.92              ( ( A2
% 9.03/4.92                = ( insert_real @ X @ B3 ) )
% 9.03/4.92             => ( member_real @ X @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_288_Set_Oset__insert,axiom,
% 9.03/4.92      ! [X: set_nat,A2: set_set_nat] :
% 9.03/4.92        ( ( member_set_nat @ X @ A2 )
% 9.03/4.92       => ~ ! [B3: set_set_nat] :
% 9.03/4.92              ( ( A2
% 9.03/4.92                = ( insert_set_nat @ X @ B3 ) )
% 9.03/4.92             => ( member_set_nat @ X @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_289_Set_Oset__insert,axiom,
% 9.03/4.92      ! [X: nat,A2: set_nat] :
% 9.03/4.92        ( ( member_nat @ X @ A2 )
% 9.03/4.92       => ~ ! [B3: set_nat] :
% 9.03/4.92              ( ( A2
% 9.03/4.92                = ( insert_nat @ X @ B3 ) )
% 9.03/4.92             => ( member_nat @ X @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_290_Set_Oset__insert,axiom,
% 9.03/4.92      ! [X: int,A2: set_int] :
% 9.03/4.92        ( ( member_int @ X @ A2 )
% 9.03/4.92       => ~ ! [B3: set_int] :
% 9.03/4.92              ( ( A2
% 9.03/4.92                = ( insert_int @ X @ B3 ) )
% 9.03/4.92             => ( member_int @ X @ B3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_291_insertI2,axiom,
% 9.03/4.92      ! [A: vEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
% 9.03/4.92        ( ( member_VEBT_VEBT @ A @ B2 )
% 9.03/4.92       => ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_292_insertI2,axiom,
% 9.03/4.92      ! [A: $o,B2: set_o,B: $o] :
% 9.03/4.92        ( ( member_o @ A @ B2 )
% 9.03/4.92       => ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_293_insertI2,axiom,
% 9.03/4.92      ! [A: complex,B2: set_complex,B: complex] :
% 9.03/4.92        ( ( member_complex @ A @ B2 )
% 9.03/4.92       => ( member_complex @ A @ ( insert_complex @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_294_insertI2,axiom,
% 9.03/4.92      ! [A: real,B2: set_real,B: real] :
% 9.03/4.92        ( ( member_real @ A @ B2 )
% 9.03/4.92       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_295_insertI2,axiom,
% 9.03/4.92      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 9.03/4.92        ( ( member_set_nat @ A @ B2 )
% 9.03/4.92       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_296_insertI2,axiom,
% 9.03/4.92      ! [A: nat,B2: set_nat,B: nat] :
% 9.03/4.92        ( ( member_nat @ A @ B2 )
% 9.03/4.92       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_297_insertI2,axiom,
% 9.03/4.92      ! [A: int,B2: set_int,B: int] :
% 9.03/4.92        ( ( member_int @ A @ B2 )
% 9.03/4.92       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_298_insertI1,axiom,
% 9.03/4.92      ! [A: vEBT_VEBT,B2: set_VEBT_VEBT] : ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ A @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_299_insertI1,axiom,
% 9.03/4.92      ! [A: $o,B2: set_o] : ( member_o @ A @ ( insert_o @ A @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_300_insertI1,axiom,
% 9.03/4.92      ! [A: complex,B2: set_complex] : ( member_complex @ A @ ( insert_complex @ A @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_301_insertI1,axiom,
% 9.03/4.92      ! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_302_insertI1,axiom,
% 9.03/4.92      ! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_303_insertI1,axiom,
% 9.03/4.92      ! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_304_insertI1,axiom,
% 9.03/4.92      ! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_305_insertE,axiom,
% 9.03/4.92      ! [A: vEBT_VEBT,B: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ A2 ) )
% 9.03/4.92       => ( ( A != B )
% 9.03/4.92         => ( member_VEBT_VEBT @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_306_insertE,axiom,
% 9.03/4.92      ! [A: $o,B: $o,A2: set_o] :
% 9.03/4.92        ( ( member_o @ A @ ( insert_o @ B @ A2 ) )
% 9.03/4.92       => ( ( A = ~ B )
% 9.03/4.92         => ( member_o @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_307_insertE,axiom,
% 9.03/4.92      ! [A: complex,B: complex,A2: set_complex] :
% 9.03/4.92        ( ( member_complex @ A @ ( insert_complex @ B @ A2 ) )
% 9.03/4.92       => ( ( A != B )
% 9.03/4.92         => ( member_complex @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_308_insertE,axiom,
% 9.03/4.92      ! [A: real,B: real,A2: set_real] :
% 9.03/4.92        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 9.03/4.92       => ( ( A != B )
% 9.03/4.92         => ( member_real @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_309_insertE,axiom,
% 9.03/4.92      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 9.03/4.92        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 9.03/4.92       => ( ( A != B )
% 9.03/4.92         => ( member_set_nat @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_310_insertE,axiom,
% 9.03/4.92      ! [A: nat,B: nat,A2: set_nat] :
% 9.03/4.92        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 9.03/4.92       => ( ( A != B )
% 9.03/4.92         => ( member_nat @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_311_insertE,axiom,
% 9.03/4.92      ! [A: int,B: int,A2: set_int] :
% 9.03/4.92        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 9.03/4.92       => ( ( A != B )
% 9.03/4.92         => ( member_int @ A @ A2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_312_nat__int__comparison_I1_J,axiom,
% 9.03/4.92      ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 9.03/4.92      = ( ^ [A3: nat,B4: nat] :
% 9.03/4.92            ( ( semiri1314217659103216013at_int @ A3 )
% 9.03/4.92            = ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_313_int__if,axiom,
% 9.03/4.92      ! [P: $o,A: nat,B: nat] :
% 9.03/4.92        ( ( P
% 9.03/4.92         => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 9.03/4.92            = ( semiri1314217659103216013at_int @ A ) ) )
% 9.03/4.92        & ( ~ P
% 9.03/4.92         => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 9.03/4.92            = ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_314_int__int__eq,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ( semiri1314217659103216013at_int @ M )
% 9.03/4.92          = ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.92        = ( M = N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_315_verit__eq__simplify_I10_J,axiom,
% 9.03/4.92      ! [X22: num] :
% 9.03/4.92        ( one
% 9.03/4.92       != ( bit0 @ X22 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_316_int__ops_I3_J,axiom,
% 9.03/4.92      ! [N: num] :
% 9.03/4.92        ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 9.03/4.92        = ( numeral_numeral_int @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_317__C2_C,axiom,
% 9.03/4.92      ord_less_eq_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ na ) ) ) @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_318__C1_C,axiom,
% 9.03/4.92      ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ na ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_319__C00_C,axiom,
% 9.03/4.92      ( ( ord_less_eq_nat @ one_one_nat @ na )
% 9.03/4.92      & ( ( suc @ na )
% 9.03/4.92        = m ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_320__C0_C,axiom,
% 9.03/4.92      ord_less_eq_nat @ na @ m ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_321_dbl__simps_I3_J,axiom,
% 9.03/4.92      ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 9.03/4.92      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_322_dbl__simps_I3_J,axiom,
% 9.03/4.92      ( ( neg_numeral_dbl_real @ one_one_real )
% 9.03/4.92      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_323_dbl__simps_I3_J,axiom,
% 9.03/4.92      ( ( neg_numeral_dbl_rat @ one_one_rat )
% 9.03/4.92      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_324_dbl__simps_I3_J,axiom,
% 9.03/4.92      ( ( neg_numeral_dbl_int @ one_one_int )
% 9.03/4.92      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_325_VEBT_Oinject_I1_J,axiom,
% 9.03/4.92      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 9.03/4.92        ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 9.03/4.92          = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 9.03/4.92        = ( ( X11 = Y11 )
% 9.03/4.92          & ( X12 = Y12 )
% 9.03/4.92          & ( X13 = Y13 )
% 9.03/4.92          & ( X14 = Y14 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_326__C3_Ohyps_C_I2_J,axiom,
% 9.03/4.92      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 9.03/4.92      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_327_valid__eq2,axiom,
% 9.03/4.92      ! [T: vEBT_VEBT,D: nat] :
% 9.03/4.92        ( ( vEBT_VEBT_valid @ T @ D )
% 9.03/4.92       => ( vEBT_invar_vebt @ T @ D ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_328_valid__eq1,axiom,
% 9.03/4.92      ! [T: vEBT_VEBT,D: nat] :
% 9.03/4.92        ( ( vEBT_invar_vebt @ T @ D )
% 9.03/4.92       => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_329_valid__eq,axiom,
% 9.03/4.92      vEBT_VEBT_valid = vEBT_invar_vebt ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_330_add__left__cancel,axiom,
% 9.03/4.92      ! [A: real,B: real,C: real] :
% 9.03/4.92        ( ( ( plus_plus_real @ A @ B )
% 9.03/4.92          = ( plus_plus_real @ A @ C ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_331_add__left__cancel,axiom,
% 9.03/4.92      ! [A: rat,B: rat,C: rat] :
% 9.03/4.92        ( ( ( plus_plus_rat @ A @ B )
% 9.03/4.92          = ( plus_plus_rat @ A @ C ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_332_add__left__cancel,axiom,
% 9.03/4.92      ! [A: nat,B: nat,C: nat] :
% 9.03/4.92        ( ( ( plus_plus_nat @ A @ B )
% 9.03/4.92          = ( plus_plus_nat @ A @ C ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_333_add__left__cancel,axiom,
% 9.03/4.92      ! [A: int,B: int,C: int] :
% 9.03/4.92        ( ( ( plus_plus_int @ A @ B )
% 9.03/4.92          = ( plus_plus_int @ A @ C ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_334_add__left__cancel,axiom,
% 9.03/4.92      ! [A: complex,B: complex,C: complex] :
% 9.03/4.92        ( ( ( plus_plus_complex @ A @ B )
% 9.03/4.92          = ( plus_plus_complex @ A @ C ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_335_add__right__cancel,axiom,
% 9.03/4.92      ! [B: real,A: real,C: real] :
% 9.03/4.92        ( ( ( plus_plus_real @ B @ A )
% 9.03/4.92          = ( plus_plus_real @ C @ A ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_336_add__right__cancel,axiom,
% 9.03/4.92      ! [B: rat,A: rat,C: rat] :
% 9.03/4.92        ( ( ( plus_plus_rat @ B @ A )
% 9.03/4.92          = ( plus_plus_rat @ C @ A ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_337_add__right__cancel,axiom,
% 9.03/4.92      ! [B: nat,A: nat,C: nat] :
% 9.03/4.92        ( ( ( plus_plus_nat @ B @ A )
% 9.03/4.92          = ( plus_plus_nat @ C @ A ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_338_add__right__cancel,axiom,
% 9.03/4.92      ! [B: int,A: int,C: int] :
% 9.03/4.92        ( ( ( plus_plus_int @ B @ A )
% 9.03/4.92          = ( plus_plus_int @ C @ A ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_339_add__right__cancel,axiom,
% 9.03/4.92      ! [B: complex,A: complex,C: complex] :
% 9.03/4.92        ( ( ( plus_plus_complex @ B @ A )
% 9.03/4.92          = ( plus_plus_complex @ C @ A ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_340_numeral__le__iff,axiom,
% 9.03/4.92      ! [M: num,N: num] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.03/4.92        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_341_numeral__le__iff,axiom,
% 9.03/4.92      ! [M: num,N: num] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.03/4.92        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_342_numeral__le__iff,axiom,
% 9.03/4.92      ! [M: num,N: num] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.03/4.92        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_343_numeral__le__iff,axiom,
% 9.03/4.92      ! [M: num,N: num] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.03/4.92        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_344_add__le__cancel__right,axiom,
% 9.03/4.92      ! [A: real,C: real,B: real] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.03/4.92        = ( ord_less_eq_real @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_345_add__le__cancel__right,axiom,
% 9.03/4.92      ! [A: rat,C: rat,B: rat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.03/4.92        = ( ord_less_eq_rat @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_346_add__le__cancel__right,axiom,
% 9.03/4.92      ! [A: nat,C: nat,B: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.03/4.92        = ( ord_less_eq_nat @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_347_add__le__cancel__right,axiom,
% 9.03/4.92      ! [A: int,C: int,B: int] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.03/4.92        = ( ord_less_eq_int @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_348_add__le__cancel__left,axiom,
% 9.03/4.92      ! [C: real,A: real,B: real] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.03/4.92        = ( ord_less_eq_real @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_349_add__le__cancel__left,axiom,
% 9.03/4.92      ! [C: rat,A: rat,B: rat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.03/4.92        = ( ord_less_eq_rat @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_350_add__le__cancel__left,axiom,
% 9.03/4.92      ! [C: nat,A: nat,B: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.03/4.92        = ( ord_less_eq_nat @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_351_add__le__cancel__left,axiom,
% 9.03/4.92      ! [C: int,A: int,B: int] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.03/4.92        = ( ord_less_eq_int @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_352_Suc__le__mono,axiom,
% 9.03/4.92      ! [N: nat,M: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 9.03/4.92        = ( ord_less_eq_nat @ N @ M ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_353_nat__add__left__cancel__le,axiom,
% 9.03/4.92      ! [K: nat,M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_354_set__n__deg__not__0,axiom,
% 9.03/4.92      ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
% 9.03/4.92        ( ! [X3: vEBT_VEBT] :
% 9.03/4.92            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.03/4.92           => ( vEBT_invar_vebt @ X3 @ N ) )
% 9.03/4.92       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.03/4.92            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.03/4.92         => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_355_of__nat__le__iff,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_356_of__nat__le__iff,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_357_of__nat__le__iff,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_358_of__nat__le__iff,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_359_dbl__simps_I5_J,axiom,
% 9.03/4.92      ! [K: num] :
% 9.03/4.92        ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 9.03/4.92        = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_360_dbl__simps_I5_J,axiom,
% 9.03/4.92      ! [K: num] :
% 9.03/4.92        ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 9.03/4.92        = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_361_dbl__simps_I5_J,axiom,
% 9.03/4.92      ! [K: num] :
% 9.03/4.92        ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 9.03/4.92        = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_362_dbl__simps_I5_J,axiom,
% 9.03/4.92      ! [K: num] :
% 9.03/4.92        ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 9.03/4.92        = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_363_numeral__le__one__iff,axiom,
% 9.03/4.92      ! [N: num] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 9.03/4.92        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_364_numeral__le__one__iff,axiom,
% 9.03/4.92      ! [N: num] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 9.03/4.92        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_365_numeral__le__one__iff,axiom,
% 9.03/4.92      ! [N: num] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 9.03/4.92        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_366_numeral__le__one__iff,axiom,
% 9.03/4.92      ! [N: num] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 9.03/4.92        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_367_ceiling__le__numeral,axiom,
% 9.03/4.92      ! [X: real,V: num] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 9.03/4.92        = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_368_ceiling__le__numeral,axiom,
% 9.03/4.92      ! [X: rat,V: num] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 9.03/4.92        = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_369_ceiling__le__one,axiom,
% 9.03/4.92      ! [X: real] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 9.03/4.92        = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_370_ceiling__le__one,axiom,
% 9.03/4.92      ! [X: rat] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 9.03/4.92        = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_371_verit__la__disequality,axiom,
% 9.03/4.92      ! [A: rat,B: rat] :
% 9.03/4.92        ( ( A = B )
% 9.03/4.92        | ~ ( ord_less_eq_rat @ A @ B )
% 9.03/4.92        | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_372_verit__la__disequality,axiom,
% 9.03/4.92      ! [A: num,B: num] :
% 9.03/4.92        ( ( A = B )
% 9.03/4.92        | ~ ( ord_less_eq_num @ A @ B )
% 9.03/4.92        | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_373_verit__la__disequality,axiom,
% 9.03/4.92      ! [A: nat,B: nat] :
% 9.03/4.92        ( ( A = B )
% 9.03/4.92        | ~ ( ord_less_eq_nat @ A @ B )
% 9.03/4.92        | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_374_verit__la__disequality,axiom,
% 9.03/4.92      ! [A: int,B: int] :
% 9.03/4.92        ( ( A = B )
% 9.03/4.92        | ~ ( ord_less_eq_int @ A @ B )
% 9.03/4.92        | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_375_verit__la__generic,axiom,
% 9.03/4.92      ! [A: int,X: int] :
% 9.03/4.92        ( ( ord_less_eq_int @ A @ X )
% 9.03/4.92        | ( A = X )
% 9.03/4.92        | ( ord_less_eq_int @ X @ A ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_376_verit__comp__simplify1_I2_J,axiom,
% 9.03/4.92      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_377_verit__comp__simplify1_I2_J,axiom,
% 9.03/4.92      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_378_verit__comp__simplify1_I2_J,axiom,
% 9.03/4.92      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_379_verit__comp__simplify1_I2_J,axiom,
% 9.03/4.92      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_380_verit__comp__simplify1_I2_J,axiom,
% 9.03/4.92      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_381_Nat_Oex__has__greatest__nat,axiom,
% 9.03/4.92      ! [P: nat > $o,K: nat,B: nat] :
% 9.03/4.92        ( ( P @ K )
% 9.03/4.92       => ( ! [Y4: nat] :
% 9.03/4.92              ( ( P @ Y4 )
% 9.03/4.92             => ( ord_less_eq_nat @ Y4 @ B ) )
% 9.03/4.92         => ? [X3: nat] :
% 9.03/4.92              ( ( P @ X3 )
% 9.03/4.92              & ! [Y5: nat] :
% 9.03/4.92                  ( ( P @ Y5 )
% 9.03/4.92                 => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_382_nat__le__linear,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.92        | ( ord_less_eq_nat @ N @ M ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_383_le__antisym,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.92         => ( M = N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_384_eq__imp__le,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( M = N )
% 9.03/4.92       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_385_le__trans,axiom,
% 9.03/4.92      ! [I: nat,J: nat,K: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92       => ( ( ord_less_eq_nat @ J @ K )
% 9.03/4.92         => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_386_le__refl,axiom,
% 9.03/4.92      ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_387_lift__Suc__antimono__le,axiom,
% 9.03/4.92      ! [F: nat > set_int,N: nat,N5: nat] :
% 9.03/4.92        ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.03/4.92         => ( ord_less_eq_set_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_388_lift__Suc__antimono__le,axiom,
% 9.03/4.92      ! [F: nat > rat,N: nat,N5: nat] :
% 9.03/4.92        ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.03/4.92         => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_389_lift__Suc__antimono__le,axiom,
% 9.03/4.92      ! [F: nat > num,N: nat,N5: nat] :
% 9.03/4.92        ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.03/4.92         => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_390_lift__Suc__antimono__le,axiom,
% 9.03/4.92      ! [F: nat > nat,N: nat,N5: nat] :
% 9.03/4.92        ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.03/4.92         => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_391_lift__Suc__antimono__le,axiom,
% 9.03/4.92      ! [F: nat > int,N: nat,N5: nat] :
% 9.03/4.92        ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.03/4.92         => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_392_lift__Suc__mono__le,axiom,
% 9.03/4.92      ! [F: nat > set_int,N: nat,N5: nat] :
% 9.03/4.92        ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.03/4.92         => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_393_lift__Suc__mono__le,axiom,
% 9.03/4.92      ! [F: nat > rat,N: nat,N5: nat] :
% 9.03/4.92        ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.03/4.92         => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_394_lift__Suc__mono__le,axiom,
% 9.03/4.92      ! [F: nat > num,N: nat,N5: nat] :
% 9.03/4.92        ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.03/4.92         => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_395_lift__Suc__mono__le,axiom,
% 9.03/4.92      ! [F: nat > nat,N: nat,N5: nat] :
% 9.03/4.92        ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.03/4.92         => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_396_lift__Suc__mono__le,axiom,
% 9.03/4.92      ! [F: nat > int,N: nat,N5: nat] :
% 9.03/4.92        ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.92       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.03/4.92         => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_397_of__nat__mono,axiom,
% 9.03/4.92      ! [I: nat,J: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_398_of__nat__mono,axiom,
% 9.03/4.92      ! [I: nat,J: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_399_of__nat__mono,axiom,
% 9.03/4.92      ! [I: nat,J: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92       => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_400_of__nat__mono,axiom,
% 9.03/4.92      ! [I: nat,J: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92       => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_401_zle__int,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_402_nat__int__comparison_I3_J,axiom,
% 9.03/4.92      ( ord_less_eq_nat
% 9.03/4.92      = ( ^ [A3: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_403_ceiling__mono,axiom,
% 9.03/4.92      ! [Y: real,X: real] :
% 9.03/4.92        ( ( ord_less_eq_real @ Y @ X )
% 9.03/4.92       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_404_ceiling__mono,axiom,
% 9.03/4.92      ! [Y: rat,X: rat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ Y @ X )
% 9.03/4.92       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_405_add__le__imp__le__right,axiom,
% 9.03/4.92      ! [A: real,C: real,B: real] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.03/4.92       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_406_add__le__imp__le__right,axiom,
% 9.03/4.92      ! [A: rat,C: rat,B: rat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.03/4.92       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_407_add__le__imp__le__right,axiom,
% 9.03/4.92      ! [A: nat,C: nat,B: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.03/4.92       => ( ord_less_eq_nat @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_408_add__le__imp__le__right,axiom,
% 9.03/4.92      ! [A: int,C: int,B: int] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.03/4.92       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_409_add__le__imp__le__left,axiom,
% 9.03/4.92      ! [C: real,A: real,B: real] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.03/4.92       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_410_add__le__imp__le__left,axiom,
% 9.03/4.92      ! [C: rat,A: rat,B: rat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.03/4.92       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_411_add__le__imp__le__left,axiom,
% 9.03/4.92      ! [C: nat,A: nat,B: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.03/4.92       => ( ord_less_eq_nat @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_412_add__le__imp__le__left,axiom,
% 9.03/4.92      ! [C: int,A: int,B: int] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.03/4.92       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_413_le__iff__add,axiom,
% 9.03/4.92      ( ord_less_eq_nat
% 9.03/4.92      = ( ^ [A3: nat,B4: nat] :
% 9.03/4.92          ? [C3: nat] :
% 9.03/4.92            ( B4
% 9.03/4.92            = ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_414_add__right__mono,axiom,
% 9.03/4.92      ! [A: real,B: real,C: real] :
% 9.03/4.92        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.92       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_415_add__right__mono,axiom,
% 9.03/4.92      ! [A: rat,B: rat,C: rat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.92       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_416_add__right__mono,axiom,
% 9.03/4.92      ! [A: nat,B: nat,C: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.92       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_417_add__right__mono,axiom,
% 9.03/4.92      ! [A: int,B: int,C: int] :
% 9.03/4.92        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.92       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_418_less__eqE,axiom,
% 9.03/4.92      ! [A: nat,B: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.92       => ~ ! [C4: nat] :
% 9.03/4.92              ( B
% 9.03/4.92             != ( plus_plus_nat @ A @ C4 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_419_add__left__mono,axiom,
% 9.03/4.92      ! [A: real,B: real,C: real] :
% 9.03/4.92        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.92       => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_420_add__left__mono,axiom,
% 9.03/4.92      ! [A: rat,B: rat,C: rat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.92       => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_421_add__left__mono,axiom,
% 9.03/4.92      ! [A: nat,B: nat,C: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.92       => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_422_add__left__mono,axiom,
% 9.03/4.92      ! [A: int,B: int,C: int] :
% 9.03/4.92        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.92       => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_423_add__mono,axiom,
% 9.03/4.92      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.92        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.92       => ( ( ord_less_eq_real @ C @ D )
% 9.03/4.92         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_424_add__mono,axiom,
% 9.03/4.92      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.92       => ( ( ord_less_eq_rat @ C @ D )
% 9.03/4.92         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_425_add__mono,axiom,
% 9.03/4.92      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.92       => ( ( ord_less_eq_nat @ C @ D )
% 9.03/4.92         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_426_add__mono,axiom,
% 9.03/4.92      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.92        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.92       => ( ( ord_less_eq_int @ C @ D )
% 9.03/4.92         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_427_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.03/4.92      ! [I: real,J: real,K: real,L: real] :
% 9.03/4.92        ( ( ( ord_less_eq_real @ I @ J )
% 9.03/4.92          & ( ord_less_eq_real @ K @ L ) )
% 9.03/4.92       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_428_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.03/4.92      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.03/4.92        ( ( ( ord_less_eq_rat @ I @ J )
% 9.03/4.92          & ( ord_less_eq_rat @ K @ L ) )
% 9.03/4.92       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_429_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.03/4.92      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.92        ( ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92          & ( ord_less_eq_nat @ K @ L ) )
% 9.03/4.92       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_430_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.03/4.92      ! [I: int,J: int,K: int,L: int] :
% 9.03/4.92        ( ( ( ord_less_eq_int @ I @ J )
% 9.03/4.92          & ( ord_less_eq_int @ K @ L ) )
% 9.03/4.92       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_431_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.03/4.92      ! [I: real,J: real,K: real,L: real] :
% 9.03/4.92        ( ( ( I = J )
% 9.03/4.92          & ( ord_less_eq_real @ K @ L ) )
% 9.03/4.92       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_432_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.03/4.92      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.03/4.92        ( ( ( I = J )
% 9.03/4.92          & ( ord_less_eq_rat @ K @ L ) )
% 9.03/4.92       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_433_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.03/4.92      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.92        ( ( ( I = J )
% 9.03/4.92          & ( ord_less_eq_nat @ K @ L ) )
% 9.03/4.92       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_434_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.03/4.92      ! [I: int,J: int,K: int,L: int] :
% 9.03/4.92        ( ( ( I = J )
% 9.03/4.92          & ( ord_less_eq_int @ K @ L ) )
% 9.03/4.92       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_435_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.03/4.92      ! [I: real,J: real,K: real,L: real] :
% 9.03/4.92        ( ( ( ord_less_eq_real @ I @ J )
% 9.03/4.92          & ( K = L ) )
% 9.03/4.92       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_436_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.03/4.92      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.03/4.92        ( ( ( ord_less_eq_rat @ I @ J )
% 9.03/4.92          & ( K = L ) )
% 9.03/4.92       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_437_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.03/4.92      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.92        ( ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92          & ( K = L ) )
% 9.03/4.92       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_438_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.03/4.92      ! [I: int,J: int,K: int,L: int] :
% 9.03/4.92        ( ( ( ord_less_eq_int @ I @ J )
% 9.03/4.92          & ( K = L ) )
% 9.03/4.92       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_439_size__neq__size__imp__neq,axiom,
% 9.03/4.92      ! [X: char,Y: char] :
% 9.03/4.92        ( ( ( size_size_char @ X )
% 9.03/4.92         != ( size_size_char @ Y ) )
% 9.03/4.92       => ( X != Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_440_size__neq__size__imp__neq,axiom,
% 9.03/4.92      ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
% 9.03/4.92        ( ( ( size_s6755466524823107622T_VEBT @ X )
% 9.03/4.92         != ( size_s6755466524823107622T_VEBT @ Y ) )
% 9.03/4.92       => ( X != Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_441_size__neq__size__imp__neq,axiom,
% 9.03/4.92      ! [X: list_o,Y: list_o] :
% 9.03/4.92        ( ( ( size_size_list_o @ X )
% 9.03/4.92         != ( size_size_list_o @ Y ) )
% 9.03/4.92       => ( X != Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_442_size__neq__size__imp__neq,axiom,
% 9.03/4.92      ! [X: list_int,Y: list_int] :
% 9.03/4.92        ( ( ( size_size_list_int @ X )
% 9.03/4.92         != ( size_size_list_int @ Y ) )
% 9.03/4.92       => ( X != Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_443_size__neq__size__imp__neq,axiom,
% 9.03/4.92      ! [X: num,Y: num] :
% 9.03/4.92        ( ( ( size_size_num @ X )
% 9.03/4.92         != ( size_size_num @ Y ) )
% 9.03/4.92       => ( X != Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_444_le__numeral__extra_I4_J,axiom,
% 9.03/4.92      ord_less_eq_real @ one_one_real @ one_one_real ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_445_le__numeral__extra_I4_J,axiom,
% 9.03/4.92      ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_446_le__numeral__extra_I4_J,axiom,
% 9.03/4.92      ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_447_le__numeral__extra_I4_J,axiom,
% 9.03/4.92      ord_less_eq_int @ one_one_int @ one_one_int ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_448_transitive__stepwise__le,axiom,
% 9.03/4.92      ! [M: nat,N: nat,R: nat > nat > $o] :
% 9.03/4.92        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.92       => ( ! [X3: nat] : ( R @ X3 @ X3 )
% 9.03/4.92         => ( ! [X3: nat,Y4: nat,Z3: nat] :
% 9.03/4.92                ( ( R @ X3 @ Y4 )
% 9.03/4.92               => ( ( R @ Y4 @ Z3 )
% 9.03/4.92                 => ( R @ X3 @ Z3 ) ) )
% 9.03/4.92           => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
% 9.03/4.92             => ( R @ M @ N ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_449_nat__induct__at__least,axiom,
% 9.03/4.92      ! [M: nat,N: nat,P: nat > $o] :
% 9.03/4.92        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.92       => ( ( P @ M )
% 9.03/4.92         => ( ! [N2: nat] :
% 9.03/4.92                ( ( ord_less_eq_nat @ M @ N2 )
% 9.03/4.92               => ( ( P @ N2 )
% 9.03/4.92                 => ( P @ ( suc @ N2 ) ) ) )
% 9.03/4.92           => ( P @ N ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_450_full__nat__induct,axiom,
% 9.03/4.92      ! [P: nat > $o,N: nat] :
% 9.03/4.92        ( ! [N2: nat] :
% 9.03/4.92            ( ! [M3: nat] :
% 9.03/4.92                ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
% 9.03/4.92               => ( P @ M3 ) )
% 9.03/4.92           => ( P @ N2 ) )
% 9.03/4.92       => ( P @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_451_not__less__eq__eq,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_452_Suc__n__not__le__n,axiom,
% 9.03/4.92      ! [N: nat] :
% 9.03/4.92        ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_453_le__Suc__eq,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.03/4.92        = ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.92          | ( M
% 9.03/4.92            = ( suc @ N ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_454_Suc__le__D,axiom,
% 9.03/4.92      ! [N: nat,M4: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
% 9.03/4.92       => ? [M5: nat] :
% 9.03/4.92            ( M4
% 9.03/4.92            = ( suc @ M5 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_455_le__SucI,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.92       => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_456_le__SucE,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.03/4.92       => ( ~ ( ord_less_eq_nat @ M @ N )
% 9.03/4.92         => ( M
% 9.03/4.92            = ( suc @ N ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_457_Suc__leD,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.03/4.92       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_458_real__arch__simple,axiom,
% 9.03/4.92      ! [X: real] :
% 9.03/4.92      ? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_459_real__arch__simple,axiom,
% 9.03/4.92      ! [X: rat] :
% 9.03/4.92      ? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_460_add__leE,axiom,
% 9.03/4.92      ! [M: nat,K: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.03/4.92       => ~ ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.92           => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_461_le__add1,axiom,
% 9.03/4.92      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_462_le__add2,axiom,
% 9.03/4.92      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_463_add__leD1,axiom,
% 9.03/4.92      ! [M: nat,K: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.03/4.92       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_464_add__leD2,axiom,
% 9.03/4.92      ! [M: nat,K: nat,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.03/4.92       => ( ord_less_eq_nat @ K @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_465_le__Suc__ex,axiom,
% 9.03/4.92      ! [K: nat,L: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ K @ L )
% 9.03/4.92       => ? [N2: nat] :
% 9.03/4.92            ( L
% 9.03/4.92            = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_466_add__le__mono,axiom,
% 9.03/4.92      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92       => ( ( ord_less_eq_nat @ K @ L )
% 9.03/4.92         => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_467_add__le__mono1,axiom,
% 9.03/4.92      ! [I: nat,J: nat,K: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_468_trans__le__add1,axiom,
% 9.03/4.92      ! [I: nat,J: nat,M: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_469_trans__le__add2,axiom,
% 9.03/4.92      ! [I: nat,J: nat,M: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.92       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_470_nat__le__iff__add,axiom,
% 9.03/4.92      ( ord_less_eq_nat
% 9.03/4.92      = ( ^ [M6: nat,N3: nat] :
% 9.03/4.92          ? [K2: nat] :
% 9.03/4.92            ( N3
% 9.03/4.92            = ( plus_plus_nat @ M6 @ K2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_471_one__le__numeral,axiom,
% 9.03/4.92      ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_472_one__le__numeral,axiom,
% 9.03/4.92      ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_473_one__le__numeral,axiom,
% 9.03/4.92      ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_474_one__le__numeral,axiom,
% 9.03/4.92      ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_475_dbl__def,axiom,
% 9.03/4.92      ( neg_numeral_dbl_real
% 9.03/4.92      = ( ^ [X4: real] : ( plus_plus_real @ X4 @ X4 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_476_dbl__def,axiom,
% 9.03/4.92      ( neg_numeral_dbl_rat
% 9.03/4.92      = ( ^ [X4: rat] : ( plus_plus_rat @ X4 @ X4 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_477_dbl__def,axiom,
% 9.03/4.92      ( neg_numeral_dbl_int
% 9.03/4.92      = ( ^ [X4: int] : ( plus_plus_int @ X4 @ X4 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_478_dbl__def,axiom,
% 9.03/4.92      ( neg_nu7009210354673126013omplex
% 9.03/4.92      = ( ^ [X4: complex] : ( plus_plus_complex @ X4 @ X4 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_479_int__ge__induct,axiom,
% 9.03/4.92      ! [K: int,I: int,P: int > $o] :
% 9.03/4.92        ( ( ord_less_eq_int @ K @ I )
% 9.03/4.92       => ( ( P @ K )
% 9.03/4.92         => ( ! [I2: int] :
% 9.03/4.92                ( ( ord_less_eq_int @ K @ I2 )
% 9.03/4.92               => ( ( P @ I2 )
% 9.03/4.92                 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 9.03/4.92           => ( P @ I ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_480_zle__iff__zadd,axiom,
% 9.03/4.92      ( ord_less_eq_int
% 9.03/4.92      = ( ^ [W2: int,Z4: int] :
% 9.03/4.92          ? [N3: nat] :
% 9.03/4.92            ( Z4
% 9.03/4.92            = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_481_one__reorient,axiom,
% 9.03/4.92      ! [X: complex] :
% 9.03/4.92        ( ( one_one_complex = X )
% 9.03/4.92        = ( X = one_one_complex ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_482_one__reorient,axiom,
% 9.03/4.92      ! [X: real] :
% 9.03/4.92        ( ( one_one_real = X )
% 9.03/4.92        = ( X = one_one_real ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_483_one__reorient,axiom,
% 9.03/4.92      ! [X: rat] :
% 9.03/4.92        ( ( one_one_rat = X )
% 9.03/4.92        = ( X = one_one_rat ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_484_one__reorient,axiom,
% 9.03/4.92      ! [X: nat] :
% 9.03/4.92        ( ( one_one_nat = X )
% 9.03/4.92        = ( X = one_one_nat ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_485_one__reorient,axiom,
% 9.03/4.92      ! [X: int] :
% 9.03/4.92        ( ( one_one_int = X )
% 9.03/4.92        = ( X = one_one_int ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_486_add__right__imp__eq,axiom,
% 9.03/4.92      ! [B: real,A: real,C: real] :
% 9.03/4.92        ( ( ( plus_plus_real @ B @ A )
% 9.03/4.92          = ( plus_plus_real @ C @ A ) )
% 9.03/4.92       => ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_487_add__right__imp__eq,axiom,
% 9.03/4.92      ! [B: rat,A: rat,C: rat] :
% 9.03/4.92        ( ( ( plus_plus_rat @ B @ A )
% 9.03/4.92          = ( plus_plus_rat @ C @ A ) )
% 9.03/4.92       => ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_488_add__right__imp__eq,axiom,
% 9.03/4.92      ! [B: nat,A: nat,C: nat] :
% 9.03/4.92        ( ( ( plus_plus_nat @ B @ A )
% 9.03/4.92          = ( plus_plus_nat @ C @ A ) )
% 9.03/4.92       => ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_489_add__right__imp__eq,axiom,
% 9.03/4.92      ! [B: int,A: int,C: int] :
% 9.03/4.92        ( ( ( plus_plus_int @ B @ A )
% 9.03/4.92          = ( plus_plus_int @ C @ A ) )
% 9.03/4.92       => ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_490_add__right__imp__eq,axiom,
% 9.03/4.92      ! [B: complex,A: complex,C: complex] :
% 9.03/4.92        ( ( ( plus_plus_complex @ B @ A )
% 9.03/4.92          = ( plus_plus_complex @ C @ A ) )
% 9.03/4.92       => ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_491_add__left__imp__eq,axiom,
% 9.03/4.92      ! [A: real,B: real,C: real] :
% 9.03/4.92        ( ( ( plus_plus_real @ A @ B )
% 9.03/4.92          = ( plus_plus_real @ A @ C ) )
% 9.03/4.92       => ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_492_add__left__imp__eq,axiom,
% 9.03/4.92      ! [A: rat,B: rat,C: rat] :
% 9.03/4.92        ( ( ( plus_plus_rat @ A @ B )
% 9.03/4.92          = ( plus_plus_rat @ A @ C ) )
% 9.03/4.92       => ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_493_add__left__imp__eq,axiom,
% 9.03/4.92      ! [A: nat,B: nat,C: nat] :
% 9.03/4.92        ( ( ( plus_plus_nat @ A @ B )
% 9.03/4.92          = ( plus_plus_nat @ A @ C ) )
% 9.03/4.92       => ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_494_add__left__imp__eq,axiom,
% 9.03/4.92      ! [A: int,B: int,C: int] :
% 9.03/4.92        ( ( ( plus_plus_int @ A @ B )
% 9.03/4.92          = ( plus_plus_int @ A @ C ) )
% 9.03/4.92       => ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_495_add__left__imp__eq,axiom,
% 9.03/4.92      ! [A: complex,B: complex,C: complex] :
% 9.03/4.92        ( ( ( plus_plus_complex @ A @ B )
% 9.03/4.92          = ( plus_plus_complex @ A @ C ) )
% 9.03/4.92       => ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_496_add_Oleft__commute,axiom,
% 9.03/4.92      ! [B: real,A: real,C: real] :
% 9.03/4.92        ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 9.03/4.92        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_497_add_Oleft__commute,axiom,
% 9.03/4.92      ! [B: rat,A: rat,C: rat] :
% 9.03/4.92        ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 9.03/4.92        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_498_add_Oleft__commute,axiom,
% 9.03/4.92      ! [B: nat,A: nat,C: nat] :
% 9.03/4.92        ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 9.03/4.92        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_499_add_Oleft__commute,axiom,
% 9.03/4.92      ! [B: int,A: int,C: int] :
% 9.03/4.92        ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 9.03/4.92        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_500_add_Oleft__commute,axiom,
% 9.03/4.92      ! [B: complex,A: complex,C: complex] :
% 9.03/4.92        ( ( plus_plus_complex @ B @ ( plus_plus_complex @ A @ C ) )
% 9.03/4.92        = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_501_add_Ocommute,axiom,
% 9.03/4.92      ( plus_plus_real
% 9.03/4.92      = ( ^ [A3: real,B4: real] : ( plus_plus_real @ B4 @ A3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_502_add_Ocommute,axiom,
% 9.03/4.92      ( plus_plus_rat
% 9.03/4.92      = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ B4 @ A3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_503_add_Ocommute,axiom,
% 9.03/4.92      ( plus_plus_nat
% 9.03/4.92      = ( ^ [A3: nat,B4: nat] : ( plus_plus_nat @ B4 @ A3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_504_add_Ocommute,axiom,
% 9.03/4.92      ( plus_plus_int
% 9.03/4.92      = ( ^ [A3: int,B4: int] : ( plus_plus_int @ B4 @ A3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_505_add_Ocommute,axiom,
% 9.03/4.92      ( plus_plus_complex
% 9.03/4.92      = ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ B4 @ A3 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_506_add_Oright__cancel,axiom,
% 9.03/4.92      ! [B: real,A: real,C: real] :
% 9.03/4.92        ( ( ( plus_plus_real @ B @ A )
% 9.03/4.92          = ( plus_plus_real @ C @ A ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_507_add_Oright__cancel,axiom,
% 9.03/4.92      ! [B: rat,A: rat,C: rat] :
% 9.03/4.92        ( ( ( plus_plus_rat @ B @ A )
% 9.03/4.92          = ( plus_plus_rat @ C @ A ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_508_add_Oright__cancel,axiom,
% 9.03/4.92      ! [B: int,A: int,C: int] :
% 9.03/4.92        ( ( ( plus_plus_int @ B @ A )
% 9.03/4.92          = ( plus_plus_int @ C @ A ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_509_add_Oright__cancel,axiom,
% 9.03/4.92      ! [B: complex,A: complex,C: complex] :
% 9.03/4.92        ( ( ( plus_plus_complex @ B @ A )
% 9.03/4.92          = ( plus_plus_complex @ C @ A ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_510_add_Oleft__cancel,axiom,
% 9.03/4.92      ! [A: real,B: real,C: real] :
% 9.03/4.92        ( ( ( plus_plus_real @ A @ B )
% 9.03/4.92          = ( plus_plus_real @ A @ C ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_511_add_Oleft__cancel,axiom,
% 9.03/4.92      ! [A: rat,B: rat,C: rat] :
% 9.03/4.92        ( ( ( plus_plus_rat @ A @ B )
% 9.03/4.92          = ( plus_plus_rat @ A @ C ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_512_add_Oleft__cancel,axiom,
% 9.03/4.92      ! [A: int,B: int,C: int] :
% 9.03/4.92        ( ( ( plus_plus_int @ A @ B )
% 9.03/4.92          = ( plus_plus_int @ A @ C ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_513_add_Oleft__cancel,axiom,
% 9.03/4.92      ! [A: complex,B: complex,C: complex] :
% 9.03/4.92        ( ( ( plus_plus_complex @ A @ B )
% 9.03/4.92          = ( plus_plus_complex @ A @ C ) )
% 9.03/4.92        = ( B = C ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_514_add_Oassoc,axiom,
% 9.03/4.92      ! [A: real,B: real,C: real] :
% 9.03/4.92        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.03/4.92        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_515_add_Oassoc,axiom,
% 9.03/4.92      ! [A: rat,B: rat,C: rat] :
% 9.03/4.92        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.03/4.92        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_516_add_Oassoc,axiom,
% 9.03/4.92      ! [A: nat,B: nat,C: nat] :
% 9.03/4.92        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.03/4.92        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_517_add_Oassoc,axiom,
% 9.03/4.92      ! [A: int,B: int,C: int] :
% 9.03/4.92        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.03/4.92        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_518_add_Oassoc,axiom,
% 9.03/4.92      ! [A: complex,B: complex,C: complex] :
% 9.03/4.92        ( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 9.03/4.92        = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_519_group__cancel_Oadd2,axiom,
% 9.03/4.92      ! [B2: real,K: real,B: real,A: real] :
% 9.03/4.92        ( ( B2
% 9.03/4.92          = ( plus_plus_real @ K @ B ) )
% 9.03/4.92       => ( ( plus_plus_real @ A @ B2 )
% 9.03/4.92          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_520_group__cancel_Oadd2,axiom,
% 9.03/4.92      ! [B2: rat,K: rat,B: rat,A: rat] :
% 9.03/4.92        ( ( B2
% 9.03/4.92          = ( plus_plus_rat @ K @ B ) )
% 9.03/4.92       => ( ( plus_plus_rat @ A @ B2 )
% 9.03/4.92          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_521_group__cancel_Oadd2,axiom,
% 9.03/4.92      ! [B2: nat,K: nat,B: nat,A: nat] :
% 9.03/4.92        ( ( B2
% 9.03/4.92          = ( plus_plus_nat @ K @ B ) )
% 9.03/4.92       => ( ( plus_plus_nat @ A @ B2 )
% 9.03/4.92          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_522_group__cancel_Oadd2,axiom,
% 9.03/4.92      ! [B2: int,K: int,B: int,A: int] :
% 9.03/4.92        ( ( B2
% 9.03/4.92          = ( plus_plus_int @ K @ B ) )
% 9.03/4.92       => ( ( plus_plus_int @ A @ B2 )
% 9.03/4.92          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_523_group__cancel_Oadd2,axiom,
% 9.03/4.92      ! [B2: complex,K: complex,B: complex,A: complex] :
% 9.03/4.92        ( ( B2
% 9.03/4.92          = ( plus_plus_complex @ K @ B ) )
% 9.03/4.92       => ( ( plus_plus_complex @ A @ B2 )
% 9.03/4.92          = ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_524_group__cancel_Oadd1,axiom,
% 9.03/4.92      ! [A2: real,K: real,A: real,B: real] :
% 9.03/4.92        ( ( A2
% 9.03/4.92          = ( plus_plus_real @ K @ A ) )
% 9.03/4.92       => ( ( plus_plus_real @ A2 @ B )
% 9.03/4.92          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_525_group__cancel_Oadd1,axiom,
% 9.03/4.92      ! [A2: rat,K: rat,A: rat,B: rat] :
% 9.03/4.92        ( ( A2
% 9.03/4.92          = ( plus_plus_rat @ K @ A ) )
% 9.03/4.92       => ( ( plus_plus_rat @ A2 @ B )
% 9.03/4.92          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_526_group__cancel_Oadd1,axiom,
% 9.03/4.92      ! [A2: nat,K: nat,A: nat,B: nat] :
% 9.03/4.92        ( ( A2
% 9.03/4.92          = ( plus_plus_nat @ K @ A ) )
% 9.03/4.92       => ( ( plus_plus_nat @ A2 @ B )
% 9.03/4.92          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_527_group__cancel_Oadd1,axiom,
% 9.03/4.92      ! [A2: int,K: int,A: int,B: int] :
% 9.03/4.92        ( ( A2
% 9.03/4.92          = ( plus_plus_int @ K @ A ) )
% 9.03/4.92       => ( ( plus_plus_int @ A2 @ B )
% 9.03/4.92          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_528_group__cancel_Oadd1,axiom,
% 9.03/4.92      ! [A2: complex,K: complex,A: complex,B: complex] :
% 9.03/4.92        ( ( A2
% 9.03/4.92          = ( plus_plus_complex @ K @ A ) )
% 9.03/4.92       => ( ( plus_plus_complex @ A2 @ B )
% 9.03/4.92          = ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_529_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.03/4.92      ! [I: real,J: real,K: real,L: real] :
% 9.03/4.92        ( ( ( I = J )
% 9.03/4.92          & ( K = L ) )
% 9.03/4.92       => ( ( plus_plus_real @ I @ K )
% 9.03/4.92          = ( plus_plus_real @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_530_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.03/4.92      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.03/4.92        ( ( ( I = J )
% 9.03/4.92          & ( K = L ) )
% 9.03/4.92       => ( ( plus_plus_rat @ I @ K )
% 9.03/4.92          = ( plus_plus_rat @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_531_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.03/4.92      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.92        ( ( ( I = J )
% 9.03/4.92          & ( K = L ) )
% 9.03/4.92       => ( ( plus_plus_nat @ I @ K )
% 9.03/4.92          = ( plus_plus_nat @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_532_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.03/4.92      ! [I: int,J: int,K: int,L: int] :
% 9.03/4.92        ( ( ( I = J )
% 9.03/4.92          & ( K = L ) )
% 9.03/4.92       => ( ( plus_plus_int @ I @ K )
% 9.03/4.92          = ( plus_plus_int @ J @ L ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_533_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.03/4.92      ! [A: real,B: real,C: real] :
% 9.03/4.92        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.03/4.92        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_534_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.03/4.92      ! [A: rat,B: rat,C: rat] :
% 9.03/4.92        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.03/4.92        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_535_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.03/4.92      ! [A: nat,B: nat,C: nat] :
% 9.03/4.92        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.03/4.92        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_536_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.03/4.92      ! [A: int,B: int,C: int] :
% 9.03/4.92        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.03/4.92        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_537_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.03/4.92      ! [A: complex,B: complex,C: complex] :
% 9.03/4.92        ( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 9.03/4.92        = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_538_ceiling__add__le,axiom,
% 9.03/4.92      ! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_539_ceiling__add__le,axiom,
% 9.03/4.92      ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_540_invar__vebt_Ointros_I2_J,axiom,
% 9.03/4.92      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 9.03/4.92        ( ! [X3: vEBT_VEBT] :
% 9.03/4.92            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.03/4.92           => ( vEBT_invar_vebt @ X3 @ N ) )
% 9.03/4.92       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.03/4.92         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.03/4.92              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.03/4.92           => ( ( M = N )
% 9.03/4.92             => ( ( Deg
% 9.03/4.92                  = ( plus_plus_nat @ N @ M ) )
% 9.03/4.92               => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
% 9.03/4.92                 => ( ! [X3: vEBT_VEBT] :
% 9.03/4.92                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.03/4.92                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
% 9.03/4.92                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_541_invar__vebt_Ointros_I3_J,axiom,
% 9.03/4.92      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 9.03/4.92        ( ! [X3: vEBT_VEBT] :
% 9.03/4.92            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.03/4.92           => ( vEBT_invar_vebt @ X3 @ N ) )
% 9.03/4.92       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.03/4.92         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.03/4.92              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.03/4.92           => ( ( M
% 9.03/4.92                = ( suc @ N ) )
% 9.03/4.92             => ( ( Deg
% 9.03/4.92                  = ( plus_plus_nat @ N @ M ) )
% 9.03/4.92               => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
% 9.03/4.92                 => ( ! [X3: vEBT_VEBT] :
% 9.03/4.92                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.03/4.92                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
% 9.03/4.92                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_542_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,I: num,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_543_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,I: num,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_544_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,I: num,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_545_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,I: num,N: nat] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 9.03/4.92        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_546_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [I: num,N: nat,X: nat] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 9.03/4.92        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_547_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [I: num,N: nat,X: nat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 9.03/4.92        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_548_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [I: num,N: nat,X: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 9.03/4.92        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_549_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [I: num,N: nat,X: nat] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 9.03/4.92        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_550_numeral__le__real__of__nat__iff,axiom,
% 9.03/4.92      ! [N: num,M: nat] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 9.03/4.92        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_551_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 9.03/4.92      ! [Y: nat,X: num,N: nat] :
% 9.03/4.92        ( ( ( semiri8010041392384452111omplex @ Y )
% 9.03/4.92          = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 9.03/4.92        = ( Y
% 9.03/4.92          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_552_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 9.03/4.92      ! [Y: nat,X: num,N: nat] :
% 9.03/4.92        ( ( ( semiri5074537144036343181t_real @ Y )
% 9.03/4.92          = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 9.03/4.92        = ( Y
% 9.03/4.92          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_553_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 9.03/4.92      ! [Y: nat,X: num,N: nat] :
% 9.03/4.92        ( ( ( semiri681578069525770553at_rat @ Y )
% 9.03/4.92          = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 9.03/4.92        = ( Y
% 9.03/4.92          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_554_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 9.03/4.92      ! [Y: nat,X: num,N: nat] :
% 9.03/4.92        ( ( ( semiri1316708129612266289at_nat @ Y )
% 9.03/4.92          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 9.03/4.92        = ( Y
% 9.03/4.92          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_555_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 9.03/4.92      ! [Y: nat,X: num,N: nat] :
% 9.03/4.92        ( ( ( semiri1314217659103216013at_int @ Y )
% 9.03/4.92          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 9.03/4.92        = ( Y
% 9.03/4.92          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_556_numeral__power__eq__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: num,N: nat,Y: nat] :
% 9.03/4.92        ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 9.03/4.92          = ( semiri8010041392384452111omplex @ Y ) )
% 9.03/4.92        = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 9.03/4.92          = Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_557_numeral__power__eq__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: num,N: nat,Y: nat] :
% 9.03/4.92        ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 9.03/4.92          = ( semiri5074537144036343181t_real @ Y ) )
% 9.03/4.92        = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 9.03/4.92          = Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_558_numeral__power__eq__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: num,N: nat,Y: nat] :
% 9.03/4.92        ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 9.03/4.92          = ( semiri681578069525770553at_rat @ Y ) )
% 9.03/4.92        = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 9.03/4.92          = Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_559_numeral__power__eq__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: num,N: nat,Y: nat] :
% 9.03/4.92        ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 9.03/4.92          = ( semiri1316708129612266289at_nat @ Y ) )
% 9.03/4.92        = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 9.03/4.92          = Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_560_numeral__power__eq__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: num,N: nat,Y: nat] :
% 9.03/4.92        ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 9.03/4.92          = ( semiri1314217659103216013at_int @ Y ) )
% 9.03/4.92        = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 9.03/4.92          = Y ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_561_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,B: nat,W: nat] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 9.03/4.92        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_562_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,B: nat,W: nat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 9.03/4.92        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_563_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,B: nat,W: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 9.03/4.92        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_564_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,B: nat,W: nat] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 9.03/4.92        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_565_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.03/4.92      ! [B: nat,W: nat,X: nat] :
% 9.03/4.92        ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 9.03/4.92        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_566_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.03/4.92      ! [B: nat,W: nat,X: nat] :
% 9.03/4.92        ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 9.03/4.92        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_567_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.03/4.92      ! [B: nat,W: nat,X: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 9.03/4.92        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_568_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.03/4.92      ! [B: nat,W: nat,X: nat] :
% 9.03/4.92        ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 9.03/4.92        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_569_ceiling__numeral__power,axiom,
% 9.03/4.92      ! [X: num,N: nat] :
% 9.03/4.92        ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 9.03/4.92        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_570_ceiling__numeral__power,axiom,
% 9.03/4.92      ! [X: num,N: nat] :
% 9.03/4.92        ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 9.03/4.92        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_571_le__log2__of__power,axiom,
% 9.03/4.92      ! [N: nat,M: nat] :
% 9.03/4.92        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 9.03/4.92       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_572_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,B: nat,W: nat] :
% 9.03/4.92        ( ( ( semiri8010041392384452111omplex @ X )
% 9.03/4.92          = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 9.03/4.92        = ( X
% 9.03/4.92          = ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_573_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,B: nat,W: nat] :
% 9.03/4.92        ( ( ( semiri5074537144036343181t_real @ X )
% 9.03/4.92          = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 9.03/4.92        = ( X
% 9.03/4.92          = ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_574_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,B: nat,W: nat] :
% 9.03/4.92        ( ( ( semiri681578069525770553at_rat @ X )
% 9.03/4.92          = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 9.03/4.92        = ( X
% 9.03/4.92          = ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_575_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,B: nat,W: nat] :
% 9.03/4.92        ( ( ( semiri1316708129612266289at_nat @ X )
% 9.03/4.92          = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 9.03/4.92        = ( X
% 9.03/4.92          = ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_576_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 9.03/4.92      ! [X: nat,B: nat,W: nat] :
% 9.03/4.92        ( ( ( semiri1314217659103216013at_int @ X )
% 9.03/4.92          = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 9.03/4.92        = ( X
% 9.03/4.92          = ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_577_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 9.03/4.92      ! [B: nat,W: nat,X: nat] :
% 9.03/4.92        ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 9.03/4.92          = ( semiri8010041392384452111omplex @ X ) )
% 9.03/4.92        = ( ( power_power_nat @ B @ W )
% 9.03/4.92          = X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_578_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 9.03/4.92      ! [B: nat,W: nat,X: nat] :
% 9.03/4.92        ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 9.03/4.92          = ( semiri5074537144036343181t_real @ X ) )
% 9.03/4.92        = ( ( power_power_nat @ B @ W )
% 9.03/4.92          = X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_579_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 9.03/4.92      ! [B: nat,W: nat,X: nat] :
% 9.03/4.92        ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
% 9.03/4.92          = ( semiri681578069525770553at_rat @ X ) )
% 9.03/4.92        = ( ( power_power_nat @ B @ W )
% 9.03/4.92          = X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_580_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 9.03/4.92      ! [B: nat,W: nat,X: nat] :
% 9.03/4.92        ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 9.03/4.92          = ( semiri1316708129612266289at_nat @ X ) )
% 9.03/4.92        = ( ( power_power_nat @ B @ W )
% 9.03/4.92          = X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_581_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 9.03/4.92      ! [B: nat,W: nat,X: nat] :
% 9.03/4.92        ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 9.03/4.92          = ( semiri1314217659103216013at_int @ X ) )
% 9.03/4.92        = ( ( power_power_nat @ B @ W )
% 9.03/4.92          = X ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_582_of__nat__power,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 9.03/4.92        = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_583_of__nat__power,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 9.03/4.92        = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_584_of__nat__power,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 9.03/4.92        = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_585_of__nat__power,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 9.03/4.92        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_586_of__nat__power,axiom,
% 9.03/4.92      ! [M: nat,N: nat] :
% 9.03/4.92        ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 9.03/4.92        = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_587_insert__subset,axiom,
% 9.03/4.92      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.92        ( ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B2 )
% 9.03/4.92        = ( ( member_VEBT_VEBT @ X @ B2 )
% 9.03/4.92          & ( ord_le4337996190870823476T_VEBT @ A2 @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_588_insert__subset,axiom,
% 9.03/4.92      ! [X: $o,A2: set_o,B2: set_o] :
% 9.03/4.92        ( ( ord_less_eq_set_o @ ( insert_o @ X @ A2 ) @ B2 )
% 9.03/4.92        = ( ( member_o @ X @ B2 )
% 9.03/4.92          & ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_589_insert__subset,axiom,
% 9.03/4.92      ! [X: complex,A2: set_complex,B2: set_complex] :
% 9.03/4.92        ( ( ord_le211207098394363844omplex @ ( insert_complex @ X @ A2 ) @ B2 )
% 9.03/4.92        = ( ( member_complex @ X @ B2 )
% 9.03/4.92          & ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_590_insert__subset,axiom,
% 9.03/4.92      ! [X: real,A2: set_real,B2: set_real] :
% 9.03/4.92        ( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 9.03/4.92        = ( ( member_real @ X @ B2 )
% 9.03/4.92          & ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_591_insert__subset,axiom,
% 9.03/4.92      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.92        ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 9.03/4.92        = ( ( member_set_nat @ X @ B2 )
% 9.03/4.92          & ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_592_insert__subset,axiom,
% 9.03/4.92      ! [X: nat,A2: set_nat,B2: set_nat] :
% 9.03/4.92        ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 9.03/4.92        = ( ( member_nat @ X @ B2 )
% 9.03/4.92          & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_593_insert__subset,axiom,
% 9.03/4.92      ! [X: int,A2: set_int,B2: set_int] :
% 9.03/4.92        ( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 9.03/4.92        = ( ( member_int @ X @ B2 )
% 9.03/4.92          & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_594_power__one,axiom,
% 9.03/4.92      ! [N: nat] :
% 9.03/4.92        ( ( power_power_rat @ one_one_rat @ N )
% 9.03/4.92        = one_one_rat ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_595_power__one,axiom,
% 9.03/4.92      ! [N: nat] :
% 9.03/4.92        ( ( power_power_nat @ one_one_nat @ N )
% 9.03/4.92        = one_one_nat ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_596_power__one,axiom,
% 9.03/4.92      ! [N: nat] :
% 9.03/4.92        ( ( power_power_real @ one_one_real @ N )
% 9.03/4.92        = one_one_real ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_597_power__one,axiom,
% 9.03/4.92      ! [N: nat] :
% 9.03/4.92        ( ( power_power_int @ one_one_int @ N )
% 9.03/4.92        = one_one_int ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_598_power__one,axiom,
% 9.03/4.92      ! [N: nat] :
% 9.03/4.92        ( ( power_power_complex @ one_one_complex @ N )
% 9.03/4.92        = one_one_complex ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_599_power__one__right,axiom,
% 9.03/4.92      ! [A: nat] :
% 9.03/4.92        ( ( power_power_nat @ A @ one_one_nat )
% 9.03/4.92        = A ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_600_power__one__right,axiom,
% 9.03/4.92      ! [A: real] :
% 9.03/4.92        ( ( power_power_real @ A @ one_one_nat )
% 9.03/4.92        = A ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_601_power__one__right,axiom,
% 9.03/4.92      ! [A: int] :
% 9.03/4.92        ( ( power_power_int @ A @ one_one_nat )
% 9.03/4.92        = A ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_602_power__one__right,axiom,
% 9.03/4.92      ! [A: complex] :
% 9.03/4.92        ( ( power_power_complex @ A @ one_one_nat )
% 9.03/4.92        = A ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_603_semiring__norm_I71_J,axiom,
% 9.03/4.92      ! [M: num,N: num] :
% 9.03/4.92        ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.03/4.92        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_604_semiring__norm_I68_J,axiom,
% 9.03/4.92      ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_605_semiring__norm_I69_J,axiom,
% 9.03/4.92      ! [M: num] :
% 9.03/4.92        ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_606_le__num__One__iff,axiom,
% 9.03/4.92      ! [X: num] :
% 9.03/4.92        ( ( ord_less_eq_num @ X @ one )
% 9.03/4.92        = ( X = one ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_607_subset__image__iff,axiom,
% 9.03/4.92      ! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
% 9.03/4.92        ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
% 9.03/4.92        = ( ? [AA: set_nat] :
% 9.03/4.92              ( ( ord_less_eq_set_nat @ AA @ A2 )
% 9.03/4.92              & ( B2
% 9.03/4.92                = ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_608_subset__image__iff,axiom,
% 9.03/4.92      ! [B2: set_nat,F: nat > nat,A2: set_nat] :
% 9.03/4.92        ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
% 9.03/4.92        = ( ? [AA: set_nat] :
% 9.03/4.92              ( ( ord_less_eq_set_nat @ AA @ A2 )
% 9.03/4.92              & ( B2
% 9.03/4.92                = ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_609_subset__image__iff,axiom,
% 9.03/4.92      ! [B2: set_nat,F: int > nat,A2: set_int] :
% 9.03/4.92        ( ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F @ A2 ) )
% 9.03/4.92        = ( ? [AA: set_int] :
% 9.03/4.92              ( ( ord_less_eq_set_int @ AA @ A2 )
% 9.03/4.92              & ( B2
% 9.03/4.92                = ( image_int_nat @ F @ AA ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_610_subset__image__iff,axiom,
% 9.03/4.92      ! [B2: set_int,F: nat > int,A2: set_nat] :
% 9.03/4.92        ( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A2 ) )
% 9.03/4.92        = ( ? [AA: set_nat] :
% 9.03/4.92              ( ( ord_less_eq_set_nat @ AA @ A2 )
% 9.03/4.92              & ( B2
% 9.03/4.92                = ( image_nat_int @ F @ AA ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_611_subset__image__iff,axiom,
% 9.03/4.92      ! [B2: set_int,F: int > int,A2: set_int] :
% 9.03/4.92        ( ( ord_less_eq_set_int @ B2 @ ( image_int_int @ F @ A2 ) )
% 9.03/4.92        = ( ? [AA: set_int] :
% 9.03/4.92              ( ( ord_less_eq_set_int @ AA @ A2 )
% 9.03/4.92              & ( B2
% 9.03/4.92                = ( image_int_int @ F @ AA ) ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_612_image__subset__iff,axiom,
% 9.03/4.92      ! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
% 9.03/4.92        ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
% 9.03/4.92        = ( ! [X4: nat] :
% 9.03/4.92              ( ( member_nat @ X4 @ A2 )
% 9.03/4.92             => ( member_set_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_613_image__subset__iff,axiom,
% 9.03/4.92      ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 9.03/4.92        ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
% 9.03/4.92        = ( ! [X4: nat] :
% 9.03/4.92              ( ( member_nat @ X4 @ A2 )
% 9.03/4.92             => ( member_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_614_image__subset__iff,axiom,
% 9.03/4.92      ! [F: int > nat,A2: set_int,B2: set_nat] :
% 9.03/4.92        ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A2 ) @ B2 )
% 9.03/4.92        = ( ! [X4: int] :
% 9.03/4.92              ( ( member_int @ X4 @ A2 )
% 9.03/4.92             => ( member_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_615_image__subset__iff,axiom,
% 9.03/4.92      ! [F: nat > int,A2: set_nat,B2: set_int] :
% 9.03/4.92        ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B2 )
% 9.03/4.92        = ( ! [X4: nat] :
% 9.03/4.92              ( ( member_nat @ X4 @ A2 )
% 9.03/4.92             => ( member_int @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_616_image__subset__iff,axiom,
% 9.03/4.92      ! [F: int > int,A2: set_int,B2: set_int] :
% 9.03/4.92        ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A2 ) @ B2 )
% 9.03/4.92        = ( ! [X4: int] :
% 9.03/4.92              ( ( member_int @ X4 @ A2 )
% 9.03/4.92             => ( member_int @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_617_subset__imageE,axiom,
% 9.03/4.92      ! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
% 9.03/4.92        ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
% 9.03/4.92       => ~ ! [C5: set_nat] :
% 9.03/4.92              ( ( ord_less_eq_set_nat @ C5 @ A2 )
% 9.03/4.92             => ( B2
% 9.03/4.92               != ( image_nat_set_nat @ F @ C5 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_618_subset__imageE,axiom,
% 9.03/4.92      ! [B2: set_nat,F: nat > nat,A2: set_nat] :
% 9.03/4.92        ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
% 9.03/4.92       => ~ ! [C5: set_nat] :
% 9.03/4.92              ( ( ord_less_eq_set_nat @ C5 @ A2 )
% 9.03/4.92             => ( B2
% 9.03/4.92               != ( image_nat_nat @ F @ C5 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_619_subset__imageE,axiom,
% 9.03/4.92      ! [B2: set_nat,F: int > nat,A2: set_int] :
% 9.03/4.92        ( ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F @ A2 ) )
% 9.03/4.92       => ~ ! [C5: set_int] :
% 9.03/4.92              ( ( ord_less_eq_set_int @ C5 @ A2 )
% 9.03/4.92             => ( B2
% 9.03/4.92               != ( image_int_nat @ F @ C5 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_620_subset__imageE,axiom,
% 9.03/4.92      ! [B2: set_int,F: nat > int,A2: set_nat] :
% 9.03/4.92        ( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A2 ) )
% 9.03/4.92       => ~ ! [C5: set_nat] :
% 9.03/4.92              ( ( ord_less_eq_set_nat @ C5 @ A2 )
% 9.03/4.92             => ( B2
% 9.03/4.92               != ( image_nat_int @ F @ C5 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_621_subset__imageE,axiom,
% 9.03/4.92      ! [B2: set_int,F: int > int,A2: set_int] :
% 9.03/4.92        ( ( ord_less_eq_set_int @ B2 @ ( image_int_int @ F @ A2 ) )
% 9.03/4.92       => ~ ! [C5: set_int] :
% 9.03/4.92              ( ( ord_less_eq_set_int @ C5 @ A2 )
% 9.03/4.92             => ( B2
% 9.03/4.92               != ( image_int_int @ F @ C5 ) ) ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_622_image__subsetI,axiom,
% 9.03/4.92      ! [A2: set_complex,F: complex > complex,B2: set_complex] :
% 9.03/4.92        ( ! [X3: complex] :
% 9.03/4.92            ( ( member_complex @ X3 @ A2 )
% 9.03/4.92           => ( member_complex @ ( F @ X3 ) @ B2 ) )
% 9.03/4.92       => ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F @ A2 ) @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_623_image__subsetI,axiom,
% 9.03/4.92      ! [A2: set_complex,F: complex > real,B2: set_real] :
% 9.03/4.92        ( ! [X3: complex] :
% 9.03/4.92            ( ( member_complex @ X3 @ A2 )
% 9.03/4.92           => ( member_real @ ( F @ X3 ) @ B2 ) )
% 9.03/4.92       => ( ord_less_eq_set_real @ ( image_complex_real @ F @ A2 ) @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_624_image__subsetI,axiom,
% 9.03/4.92      ! [A2: set_complex,F: complex > nat,B2: set_nat] :
% 9.03/4.92        ( ! [X3: complex] :
% 9.03/4.92            ( ( member_complex @ X3 @ A2 )
% 9.03/4.92           => ( member_nat @ ( F @ X3 ) @ B2 ) )
% 9.03/4.92       => ( ord_less_eq_set_nat @ ( image_complex_nat @ F @ A2 ) @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_625_image__subsetI,axiom,
% 9.03/4.92      ! [A2: set_real,F: real > complex,B2: set_complex] :
% 9.03/4.92        ( ! [X3: real] :
% 9.03/4.92            ( ( member_real @ X3 @ A2 )
% 9.03/4.92           => ( member_complex @ ( F @ X3 ) @ B2 ) )
% 9.03/4.92       => ( ord_le211207098394363844omplex @ ( image_real_complex @ F @ A2 ) @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_626_image__subsetI,axiom,
% 9.03/4.92      ! [A2: set_real,F: real > real,B2: set_real] :
% 9.03/4.92        ( ! [X3: real] :
% 9.03/4.92            ( ( member_real @ X3 @ A2 )
% 9.03/4.92           => ( member_real @ ( F @ X3 ) @ B2 ) )
% 9.03/4.92       => ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_627_image__subsetI,axiom,
% 9.03/4.92      ! [A2: set_real,F: real > nat,B2: set_nat] :
% 9.03/4.92        ( ! [X3: real] :
% 9.03/4.92            ( ( member_real @ X3 @ A2 )
% 9.03/4.92           => ( member_nat @ ( F @ X3 ) @ B2 ) )
% 9.03/4.92       => ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A2 ) @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_628_image__subsetI,axiom,
% 9.03/4.92      ! [A2: set_nat,F: nat > complex,B2: set_complex] :
% 9.03/4.92        ( ! [X3: nat] :
% 9.03/4.92            ( ( member_nat @ X3 @ A2 )
% 9.03/4.92           => ( member_complex @ ( F @ X3 ) @ B2 ) )
% 9.03/4.92       => ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A2 ) @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_629_image__subsetI,axiom,
% 9.03/4.92      ! [A2: set_nat,F: nat > real,B2: set_real] :
% 9.03/4.92        ( ! [X3: nat] :
% 9.03/4.92            ( ( member_nat @ X3 @ A2 )
% 9.03/4.92           => ( member_real @ ( F @ X3 ) @ B2 ) )
% 9.03/4.92       => ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_630_image__subsetI,axiom,
% 9.03/4.92      ! [A2: set_nat,F: nat > nat,B2: set_nat] :
% 9.03/4.92        ( ! [X3: nat] :
% 9.03/4.92            ( ( member_nat @ X3 @ A2 )
% 9.03/4.92           => ( member_nat @ ( F @ X3 ) @ B2 ) )
% 9.03/4.92       => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% 9.03/4.92  
% 9.03/4.92  thf(fact_631_image__subsetI,axiom,
% 9.03/4.92      ! [A2: set_int,F: int > complex,B2: set_complex] :
% 9.03/4.92        ( ! [X3: int] :
% 9.03/4.92            ( ( member_int @ X3 @ A2 )
% 9.03/4.93           => ( member_complex @ ( F @ X3 ) @ B2 ) )
% 9.03/4.93       => ( ord_le211207098394363844omplex @ ( image_int_complex @ F @ A2 ) @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_632_image__mono,axiom,
% 9.03/4.93      ! [A2: set_nat,B2: set_nat,F: nat > set_nat] :
% 9.03/4.93        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.03/4.93       => ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_633_image__mono,axiom,
% 9.03/4.93      ! [A2: set_nat,B2: set_nat,F: nat > nat] :
% 9.03/4.93        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.03/4.93       => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_634_image__mono,axiom,
% 9.03/4.93      ! [A2: set_nat,B2: set_nat,F: nat > int] :
% 9.03/4.93        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.03/4.93       => ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_635_image__mono,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int,F: int > nat] :
% 9.03/4.93        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.93       => ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A2 ) @ ( image_int_nat @ F @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_636_image__mono,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int,F: int > int] :
% 9.03/4.93        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.93       => ( ord_less_eq_set_int @ ( image_int_int @ F @ A2 ) @ ( image_int_int @ F @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_637_subset__insertI2,axiom,
% 9.03/4.93      ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
% 9.03/4.93        ( ( ord_le4337996190870823476T_VEBT @ A2 @ B2 )
% 9.03/4.93       => ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_638_subset__insertI2,axiom,
% 9.03/4.93      ! [A2: set_nat,B2: set_nat,B: nat] :
% 9.03/4.93        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.03/4.93       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_639_subset__insertI2,axiom,
% 9.03/4.93      ! [A2: set_real,B2: set_real,B: real] :
% 9.03/4.93        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.03/4.93       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_640_subset__insertI2,axiom,
% 9.03/4.93      ! [A2: set_o,B2: set_o,B: $o] :
% 9.03/4.93        ( ( ord_less_eq_set_o @ A2 @ B2 )
% 9.03/4.93       => ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_641_subset__insertI2,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int,B: int] :
% 9.03/4.93        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.93       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_642_subset__insertI,axiom,
% 9.03/4.93      ! [B2: set_VEBT_VEBT,A: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ B2 @ ( insert_VEBT_VEBT @ A @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_643_subset__insertI,axiom,
% 9.03/4.93      ! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_644_subset__insertI,axiom,
% 9.03/4.93      ! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_645_subset__insertI,axiom,
% 9.03/4.93      ! [B2: set_o,A: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o @ A @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_646_subset__insertI,axiom,
% 9.03/4.93      ! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_647_subset__insert,axiom,
% 9.03/4.93      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.93        ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 9.03/4.93       => ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B2 ) )
% 9.03/4.93          = ( ord_le4337996190870823476T_VEBT @ A2 @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_648_subset__insert,axiom,
% 9.03/4.93      ! [X: $o,A2: set_o,B2: set_o] :
% 9.03/4.93        ( ~ ( member_o @ X @ A2 )
% 9.03/4.93       => ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
% 9.03/4.93          = ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_649_subset__insert,axiom,
% 9.03/4.93      ! [X: complex,A2: set_complex,B2: set_complex] :
% 9.03/4.93        ( ~ ( member_complex @ X @ A2 )
% 9.03/4.93       => ( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B2 ) )
% 9.03/4.93          = ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_650_subset__insert,axiom,
% 9.03/4.93      ! [X: real,A2: set_real,B2: set_real] :
% 9.03/4.93        ( ~ ( member_real @ X @ A2 )
% 9.03/4.93       => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 9.03/4.93          = ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_651_subset__insert,axiom,
% 9.03/4.93      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.93        ( ~ ( member_set_nat @ X @ A2 )
% 9.03/4.93       => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 9.03/4.93          = ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_652_subset__insert,axiom,
% 9.03/4.93      ! [X: nat,A2: set_nat,B2: set_nat] :
% 9.03/4.93        ( ~ ( member_nat @ X @ A2 )
% 9.03/4.93       => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 9.03/4.93          = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_653_subset__insert,axiom,
% 9.03/4.93      ! [X: int,A2: set_int,B2: set_int] :
% 9.03/4.93        ( ~ ( member_int @ X @ A2 )
% 9.03/4.93       => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 9.03/4.93          = ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_654_insert__mono,axiom,
% 9.03/4.93      ! [C6: set_VEBT_VEBT,D2: set_VEBT_VEBT,A: vEBT_VEBT] :
% 9.03/4.93        ( ( ord_le4337996190870823476T_VEBT @ C6 @ D2 )
% 9.03/4.93       => ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ A @ C6 ) @ ( insert_VEBT_VEBT @ A @ D2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_655_insert__mono,axiom,
% 9.03/4.93      ! [C6: set_nat,D2: set_nat,A: nat] :
% 9.03/4.93        ( ( ord_less_eq_set_nat @ C6 @ D2 )
% 9.03/4.93       => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C6 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_656_insert__mono,axiom,
% 9.03/4.93      ! [C6: set_real,D2: set_real,A: real] :
% 9.03/4.93        ( ( ord_less_eq_set_real @ C6 @ D2 )
% 9.03/4.93       => ( ord_less_eq_set_real @ ( insert_real @ A @ C6 ) @ ( insert_real @ A @ D2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_657_insert__mono,axiom,
% 9.03/4.93      ! [C6: set_o,D2: set_o,A: $o] :
% 9.03/4.93        ( ( ord_less_eq_set_o @ C6 @ D2 )
% 9.03/4.93       => ( ord_less_eq_set_o @ ( insert_o @ A @ C6 ) @ ( insert_o @ A @ D2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_658_insert__mono,axiom,
% 9.03/4.93      ! [C6: set_int,D2: set_int,A: int] :
% 9.03/4.93        ( ( ord_less_eq_set_int @ C6 @ D2 )
% 9.03/4.93       => ( ord_less_eq_set_int @ ( insert_int @ A @ C6 ) @ ( insert_int @ A @ D2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_659_two__realpow__ge__one,axiom,
% 9.03/4.93      ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_660_complete__real,axiom,
% 9.03/4.93      ! [S2: set_real] :
% 9.03/4.93        ( ? [X2: real] : ( member_real @ X2 @ S2 )
% 9.03/4.93       => ( ? [Z5: real] :
% 9.03/4.93            ! [X3: real] :
% 9.03/4.93              ( ( member_real @ X3 @ S2 )
% 9.03/4.93             => ( ord_less_eq_real @ X3 @ Z5 ) )
% 9.03/4.93         => ? [Y4: real] :
% 9.03/4.93              ( ! [X2: real] :
% 9.03/4.93                  ( ( member_real @ X2 @ S2 )
% 9.03/4.93                 => ( ord_less_eq_real @ X2 @ Y4 ) )
% 9.03/4.93              & ! [Z5: real] :
% 9.03/4.93                  ( ! [X3: real] :
% 9.03/4.93                      ( ( member_real @ X3 @ S2 )
% 9.03/4.93                     => ( ord_less_eq_real @ X3 @ Z5 ) )
% 9.03/4.93                 => ( ord_less_eq_real @ Y4 @ Z5 ) ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_661_one__le__power,axiom,
% 9.03/4.93      ! [A: real,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_real @ one_one_real @ A )
% 9.03/4.93       => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_662_one__le__power,axiom,
% 9.03/4.93      ! [A: rat,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.03/4.93       => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_663_one__le__power,axiom,
% 9.03/4.93      ! [A: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.03/4.93       => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_664_one__le__power,axiom,
% 9.03/4.93      ! [A: int,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_int @ one_one_int @ A )
% 9.03/4.93       => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_665_power__increasing,axiom,
% 9.03/4.93      ! [N: nat,N4: nat,A: real] :
% 9.03/4.93        ( ( ord_less_eq_nat @ N @ N4 )
% 9.03/4.93       => ( ( ord_less_eq_real @ one_one_real @ A )
% 9.03/4.93         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_666_power__increasing,axiom,
% 9.03/4.93      ! [N: nat,N4: nat,A: rat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ N @ N4 )
% 9.03/4.93       => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.03/4.93         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_667_power__increasing,axiom,
% 9.03/4.93      ! [N: nat,N4: nat,A: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ N @ N4 )
% 9.03/4.93       => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.03/4.93         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_668_power__increasing,axiom,
% 9.03/4.93      ! [N: nat,N4: nat,A: int] :
% 9.03/4.93        ( ( ord_less_eq_nat @ N @ N4 )
% 9.03/4.93       => ( ( ord_less_eq_int @ one_one_int @ A )
% 9.03/4.93         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_669_one__power2,axiom,
% 9.03/4.93      ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93      = one_one_rat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_670_one__power2,axiom,
% 9.03/4.93      ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93      = one_one_nat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_671_one__power2,axiom,
% 9.03/4.93      ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93      = one_one_real ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_672_one__power2,axiom,
% 9.03/4.93      ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93      = one_one_int ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_673_one__power2,axiom,
% 9.03/4.93      ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93      = one_one_complex ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_674_self__le__ge2__pow,axiom,
% 9.03/4.93      ! [K: nat,M: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.03/4.93       => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_675_power2__nat__le__eq__le,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_676_power2__nat__le__imp__le,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 9.03/4.93       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_677_log2__of__power__eq,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( M
% 9.03/4.93          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/4.93       => ( ( semiri5074537144036343181t_real @ N )
% 9.03/4.93          = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_678_enat__ord__number_I1_J,axiom,
% 9.03/4.93      ! [M: num,N: num] :
% 9.03/4.93        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.03/4.93        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_679_log__ceil__idem,axiom,
% 9.03/4.93      ! [X: real] :
% 9.03/4.93        ( ( ord_less_eq_real @ one_one_real @ X )
% 9.03/4.93       => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 9.03/4.93          = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_680_ceiling__log__nat__eq__if,axiom,
% 9.03/4.93      ! [B: nat,N: nat,K: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 9.03/4.93       => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 9.03/4.93         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.03/4.93           => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.03/4.93              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_681_pow__sum,axiom,
% 9.03/4.93      ! [A: nat,B: nat] :
% 9.03/4.93        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.03/4.93        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_682_power__numeral,axiom,
% 9.03/4.93      ! [K: num,L: num] :
% 9.03/4.93        ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
% 9.03/4.93        = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_683_power__numeral,axiom,
% 9.03/4.93      ! [K: num,L: num] :
% 9.03/4.93        ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
% 9.03/4.93        = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_684_power__numeral,axiom,
% 9.03/4.93      ! [K: num,L: num] :
% 9.03/4.93        ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
% 9.03/4.93        = ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_685_power__numeral,axiom,
% 9.03/4.93      ! [K: num,L: num] :
% 9.03/4.93        ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
% 9.03/4.93        = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_686_power__numeral,axiom,
% 9.03/4.93      ! [K: num,L: num] :
% 9.03/4.93        ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
% 9.03/4.93        = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_687_dual__order_Orefl,axiom,
% 9.03/4.93      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_688_dual__order_Orefl,axiom,
% 9.03/4.93      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_689_dual__order_Orefl,axiom,
% 9.03/4.93      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_690_dual__order_Orefl,axiom,
% 9.03/4.93      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_691_dual__order_Orefl,axiom,
% 9.03/4.93      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_692_order__refl,axiom,
% 9.03/4.93      ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_693_order__refl,axiom,
% 9.03/4.93      ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_694_order__refl,axiom,
% 9.03/4.93      ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_695_order__refl,axiom,
% 9.03/4.93      ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_696_order__refl,axiom,
% 9.03/4.93      ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_697_ex__power__ivl2,axiom,
% 9.03/4.93      ! [B: nat,K: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.03/4.93       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.03/4.93         => ? [N2: nat] :
% 9.03/4.93              ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 9.03/4.93              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_698_ex__power__ivl1,axiom,
% 9.03/4.93      ! [B: nat,K: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.03/4.93       => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 9.03/4.93         => ? [N2: nat] :
% 9.03/4.93              ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 9.03/4.93              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_699_ceiling__log__nat__eq__powr__iff,axiom,
% 9.03/4.93      ! [B: nat,K: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.03/4.93       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.93         => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.03/4.93              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 9.03/4.93            = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 9.03/4.93              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_700_deg__not__0,axiom,
% 9.03/4.93      ! [T: vEBT_VEBT,N: nat] :
% 9.03/4.93        ( ( vEBT_invar_vebt @ T @ N )
% 9.03/4.93       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_701_subset__antisym,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int] :
% 9.03/4.93        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.93       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.03/4.93         => ( A2 = B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_702_subsetI,axiom,
% 9.03/4.93      ! [A2: set_complex,B2: set_complex] :
% 9.03/4.93        ( ! [X3: complex] :
% 9.03/4.93            ( ( member_complex @ X3 @ A2 )
% 9.03/4.93           => ( member_complex @ X3 @ B2 ) )
% 9.03/4.93       => ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_703_subsetI,axiom,
% 9.03/4.93      ! [A2: set_real,B2: set_real] :
% 9.03/4.93        ( ! [X3: real] :
% 9.03/4.93            ( ( member_real @ X3 @ A2 )
% 9.03/4.93           => ( member_real @ X3 @ B2 ) )
% 9.03/4.93       => ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_704_subsetI,axiom,
% 9.03/4.93      ! [A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.93        ( ! [X3: set_nat] :
% 9.03/4.93            ( ( member_set_nat @ X3 @ A2 )
% 9.03/4.93           => ( member_set_nat @ X3 @ B2 ) )
% 9.03/4.93       => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_705_subsetI,axiom,
% 9.03/4.93      ! [A2: set_nat,B2: set_nat] :
% 9.03/4.93        ( ! [X3: nat] :
% 9.03/4.93            ( ( member_nat @ X3 @ A2 )
% 9.03/4.93           => ( member_nat @ X3 @ B2 ) )
% 9.03/4.93       => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_706_subsetI,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int] :
% 9.03/4.93        ( ! [X3: int] :
% 9.03/4.93            ( ( member_int @ X3 @ A2 )
% 9.03/4.93           => ( member_int @ X3 @ B2 ) )
% 9.03/4.93       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_707_of__int__eq__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ( ring_1_of_int_real @ W )
% 9.03/4.93          = ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = ( W = Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_708_of__int__eq__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ( ring_1_of_int_rat @ W )
% 9.03/4.93          = ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = ( W = Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_709_of__int__eq__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ( ring_18347121197199848620nteger @ W )
% 9.03/4.93          = ( ring_18347121197199848620nteger @ Z ) )
% 9.03/4.93        = ( W = Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_710_of__int__eq__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ( ring_17405671764205052669omplex @ W )
% 9.03/4.93          = ( ring_17405671764205052669omplex @ Z ) )
% 9.03/4.93        = ( W = Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_711_le__zero__eq,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 9.03/4.93        = ( N = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_712_not__gr__zero,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 9.03/4.93        = ( N = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_713_numeral__less__iff,axiom,
% 9.03/4.93      ! [M: num,N: num] :
% 9.03/4.93        ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.03/4.93        = ( ord_less_num @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_714_numeral__less__iff,axiom,
% 9.03/4.93      ! [M: num,N: num] :
% 9.03/4.93        ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.03/4.93        = ( ord_less_num @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_715_numeral__less__iff,axiom,
% 9.03/4.93      ! [M: num,N: num] :
% 9.03/4.93        ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.03/4.93        = ( ord_less_num @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_716_numeral__less__iff,axiom,
% 9.03/4.93      ! [M: num,N: num] :
% 9.03/4.93        ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.03/4.93        = ( ord_less_num @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_717_add__0,axiom,
% 9.03/4.93      ! [A: complex] :
% 9.03/4.93        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_718_add__0,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_719_add__0,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_720_add__0,axiom,
% 9.03/4.93      ! [A: nat] :
% 9.03/4.93        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_721_add__0,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_722_zero__eq__add__iff__both__eq__0,axiom,
% 9.03/4.93      ! [X: nat,Y: nat] :
% 9.03/4.93        ( ( zero_zero_nat
% 9.03/4.93          = ( plus_plus_nat @ X @ Y ) )
% 9.03/4.93        = ( ( X = zero_zero_nat )
% 9.03/4.93          & ( Y = zero_zero_nat ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_723_add__eq__0__iff__both__eq__0,axiom,
% 9.03/4.93      ! [X: nat,Y: nat] :
% 9.03/4.93        ( ( ( plus_plus_nat @ X @ Y )
% 9.03/4.93          = zero_zero_nat )
% 9.03/4.93        = ( ( X = zero_zero_nat )
% 9.03/4.93          & ( Y = zero_zero_nat ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_724_add__cancel__right__right,axiom,
% 9.03/4.93      ! [A: complex,B: complex] :
% 9.03/4.93        ( ( A
% 9.03/4.93          = ( plus_plus_complex @ A @ B ) )
% 9.03/4.93        = ( B = zero_zero_complex ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_725_add__cancel__right__right,axiom,
% 9.03/4.93      ! [A: real,B: real] :
% 9.03/4.93        ( ( A
% 9.03/4.93          = ( plus_plus_real @ A @ B ) )
% 9.03/4.93        = ( B = zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_726_add__cancel__right__right,axiom,
% 9.03/4.93      ! [A: rat,B: rat] :
% 9.03/4.93        ( ( A
% 9.03/4.93          = ( plus_plus_rat @ A @ B ) )
% 9.03/4.93        = ( B = zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_727_add__cancel__right__right,axiom,
% 9.03/4.93      ! [A: nat,B: nat] :
% 9.03/4.93        ( ( A
% 9.03/4.93          = ( plus_plus_nat @ A @ B ) )
% 9.03/4.93        = ( B = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_728_add__cancel__right__right,axiom,
% 9.03/4.93      ! [A: int,B: int] :
% 9.03/4.93        ( ( A
% 9.03/4.93          = ( plus_plus_int @ A @ B ) )
% 9.03/4.93        = ( B = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_729_add__cancel__right__left,axiom,
% 9.03/4.93      ! [A: complex,B: complex] :
% 9.03/4.93        ( ( A
% 9.03/4.93          = ( plus_plus_complex @ B @ A ) )
% 9.03/4.93        = ( B = zero_zero_complex ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_730_add__cancel__right__left,axiom,
% 9.03/4.93      ! [A: real,B: real] :
% 9.03/4.93        ( ( A
% 9.03/4.93          = ( plus_plus_real @ B @ A ) )
% 9.03/4.93        = ( B = zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_731_add__cancel__right__left,axiom,
% 9.03/4.93      ! [A: rat,B: rat] :
% 9.03/4.93        ( ( A
% 9.03/4.93          = ( plus_plus_rat @ B @ A ) )
% 9.03/4.93        = ( B = zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_732_add__cancel__right__left,axiom,
% 9.03/4.93      ! [A: nat,B: nat] :
% 9.03/4.93        ( ( A
% 9.03/4.93          = ( plus_plus_nat @ B @ A ) )
% 9.03/4.93        = ( B = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_733_add__cancel__right__left,axiom,
% 9.03/4.93      ! [A: int,B: int] :
% 9.03/4.93        ( ( A
% 9.03/4.93          = ( plus_plus_int @ B @ A ) )
% 9.03/4.93        = ( B = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_734_add__cancel__left__right,axiom,
% 9.03/4.93      ! [A: complex,B: complex] :
% 9.03/4.93        ( ( ( plus_plus_complex @ A @ B )
% 9.03/4.93          = A )
% 9.03/4.93        = ( B = zero_zero_complex ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_735_add__cancel__left__right,axiom,
% 9.03/4.93      ! [A: real,B: real] :
% 9.03/4.93        ( ( ( plus_plus_real @ A @ B )
% 9.03/4.93          = A )
% 9.03/4.93        = ( B = zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_736_add__cancel__left__right,axiom,
% 9.03/4.93      ! [A: rat,B: rat] :
% 9.03/4.93        ( ( ( plus_plus_rat @ A @ B )
% 9.03/4.93          = A )
% 9.03/4.93        = ( B = zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_737_add__cancel__left__right,axiom,
% 9.03/4.93      ! [A: nat,B: nat] :
% 9.03/4.93        ( ( ( plus_plus_nat @ A @ B )
% 9.03/4.93          = A )
% 9.03/4.93        = ( B = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_738_add__cancel__left__right,axiom,
% 9.03/4.93      ! [A: int,B: int] :
% 9.03/4.93        ( ( ( plus_plus_int @ A @ B )
% 9.03/4.93          = A )
% 9.03/4.93        = ( B = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_739_add__cancel__left__left,axiom,
% 9.03/4.93      ! [B: complex,A: complex] :
% 9.03/4.93        ( ( ( plus_plus_complex @ B @ A )
% 9.03/4.93          = A )
% 9.03/4.93        = ( B = zero_zero_complex ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_740_add__cancel__left__left,axiom,
% 9.03/4.93      ! [B: real,A: real] :
% 9.03/4.93        ( ( ( plus_plus_real @ B @ A )
% 9.03/4.93          = A )
% 9.03/4.93        = ( B = zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_741_add__cancel__left__left,axiom,
% 9.03/4.93      ! [B: rat,A: rat] :
% 9.03/4.93        ( ( ( plus_plus_rat @ B @ A )
% 9.03/4.93          = A )
% 9.03/4.93        = ( B = zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_742_add__cancel__left__left,axiom,
% 9.03/4.93      ! [B: nat,A: nat] :
% 9.03/4.93        ( ( ( plus_plus_nat @ B @ A )
% 9.03/4.93          = A )
% 9.03/4.93        = ( B = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_743_add__cancel__left__left,axiom,
% 9.03/4.93      ! [B: int,A: int] :
% 9.03/4.93        ( ( ( plus_plus_int @ B @ A )
% 9.03/4.93          = A )
% 9.03/4.93        = ( B = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_744_double__zero__sym,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( zero_zero_real
% 9.03/4.93          = ( plus_plus_real @ A @ A ) )
% 9.03/4.93        = ( A = zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_745_double__zero__sym,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( zero_zero_rat
% 9.03/4.93          = ( plus_plus_rat @ A @ A ) )
% 9.03/4.93        = ( A = zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_746_double__zero__sym,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( zero_zero_int
% 9.03/4.93          = ( plus_plus_int @ A @ A ) )
% 9.03/4.93        = ( A = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_747_add_Oright__neutral,axiom,
% 9.03/4.93      ! [A: complex] :
% 9.03/4.93        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_748_add_Oright__neutral,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_749_add_Oright__neutral,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_750_add_Oright__neutral,axiom,
% 9.03/4.93      ! [A: nat] :
% 9.03/4.93        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_751_add_Oright__neutral,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_752_double__eq__0__iff,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( ( plus_plus_real @ A @ A )
% 9.03/4.93          = zero_zero_real )
% 9.03/4.93        = ( A = zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_753_double__eq__0__iff,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( ( plus_plus_rat @ A @ A )
% 9.03/4.93          = zero_zero_rat )
% 9.03/4.93        = ( A = zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_754_double__eq__0__iff,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( ( plus_plus_int @ A @ A )
% 9.03/4.93          = zero_zero_int )
% 9.03/4.93        = ( A = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_755_add__less__cancel__right,axiom,
% 9.03/4.93      ! [A: real,C: real,B: real] :
% 9.03/4.93        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.03/4.93        = ( ord_less_real @ A @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_756_add__less__cancel__right,axiom,
% 9.03/4.93      ! [A: rat,C: rat,B: rat] :
% 9.03/4.93        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.03/4.93        = ( ord_less_rat @ A @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_757_add__less__cancel__right,axiom,
% 9.03/4.93      ! [A: nat,C: nat,B: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.03/4.93        = ( ord_less_nat @ A @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_758_add__less__cancel__right,axiom,
% 9.03/4.93      ! [A: int,C: int,B: int] :
% 9.03/4.93        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.03/4.93        = ( ord_less_int @ A @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_759_add__less__cancel__left,axiom,
% 9.03/4.93      ! [C: real,A: real,B: real] :
% 9.03/4.93        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.03/4.93        = ( ord_less_real @ A @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_760_add__less__cancel__left,axiom,
% 9.03/4.93      ! [C: rat,A: rat,B: rat] :
% 9.03/4.93        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.03/4.93        = ( ord_less_rat @ A @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_761_add__less__cancel__left,axiom,
% 9.03/4.93      ! [C: nat,A: nat,B: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.03/4.93        = ( ord_less_nat @ A @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_762_add__less__cancel__left,axiom,
% 9.03/4.93      ! [C: int,A: int,B: int] :
% 9.03/4.93        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.03/4.93        = ( ord_less_int @ A @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_763_Suc__less__eq,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.03/4.93        = ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_764_Suc__mono,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ M @ N )
% 9.03/4.93       => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_765_lessI,axiom,
% 9.03/4.93      ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_766_less__nat__zero__code,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_767_neq0__conv,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( N != zero_zero_nat )
% 9.03/4.93        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_768_bot__nat__0_Onot__eq__extremum,axiom,
% 9.03/4.93      ! [A: nat] :
% 9.03/4.93        ( ( A != zero_zero_nat )
% 9.03/4.93        = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_769_of__int__eq__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ( ring_1_of_int_int @ Z )
% 9.03/4.93          = zero_zero_int )
% 9.03/4.93        = ( Z = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_770_of__int__eq__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ( ring_1_of_int_real @ Z )
% 9.03/4.93          = zero_zero_real )
% 9.03/4.93        = ( Z = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_771_of__int__eq__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ( ring_1_of_int_rat @ Z )
% 9.03/4.93          = zero_zero_rat )
% 9.03/4.93        = ( Z = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_772_of__int__eq__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ( ring_18347121197199848620nteger @ Z )
% 9.03/4.93          = zero_z3403309356797280102nteger )
% 9.03/4.93        = ( Z = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_773_of__int__eq__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ( ring_17405671764205052669omplex @ Z )
% 9.03/4.93          = zero_zero_complex )
% 9.03/4.93        = ( Z = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_774_of__int__0__eq__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( zero_zero_int
% 9.03/4.93          = ( ring_1_of_int_int @ Z ) )
% 9.03/4.93        = ( Z = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_775_of__int__0__eq__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( zero_zero_real
% 9.03/4.93          = ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = ( Z = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_776_of__int__0__eq__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( zero_zero_rat
% 9.03/4.93          = ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = ( Z = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_777_of__int__0__eq__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( zero_z3403309356797280102nteger
% 9.03/4.93          = ( ring_18347121197199848620nteger @ Z ) )
% 9.03/4.93        = ( Z = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_778_of__int__0__eq__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( zero_zero_complex
% 9.03/4.93          = ( ring_17405671764205052669omplex @ Z ) )
% 9.03/4.93        = ( Z = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_779_of__int__0,axiom,
% 9.03/4.93      ( ( ring_1_of_int_int @ zero_zero_int )
% 9.03/4.93      = zero_zero_int ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_780_of__int__0,axiom,
% 9.03/4.93      ( ( ring_1_of_int_real @ zero_zero_int )
% 9.03/4.93      = zero_zero_real ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_781_of__int__0,axiom,
% 9.03/4.93      ( ( ring_1_of_int_rat @ zero_zero_int )
% 9.03/4.93      = zero_zero_rat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_782_of__int__0,axiom,
% 9.03/4.93      ( ( ring_18347121197199848620nteger @ zero_zero_int )
% 9.03/4.93      = zero_z3403309356797280102nteger ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_783_of__int__0,axiom,
% 9.03/4.93      ( ( ring_17405671764205052669omplex @ zero_zero_int )
% 9.03/4.93      = zero_zero_complex ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_784_le0,axiom,
% 9.03/4.93      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_785_bot__nat__0_Oextremum,axiom,
% 9.03/4.93      ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_786_of__int__less__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) )
% 9.03/4.93        = ( ord_less_int @ W @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_787_of__int__less__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = ( ord_less_int @ W @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_788_of__int__less__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = ( ord_less_int @ W @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_789_of__int__less__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 9.03/4.93        = ( ord_less_int @ W @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_790_add__is__0,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( ( plus_plus_nat @ M @ N )
% 9.03/4.93          = zero_zero_nat )
% 9.03/4.93        = ( ( M = zero_zero_nat )
% 9.03/4.93          & ( N = zero_zero_nat ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_791_Nat_Oadd__0__right,axiom,
% 9.03/4.93      ! [M: nat] :
% 9.03/4.93        ( ( plus_plus_nat @ M @ zero_zero_nat )
% 9.03/4.93        = M ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_792_nat__add__left__cancel__less,axiom,
% 9.03/4.93      ! [K: nat,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.03/4.93        = ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_793_ceiling__zero,axiom,
% 9.03/4.93      ( ( archim7802044766580827645g_real @ zero_zero_real )
% 9.03/4.93      = zero_zero_int ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_794_ceiling__zero,axiom,
% 9.03/4.93      ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 9.03/4.93      = zero_zero_int ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_795_ceiling__of__int,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( archim7802044766580827645g_real @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = Z ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_796_ceiling__of__int,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( archim2889992004027027881ng_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = Z ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_797_of__int__ceiling__cancel,axiom,
% 9.03/4.93      ! [X: real] :
% 9.03/4.93        ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
% 9.03/4.93          = X )
% 9.03/4.93        = ( ? [N3: int] :
% 9.03/4.93              ( X
% 9.03/4.93              = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_798_of__int__ceiling__cancel,axiom,
% 9.03/4.93      ! [X: rat] :
% 9.03/4.93        ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/4.93          = X )
% 9.03/4.93        = ( ? [N3: int] :
% 9.03/4.93              ( X
% 9.03/4.93              = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_799_dbl__simps_I2_J,axiom,
% 9.03/4.93      ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 9.03/4.93      = zero_zero_complex ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_800_dbl__simps_I2_J,axiom,
% 9.03/4.93      ( ( neg_numeral_dbl_real @ zero_zero_real )
% 9.03/4.93      = zero_zero_real ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_801_dbl__simps_I2_J,axiom,
% 9.03/4.93      ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 9.03/4.93      = zero_zero_rat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_802_dbl__simps_I2_J,axiom,
% 9.03/4.93      ( ( neg_numeral_dbl_int @ zero_zero_int )
% 9.03/4.93      = zero_zero_int ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_803_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 9.03/4.93        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_804_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 9.03/4.93        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_805_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 9.03/4.93        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_806_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 9.03/4.93        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_807_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 9.03/4.93        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_808_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 9.03/4.93        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_809_le__add__same__cancel2,axiom,
% 9.03/4.93      ! [A: real,B: real] :
% 9.03/4.93        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 9.03/4.93        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_810_le__add__same__cancel2,axiom,
% 9.03/4.93      ! [A: rat,B: rat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 9.03/4.93        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_811_le__add__same__cancel2,axiom,
% 9.03/4.93      ! [A: nat,B: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 9.03/4.93        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_812_le__add__same__cancel2,axiom,
% 9.03/4.93      ! [A: int,B: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 9.03/4.93        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_813_le__add__same__cancel1,axiom,
% 9.03/4.93      ! [A: real,B: real] :
% 9.03/4.93        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 9.03/4.93        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_814_le__add__same__cancel1,axiom,
% 9.03/4.93      ! [A: rat,B: rat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 9.03/4.93        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_815_le__add__same__cancel1,axiom,
% 9.03/4.93      ! [A: nat,B: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.03/4.93        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_816_le__add__same__cancel1,axiom,
% 9.03/4.93      ! [A: int,B: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 9.03/4.93        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_817_add__le__same__cancel2,axiom,
% 9.03/4.93      ! [A: real,B: real] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.03/4.93        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_818_add__le__same__cancel2,axiom,
% 9.03/4.93      ! [A: rat,B: rat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.03/4.93        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_819_add__le__same__cancel2,axiom,
% 9.03/4.93      ! [A: nat,B: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.03/4.93        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_820_add__le__same__cancel2,axiom,
% 9.03/4.93      ! [A: int,B: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.03/4.93        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_821_add__le__same__cancel1,axiom,
% 9.03/4.93      ! [B: real,A: real] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 9.03/4.93        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_822_add__le__same__cancel1,axiom,
% 9.03/4.93      ! [B: rat,A: rat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 9.03/4.93        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_823_add__le__same__cancel1,axiom,
% 9.03/4.93      ! [B: nat,A: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.03/4.93        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_824_add__le__same__cancel1,axiom,
% 9.03/4.93      ! [B: int,A: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.03/4.93        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_825_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 9.03/4.93        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_826_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 9.03/4.93        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_827_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 9.03/4.93        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_828_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 9.03/4.93        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_829_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 9.03/4.93        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_830_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 9.03/4.93        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_831_less__add__same__cancel2,axiom,
% 9.03/4.93      ! [A: real,B: real] :
% 9.03/4.93        ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 9.03/4.93        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_832_less__add__same__cancel2,axiom,
% 9.03/4.93      ! [A: rat,B: rat] :
% 9.03/4.93        ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 9.03/4.93        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_833_less__add__same__cancel2,axiom,
% 9.03/4.93      ! [A: nat,B: nat] :
% 9.03/4.93        ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 9.03/4.93        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_834_less__add__same__cancel2,axiom,
% 9.03/4.93      ! [A: int,B: int] :
% 9.03/4.93        ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 9.03/4.93        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_835_less__add__same__cancel1,axiom,
% 9.03/4.93      ! [A: real,B: real] :
% 9.03/4.93        ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 9.03/4.93        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_836_less__add__same__cancel1,axiom,
% 9.03/4.93      ! [A: rat,B: rat] :
% 9.03/4.93        ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 9.03/4.93        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_837_less__add__same__cancel1,axiom,
% 9.03/4.93      ! [A: nat,B: nat] :
% 9.03/4.93        ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.03/4.93        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_838_less__add__same__cancel1,axiom,
% 9.03/4.93      ! [A: int,B: int] :
% 9.03/4.93        ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 9.03/4.93        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_839_add__less__same__cancel2,axiom,
% 9.03/4.93      ! [A: real,B: real] :
% 9.03/4.93        ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.03/4.93        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_840_add__less__same__cancel2,axiom,
% 9.03/4.93      ! [A: rat,B: rat] :
% 9.03/4.93        ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.03/4.93        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_841_add__less__same__cancel2,axiom,
% 9.03/4.93      ! [A: nat,B: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.03/4.93        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_842_add__less__same__cancel2,axiom,
% 9.03/4.93      ! [A: int,B: int] :
% 9.03/4.93        ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.03/4.93        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_843_add__less__same__cancel1,axiom,
% 9.03/4.93      ! [B: real,A: real] :
% 9.03/4.93        ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 9.03/4.93        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_844_add__less__same__cancel1,axiom,
% 9.03/4.93      ! [B: rat,A: rat] :
% 9.03/4.93        ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 9.03/4.93        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_845_add__less__same__cancel1,axiom,
% 9.03/4.93      ! [B: nat,A: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.03/4.93        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_846_add__less__same__cancel1,axiom,
% 9.03/4.93      ! [B: int,A: int] :
% 9.03/4.93        ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.03/4.93        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_847_image__add__0,axiom,
% 9.03/4.93      ! [S2: set_complex] :
% 9.03/4.93        ( ( image_1468599708987790691omplex @ ( plus_plus_complex @ zero_zero_complex ) @ S2 )
% 9.03/4.93        = S2 ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_848_image__add__0,axiom,
% 9.03/4.93      ! [S2: set_real] :
% 9.03/4.93        ( ( image_real_real @ ( plus_plus_real @ zero_zero_real ) @ S2 )
% 9.03/4.93        = S2 ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_849_image__add__0,axiom,
% 9.03/4.93      ! [S2: set_rat] :
% 9.03/4.93        ( ( image_rat_rat @ ( plus_plus_rat @ zero_zero_rat ) @ S2 )
% 9.03/4.93        = S2 ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_850_image__add__0,axiom,
% 9.03/4.93      ! [S2: set_nat] :
% 9.03/4.93        ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S2 )
% 9.03/4.93        = S2 ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_851_image__add__0,axiom,
% 9.03/4.93      ! [S2: set_int] :
% 9.03/4.93        ( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S2 )
% 9.03/4.93        = S2 ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_852_power__inject__exp,axiom,
% 9.03/4.93      ! [A: real,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.93       => ( ( ( power_power_real @ A @ M )
% 9.03/4.93            = ( power_power_real @ A @ N ) )
% 9.03/4.93          = ( M = N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_853_power__inject__exp,axiom,
% 9.03/4.93      ! [A: rat,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_rat @ one_one_rat @ A )
% 9.03/4.93       => ( ( ( power_power_rat @ A @ M )
% 9.03/4.93            = ( power_power_rat @ A @ N ) )
% 9.03/4.93          = ( M = N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_854_power__inject__exp,axiom,
% 9.03/4.93      ! [A: nat,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ one_one_nat @ A )
% 9.03/4.93       => ( ( ( power_power_nat @ A @ M )
% 9.03/4.93            = ( power_power_nat @ A @ N ) )
% 9.03/4.93          = ( M = N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_855_power__inject__exp,axiom,
% 9.03/4.93      ! [A: int,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_int @ one_one_int @ A )
% 9.03/4.93       => ( ( ( power_power_int @ A @ M )
% 9.03/4.93            = ( power_power_int @ A @ N ) )
% 9.03/4.93          = ( M = N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_856_power__0__Suc,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 9.03/4.93        = zero_zero_rat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_857_power__0__Suc,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.03/4.93        = zero_zero_nat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_858_power__0__Suc,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 9.03/4.93        = zero_zero_real ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_859_power__0__Suc,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 9.03/4.93        = zero_zero_int ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_860_power__0__Suc,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 9.03/4.93        = zero_zero_complex ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_861_power__zero__numeral,axiom,
% 9.03/4.93      ! [K: num] :
% 9.03/4.93        ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 9.03/4.93        = zero_zero_rat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_862_power__zero__numeral,axiom,
% 9.03/4.93      ! [K: num] :
% 9.03/4.93        ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 9.03/4.93        = zero_zero_nat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_863_power__zero__numeral,axiom,
% 9.03/4.93      ! [K: num] :
% 9.03/4.93        ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 9.03/4.93        = zero_zero_real ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_864_power__zero__numeral,axiom,
% 9.03/4.93      ! [K: num] :
% 9.03/4.93        ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 9.03/4.93        = zero_zero_int ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_865_power__zero__numeral,axiom,
% 9.03/4.93      ! [K: num] :
% 9.03/4.93        ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 9.03/4.93        = zero_zero_complex ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_866_of__nat__0,axiom,
% 9.03/4.93      ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 9.03/4.93      = zero_zero_complex ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_867_of__nat__0,axiom,
% 9.03/4.93      ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 9.03/4.93      = zero_zero_real ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_868_of__nat__0,axiom,
% 9.03/4.93      ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 9.03/4.93      = zero_zero_rat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_869_of__nat__0,axiom,
% 9.03/4.93      ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 9.03/4.93      = zero_zero_nat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_870_of__nat__0,axiom,
% 9.03/4.93      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 9.03/4.93      = zero_zero_int ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_871_of__nat__0__eq__iff,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( zero_zero_complex
% 9.03/4.93          = ( semiri8010041392384452111omplex @ N ) )
% 9.03/4.93        = ( zero_zero_nat = N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_872_of__nat__0__eq__iff,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( zero_zero_real
% 9.03/4.93          = ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.93        = ( zero_zero_nat = N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_873_of__nat__0__eq__iff,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( zero_zero_rat
% 9.03/4.93          = ( semiri681578069525770553at_rat @ N ) )
% 9.03/4.93        = ( zero_zero_nat = N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_874_of__nat__0__eq__iff,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( zero_zero_nat
% 9.03/4.93          = ( semiri1316708129612266289at_nat @ N ) )
% 9.03/4.93        = ( zero_zero_nat = N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_875_of__nat__0__eq__iff,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( zero_zero_int
% 9.03/4.93          = ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.93        = ( zero_zero_nat = N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_876_of__nat__eq__0__iff,axiom,
% 9.03/4.93      ! [M: nat] :
% 9.03/4.93        ( ( ( semiri8010041392384452111omplex @ M )
% 9.03/4.93          = zero_zero_complex )
% 9.03/4.93        = ( M = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_877_of__nat__eq__0__iff,axiom,
% 9.03/4.93      ! [M: nat] :
% 9.03/4.93        ( ( ( semiri5074537144036343181t_real @ M )
% 9.03/4.93          = zero_zero_real )
% 9.03/4.93        = ( M = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_878_of__nat__eq__0__iff,axiom,
% 9.03/4.93      ! [M: nat] :
% 9.03/4.93        ( ( ( semiri681578069525770553at_rat @ M )
% 9.03/4.93          = zero_zero_rat )
% 9.03/4.93        = ( M = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_879_of__nat__eq__0__iff,axiom,
% 9.03/4.93      ! [M: nat] :
% 9.03/4.93        ( ( ( semiri1316708129612266289at_nat @ M )
% 9.03/4.93          = zero_zero_nat )
% 9.03/4.93        = ( M = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_880_of__nat__eq__0__iff,axiom,
% 9.03/4.93      ! [M: nat] :
% 9.03/4.93        ( ( ( semiri1314217659103216013at_int @ M )
% 9.03/4.93          = zero_zero_int )
% 9.03/4.93        = ( M = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_881_of__int__less__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ zero_z3403309356797280102nteger )
% 9.03/4.93        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_882_of__int__less__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 9.03/4.93        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_883_of__int__less__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 9.03/4.93        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_884_of__int__less__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 9.03/4.93        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_885_of__int__0__less__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) )
% 9.03/4.93        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_886_of__int__0__less__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_887_of__int__0__less__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_888_of__int__0__less__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 9.03/4.93        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_889_of__nat__less__iff,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.93        = ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_890_of__nat__less__iff,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.03/4.93        = ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_891_of__nat__less__iff,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.03/4.93        = ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_892_of__nat__less__iff,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.93        = ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_893_power__Suc0__right,axiom,
% 9.03/4.93      ! [A: nat] :
% 9.03/4.93        ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_894_power__Suc0__right,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_895_power__Suc0__right,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_896_power__Suc0__right,axiom,
% 9.03/4.93      ! [A: complex] :
% 9.03/4.93        ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 9.03/4.93        = A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_897_zero__less__Suc,axiom,
% 9.03/4.93      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_898_less__Suc0,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.03/4.93        = ( N = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_899_add__gr__0,axiom,
% 9.03/4.93      ! [M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.93        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.93          | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_900_zero__less__ceiling,axiom,
% 9.03/4.93      ! [X: real] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 9.03/4.93        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_901_zero__less__ceiling,axiom,
% 9.03/4.93      ! [X: rat] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/4.93        = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_902_of__int__le__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ W @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_903_of__int__le__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ W @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_904_of__int__le__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ W @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_905_of__int__le__iff,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ W @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_906_of__int__eq__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ( ring_18347121197199848620nteger @ Z )
% 9.03/4.93          = ( numera6620942414471956472nteger @ N ) )
% 9.03/4.93        = ( Z
% 9.03/4.93          = ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_907_of__int__eq__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ( ring_17405671764205052669omplex @ Z )
% 9.03/4.93          = ( numera6690914467698888265omplex @ N ) )
% 9.03/4.93        = ( Z
% 9.03/4.93          = ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_908_of__int__eq__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ( ring_1_of_int_real @ Z )
% 9.03/4.93          = ( numeral_numeral_real @ N ) )
% 9.03/4.93        = ( Z
% 9.03/4.93          = ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_909_of__int__eq__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ( ring_1_of_int_rat @ Z )
% 9.03/4.93          = ( numeral_numeral_rat @ N ) )
% 9.03/4.93        = ( Z
% 9.03/4.93          = ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_910_of__int__eq__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ( ring_1_of_int_int @ Z )
% 9.03/4.93          = ( numeral_numeral_int @ N ) )
% 9.03/4.93        = ( Z
% 9.03/4.93          = ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_911_of__int__numeral,axiom,
% 9.03/4.93      ! [K: num] :
% 9.03/4.93        ( ( ring_18347121197199848620nteger @ ( numeral_numeral_int @ K ) )
% 9.03/4.93        = ( numera6620942414471956472nteger @ K ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_912_of__int__numeral,axiom,
% 9.03/4.93      ! [K: num] :
% 9.03/4.93        ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 9.03/4.93        = ( numera6690914467698888265omplex @ K ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_913_of__int__numeral,axiom,
% 9.03/4.93      ! [K: num] :
% 9.03/4.93        ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 9.03/4.93        = ( numeral_numeral_real @ K ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_914_of__int__numeral,axiom,
% 9.03/4.93      ! [K: num] :
% 9.03/4.93        ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 9.03/4.93        = ( numeral_numeral_rat @ K ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_915_of__int__numeral,axiom,
% 9.03/4.93      ! [K: num] :
% 9.03/4.93        ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 9.03/4.93        = ( numeral_numeral_int @ K ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_916_less__one,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ N @ one_one_nat )
% 9.03/4.93        = ( N = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_917_nat__power__eq__Suc__0__iff,axiom,
% 9.03/4.93      ! [X: nat,M: nat] :
% 9.03/4.93        ( ( ( power_power_nat @ X @ M )
% 9.03/4.93          = ( suc @ zero_zero_nat ) )
% 9.03/4.93        = ( ( M = zero_zero_nat )
% 9.03/4.93          | ( X
% 9.03/4.93            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_918_power__Suc__0,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.03/4.93        = ( suc @ zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_919_of__int__eq__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ( ring_1_of_int_int @ Z )
% 9.03/4.93          = one_one_int )
% 9.03/4.93        = ( Z = one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_920_of__int__eq__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ( ring_1_of_int_real @ Z )
% 9.03/4.93          = one_one_real )
% 9.03/4.93        = ( Z = one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_921_of__int__eq__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ( ring_1_of_int_rat @ Z )
% 9.03/4.93          = one_one_rat )
% 9.03/4.93        = ( Z = one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_922_of__int__eq__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ( ring_18347121197199848620nteger @ Z )
% 9.03/4.93          = one_one_Code_integer )
% 9.03/4.93        = ( Z = one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_923_of__int__eq__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ( ring_17405671764205052669omplex @ Z )
% 9.03/4.93          = one_one_complex )
% 9.03/4.93        = ( Z = one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_924_of__int__1,axiom,
% 9.03/4.93      ( ( ring_1_of_int_int @ one_one_int )
% 9.03/4.93      = one_one_int ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_925_of__int__1,axiom,
% 9.03/4.93      ( ( ring_1_of_int_real @ one_one_int )
% 9.03/4.93      = one_one_real ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_926_of__int__1,axiom,
% 9.03/4.93      ( ( ring_1_of_int_rat @ one_one_int )
% 9.03/4.93      = one_one_rat ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_927_of__int__1,axiom,
% 9.03/4.93      ( ( ring_18347121197199848620nteger @ one_one_int )
% 9.03/4.93      = one_one_Code_integer ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_928_of__int__1,axiom,
% 9.03/4.93      ( ( ring_17405671764205052669omplex @ one_one_int )
% 9.03/4.93      = one_one_complex ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_929_nat__zero__less__power__iff,axiom,
% 9.03/4.93      ! [X: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 9.03/4.93        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.03/4.93          | ( N = zero_zero_nat ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_930_of__int__add,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 9.03/4.93        = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_931_of__int__add,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 9.03/4.93        = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_932_of__int__add,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 9.03/4.93        = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_933_of__int__add,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ring_18347121197199848620nteger @ ( plus_plus_int @ W @ Z ) )
% 9.03/4.93        = ( plus_p5714425477246183910nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_934_of__int__add,axiom,
% 9.03/4.93      ! [W: int,Z: int] :
% 9.03/4.93        ( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W @ Z ) )
% 9.03/4.93        = ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_935_of__int__of__nat__eq,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ring_18347121197199848620nteger @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.93        = ( semiri4939895301339042750nteger @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_936_of__int__of__nat__eq,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.93        = ( semiri8010041392384452111omplex @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_937_of__int__of__nat__eq,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.93        = ( semiri5074537144036343181t_real @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_938_of__int__of__nat__eq,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.93        = ( semiri681578069525770553at_rat @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_939_of__int__of__nat__eq,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.93        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_940_of__int__power__eq__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ( ring_1_of_int_rat @ X )
% 9.03/4.93          = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 9.03/4.93        = ( X
% 9.03/4.93          = ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_941_of__int__power__eq__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ( ring_18347121197199848620nteger @ X )
% 9.03/4.93          = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
% 9.03/4.93        = ( X
% 9.03/4.93          = ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_942_of__int__power__eq__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ( ring_1_of_int_real @ X )
% 9.03/4.93          = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 9.03/4.93        = ( X
% 9.03/4.93          = ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_943_of__int__power__eq__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ( ring_1_of_int_int @ X )
% 9.03/4.93          = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 9.03/4.93        = ( X
% 9.03/4.93          = ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_944_of__int__power__eq__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ( ring_17405671764205052669omplex @ X )
% 9.03/4.93          = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 9.03/4.93        = ( X
% 9.03/4.93          = ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_945_of__int__eq__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 9.03/4.93          = ( ring_1_of_int_rat @ X ) )
% 9.03/4.93        = ( ( power_power_int @ B @ W )
% 9.03/4.93          = X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_946_of__int__eq__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W )
% 9.03/4.93          = ( ring_18347121197199848620nteger @ X ) )
% 9.03/4.93        = ( ( power_power_int @ B @ W )
% 9.03/4.93          = X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_947_of__int__eq__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 9.03/4.93          = ( ring_1_of_int_real @ X ) )
% 9.03/4.93        = ( ( power_power_int @ B @ W )
% 9.03/4.93          = X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_948_of__int__eq__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 9.03/4.93          = ( ring_1_of_int_int @ X ) )
% 9.03/4.93        = ( ( power_power_int @ B @ W )
% 9.03/4.93          = X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_949_of__int__eq__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 9.03/4.93          = ( ring_17405671764205052669omplex @ X ) )
% 9.03/4.93        = ( ( power_power_int @ B @ W )
% 9.03/4.93          = X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_950_of__int__power,axiom,
% 9.03/4.93      ! [Z: int,N: nat] :
% 9.03/4.93        ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
% 9.03/4.93        = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_951_of__int__power,axiom,
% 9.03/4.93      ! [Z: int,N: nat] :
% 9.03/4.93        ( ( ring_18347121197199848620nteger @ ( power_power_int @ Z @ N ) )
% 9.03/4.93        = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ Z ) @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_952_of__int__power,axiom,
% 9.03/4.93      ! [Z: int,N: nat] :
% 9.03/4.93        ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 9.03/4.93        = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_953_of__int__power,axiom,
% 9.03/4.93      ! [Z: int,N: nat] :
% 9.03/4.93        ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 9.03/4.93        = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_954_of__int__power,axiom,
% 9.03/4.93      ! [Z: int,N: nat] :
% 9.03/4.93        ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
% 9.03/4.93        = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_955_one__less__numeral__iff,axiom,
% 9.03/4.93      ! [N: num] :
% 9.03/4.93        ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 9.03/4.93        = ( ord_less_num @ one @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_956_one__less__numeral__iff,axiom,
% 9.03/4.93      ! [N: num] :
% 9.03/4.93        ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 9.03/4.93        = ( ord_less_num @ one @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_957_one__less__numeral__iff,axiom,
% 9.03/4.93      ! [N: num] :
% 9.03/4.93        ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 9.03/4.93        = ( ord_less_num @ one @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_958_one__less__numeral__iff,axiom,
% 9.03/4.93      ! [N: num] :
% 9.03/4.93        ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 9.03/4.93        = ( ord_less_num @ one @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_959_of__nat__le__0__iff,axiom,
% 9.03/4.93      ! [M: nat] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 9.03/4.93        = ( M = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_960_of__nat__le__0__iff,axiom,
% 9.03/4.93      ! [M: nat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 9.03/4.93        = ( M = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_961_of__nat__le__0__iff,axiom,
% 9.03/4.93      ! [M: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 9.03/4.93        = ( M = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_962_of__nat__le__0__iff,axiom,
% 9.03/4.93      ! [M: nat] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 9.03/4.93        = ( M = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_963_power__strict__increasing__iff,axiom,
% 9.03/4.93      ! [B: real,X: nat,Y: nat] :
% 9.03/4.93        ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.93       => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 9.03/4.93          = ( ord_less_nat @ X @ Y ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_964_power__strict__increasing__iff,axiom,
% 9.03/4.93      ! [B: rat,X: nat,Y: nat] :
% 9.03/4.93        ( ( ord_less_rat @ one_one_rat @ B )
% 9.03/4.93       => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 9.03/4.93          = ( ord_less_nat @ X @ Y ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_965_power__strict__increasing__iff,axiom,
% 9.03/4.93      ! [B: nat,X: nat,Y: nat] :
% 9.03/4.93        ( ( ord_less_nat @ one_one_nat @ B )
% 9.03/4.93       => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 9.03/4.93          = ( ord_less_nat @ X @ Y ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_966_power__strict__increasing__iff,axiom,
% 9.03/4.93      ! [B: int,X: nat,Y: nat] :
% 9.03/4.93        ( ( ord_less_int @ one_one_int @ B )
% 9.03/4.93       => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 9.03/4.93          = ( ord_less_nat @ X @ Y ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_967_power__eq__0__iff,axiom,
% 9.03/4.93      ! [A: rat,N: nat] :
% 9.03/4.93        ( ( ( power_power_rat @ A @ N )
% 9.03/4.93          = zero_zero_rat )
% 9.03/4.93        = ( ( A = zero_zero_rat )
% 9.03/4.93          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_968_power__eq__0__iff,axiom,
% 9.03/4.93      ! [A: nat,N: nat] :
% 9.03/4.93        ( ( ( power_power_nat @ A @ N )
% 9.03/4.93          = zero_zero_nat )
% 9.03/4.93        = ( ( A = zero_zero_nat )
% 9.03/4.93          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_969_power__eq__0__iff,axiom,
% 9.03/4.93      ! [A: real,N: nat] :
% 9.03/4.93        ( ( ( power_power_real @ A @ N )
% 9.03/4.93          = zero_zero_real )
% 9.03/4.93        = ( ( A = zero_zero_real )
% 9.03/4.93          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_970_power__eq__0__iff,axiom,
% 9.03/4.93      ! [A: int,N: nat] :
% 9.03/4.93        ( ( ( power_power_int @ A @ N )
% 9.03/4.93          = zero_zero_int )
% 9.03/4.93        = ( ( A = zero_zero_int )
% 9.03/4.93          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_971_power__eq__0__iff,axiom,
% 9.03/4.93      ! [A: complex,N: nat] :
% 9.03/4.93        ( ( ( power_power_complex @ A @ N )
% 9.03/4.93          = zero_zero_complex )
% 9.03/4.93        = ( ( A = zero_zero_complex )
% 9.03/4.93          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_972_of__int__le__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_973_of__int__le__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ zero_z3403309356797280102nteger )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_974_of__int__le__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_975_of__int__le__0__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_976_of__int__0__le__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_977_of__int__0__le__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_978_of__int__0__le__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_979_of__int__0__le__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_980_of__int__less__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
% 9.03/4.93        = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_981_of__int__less__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 9.03/4.93        = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_982_of__int__less__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 9.03/4.93        = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_983_of__int__less__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 9.03/4.93        = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_984_of__int__numeral__less__iff,axiom,
% 9.03/4.93      ! [N: num,Z: int] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
% 9.03/4.93        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_985_of__int__numeral__less__iff,axiom,
% 9.03/4.93      ! [N: num,Z: int] :
% 9.03/4.93        ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_986_of__int__numeral__less__iff,axiom,
% 9.03/4.93      ! [N: num,Z: int] :
% 9.03/4.93        ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_987_of__int__numeral__less__iff,axiom,
% 9.03/4.93      ! [N: num,Z: int] :
% 9.03/4.93        ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 9.03/4.93        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_988_of__int__less__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
% 9.03/4.93        = ( ord_less_int @ Z @ one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_989_of__int__less__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 9.03/4.93        = ( ord_less_int @ Z @ one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_990_of__int__less__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 9.03/4.93        = ( ord_less_int @ Z @ one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_991_of__int__less__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 9.03/4.93        = ( ord_less_int @ Z @ one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_992_of__int__1__less__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
% 9.03/4.93        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_993_of__int__1__less__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_994_of__int__1__less__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_995_of__int__1__less__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 9.03/4.93        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_996_ceiling__le__zero,axiom,
% 9.03/4.93      ! [X: real] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 9.03/4.93        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_997_ceiling__le__zero,axiom,
% 9.03/4.93      ! [X: rat] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 9.03/4.93        = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_998_numeral__less__ceiling,axiom,
% 9.03/4.93      ! [V: num,X: real] :
% 9.03/4.93        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 9.03/4.93        = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_999_numeral__less__ceiling,axiom,
% 9.03/4.93      ! [V: num,X: rat] :
% 9.03/4.93        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/4.93        = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1000_one__less__ceiling,axiom,
% 9.03/4.93      ! [X: real] :
% 9.03/4.93        ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 9.03/4.93        = ( ord_less_real @ one_one_real @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1001_one__less__ceiling,axiom,
% 9.03/4.93      ! [X: rat] :
% 9.03/4.93        ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/4.93        = ( ord_less_rat @ one_one_rat @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1002_of__int__less__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) @ ( ring_18347121197199848620nteger @ X ) )
% 9.03/4.93        = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1003_of__int__less__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 9.03/4.93        = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1004_of__int__less__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 9.03/4.93        = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1005_of__int__less__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 9.03/4.93        = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1006_of__int__power__less__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1007_of__int__power__less__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1008_of__int__power__less__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1009_of__int__power__less__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1010_ceiling__add__of__int,axiom,
% 9.03/4.93      ! [X: real,Z: int] :
% 9.03/4.93        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 9.03/4.93        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1011_ceiling__add__of__int,axiom,
% 9.03/4.93      ! [X: rat,Z: int] :
% 9.03/4.93        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
% 9.03/4.93        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1012_power__strict__decreasing__iff,axiom,
% 9.03/4.93      ! [B: real,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.93       => ( ( ord_less_real @ B @ one_one_real )
% 9.03/4.93         => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 9.03/4.93            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1013_power__strict__decreasing__iff,axiom,
% 9.03/4.93      ! [B: rat,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.03/4.93       => ( ( ord_less_rat @ B @ one_one_rat )
% 9.03/4.93         => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 9.03/4.93            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1014_power__strict__decreasing__iff,axiom,
% 9.03/4.93      ! [B: nat,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.03/4.93       => ( ( ord_less_nat @ B @ one_one_nat )
% 9.03/4.93         => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 9.03/4.93            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1015_power__strict__decreasing__iff,axiom,
% 9.03/4.93      ! [B: int,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.93       => ( ( ord_less_int @ B @ one_one_int )
% 9.03/4.93         => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 9.03/4.93            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1016_power__increasing__iff,axiom,
% 9.03/4.93      ! [B: real,X: nat,Y: nat] :
% 9.03/4.93        ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.93       => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 9.03/4.93          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1017_power__increasing__iff,axiom,
% 9.03/4.93      ! [B: rat,X: nat,Y: nat] :
% 9.03/4.93        ( ( ord_less_rat @ one_one_rat @ B )
% 9.03/4.93       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 9.03/4.93          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1018_power__increasing__iff,axiom,
% 9.03/4.93      ! [B: nat,X: nat,Y: nat] :
% 9.03/4.93        ( ( ord_less_nat @ one_one_nat @ B )
% 9.03/4.93       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 9.03/4.93          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1019_power__increasing__iff,axiom,
% 9.03/4.93      ! [B: int,X: nat,Y: nat] :
% 9.03/4.93        ( ( ord_less_int @ one_one_int @ B )
% 9.03/4.93       => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 9.03/4.93          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1020_power__mono__iff,axiom,
% 9.03/4.93      ! [A: real,B: real,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.93       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.93         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.93           => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 9.03/4.93              = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1021_power__mono__iff,axiom,
% 9.03/4.93      ! [A: rat,B: rat,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.93       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.93         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.93           => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 9.03/4.93              = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1022_power__mono__iff,axiom,
% 9.03/4.93      ! [A: nat,B: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.93       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.93         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.93           => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 9.03/4.93              = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1023_power__mono__iff,axiom,
% 9.03/4.93      ! [A: int,B: int,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.93       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.93         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.93           => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 9.03/4.93              = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1024_zero__eq__power2,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93          = zero_zero_rat )
% 9.03/4.93        = ( A = zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1025_zero__eq__power2,axiom,
% 9.03/4.93      ! [A: nat] :
% 9.03/4.93        ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93          = zero_zero_nat )
% 9.03/4.93        = ( A = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1026_zero__eq__power2,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93          = zero_zero_real )
% 9.03/4.93        = ( A = zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1027_zero__eq__power2,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93          = zero_zero_int )
% 9.03/4.93        = ( A = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1028_zero__eq__power2,axiom,
% 9.03/4.93      ! [A: complex] :
% 9.03/4.93        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93          = zero_zero_complex )
% 9.03/4.93        = ( A = zero_zero_complex ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1029_of__nat__0__less__iff,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.93        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1030_of__nat__0__less__iff,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.03/4.93        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1031_of__nat__0__less__iff,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 9.03/4.93        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1032_of__nat__0__less__iff,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.93        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1033_of__int__le__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1034_of__int__le__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1035_of__int__le__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1036_of__int__le__numeral__iff,axiom,
% 9.03/4.93      ! [Z: int,N: num] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1037_of__int__numeral__le__iff,axiom,
% 9.03/4.93      ! [N: num,Z: int] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1038_of__int__numeral__le__iff,axiom,
% 9.03/4.93      ! [N: num,Z: int] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1039_of__int__numeral__le__iff,axiom,
% 9.03/4.93      ! [N: num,Z: int] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1040_of__int__numeral__le__iff,axiom,
% 9.03/4.93      ! [N: num,Z: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1041_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.03/4.93      ! [X: nat,B: nat,W: nat] :
% 9.03/4.93        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1042_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.03/4.93      ! [X: nat,B: nat,W: nat] :
% 9.03/4.93        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1043_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.03/4.93      ! [X: nat,B: nat,W: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1044_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.03/4.93      ! [X: nat,B: nat,W: nat] :
% 9.03/4.93        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1045_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: nat,W: nat,X: nat] :
% 9.03/4.93        ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 9.03/4.93        = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1046_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: nat,W: nat,X: nat] :
% 9.03/4.93        ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 9.03/4.93        = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1047_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: nat,W: nat,X: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 9.03/4.93        = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1048_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: nat,W: nat,X: nat] :
% 9.03/4.93        ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 9.03/4.93        = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1049_of__int__le__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1050_of__int__le__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1051_of__int__le__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1052_of__int__le__1__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 9.03/4.93        = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1053_of__int__1__le__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1054_of__int__1__le__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1055_of__int__1__le__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1056_of__int__1__le__iff,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 9.03/4.93        = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1057_one__le__ceiling,axiom,
% 9.03/4.93      ! [X: real] :
% 9.03/4.93        ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 9.03/4.93        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1058_one__le__ceiling,axiom,
% 9.03/4.93      ! [X: rat] :
% 9.03/4.93        ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/4.93        = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1059_of__int__power__le__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1060_of__int__power__le__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1061_of__int__power__le__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1062_of__int__power__le__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: int,B: int,W: nat] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 9.03/4.93        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1063_of__int__le__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) @ ( ring_18347121197199848620nteger @ X ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1064_of__int__le__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1065_of__int__le__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1066_of__int__le__of__int__power__cancel__iff,axiom,
% 9.03/4.93      ! [B: int,W: nat,X: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1067_numeral__power__eq__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,Y: int] :
% 9.03/4.93        ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
% 9.03/4.93          = ( ring_18347121197199848620nteger @ Y ) )
% 9.03/4.93        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 9.03/4.93          = Y ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1068_numeral__power__eq__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,Y: int] :
% 9.03/4.93        ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 9.03/4.93          = ( ring_17405671764205052669omplex @ Y ) )
% 9.03/4.93        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 9.03/4.93          = Y ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1069_numeral__power__eq__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,Y: int] :
% 9.03/4.93        ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 9.03/4.93          = ( ring_1_of_int_real @ Y ) )
% 9.03/4.93        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 9.03/4.93          = Y ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1070_numeral__power__eq__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,Y: int] :
% 9.03/4.93        ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 9.03/4.93          = ( ring_1_of_int_rat @ Y ) )
% 9.03/4.93        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 9.03/4.93          = Y ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1071_numeral__power__eq__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,Y: int] :
% 9.03/4.93        ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 9.03/4.93          = ( ring_1_of_int_int @ Y ) )
% 9.03/4.93        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 9.03/4.93          = Y ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1072_of__int__eq__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [Y: int,X: num,N: nat] :
% 9.03/4.93        ( ( ( ring_18347121197199848620nteger @ Y )
% 9.03/4.93          = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
% 9.03/4.93        = ( Y
% 9.03/4.93          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1073_of__int__eq__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [Y: int,X: num,N: nat] :
% 9.03/4.93        ( ( ( ring_17405671764205052669omplex @ Y )
% 9.03/4.93          = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 9.03/4.93        = ( Y
% 9.03/4.93          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1074_of__int__eq__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [Y: int,X: num,N: nat] :
% 9.03/4.93        ( ( ( ring_1_of_int_real @ Y )
% 9.03/4.93          = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 9.03/4.93        = ( Y
% 9.03/4.93          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1075_of__int__eq__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [Y: int,X: num,N: nat] :
% 9.03/4.93        ( ( ( ring_1_of_int_rat @ Y )
% 9.03/4.93          = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 9.03/4.93        = ( Y
% 9.03/4.93          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1076_of__int__eq__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [Y: int,X: num,N: nat] :
% 9.03/4.93        ( ( ( ring_1_of_int_int @ Y )
% 9.03/4.93          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 9.03/4.93        = ( Y
% 9.03/4.93          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1077_power__decreasing__iff,axiom,
% 9.03/4.93      ! [B: real,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.93       => ( ( ord_less_real @ B @ one_one_real )
% 9.03/4.93         => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 9.03/4.93            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1078_power__decreasing__iff,axiom,
% 9.03/4.93      ! [B: rat,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.03/4.93       => ( ( ord_less_rat @ B @ one_one_rat )
% 9.03/4.93         => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 9.03/4.93            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1079_power__decreasing__iff,axiom,
% 9.03/4.93      ! [B: nat,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.03/4.93       => ( ( ord_less_nat @ B @ one_one_nat )
% 9.03/4.93         => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 9.03/4.93            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1080_power__decreasing__iff,axiom,
% 9.03/4.93      ! [B: int,M: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.93       => ( ( ord_less_int @ B @ one_one_int )
% 9.03/4.93         => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 9.03/4.93            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1081_power2__less__eq__zero__iff,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 9.03/4.93        = ( A = zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1082_power2__less__eq__zero__iff,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 9.03/4.93        = ( A = zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1083_power2__less__eq__zero__iff,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 9.03/4.93        = ( A = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1084_power2__eq__iff__nonneg,axiom,
% 9.03/4.93      ! [X: real,Y: real] :
% 9.03/4.93        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.93       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.93         => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93              = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93            = ( X = Y ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1085_power2__eq__iff__nonneg,axiom,
% 9.03/4.93      ! [X: rat,Y: rat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.93       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.93         => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93              = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93            = ( X = Y ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1086_power2__eq__iff__nonneg,axiom,
% 9.03/4.93      ! [X: nat,Y: nat] :
% 9.03/4.93        ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 9.03/4.93       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 9.03/4.93         => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93              = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93            = ( X = Y ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1087_power2__eq__iff__nonneg,axiom,
% 9.03/4.93      ! [X: int,Y: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.03/4.93       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.03/4.93         => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.93              = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93            = ( X = Y ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1088_zero__less__power2,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93        = ( A != zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1089_zero__less__power2,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93        = ( A != zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1090_zero__less__power2,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93        = ( A != zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1091_sum__power2__eq__zero__iff,axiom,
% 9.03/4.93      ! [X: rat,Y: rat] :
% 9.03/4.93        ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93          = zero_zero_rat )
% 9.03/4.93        = ( ( X = zero_zero_rat )
% 9.03/4.93          & ( Y = zero_zero_rat ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1092_sum__power2__eq__zero__iff,axiom,
% 9.03/4.93      ! [X: real,Y: real] :
% 9.03/4.93        ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93          = zero_zero_real )
% 9.03/4.93        = ( ( X = zero_zero_real )
% 9.03/4.93          & ( Y = zero_zero_real ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1093_sum__power2__eq__zero__iff,axiom,
% 9.03/4.93      ! [X: int,Y: int] :
% 9.03/4.93        ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.93          = zero_zero_int )
% 9.03/4.93        = ( ( X = zero_zero_int )
% 9.03/4.93          & ( Y = zero_zero_int ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1094_of__nat__zero__less__power__iff,axiom,
% 9.03/4.93      ! [X: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
% 9.03/4.93        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.03/4.93          | ( N = zero_zero_nat ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1095_of__nat__zero__less__power__iff,axiom,
% 9.03/4.93      ! [X: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
% 9.03/4.93        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.03/4.93          | ( N = zero_zero_nat ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1096_of__nat__zero__less__power__iff,axiom,
% 9.03/4.93      ! [X: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
% 9.03/4.93        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.03/4.93          | ( N = zero_zero_nat ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1097_of__nat__zero__less__power__iff,axiom,
% 9.03/4.93      ! [X: nat,N: nat] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
% 9.03/4.93        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.03/4.93          | ( N = zero_zero_nat ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1098_numeral__power__less__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,A: int] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 9.03/4.93        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1099_numeral__power__less__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,A: int] :
% 9.03/4.93        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 9.03/4.93        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1100_numeral__power__less__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,A: int] :
% 9.03/4.93        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 9.03/4.93        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1101_numeral__power__less__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,A: int] :
% 9.03/4.93        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 9.03/4.93        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1102_of__int__less__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [A: int,X: num,N: nat] :
% 9.03/4.93        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
% 9.03/4.93        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1103_of__int__less__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [A: int,X: num,N: nat] :
% 9.03/4.93        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 9.03/4.93        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1104_of__int__less__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [A: int,X: num,N: nat] :
% 9.03/4.93        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 9.03/4.93        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1105_of__int__less__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [A: int,X: num,N: nat] :
% 9.03/4.93        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 9.03/4.93        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1106_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [X: nat,I: num,N: nat] :
% 9.03/4.93        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 9.03/4.93        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1107_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [X: nat,I: num,N: nat] :
% 9.03/4.93        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 9.03/4.93        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1108_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [X: nat,I: num,N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 9.03/4.93        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1109_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [X: nat,I: num,N: nat] :
% 9.03/4.93        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 9.03/4.93        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1110_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.03/4.93      ! [I: num,N: nat,X: nat] :
% 9.03/4.93        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 9.03/4.93        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1111_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.03/4.93      ! [I: num,N: nat,X: nat] :
% 9.03/4.93        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 9.03/4.93        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1112_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.03/4.93      ! [I: num,N: nat,X: nat] :
% 9.03/4.93        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 9.03/4.93        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1113_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.03/4.93      ! [I: num,N: nat,X: nat] :
% 9.03/4.93        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 9.03/4.93        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1114_numeral__power__le__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,A: int] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1115_numeral__power__le__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,A: int] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1116_numeral__power__le__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,A: int] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1117_numeral__power__le__of__int__cancel__iff,axiom,
% 9.03/4.93      ! [X: num,N: nat,A: int] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 9.03/4.93        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1118_of__int__le__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [A: int,X: num,N: nat] :
% 9.03/4.93        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
% 9.03/4.93        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1119_of__int__le__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [A: int,X: num,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 9.03/4.93        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1120_of__int__le__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [A: int,X: num,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 9.03/4.93        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1121_of__int__le__numeral__power__cancel__iff,axiom,
% 9.03/4.93      ! [A: int,X: num,N: nat] :
% 9.03/4.93        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 9.03/4.93        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1122_Collect__mono__iff,axiom,
% 9.03/4.93      ! [P: complex > $o,Q: complex > $o] :
% 9.03/4.93        ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
% 9.03/4.93        = ( ! [X4: complex] :
% 9.03/4.93              ( ( P @ X4 )
% 9.03/4.93             => ( Q @ X4 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1123_Collect__mono__iff,axiom,
% 9.03/4.93      ! [P: real > $o,Q: real > $o] :
% 9.03/4.93        ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 9.03/4.93        = ( ! [X4: real] :
% 9.03/4.93              ( ( P @ X4 )
% 9.03/4.93             => ( Q @ X4 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1124_Collect__mono__iff,axiom,
% 9.03/4.93      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.03/4.93        ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 9.03/4.93        = ( ! [X4: set_nat] :
% 9.03/4.93              ( ( P @ X4 )
% 9.03/4.93             => ( Q @ X4 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1125_Collect__mono__iff,axiom,
% 9.03/4.93      ! [P: nat > $o,Q: nat > $o] :
% 9.03/4.93        ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 9.03/4.93        = ( ! [X4: nat] :
% 9.03/4.93              ( ( P @ X4 )
% 9.03/4.93             => ( Q @ X4 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1126_Collect__mono__iff,axiom,
% 9.03/4.93      ! [P: int > $o,Q: int > $o] :
% 9.03/4.93        ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 9.03/4.93        = ( ! [X4: int] :
% 9.03/4.93              ( ( P @ X4 )
% 9.03/4.93             => ( Q @ X4 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1127_set__eq__subset,axiom,
% 9.03/4.93      ( ( ^ [Y3: set_int,Z2: set_int] : ( Y3 = Z2 ) )
% 9.03/4.93      = ( ^ [A4: set_int,B5: set_int] :
% 9.03/4.93            ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.03/4.93            & ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1128_subset__trans,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int,C6: set_int] :
% 9.03/4.93        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.93       => ( ( ord_less_eq_set_int @ B2 @ C6 )
% 9.03/4.93         => ( ord_less_eq_set_int @ A2 @ C6 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1129_Collect__mono,axiom,
% 9.03/4.93      ! [P: complex > $o,Q: complex > $o] :
% 9.03/4.93        ( ! [X3: complex] :
% 9.03/4.93            ( ( P @ X3 )
% 9.03/4.93           => ( Q @ X3 ) )
% 9.03/4.93       => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1130_Collect__mono,axiom,
% 9.03/4.93      ! [P: real > $o,Q: real > $o] :
% 9.03/4.93        ( ! [X3: real] :
% 9.03/4.93            ( ( P @ X3 )
% 9.03/4.93           => ( Q @ X3 ) )
% 9.03/4.93       => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1131_Collect__mono,axiom,
% 9.03/4.93      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.03/4.93        ( ! [X3: set_nat] :
% 9.03/4.93            ( ( P @ X3 )
% 9.03/4.93           => ( Q @ X3 ) )
% 9.03/4.93       => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1132_Collect__mono,axiom,
% 9.03/4.93      ! [P: nat > $o,Q: nat > $o] :
% 9.03/4.93        ( ! [X3: nat] :
% 9.03/4.93            ( ( P @ X3 )
% 9.03/4.93           => ( Q @ X3 ) )
% 9.03/4.93       => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1133_Collect__mono,axiom,
% 9.03/4.93      ! [P: int > $o,Q: int > $o] :
% 9.03/4.93        ( ! [X3: int] :
% 9.03/4.93            ( ( P @ X3 )
% 9.03/4.93           => ( Q @ X3 ) )
% 9.03/4.93       => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1134_subset__refl,axiom,
% 9.03/4.93      ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1135_subset__iff,axiom,
% 9.03/4.93      ( ord_le211207098394363844omplex
% 9.03/4.93      = ( ^ [A4: set_complex,B5: set_complex] :
% 9.03/4.93          ! [T2: complex] :
% 9.03/4.93            ( ( member_complex @ T2 @ A4 )
% 9.03/4.93           => ( member_complex @ T2 @ B5 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1136_subset__iff,axiom,
% 9.03/4.93      ( ord_less_eq_set_real
% 9.03/4.93      = ( ^ [A4: set_real,B5: set_real] :
% 9.03/4.93          ! [T2: real] :
% 9.03/4.93            ( ( member_real @ T2 @ A4 )
% 9.03/4.93           => ( member_real @ T2 @ B5 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1137_subset__iff,axiom,
% 9.03/4.93      ( ord_le6893508408891458716et_nat
% 9.03/4.93      = ( ^ [A4: set_set_nat,B5: set_set_nat] :
% 9.03/4.93          ! [T2: set_nat] :
% 9.03/4.93            ( ( member_set_nat @ T2 @ A4 )
% 9.03/4.93           => ( member_set_nat @ T2 @ B5 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1138_subset__iff,axiom,
% 9.03/4.93      ( ord_less_eq_set_nat
% 9.03/4.93      = ( ^ [A4: set_nat,B5: set_nat] :
% 9.03/4.93          ! [T2: nat] :
% 9.03/4.93            ( ( member_nat @ T2 @ A4 )
% 9.03/4.93           => ( member_nat @ T2 @ B5 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1139_subset__iff,axiom,
% 9.03/4.93      ( ord_less_eq_set_int
% 9.03/4.93      = ( ^ [A4: set_int,B5: set_int] :
% 9.03/4.93          ! [T2: int] :
% 9.03/4.93            ( ( member_int @ T2 @ A4 )
% 9.03/4.93           => ( member_int @ T2 @ B5 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1140_equalityD2,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int] :
% 9.03/4.93        ( ( A2 = B2 )
% 9.03/4.93       => ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1141_equalityD1,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int] :
% 9.03/4.93        ( ( A2 = B2 )
% 9.03/4.93       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1142_subset__eq,axiom,
% 9.03/4.93      ( ord_le211207098394363844omplex
% 9.03/4.93      = ( ^ [A4: set_complex,B5: set_complex] :
% 9.03/4.93          ! [X4: complex] :
% 9.03/4.93            ( ( member_complex @ X4 @ A4 )
% 9.03/4.93           => ( member_complex @ X4 @ B5 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1143_subset__eq,axiom,
% 9.03/4.93      ( ord_less_eq_set_real
% 9.03/4.93      = ( ^ [A4: set_real,B5: set_real] :
% 9.03/4.93          ! [X4: real] :
% 9.03/4.93            ( ( member_real @ X4 @ A4 )
% 9.03/4.93           => ( member_real @ X4 @ B5 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1144_subset__eq,axiom,
% 9.03/4.93      ( ord_le6893508408891458716et_nat
% 9.03/4.93      = ( ^ [A4: set_set_nat,B5: set_set_nat] :
% 9.03/4.93          ! [X4: set_nat] :
% 9.03/4.93            ( ( member_set_nat @ X4 @ A4 )
% 9.03/4.93           => ( member_set_nat @ X4 @ B5 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1145_subset__eq,axiom,
% 9.03/4.93      ( ord_less_eq_set_nat
% 9.03/4.93      = ( ^ [A4: set_nat,B5: set_nat] :
% 9.03/4.93          ! [X4: nat] :
% 9.03/4.93            ( ( member_nat @ X4 @ A4 )
% 9.03/4.93           => ( member_nat @ X4 @ B5 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1146_subset__eq,axiom,
% 9.03/4.93      ( ord_less_eq_set_int
% 9.03/4.93      = ( ^ [A4: set_int,B5: set_int] :
% 9.03/4.93          ! [X4: int] :
% 9.03/4.93            ( ( member_int @ X4 @ A4 )
% 9.03/4.93           => ( member_int @ X4 @ B5 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1147_equalityE,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int] :
% 9.03/4.93        ( ( A2 = B2 )
% 9.03/4.93       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.93           => ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1148_subsetD,axiom,
% 9.03/4.93      ! [A2: set_complex,B2: set_complex,C: complex] :
% 9.03/4.93        ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.03/4.93       => ( ( member_complex @ C @ A2 )
% 9.03/4.93         => ( member_complex @ C @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1149_subsetD,axiom,
% 9.03/4.93      ! [A2: set_real,B2: set_real,C: real] :
% 9.03/4.93        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.03/4.93       => ( ( member_real @ C @ A2 )
% 9.03/4.93         => ( member_real @ C @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1150_subsetD,axiom,
% 9.03/4.93      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 9.03/4.93        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 9.03/4.93       => ( ( member_set_nat @ C @ A2 )
% 9.03/4.93         => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1151_subsetD,axiom,
% 9.03/4.93      ! [A2: set_nat,B2: set_nat,C: nat] :
% 9.03/4.93        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.03/4.93       => ( ( member_nat @ C @ A2 )
% 9.03/4.93         => ( member_nat @ C @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1152_subsetD,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int,C: int] :
% 9.03/4.93        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.93       => ( ( member_int @ C @ A2 )
% 9.03/4.93         => ( member_int @ C @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1153_in__mono,axiom,
% 9.03/4.93      ! [A2: set_complex,B2: set_complex,X: complex] :
% 9.03/4.93        ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.03/4.93       => ( ( member_complex @ X @ A2 )
% 9.03/4.93         => ( member_complex @ X @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1154_in__mono,axiom,
% 9.03/4.93      ! [A2: set_real,B2: set_real,X: real] :
% 9.03/4.93        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.03/4.93       => ( ( member_real @ X @ A2 )
% 9.03/4.93         => ( member_real @ X @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1155_in__mono,axiom,
% 9.03/4.93      ! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
% 9.03/4.93        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 9.03/4.93       => ( ( member_set_nat @ X @ A2 )
% 9.03/4.93         => ( member_set_nat @ X @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1156_in__mono,axiom,
% 9.03/4.93      ! [A2: set_nat,B2: set_nat,X: nat] :
% 9.03/4.93        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.03/4.93       => ( ( member_nat @ X @ A2 )
% 9.03/4.93         => ( member_nat @ X @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1157_in__mono,axiom,
% 9.03/4.93      ! [A2: set_int,B2: set_int,X: int] :
% 9.03/4.93        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.93       => ( ( member_int @ X @ A2 )
% 9.03/4.93         => ( member_int @ X @ B2 ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1158_zero__power,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.93       => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.03/4.93          = zero_zero_rat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1159_zero__power,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.93       => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.03/4.93          = zero_zero_nat ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1160_zero__power,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.93       => ( ( power_power_real @ zero_zero_real @ N )
% 9.03/4.93          = zero_zero_real ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1161_zero__power,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.93       => ( ( power_power_int @ zero_zero_int @ N )
% 9.03/4.93          = zero_zero_int ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1162_zero__power,axiom,
% 9.03/4.93      ! [N: nat] :
% 9.03/4.93        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.93       => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.03/4.93          = zero_zero_complex ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1163_field__lbound__gt__zero,axiom,
% 9.03/4.93      ! [D1: real,D22: real] :
% 9.03/4.93        ( ( ord_less_real @ zero_zero_real @ D1 )
% 9.03/4.93       => ( ( ord_less_real @ zero_zero_real @ D22 )
% 9.03/4.93         => ? [E: real] :
% 9.03/4.93              ( ( ord_less_real @ zero_zero_real @ E )
% 9.03/4.93              & ( ord_less_real @ E @ D1 )
% 9.03/4.93              & ( ord_less_real @ E @ D22 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1164_field__lbound__gt__zero,axiom,
% 9.03/4.93      ! [D1: rat,D22: rat] :
% 9.03/4.93        ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 9.03/4.93       => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 9.03/4.93         => ? [E: rat] :
% 9.03/4.93              ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.03/4.93              & ( ord_less_rat @ E @ D1 )
% 9.03/4.93              & ( ord_less_rat @ E @ D22 ) ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1165_of__int__pos,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.03/4.93       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1166_of__int__pos,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.03/4.93       => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1167_of__int__pos,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.03/4.93       => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1168_of__int__pos,axiom,
% 9.03/4.93      ! [Z: int] :
% 9.03/4.93        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.03/4.93       => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1169_verit__comp__simplify1_I1_J,axiom,
% 9.03/4.93      ! [A: real] :
% 9.03/4.93        ~ ( ord_less_real @ A @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1170_verit__comp__simplify1_I1_J,axiom,
% 9.03/4.93      ! [A: rat] :
% 9.03/4.93        ~ ( ord_less_rat @ A @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1171_verit__comp__simplify1_I1_J,axiom,
% 9.03/4.93      ! [A: num] :
% 9.03/4.93        ~ ( ord_less_num @ A @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1172_verit__comp__simplify1_I1_J,axiom,
% 9.03/4.93      ! [A: nat] :
% 9.03/4.93        ~ ( ord_less_nat @ A @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1173_verit__comp__simplify1_I1_J,axiom,
% 9.03/4.93      ! [A: int] :
% 9.03/4.93        ~ ( ord_less_int @ A @ A ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1174_lt__ex,axiom,
% 9.03/4.93      ! [X: real] :
% 9.03/4.93      ? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1175_lt__ex,axiom,
% 9.03/4.93      ! [X: rat] :
% 9.03/4.93      ? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1176_lt__ex,axiom,
% 9.03/4.93      ! [X: int] :
% 9.03/4.93      ? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1177_gt__ex,axiom,
% 9.03/4.93      ! [X: real] :
% 9.03/4.93      ? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1178_gt__ex,axiom,
% 9.03/4.93      ! [X: rat] :
% 9.03/4.93      ? [X_12: rat] : ( ord_less_rat @ X @ X_12 ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1179_gt__ex,axiom,
% 9.03/4.93      ! [X: nat] :
% 9.03/4.93      ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1180_gt__ex,axiom,
% 9.03/4.93      ! [X: int] :
% 9.03/4.93      ? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1181_dense,axiom,
% 9.03/4.93      ! [X: real,Y: real] :
% 9.03/4.93        ( ( ord_less_real @ X @ Y )
% 9.03/4.93       => ? [Z3: real] :
% 9.03/4.93            ( ( ord_less_real @ X @ Z3 )
% 9.03/4.93            & ( ord_less_real @ Z3 @ Y ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1182_dense,axiom,
% 9.03/4.93      ! [X: rat,Y: rat] :
% 9.03/4.93        ( ( ord_less_rat @ X @ Y )
% 9.03/4.93       => ? [Z3: rat] :
% 9.03/4.93            ( ( ord_less_rat @ X @ Z3 )
% 9.03/4.93            & ( ord_less_rat @ Z3 @ Y ) ) ) ).
% 9.03/4.93  
% 9.03/4.93  thf(fact_1183_less__imp__neq,axiom,
% 9.03/4.93      ! [X: real,Y: real] :
% 9.03/4.93        ( ( ord_less_real @ X @ Y )
% 9.03/4.93       => ( X != Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1184_less__imp__neq,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( X != Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1185_less__imp__neq,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94       => ( X != Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1186_less__imp__neq,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ( X != Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1187_less__imp__neq,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94       => ( X != Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1188_order_Oasym,axiom,
% 9.03/4.94      ! [A: real,B: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ~ ( ord_less_real @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1189_order_Oasym,axiom,
% 9.03/4.94      ! [A: rat,B: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ~ ( ord_less_rat @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1190_order_Oasym,axiom,
% 9.03/4.94      ! [A: num,B: num] :
% 9.03/4.94        ( ( ord_less_num @ A @ B )
% 9.03/4.94       => ~ ( ord_less_num @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1191_order_Oasym,axiom,
% 9.03/4.94      ! [A: nat,B: nat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ~ ( ord_less_nat @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1192_order_Oasym,axiom,
% 9.03/4.94      ! [A: int,B: int] :
% 9.03/4.94        ( ( ord_less_int @ A @ B )
% 9.03/4.94       => ~ ( ord_less_int @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1193_ord__eq__less__trans,axiom,
% 9.03/4.94      ! [A: real,B: real,C: real] :
% 9.03/4.94        ( ( A = B )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ord_less_real @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1194_ord__eq__less__trans,axiom,
% 9.03/4.94      ! [A: rat,B: rat,C: rat] :
% 9.03/4.94        ( ( A = B )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1195_ord__eq__less__trans,axiom,
% 9.03/4.94      ! [A: num,B: num,C: num] :
% 9.03/4.94        ( ( A = B )
% 9.03/4.94       => ( ( ord_less_num @ B @ C )
% 9.03/4.94         => ( ord_less_num @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1196_ord__eq__less__trans,axiom,
% 9.03/4.94      ! [A: nat,B: nat,C: nat] :
% 9.03/4.94        ( ( A = B )
% 9.03/4.94       => ( ( ord_less_nat @ B @ C )
% 9.03/4.94         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1197_ord__eq__less__trans,axiom,
% 9.03/4.94      ! [A: int,B: int,C: int] :
% 9.03/4.94        ( ( A = B )
% 9.03/4.94       => ( ( ord_less_int @ B @ C )
% 9.03/4.94         => ( ord_less_int @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1198_ord__less__eq__trans,axiom,
% 9.03/4.94      ! [A: real,B: real,C: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( B = C )
% 9.03/4.94         => ( ord_less_real @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1199_ord__less__eq__trans,axiom,
% 9.03/4.94      ! [A: rat,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( B = C )
% 9.03/4.94         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1200_ord__less__eq__trans,axiom,
% 9.03/4.94      ! [A: num,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_num @ A @ B )
% 9.03/4.94       => ( ( B = C )
% 9.03/4.94         => ( ord_less_num @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1201_ord__less__eq__trans,axiom,
% 9.03/4.94      ! [A: nat,B: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ( ( B = C )
% 9.03/4.94         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1202_ord__less__eq__trans,axiom,
% 9.03/4.94      ! [A: int,B: int,C: int] :
% 9.03/4.94        ( ( ord_less_int @ A @ B )
% 9.03/4.94       => ( ( B = C )
% 9.03/4.94         => ( ord_less_int @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1203_less__induct,axiom,
% 9.03/4.94      ! [P: nat > $o,A: nat] :
% 9.03/4.94        ( ! [X3: nat] :
% 9.03/4.94            ( ! [Y5: nat] :
% 9.03/4.94                ( ( ord_less_nat @ Y5 @ X3 )
% 9.03/4.94               => ( P @ Y5 ) )
% 9.03/4.94           => ( P @ X3 ) )
% 9.03/4.94       => ( P @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1204_antisym__conv3,axiom,
% 9.03/4.94      ! [Y: real,X: real] :
% 9.03/4.94        ( ~ ( ord_less_real @ Y @ X )
% 9.03/4.94       => ( ( ~ ( ord_less_real @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1205_antisym__conv3,axiom,
% 9.03/4.94      ! [Y: rat,X: rat] :
% 9.03/4.94        ( ~ ( ord_less_rat @ Y @ X )
% 9.03/4.94       => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1206_antisym__conv3,axiom,
% 9.03/4.94      ! [Y: num,X: num] :
% 9.03/4.94        ( ~ ( ord_less_num @ Y @ X )
% 9.03/4.94       => ( ( ~ ( ord_less_num @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1207_antisym__conv3,axiom,
% 9.03/4.94      ! [Y: nat,X: nat] :
% 9.03/4.94        ( ~ ( ord_less_nat @ Y @ X )
% 9.03/4.94       => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1208_antisym__conv3,axiom,
% 9.03/4.94      ! [Y: int,X: int] :
% 9.03/4.94        ( ~ ( ord_less_int @ Y @ X )
% 9.03/4.94       => ( ( ~ ( ord_less_int @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1209_linorder__cases,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ~ ( ord_less_real @ X @ Y )
% 9.03/4.94       => ( ( X != Y )
% 9.03/4.94         => ( ord_less_real @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1210_linorder__cases,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ~ ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( ( X != Y )
% 9.03/4.94         => ( ord_less_rat @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1211_linorder__cases,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ~ ( ord_less_num @ X @ Y )
% 9.03/4.94       => ( ( X != Y )
% 9.03/4.94         => ( ord_less_num @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1212_linorder__cases,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ~ ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ( ( X != Y )
% 9.03/4.94         => ( ord_less_nat @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1213_linorder__cases,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ~ ( ord_less_int @ X @ Y )
% 9.03/4.94       => ( ( X != Y )
% 9.03/4.94         => ( ord_less_int @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1214_dual__order_Oasym,axiom,
% 9.03/4.94      ! [B: real,A: real] :
% 9.03/4.94        ( ( ord_less_real @ B @ A )
% 9.03/4.94       => ~ ( ord_less_real @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1215_dual__order_Oasym,axiom,
% 9.03/4.94      ! [B: rat,A: rat] :
% 9.03/4.94        ( ( ord_less_rat @ B @ A )
% 9.03/4.94       => ~ ( ord_less_rat @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1216_dual__order_Oasym,axiom,
% 9.03/4.94      ! [B: num,A: num] :
% 9.03/4.94        ( ( ord_less_num @ B @ A )
% 9.03/4.94       => ~ ( ord_less_num @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1217_dual__order_Oasym,axiom,
% 9.03/4.94      ! [B: nat,A: nat] :
% 9.03/4.94        ( ( ord_less_nat @ B @ A )
% 9.03/4.94       => ~ ( ord_less_nat @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1218_dual__order_Oasym,axiom,
% 9.03/4.94      ! [B: int,A: int] :
% 9.03/4.94        ( ( ord_less_int @ B @ A )
% 9.03/4.94       => ~ ( ord_less_int @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1219_dual__order_Oirrefl,axiom,
% 9.03/4.94      ! [A: real] :
% 9.03/4.94        ~ ( ord_less_real @ A @ A ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1220_dual__order_Oirrefl,axiom,
% 9.03/4.94      ! [A: rat] :
% 9.03/4.94        ~ ( ord_less_rat @ A @ A ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1221_dual__order_Oirrefl,axiom,
% 9.03/4.94      ! [A: num] :
% 9.03/4.94        ~ ( ord_less_num @ A @ A ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1222_dual__order_Oirrefl,axiom,
% 9.03/4.94      ! [A: nat] :
% 9.03/4.94        ~ ( ord_less_nat @ A @ A ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1223_dual__order_Oirrefl,axiom,
% 9.03/4.94      ! [A: int] :
% 9.03/4.94        ~ ( ord_less_int @ A @ A ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1224_exists__least__iff,axiom,
% 9.03/4.94      ( ( ^ [P2: nat > $o] :
% 9.03/4.94          ? [X5: nat] : ( P2 @ X5 ) )
% 9.03/4.94      = ( ^ [P3: nat > $o] :
% 9.03/4.94          ? [N3: nat] :
% 9.03/4.94            ( ( P3 @ N3 )
% 9.03/4.94            & ! [M6: nat] :
% 9.03/4.94                ( ( ord_less_nat @ M6 @ N3 )
% 9.03/4.94               => ~ ( P3 @ M6 ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1225_linorder__less__wlog,axiom,
% 9.03/4.94      ! [P: real > real > $o,A: real,B: real] :
% 9.03/4.94        ( ! [A5: real,B6: real] :
% 9.03/4.94            ( ( ord_less_real @ A5 @ B6 )
% 9.03/4.94           => ( P @ A5 @ B6 ) )
% 9.03/4.94       => ( ! [A5: real] : ( P @ A5 @ A5 )
% 9.03/4.94         => ( ! [A5: real,B6: real] :
% 9.03/4.94                ( ( P @ B6 @ A5 )
% 9.03/4.94               => ( P @ A5 @ B6 ) )
% 9.03/4.94           => ( P @ A @ B ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1226_linorder__less__wlog,axiom,
% 9.03/4.94      ! [P: rat > rat > $o,A: rat,B: rat] :
% 9.03/4.94        ( ! [A5: rat,B6: rat] :
% 9.03/4.94            ( ( ord_less_rat @ A5 @ B6 )
% 9.03/4.94           => ( P @ A5 @ B6 ) )
% 9.03/4.94       => ( ! [A5: rat] : ( P @ A5 @ A5 )
% 9.03/4.94         => ( ! [A5: rat,B6: rat] :
% 9.03/4.94                ( ( P @ B6 @ A5 )
% 9.03/4.94               => ( P @ A5 @ B6 ) )
% 9.03/4.94           => ( P @ A @ B ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1227_linorder__less__wlog,axiom,
% 9.03/4.94      ! [P: num > num > $o,A: num,B: num] :
% 9.03/4.94        ( ! [A5: num,B6: num] :
% 9.03/4.94            ( ( ord_less_num @ A5 @ B6 )
% 9.03/4.94           => ( P @ A5 @ B6 ) )
% 9.03/4.94       => ( ! [A5: num] : ( P @ A5 @ A5 )
% 9.03/4.94         => ( ! [A5: num,B6: num] :
% 9.03/4.94                ( ( P @ B6 @ A5 )
% 9.03/4.94               => ( P @ A5 @ B6 ) )
% 9.03/4.94           => ( P @ A @ B ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1228_linorder__less__wlog,axiom,
% 9.03/4.94      ! [P: nat > nat > $o,A: nat,B: nat] :
% 9.03/4.94        ( ! [A5: nat,B6: nat] :
% 9.03/4.94            ( ( ord_less_nat @ A5 @ B6 )
% 9.03/4.94           => ( P @ A5 @ B6 ) )
% 9.03/4.94       => ( ! [A5: nat] : ( P @ A5 @ A5 )
% 9.03/4.94         => ( ! [A5: nat,B6: nat] :
% 9.03/4.94                ( ( P @ B6 @ A5 )
% 9.03/4.94               => ( P @ A5 @ B6 ) )
% 9.03/4.94           => ( P @ A @ B ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1229_linorder__less__wlog,axiom,
% 9.03/4.94      ! [P: int > int > $o,A: int,B: int] :
% 9.03/4.94        ( ! [A5: int,B6: int] :
% 9.03/4.94            ( ( ord_less_int @ A5 @ B6 )
% 9.03/4.94           => ( P @ A5 @ B6 ) )
% 9.03/4.94       => ( ! [A5: int] : ( P @ A5 @ A5 )
% 9.03/4.94         => ( ! [A5: int,B6: int] :
% 9.03/4.94                ( ( P @ B6 @ A5 )
% 9.03/4.94               => ( P @ A5 @ B6 ) )
% 9.03/4.94           => ( P @ A @ B ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1230_order_Ostrict__trans,axiom,
% 9.03/4.94      ! [A: real,B: real,C: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ord_less_real @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1231_order_Ostrict__trans,axiom,
% 9.03/4.94      ! [A: rat,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1232_order_Ostrict__trans,axiom,
% 9.03/4.94      ! [A: num,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_num @ A @ B )
% 9.03/4.94       => ( ( ord_less_num @ B @ C )
% 9.03/4.94         => ( ord_less_num @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1233_order_Ostrict__trans,axiom,
% 9.03/4.94      ! [A: nat,B: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ( ( ord_less_nat @ B @ C )
% 9.03/4.94         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1234_order_Ostrict__trans,axiom,
% 9.03/4.94      ! [A: int,B: int,C: int] :
% 9.03/4.94        ( ( ord_less_int @ A @ B )
% 9.03/4.94       => ( ( ord_less_int @ B @ C )
% 9.03/4.94         => ( ord_less_int @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1235_not__less__iff__gr__or__eq,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ~ ( ord_less_real @ X @ Y ) )
% 9.03/4.94        = ( ( ord_less_real @ Y @ X )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1236_not__less__iff__gr__or__eq,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ~ ( ord_less_rat @ X @ Y ) )
% 9.03/4.94        = ( ( ord_less_rat @ Y @ X )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1237_not__less__iff__gr__or__eq,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ~ ( ord_less_num @ X @ Y ) )
% 9.03/4.94        = ( ( ord_less_num @ Y @ X )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1238_not__less__iff__gr__or__eq,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ~ ( ord_less_nat @ X @ Y ) )
% 9.03/4.94        = ( ( ord_less_nat @ Y @ X )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1239_not__less__iff__gr__or__eq,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ~ ( ord_less_int @ X @ Y ) )
% 9.03/4.94        = ( ( ord_less_int @ Y @ X )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1240_dual__order_Ostrict__trans,axiom,
% 9.03/4.94      ! [B: real,A: real,C: real] :
% 9.03/4.94        ( ( ord_less_real @ B @ A )
% 9.03/4.94       => ( ( ord_less_real @ C @ B )
% 9.03/4.94         => ( ord_less_real @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1241_dual__order_Ostrict__trans,axiom,
% 9.03/4.94      ! [B: rat,A: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ B @ A )
% 9.03/4.94       => ( ( ord_less_rat @ C @ B )
% 9.03/4.94         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1242_dual__order_Ostrict__trans,axiom,
% 9.03/4.94      ! [B: num,A: num,C: num] :
% 9.03/4.94        ( ( ord_less_num @ B @ A )
% 9.03/4.94       => ( ( ord_less_num @ C @ B )
% 9.03/4.94         => ( ord_less_num @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1243_dual__order_Ostrict__trans,axiom,
% 9.03/4.94      ! [B: nat,A: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_nat @ B @ A )
% 9.03/4.94       => ( ( ord_less_nat @ C @ B )
% 9.03/4.94         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1244_dual__order_Ostrict__trans,axiom,
% 9.03/4.94      ! [B: int,A: int,C: int] :
% 9.03/4.94        ( ( ord_less_int @ B @ A )
% 9.03/4.94       => ( ( ord_less_int @ C @ B )
% 9.03/4.94         => ( ord_less_int @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1245_order_Ostrict__implies__not__eq,axiom,
% 9.03/4.94      ! [A: real,B: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( A != B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1246_order_Ostrict__implies__not__eq,axiom,
% 9.03/4.94      ! [A: rat,B: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( A != B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1247_order_Ostrict__implies__not__eq,axiom,
% 9.03/4.94      ! [A: num,B: num] :
% 9.03/4.94        ( ( ord_less_num @ A @ B )
% 9.03/4.94       => ( A != B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1248_order_Ostrict__implies__not__eq,axiom,
% 9.03/4.94      ! [A: nat,B: nat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ( A != B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1249_order_Ostrict__implies__not__eq,axiom,
% 9.03/4.94      ! [A: int,B: int] :
% 9.03/4.94        ( ( ord_less_int @ A @ B )
% 9.03/4.94       => ( A != B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1250_dual__order_Ostrict__implies__not__eq,axiom,
% 9.03/4.94      ! [B: real,A: real] :
% 9.03/4.94        ( ( ord_less_real @ B @ A )
% 9.03/4.94       => ( A != B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1251_dual__order_Ostrict__implies__not__eq,axiom,
% 9.03/4.94      ! [B: rat,A: rat] :
% 9.03/4.94        ( ( ord_less_rat @ B @ A )
% 9.03/4.94       => ( A != B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1252_dual__order_Ostrict__implies__not__eq,axiom,
% 9.03/4.94      ! [B: num,A: num] :
% 9.03/4.94        ( ( ord_less_num @ B @ A )
% 9.03/4.94       => ( A != B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1253_dual__order_Ostrict__implies__not__eq,axiom,
% 9.03/4.94      ! [B: nat,A: nat] :
% 9.03/4.94        ( ( ord_less_nat @ B @ A )
% 9.03/4.94       => ( A != B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1254_dual__order_Ostrict__implies__not__eq,axiom,
% 9.03/4.94      ! [B: int,A: int] :
% 9.03/4.94        ( ( ord_less_int @ B @ A )
% 9.03/4.94       => ( A != B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1255_linorder__neqE,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94       => ( ~ ( ord_less_real @ X @ Y )
% 9.03/4.94         => ( ord_less_real @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1256_linorder__neqE,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94       => ( ~ ( ord_less_rat @ X @ Y )
% 9.03/4.94         => ( ord_less_rat @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1257_linorder__neqE,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94       => ( ~ ( ord_less_num @ X @ Y )
% 9.03/4.94         => ( ord_less_num @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1258_linorder__neqE,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94       => ( ~ ( ord_less_nat @ X @ Y )
% 9.03/4.94         => ( ord_less_nat @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1259_linorder__neqE,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94       => ( ~ ( ord_less_int @ X @ Y )
% 9.03/4.94         => ( ord_less_int @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1260_order__less__asym,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_real @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1261_order__less__asym,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1262_order__less__asym,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_num @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1263_order__less__asym,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1264_order__less__asym,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_int @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1265_linorder__neq__iff,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94        = ( ( ord_less_real @ X @ Y )
% 9.03/4.94          | ( ord_less_real @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1266_linorder__neq__iff,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94        = ( ( ord_less_rat @ X @ Y )
% 9.03/4.94          | ( ord_less_rat @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1267_linorder__neq__iff,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94        = ( ( ord_less_num @ X @ Y )
% 9.03/4.94          | ( ord_less_num @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1268_linorder__neq__iff,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94        = ( ( ord_less_nat @ X @ Y )
% 9.03/4.94          | ( ord_less_nat @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1269_linorder__neq__iff,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94        = ( ( ord_less_int @ X @ Y )
% 9.03/4.94          | ( ord_less_int @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1270_order__less__asym_H,axiom,
% 9.03/4.94      ! [A: real,B: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ~ ( ord_less_real @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1271_order__less__asym_H,axiom,
% 9.03/4.94      ! [A: rat,B: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ~ ( ord_less_rat @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1272_order__less__asym_H,axiom,
% 9.03/4.94      ! [A: num,B: num] :
% 9.03/4.94        ( ( ord_less_num @ A @ B )
% 9.03/4.94       => ~ ( ord_less_num @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1273_order__less__asym_H,axiom,
% 9.03/4.94      ! [A: nat,B: nat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ~ ( ord_less_nat @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1274_order__less__asym_H,axiom,
% 9.03/4.94      ! [A: int,B: int] :
% 9.03/4.94        ( ( ord_less_int @ A @ B )
% 9.03/4.94       => ~ ( ord_less_int @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1275_order__less__trans,axiom,
% 9.03/4.94      ! [X: real,Y: real,Z: real] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94       => ( ( ord_less_real @ Y @ Z )
% 9.03/4.94         => ( ord_less_real @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1276_order__less__trans,axiom,
% 9.03/4.94      ! [X: rat,Y: rat,Z: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_rat @ Y @ Z )
% 9.03/4.94         => ( ord_less_rat @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1277_order__less__trans,axiom,
% 9.03/4.94      ! [X: num,Y: num,Z: num] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94       => ( ( ord_less_num @ Y @ Z )
% 9.03/4.94         => ( ord_less_num @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1278_order__less__trans,axiom,
% 9.03/4.94      ! [X: nat,Y: nat,Z: nat] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_nat @ Y @ Z )
% 9.03/4.94         => ( ord_less_nat @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1279_order__less__trans,axiom,
% 9.03/4.94      ! [X: int,Y: int,Z: int] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94       => ( ( ord_less_int @ Y @ Z )
% 9.03/4.94         => ( ord_less_int @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1280_ord__eq__less__subst,axiom,
% 9.03/4.94      ! [A: real,F: real > real,B: real,C: real] :
% 9.03/4.94        ( ( A
% 9.03/4.94          = ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1281_ord__eq__less__subst,axiom,
% 9.03/4.94      ! [A: rat,F: real > rat,B: real,C: real] :
% 9.03/4.94        ( ( A
% 9.03/4.94          = ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1282_ord__eq__less__subst,axiom,
% 9.03/4.94      ! [A: num,F: real > num,B: real,C: real] :
% 9.03/4.94        ( ( A
% 9.03/4.94          = ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1283_ord__eq__less__subst,axiom,
% 9.03/4.94      ! [A: nat,F: real > nat,B: real,C: real] :
% 9.03/4.94        ( ( A
% 9.03/4.94          = ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1284_ord__eq__less__subst,axiom,
% 9.03/4.94      ! [A: int,F: real > int,B: real,C: real] :
% 9.03/4.94        ( ( A
% 9.03/4.94          = ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1285_ord__eq__less__subst,axiom,
% 9.03/4.94      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.03/4.94        ( ( A
% 9.03/4.94          = ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1286_ord__eq__less__subst,axiom,
% 9.03/4.94      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.03/4.94        ( ( A
% 9.03/4.94          = ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1287_ord__eq__less__subst,axiom,
% 9.03/4.94      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.03/4.94        ( ( A
% 9.03/4.94          = ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1288_ord__eq__less__subst,axiom,
% 9.03/4.94      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.03/4.94        ( ( A
% 9.03/4.94          = ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1289_ord__eq__less__subst,axiom,
% 9.03/4.94      ! [A: int,F: rat > int,B: rat,C: rat] :
% 9.03/4.94        ( ( A
% 9.03/4.94          = ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1290_ord__less__eq__subst,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > real,C: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ( F @ B )
% 9.03/4.94            = C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1291_ord__less__eq__subst,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ( F @ B )
% 9.03/4.94            = C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1292_ord__less__eq__subst,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > num,C: num] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ( F @ B )
% 9.03/4.94            = C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1293_ord__less__eq__subst,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > nat,C: nat] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ( F @ B )
% 9.03/4.94            = C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1294_ord__less__eq__subst,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > int,C: int] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ( F @ B )
% 9.03/4.94            = C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1295_ord__less__eq__subst,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ( F @ B )
% 9.03/4.94            = C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1296_ord__less__eq__subst,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ( F @ B )
% 9.03/4.94            = C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1297_ord__less__eq__subst,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ( F @ B )
% 9.03/4.94            = C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1298_ord__less__eq__subst,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ( F @ B )
% 9.03/4.94            = C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1299_ord__less__eq__subst,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ( F @ B )
% 9.03/4.94            = C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1300_order__less__irrefl,axiom,
% 9.03/4.94      ! [X: real] :
% 9.03/4.94        ~ ( ord_less_real @ X @ X ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1301_order__less__irrefl,axiom,
% 9.03/4.94      ! [X: rat] :
% 9.03/4.94        ~ ( ord_less_rat @ X @ X ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1302_order__less__irrefl,axiom,
% 9.03/4.94      ! [X: num] :
% 9.03/4.94        ~ ( ord_less_num @ X @ X ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1303_order__less__irrefl,axiom,
% 9.03/4.94      ! [X: nat] :
% 9.03/4.94        ~ ( ord_less_nat @ X @ X ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1304_order__less__irrefl,axiom,
% 9.03/4.94      ! [X: int] :
% 9.03/4.94        ~ ( ord_less_int @ X @ X ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1305_order__less__subst1,axiom,
% 9.03/4.94      ! [A: real,F: real > real,B: real,C: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1306_order__less__subst1,axiom,
% 9.03/4.94      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1307_order__less__subst1,axiom,
% 9.03/4.94      ! [A: real,F: num > real,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_num @ B @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1308_order__less__subst1,axiom,
% 9.03/4.94      ! [A: real,F: nat > real,B: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_nat @ B @ C )
% 9.03/4.94         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.94                ( ( ord_less_nat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1309_order__less__subst1,axiom,
% 9.03/4.94      ! [A: real,F: int > real,B: int,C: int] :
% 9.03/4.94        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_int @ B @ C )
% 9.03/4.94         => ( ! [X3: int,Y4: int] :
% 9.03/4.94                ( ( ord_less_int @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1310_order__less__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: real > rat,B: real,C: real] :
% 9.03/4.94        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1311_order__less__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1312_order__less__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_num @ B @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1313_order__less__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_nat @ B @ C )
% 9.03/4.94         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.94                ( ( ord_less_nat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1314_order__less__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: int > rat,B: int,C: int] :
% 9.03/4.94        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_int @ B @ C )
% 9.03/4.94         => ( ! [X3: int,Y4: int] :
% 9.03/4.94                ( ( ord_less_int @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1315_order__less__subst2,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > real,C: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1316_order__less__subst2,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1317_order__less__subst2,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > num,C: num] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1318_order__less__subst2,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > nat,C: nat] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1319_order__less__subst2,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > int,C: int] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1320_order__less__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1321_order__less__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1322_order__less__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1323_order__less__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1324_order__less__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1325_order__less__not__sym,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_real @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1326_order__less__not__sym,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1327_order__less__not__sym,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_num @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1328_order__less__not__sym,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1329_order__less__not__sym,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_int @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1330_order__less__imp__triv,axiom,
% 9.03/4.94      ! [X: real,Y: real,P: $o] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94       => ( ( ord_less_real @ Y @ X )
% 9.03/4.94         => P ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1331_order__less__imp__triv,axiom,
% 9.03/4.94      ! [X: rat,Y: rat,P: $o] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_rat @ Y @ X )
% 9.03/4.94         => P ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1332_order__less__imp__triv,axiom,
% 9.03/4.94      ! [X: num,Y: num,P: $o] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94       => ( ( ord_less_num @ Y @ X )
% 9.03/4.94         => P ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1333_order__less__imp__triv,axiom,
% 9.03/4.94      ! [X: nat,Y: nat,P: $o] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_nat @ Y @ X )
% 9.03/4.94         => P ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1334_order__less__imp__triv,axiom,
% 9.03/4.94      ! [X: int,Y: int,P: $o] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94       => ( ( ord_less_int @ Y @ X )
% 9.03/4.94         => P ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1335_linorder__less__linear,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94        | ( X = Y )
% 9.03/4.94        | ( ord_less_real @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1336_linorder__less__linear,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94        | ( X = Y )
% 9.03/4.94        | ( ord_less_rat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1337_linorder__less__linear,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94        | ( X = Y )
% 9.03/4.94        | ( ord_less_num @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1338_linorder__less__linear,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94        | ( X = Y )
% 9.03/4.94        | ( ord_less_nat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1339_linorder__less__linear,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94        | ( X = Y )
% 9.03/4.94        | ( ord_less_int @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1340_order__less__imp__not__eq,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94       => ( X != Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1341_order__less__imp__not__eq,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( X != Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1342_order__less__imp__not__eq,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94       => ( X != Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1343_order__less__imp__not__eq,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ( X != Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1344_order__less__imp__not__eq,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94       => ( X != Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1345_order__less__imp__not__eq2,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94       => ( Y != X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1346_order__less__imp__not__eq2,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( Y != X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1347_order__less__imp__not__eq2,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94       => ( Y != X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1348_order__less__imp__not__eq2,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ( Y != X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1349_order__less__imp__not__eq2,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94       => ( Y != X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1350_order__less__imp__not__less,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_real @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1351_order__less__imp__not__less,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1352_order__less__imp__not__less,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_num @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1353_order__less__imp__not__less,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1354_order__less__imp__not__less,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94       => ~ ( ord_less_int @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1355_less__ceiling__iff,axiom,
% 9.03/4.94      ! [Z: int,X: real] :
% 9.03/4.94        ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
% 9.03/4.94        = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1356_less__ceiling__iff,axiom,
% 9.03/4.94      ! [Z: int,X: rat] :
% 9.03/4.94        ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/4.94        = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1357_ex__of__int__less,axiom,
% 9.03/4.94      ! [X: real] :
% 9.03/4.94      ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1358_ex__of__int__less,axiom,
% 9.03/4.94      ! [X: rat] :
% 9.03/4.94      ? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1359_ex__less__of__int,axiom,
% 9.03/4.94      ! [X: real] :
% 9.03/4.94      ? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1360_ex__less__of__int,axiom,
% 9.03/4.94      ! [X: rat] :
% 9.03/4.94      ? [Z3: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1361_linorder__neqE__nat,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( X != Y )
% 9.03/4.94       => ( ~ ( ord_less_nat @ X @ Y )
% 9.03/4.94         => ( ord_less_nat @ Y @ X ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1362_infinite__descent0,axiom,
% 9.03/4.94      ! [P: nat > $o,N: nat] :
% 9.03/4.94        ( ( P @ zero_zero_nat )
% 9.03/4.94       => ( ! [N2: nat] :
% 9.03/4.94              ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.03/4.94             => ( ~ ( P @ N2 )
% 9.03/4.94               => ? [M3: nat] :
% 9.03/4.94                    ( ( ord_less_nat @ M3 @ N2 )
% 9.03/4.94                    & ~ ( P @ M3 ) ) ) )
% 9.03/4.94         => ( P @ N ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1363_infinite__descent,axiom,
% 9.03/4.94      ! [P: nat > $o,N: nat] :
% 9.03/4.94        ( ! [N2: nat] :
% 9.03/4.94            ( ~ ( P @ N2 )
% 9.03/4.94           => ? [M3: nat] :
% 9.03/4.94                ( ( ord_less_nat @ M3 @ N2 )
% 9.03/4.94                & ~ ( P @ M3 ) ) )
% 9.03/4.94       => ( P @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1364_nat__less__induct,axiom,
% 9.03/4.94      ! [P: nat > $o,N: nat] :
% 9.03/4.94        ( ! [N2: nat] :
% 9.03/4.94            ( ! [M3: nat] :
% 9.03/4.94                ( ( ord_less_nat @ M3 @ N2 )
% 9.03/4.94               => ( P @ M3 ) )
% 9.03/4.94           => ( P @ N2 ) )
% 9.03/4.94       => ( P @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1365_less__irrefl__nat,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ~ ( ord_less_nat @ N @ N ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1366_gr__implies__not0,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ M @ N )
% 9.03/4.94       => ( N != zero_zero_nat ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1367_less__not__refl3,axiom,
% 9.03/4.94      ! [S: nat,T: nat] :
% 9.03/4.94        ( ( ord_less_nat @ S @ T )
% 9.03/4.94       => ( S != T ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1368_less__not__refl2,axiom,
% 9.03/4.94      ! [N: nat,M: nat] :
% 9.03/4.94        ( ( ord_less_nat @ N @ M )
% 9.03/4.94       => ( M != N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1369_less__not__refl,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ~ ( ord_less_nat @ N @ N ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1370_nat__neq__iff,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( M != N )
% 9.03/4.94        = ( ( ord_less_nat @ M @ N )
% 9.03/4.94          | ( ord_less_nat @ N @ M ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1371_less__zeroE,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1372_not__less0,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1373_not__gr0,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 9.03/4.94        = ( N = zero_zero_nat ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1374_gr0I,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ( ( N != zero_zero_nat )
% 9.03/4.94       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1375_bot__nat__0_Oextremum__strict,axiom,
% 9.03/4.94      ! [A: nat] :
% 9.03/4.94        ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1376_less__numeral__extra_I3_J,axiom,
% 9.03/4.94      ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1377_less__numeral__extra_I3_J,axiom,
% 9.03/4.94      ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1378_less__numeral__extra_I3_J,axiom,
% 9.03/4.94      ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1379_less__numeral__extra_I3_J,axiom,
% 9.03/4.94      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1380_zero__reorient,axiom,
% 9.03/4.94      ! [X: complex] :
% 9.03/4.94        ( ( zero_zero_complex = X )
% 9.03/4.94        = ( X = zero_zero_complex ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1381_zero__reorient,axiom,
% 9.03/4.94      ! [X: real] :
% 9.03/4.94        ( ( zero_zero_real = X )
% 9.03/4.94        = ( X = zero_zero_real ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1382_zero__reorient,axiom,
% 9.03/4.94      ! [X: rat] :
% 9.03/4.94        ( ( zero_zero_rat = X )
% 9.03/4.94        = ( X = zero_zero_rat ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1383_zero__reorient,axiom,
% 9.03/4.94      ! [X: nat] :
% 9.03/4.94        ( ( zero_zero_nat = X )
% 9.03/4.94        = ( X = zero_zero_nat ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1384_zero__reorient,axiom,
% 9.03/4.94      ! [X: int] :
% 9.03/4.94        ( ( zero_zero_int = X )
% 9.03/4.94        = ( X = zero_zero_int ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1385_zero__less__iff__neq__zero,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.94        = ( N != zero_zero_nat ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1386_gr__implies__not__zero,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ M @ N )
% 9.03/4.94       => ( N != zero_zero_nat ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1387_not__less__zero,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1388_gr__zeroI,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ( ( N != zero_zero_nat )
% 9.03/4.94       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1389_power__strict__mono,axiom,
% 9.03/4.94      ! [A: real,B: real,N: nat] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.94         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.94           => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1390_power__strict__mono,axiom,
% 9.03/4.94      ! [A: rat,B: rat,N: nat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.94         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.94           => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1391_power__strict__mono,axiom,
% 9.03/4.94      ! [A: nat,B: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.94         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.94           => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1392_power__strict__mono,axiom,
% 9.03/4.94      ! [A: int,B: int,N: nat] :
% 9.03/4.94        ( ( ord_less_int @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.94         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.94           => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1393_lift__Suc__mono__less__iff,axiom,
% 9.03/4.94      ! [F: nat > real,N: nat,M: nat] :
% 9.03/4.94        ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.94       => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 9.03/4.94          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1394_lift__Suc__mono__less__iff,axiom,
% 9.03/4.94      ! [F: nat > rat,N: nat,M: nat] :
% 9.03/4.94        ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.94       => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 9.03/4.94          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1395_lift__Suc__mono__less__iff,axiom,
% 9.03/4.94      ! [F: nat > num,N: nat,M: nat] :
% 9.03/4.94        ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.94       => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 9.03/4.94          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1396_lift__Suc__mono__less__iff,axiom,
% 9.03/4.94      ! [F: nat > nat,N: nat,M: nat] :
% 9.03/4.94        ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.94       => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 9.03/4.94          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1397_lift__Suc__mono__less__iff,axiom,
% 9.03/4.94      ! [F: nat > int,N: nat,M: nat] :
% 9.03/4.94        ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.94       => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 9.03/4.94          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1398_lift__Suc__mono__less,axiom,
% 9.03/4.94      ! [F: nat > real,N: nat,N5: nat] :
% 9.03/4.94        ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.94       => ( ( ord_less_nat @ N @ N5 )
% 9.03/4.94         => ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1399_lift__Suc__mono__less,axiom,
% 9.03/4.94      ! [F: nat > rat,N: nat,N5: nat] :
% 9.03/4.94        ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.94       => ( ( ord_less_nat @ N @ N5 )
% 9.03/4.94         => ( ord_less_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1400_lift__Suc__mono__less,axiom,
% 9.03/4.94      ! [F: nat > num,N: nat,N5: nat] :
% 9.03/4.94        ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.94       => ( ( ord_less_nat @ N @ N5 )
% 9.03/4.94         => ( ord_less_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1401_lift__Suc__mono__less,axiom,
% 9.03/4.94      ! [F: nat > nat,N: nat,N5: nat] :
% 9.03/4.94        ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.94       => ( ( ord_less_nat @ N @ N5 )
% 9.03/4.94         => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1402_lift__Suc__mono__less,axiom,
% 9.03/4.94      ! [F: nat > int,N: nat,N5: nat] :
% 9.03/4.94        ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.03/4.94       => ( ( ord_less_nat @ N @ N5 )
% 9.03/4.94         => ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1403_zero__less__numeral,axiom,
% 9.03/4.94      ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1404_zero__less__numeral,axiom,
% 9.03/4.94      ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1405_zero__less__numeral,axiom,
% 9.03/4.94      ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1406_zero__less__numeral,axiom,
% 9.03/4.94      ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1407_not__numeral__less__zero,axiom,
% 9.03/4.94      ! [N: num] :
% 9.03/4.94        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1408_not__numeral__less__zero,axiom,
% 9.03/4.94      ! [N: num] :
% 9.03/4.94        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1409_not__numeral__less__zero,axiom,
% 9.03/4.94      ! [N: num] :
% 9.03/4.94        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1410_not__numeral__less__zero,axiom,
% 9.03/4.94      ! [N: num] :
% 9.03/4.94        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1411_less__numeral__extra_I1_J,axiom,
% 9.03/4.94      ord_less_real @ zero_zero_real @ one_one_real ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1412_less__numeral__extra_I1_J,axiom,
% 9.03/4.94      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1413_less__numeral__extra_I1_J,axiom,
% 9.03/4.94      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1414_less__numeral__extra_I1_J,axiom,
% 9.03/4.94      ord_less_int @ zero_zero_int @ one_one_int ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1415_pos__add__strict,axiom,
% 9.03/4.94      ! [A: real,B: real,C: real] :
% 9.03/4.94        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1416_pos__add__strict,axiom,
% 9.03/4.94      ! [A: rat,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1417_pos__add__strict,axiom,
% 9.03/4.94      ! [A: nat,B: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.94       => ( ( ord_less_nat @ B @ C )
% 9.03/4.94         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1418_pos__add__strict,axiom,
% 9.03/4.94      ! [A: int,B: int,C: int] :
% 9.03/4.94        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.94       => ( ( ord_less_int @ B @ C )
% 9.03/4.94         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1419_canonically__ordered__monoid__add__class_OlessE,axiom,
% 9.03/4.94      ! [A: nat,B: nat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ~ ! [C4: nat] :
% 9.03/4.94              ( ( B
% 9.03/4.94                = ( plus_plus_nat @ A @ C4 ) )
% 9.03/4.94             => ( C4 = zero_zero_nat ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1420_add__pos__pos,axiom,
% 9.03/4.94      ! [A: real,B: real] :
% 9.03/4.94        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.94       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.94         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1421_add__pos__pos,axiom,
% 9.03/4.94      ! [A: rat,B: rat] :
% 9.03/4.94        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.94       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.03/4.94         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1422_add__pos__pos,axiom,
% 9.03/4.94      ! [A: nat,B: nat] :
% 9.03/4.94        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.94       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.03/4.94         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1423_add__pos__pos,axiom,
% 9.03/4.94      ! [A: int,B: int] :
% 9.03/4.94        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.94       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.94         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1424_add__neg__neg,axiom,
% 9.03/4.94      ! [A: real,B: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.94       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.03/4.94         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1425_add__neg__neg,axiom,
% 9.03/4.94      ! [A: rat,B: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.94       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.03/4.94         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1426_add__neg__neg,axiom,
% 9.03/4.94      ! [A: nat,B: nat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.03/4.94       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.03/4.94         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1427_add__neg__neg,axiom,
% 9.03/4.94      ! [A: int,B: int] :
% 9.03/4.94        ( ( ord_less_int @ A @ zero_zero_int )
% 9.03/4.94       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.03/4.94         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1428_less__imp__of__nat__less,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ M @ N )
% 9.03/4.94       => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1429_less__imp__of__nat__less,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ M @ N )
% 9.03/4.94       => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1430_less__imp__of__nat__less,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ M @ N )
% 9.03/4.94       => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1431_less__imp__of__nat__less,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ M @ N )
% 9.03/4.94       => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1432_of__nat__less__imp__less,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.94       => ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1433_of__nat__less__imp__less,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.03/4.94       => ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1434_of__nat__less__imp__less,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.03/4.94       => ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1435_of__nat__less__imp__less,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.94       => ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1436_zero__less__power,axiom,
% 9.03/4.94      ! [A: real,N: nat] :
% 9.03/4.94        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.94       => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1437_zero__less__power,axiom,
% 9.03/4.94      ! [A: rat,N: nat] :
% 9.03/4.94        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.94       => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1438_zero__less__power,axiom,
% 9.03/4.94      ! [A: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.94       => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1439_zero__less__power,axiom,
% 9.03/4.94      ! [A: int,N: nat] :
% 9.03/4.94        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.94       => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1440_of__nat__less__0__iff,axiom,
% 9.03/4.94      ! [M: nat] :
% 9.03/4.94        ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1441_of__nat__less__0__iff,axiom,
% 9.03/4.94      ! [M: nat] :
% 9.03/4.94        ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1442_of__nat__less__0__iff,axiom,
% 9.03/4.94      ! [M: nat] :
% 9.03/4.94        ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1443_of__nat__less__0__iff,axiom,
% 9.03/4.94      ! [M: nat] :
% 9.03/4.94        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1444_less__Suc__eq__0__disj,axiom,
% 9.03/4.94      ! [M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.03/4.94        = ( ( M = zero_zero_nat )
% 9.03/4.94          | ? [J2: nat] :
% 9.03/4.94              ( ( M
% 9.03/4.94                = ( suc @ J2 ) )
% 9.03/4.94              & ( ord_less_nat @ J2 @ N ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1445_gr0__implies__Suc,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.94       => ? [M5: nat] :
% 9.03/4.94            ( N
% 9.03/4.94            = ( suc @ M5 ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1446_All__less__Suc2,axiom,
% 9.03/4.94      ! [N: nat,P: nat > $o] :
% 9.03/4.94        ( ( ! [I3: nat] :
% 9.03/4.94              ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 9.03/4.94             => ( P @ I3 ) ) )
% 9.03/4.94        = ( ( P @ zero_zero_nat )
% 9.03/4.94          & ! [I3: nat] :
% 9.03/4.94              ( ( ord_less_nat @ I3 @ N )
% 9.03/4.94             => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1447_gr0__conv__Suc,axiom,
% 9.03/4.94      ! [N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.94        = ( ? [M6: nat] :
% 9.03/4.94              ( N
% 9.03/4.94              = ( suc @ M6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1448_Ex__less__Suc2,axiom,
% 9.03/4.94      ! [N: nat,P: nat > $o] :
% 9.03/4.94        ( ( ? [I3: nat] :
% 9.03/4.94              ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 9.03/4.94              & ( P @ I3 ) ) )
% 9.03/4.94        = ( ( P @ zero_zero_nat )
% 9.03/4.94          | ? [I3: nat] :
% 9.03/4.94              ( ( ord_less_nat @ I3 @ N )
% 9.03/4.94              & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1449_ex__least__nat__le,axiom,
% 9.03/4.94      ! [P: nat > $o,N: nat] :
% 9.03/4.94        ( ( P @ N )
% 9.03/4.94       => ( ~ ( P @ zero_zero_nat )
% 9.03/4.94         => ? [K3: nat] :
% 9.03/4.94              ( ( ord_less_eq_nat @ K3 @ N )
% 9.03/4.94              & ! [I4: nat] :
% 9.03/4.94                  ( ( ord_less_nat @ I4 @ K3 )
% 9.03/4.94                 => ~ ( P @ I4 ) )
% 9.03/4.94              & ( P @ K3 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1450_less__imp__add__positive,axiom,
% 9.03/4.94      ! [I: nat,J: nat] :
% 9.03/4.94        ( ( ord_less_nat @ I @ J )
% 9.03/4.94       => ? [K3: nat] :
% 9.03/4.94            ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 9.03/4.94            & ( ( plus_plus_nat @ I @ K3 )
% 9.03/4.94              = J ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1451_leD,axiom,
% 9.03/4.94      ! [Y: real,X: real] :
% 9.03/4.94        ( ( ord_less_eq_real @ Y @ X )
% 9.03/4.94       => ~ ( ord_less_real @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1452_leD,axiom,
% 9.03/4.94      ! [Y: set_int,X: set_int] :
% 9.03/4.94        ( ( ord_less_eq_set_int @ Y @ X )
% 9.03/4.94       => ~ ( ord_less_set_int @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1453_leD,axiom,
% 9.03/4.94      ! [Y: rat,X: rat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ Y @ X )
% 9.03/4.94       => ~ ( ord_less_rat @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1454_leD,axiom,
% 9.03/4.94      ! [Y: num,X: num] :
% 9.03/4.94        ( ( ord_less_eq_num @ Y @ X )
% 9.03/4.94       => ~ ( ord_less_num @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1455_leD,axiom,
% 9.03/4.94      ! [Y: nat,X: nat] :
% 9.03/4.94        ( ( ord_less_eq_nat @ Y @ X )
% 9.03/4.94       => ~ ( ord_less_nat @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1456_leD,axiom,
% 9.03/4.94      ! [Y: int,X: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ Y @ X )
% 9.03/4.94       => ~ ( ord_less_int @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1457_leI,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ~ ( ord_less_real @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_real @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1458_leI,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ~ ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_rat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1459_leI,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ~ ( ord_less_num @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_num @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1460_leI,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ~ ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_nat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1461_leI,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ~ ( ord_less_int @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_int @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1462_nless__le,axiom,
% 9.03/4.94      ! [A: real,B: real] :
% 9.03/4.94        ( ( ~ ( ord_less_real @ A @ B ) )
% 9.03/4.94        = ( ~ ( ord_less_eq_real @ A @ B )
% 9.03/4.94          | ( A = B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1463_nless__le,axiom,
% 9.03/4.94      ! [A: set_int,B: set_int] :
% 9.03/4.94        ( ( ~ ( ord_less_set_int @ A @ B ) )
% 9.03/4.94        = ( ~ ( ord_less_eq_set_int @ A @ B )
% 9.03/4.94          | ( A = B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1464_nless__le,axiom,
% 9.03/4.94      ! [A: rat,B: rat] :
% 9.03/4.94        ( ( ~ ( ord_less_rat @ A @ B ) )
% 9.03/4.94        = ( ~ ( ord_less_eq_rat @ A @ B )
% 9.03/4.94          | ( A = B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1465_nless__le,axiom,
% 9.03/4.94      ! [A: num,B: num] :
% 9.03/4.94        ( ( ~ ( ord_less_num @ A @ B ) )
% 9.03/4.94        = ( ~ ( ord_less_eq_num @ A @ B )
% 9.03/4.94          | ( A = B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1466_nless__le,axiom,
% 9.03/4.94      ! [A: nat,B: nat] :
% 9.03/4.94        ( ( ~ ( ord_less_nat @ A @ B ) )
% 9.03/4.94        = ( ~ ( ord_less_eq_nat @ A @ B )
% 9.03/4.94          | ( A = B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1467_nless__le,axiom,
% 9.03/4.94      ! [A: int,B: int] :
% 9.03/4.94        ( ( ~ ( ord_less_int @ A @ B ) )
% 9.03/4.94        = ( ~ ( ord_less_eq_int @ A @ B )
% 9.03/4.94          | ( A = B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1468_antisym__conv1,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ~ ( ord_less_real @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1469_antisym__conv1,axiom,
% 9.03/4.94      ! [X: set_int,Y: set_int] :
% 9.03/4.94        ( ~ ( ord_less_set_int @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_set_int @ X @ Y )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1470_antisym__conv1,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ~ ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1471_antisym__conv1,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ~ ( ord_less_num @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_num @ X @ Y )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1472_antisym__conv1,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ~ ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_nat @ X @ Y )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1473_antisym__conv1,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ~ ( ord_less_int @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_int @ X @ Y )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1474_antisym__conv2,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.94       => ( ( ~ ( ord_less_real @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1475_antisym__conv2,axiom,
% 9.03/4.94      ! [X: set_int,Y: set_int] :
% 9.03/4.94        ( ( ord_less_eq_set_int @ X @ Y )
% 9.03/4.94       => ( ( ~ ( ord_less_set_int @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1476_antisym__conv2,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.94       => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1477_antisym__conv2,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_eq_num @ X @ Y )
% 9.03/4.94       => ( ( ~ ( ord_less_num @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1478_antisym__conv2,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_eq_nat @ X @ Y )
% 9.03/4.94       => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1479_antisym__conv2,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ X @ Y )
% 9.03/4.94       => ( ( ~ ( ord_less_int @ X @ Y ) )
% 9.03/4.94          = ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1480_dense__ge,axiom,
% 9.03/4.94      ! [Z: real,Y: real] :
% 9.03/4.94        ( ! [X3: real] :
% 9.03/4.94            ( ( ord_less_real @ Z @ X3 )
% 9.03/4.94           => ( ord_less_eq_real @ Y @ X3 ) )
% 9.03/4.94       => ( ord_less_eq_real @ Y @ Z ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1481_dense__ge,axiom,
% 9.03/4.94      ! [Z: rat,Y: rat] :
% 9.03/4.94        ( ! [X3: rat] :
% 9.03/4.94            ( ( ord_less_rat @ Z @ X3 )
% 9.03/4.94           => ( ord_less_eq_rat @ Y @ X3 ) )
% 9.03/4.94       => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1482_dense__le,axiom,
% 9.03/4.94      ! [Y: real,Z: real] :
% 9.03/4.94        ( ! [X3: real] :
% 9.03/4.94            ( ( ord_less_real @ X3 @ Y )
% 9.03/4.94           => ( ord_less_eq_real @ X3 @ Z ) )
% 9.03/4.94       => ( ord_less_eq_real @ Y @ Z ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1483_dense__le,axiom,
% 9.03/4.94      ! [Y: rat,Z: rat] :
% 9.03/4.94        ( ! [X3: rat] :
% 9.03/4.94            ( ( ord_less_rat @ X3 @ Y )
% 9.03/4.94           => ( ord_less_eq_rat @ X3 @ Z ) )
% 9.03/4.94       => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1484_less__le__not__le,axiom,
% 9.03/4.94      ( ord_less_real
% 9.03/4.94      = ( ^ [X4: real,Y6: real] :
% 9.03/4.94            ( ( ord_less_eq_real @ X4 @ Y6 )
% 9.03/4.94            & ~ ( ord_less_eq_real @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1485_less__le__not__le,axiom,
% 9.03/4.94      ( ord_less_set_int
% 9.03/4.94      = ( ^ [X4: set_int,Y6: set_int] :
% 9.03/4.94            ( ( ord_less_eq_set_int @ X4 @ Y6 )
% 9.03/4.94            & ~ ( ord_less_eq_set_int @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1486_less__le__not__le,axiom,
% 9.03/4.94      ( ord_less_rat
% 9.03/4.94      = ( ^ [X4: rat,Y6: rat] :
% 9.03/4.94            ( ( ord_less_eq_rat @ X4 @ Y6 )
% 9.03/4.94            & ~ ( ord_less_eq_rat @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1487_less__le__not__le,axiom,
% 9.03/4.94      ( ord_less_num
% 9.03/4.94      = ( ^ [X4: num,Y6: num] :
% 9.03/4.94            ( ( ord_less_eq_num @ X4 @ Y6 )
% 9.03/4.94            & ~ ( ord_less_eq_num @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1488_less__le__not__le,axiom,
% 9.03/4.94      ( ord_less_nat
% 9.03/4.94      = ( ^ [X4: nat,Y6: nat] :
% 9.03/4.94            ( ( ord_less_eq_nat @ X4 @ Y6 )
% 9.03/4.94            & ~ ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1489_less__le__not__le,axiom,
% 9.03/4.94      ( ord_less_int
% 9.03/4.94      = ( ^ [X4: int,Y6: int] :
% 9.03/4.94            ( ( ord_less_eq_int @ X4 @ Y6 )
% 9.03/4.94            & ~ ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1490_not__le__imp__less,axiom,
% 9.03/4.94      ! [Y: real,X: real] :
% 9.03/4.94        ( ~ ( ord_less_eq_real @ Y @ X )
% 9.03/4.94       => ( ord_less_real @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1491_not__le__imp__less,axiom,
% 9.03/4.94      ! [Y: rat,X: rat] :
% 9.03/4.94        ( ~ ( ord_less_eq_rat @ Y @ X )
% 9.03/4.94       => ( ord_less_rat @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1492_not__le__imp__less,axiom,
% 9.03/4.94      ! [Y: num,X: num] :
% 9.03/4.94        ( ~ ( ord_less_eq_num @ Y @ X )
% 9.03/4.94       => ( ord_less_num @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1493_not__le__imp__less,axiom,
% 9.03/4.94      ! [Y: nat,X: nat] :
% 9.03/4.94        ( ~ ( ord_less_eq_nat @ Y @ X )
% 9.03/4.94       => ( ord_less_nat @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1494_not__le__imp__less,axiom,
% 9.03/4.94      ! [Y: int,X: int] :
% 9.03/4.94        ( ~ ( ord_less_eq_int @ Y @ X )
% 9.03/4.94       => ( ord_less_int @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1495_order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_real
% 9.03/4.94      = ( ^ [A3: real,B4: real] :
% 9.03/4.94            ( ( ord_less_real @ A3 @ B4 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1496_order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_set_int
% 9.03/4.94      = ( ^ [A3: set_int,B4: set_int] :
% 9.03/4.94            ( ( ord_less_set_int @ A3 @ B4 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1497_order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_rat
% 9.03/4.94      = ( ^ [A3: rat,B4: rat] :
% 9.03/4.94            ( ( ord_less_rat @ A3 @ B4 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1498_order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_num
% 9.03/4.94      = ( ^ [A3: num,B4: num] :
% 9.03/4.94            ( ( ord_less_num @ A3 @ B4 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1499_order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_nat
% 9.03/4.94      = ( ^ [A3: nat,B4: nat] :
% 9.03/4.94            ( ( ord_less_nat @ A3 @ B4 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1500_order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_int
% 9.03/4.94      = ( ^ [A3: int,B4: int] :
% 9.03/4.94            ( ( ord_less_int @ A3 @ B4 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1501_order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_real
% 9.03/4.94      = ( ^ [A3: real,B4: real] :
% 9.03/4.94            ( ( ord_less_eq_real @ A3 @ B4 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1502_order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_set_int
% 9.03/4.94      = ( ^ [A3: set_int,B4: set_int] :
% 9.03/4.94            ( ( ord_less_eq_set_int @ A3 @ B4 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1503_order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_rat
% 9.03/4.94      = ( ^ [A3: rat,B4: rat] :
% 9.03/4.94            ( ( ord_less_eq_rat @ A3 @ B4 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1504_order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_num
% 9.03/4.94      = ( ^ [A3: num,B4: num] :
% 9.03/4.94            ( ( ord_less_eq_num @ A3 @ B4 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1505_order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_nat
% 9.03/4.94      = ( ^ [A3: nat,B4: nat] :
% 9.03/4.94            ( ( ord_less_eq_nat @ A3 @ B4 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1506_order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_int
% 9.03/4.94      = ( ^ [A3: int,B4: int] :
% 9.03/4.94            ( ( ord_less_eq_int @ A3 @ B4 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1507_order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [A: real,B: real,C: real] :
% 9.03/4.94        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ord_less_real @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1508_order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [A: set_int,B: set_int,C: set_int] :
% 9.03/4.94        ( ( ord_less_eq_set_int @ A @ B )
% 9.03/4.94       => ( ( ord_less_set_int @ B @ C )
% 9.03/4.94         => ( ord_less_set_int @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1509_order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [A: rat,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1510_order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [A: num,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.94       => ( ( ord_less_num @ B @ C )
% 9.03/4.94         => ( ord_less_num @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1511_order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [A: nat,B: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.94       => ( ( ord_less_nat @ B @ C )
% 9.03/4.94         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1512_order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [A: int,B: int,C: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.94       => ( ( ord_less_int @ B @ C )
% 9.03/4.94         => ( ord_less_int @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1513_order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [A: real,B: real,C: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_real @ B @ C )
% 9.03/4.94         => ( ord_less_real @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1514_order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [A: set_int,B: set_int,C: set_int] :
% 9.03/4.94        ( ( ord_less_set_int @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_set_int @ B @ C )
% 9.03/4.94         => ( ord_less_set_int @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1515_order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.94         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1516_order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [A: num,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_num @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.94         => ( ord_less_num @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1517_order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [A: nat,B: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_nat @ B @ C )
% 9.03/4.94         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1518_order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [A: int,B: int,C: int] :
% 9.03/4.94        ( ( ord_less_int @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_int @ B @ C )
% 9.03/4.94         => ( ord_less_int @ A @ C ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1519_order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_real
% 9.03/4.94      = ( ^ [A3: real,B4: real] :
% 9.03/4.94            ( ( ord_less_eq_real @ A3 @ B4 )
% 9.03/4.94            & ~ ( ord_less_eq_real @ B4 @ A3 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1520_order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_set_int
% 9.03/4.94      = ( ^ [A3: set_int,B4: set_int] :
% 9.03/4.94            ( ( ord_less_eq_set_int @ A3 @ B4 )
% 9.03/4.94            & ~ ( ord_less_eq_set_int @ B4 @ A3 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1521_order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_rat
% 9.03/4.94      = ( ^ [A3: rat,B4: rat] :
% 9.03/4.94            ( ( ord_less_eq_rat @ A3 @ B4 )
% 9.03/4.94            & ~ ( ord_less_eq_rat @ B4 @ A3 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1522_order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_num
% 9.03/4.94      = ( ^ [A3: num,B4: num] :
% 9.03/4.94            ( ( ord_less_eq_num @ A3 @ B4 )
% 9.03/4.94            & ~ ( ord_less_eq_num @ B4 @ A3 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1523_order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_nat
% 9.03/4.94      = ( ^ [A3: nat,B4: nat] :
% 9.03/4.94            ( ( ord_less_eq_nat @ A3 @ B4 )
% 9.03/4.94            & ~ ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1524_order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_int
% 9.03/4.94      = ( ^ [A3: int,B4: int] :
% 9.03/4.94            ( ( ord_less_eq_int @ A3 @ B4 )
% 9.03/4.94            & ~ ( ord_less_eq_int @ B4 @ A3 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1525_dense__ge__bounded,axiom,
% 9.03/4.94      ! [Z: real,X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_real @ Z @ X )
% 9.03/4.94       => ( ! [W3: real] :
% 9.03/4.94              ( ( ord_less_real @ Z @ W3 )
% 9.03/4.94             => ( ( ord_less_real @ W3 @ X )
% 9.03/4.94               => ( ord_less_eq_real @ Y @ W3 ) ) )
% 9.03/4.94         => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1526_dense__ge__bounded,axiom,
% 9.03/4.94      ! [Z: rat,X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_rat @ Z @ X )
% 9.03/4.94       => ( ! [W3: rat] :
% 9.03/4.94              ( ( ord_less_rat @ Z @ W3 )
% 9.03/4.94             => ( ( ord_less_rat @ W3 @ X )
% 9.03/4.94               => ( ord_less_eq_rat @ Y @ W3 ) ) )
% 9.03/4.94         => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1527_dense__le__bounded,axiom,
% 9.03/4.94      ! [X: real,Y: real,Z: real] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94       => ( ! [W3: real] :
% 9.03/4.94              ( ( ord_less_real @ X @ W3 )
% 9.03/4.94             => ( ( ord_less_real @ W3 @ Y )
% 9.03/4.94               => ( ord_less_eq_real @ W3 @ Z ) ) )
% 9.03/4.94         => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1528_dense__le__bounded,axiom,
% 9.03/4.94      ! [X: rat,Y: rat,Z: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( ! [W3: rat] :
% 9.03/4.94              ( ( ord_less_rat @ X @ W3 )
% 9.03/4.94             => ( ( ord_less_rat @ W3 @ Y )
% 9.03/4.94               => ( ord_less_eq_rat @ W3 @ Z ) ) )
% 9.03/4.94         => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1529_dual__order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_real
% 9.03/4.94      = ( ^ [B4: real,A3: real] :
% 9.03/4.94            ( ( ord_less_real @ B4 @ A3 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1530_dual__order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_set_int
% 9.03/4.94      = ( ^ [B4: set_int,A3: set_int] :
% 9.03/4.94            ( ( ord_less_set_int @ B4 @ A3 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1531_dual__order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_rat
% 9.03/4.94      = ( ^ [B4: rat,A3: rat] :
% 9.03/4.94            ( ( ord_less_rat @ B4 @ A3 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1532_dual__order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_num
% 9.03/4.94      = ( ^ [B4: num,A3: num] :
% 9.03/4.94            ( ( ord_less_num @ B4 @ A3 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1533_dual__order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_nat
% 9.03/4.94      = ( ^ [B4: nat,A3: nat] :
% 9.03/4.94            ( ( ord_less_nat @ B4 @ A3 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1534_dual__order_Oorder__iff__strict,axiom,
% 9.03/4.94      ( ord_less_eq_int
% 9.03/4.94      = ( ^ [B4: int,A3: int] :
% 9.03/4.94            ( ( ord_less_int @ B4 @ A3 )
% 9.03/4.94            | ( A3 = B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1535_dual__order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_real
% 9.03/4.94      = ( ^ [B4: real,A3: real] :
% 9.03/4.94            ( ( ord_less_eq_real @ B4 @ A3 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1536_dual__order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_set_int
% 9.03/4.94      = ( ^ [B4: set_int,A3: set_int] :
% 9.03/4.94            ( ( ord_less_eq_set_int @ B4 @ A3 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1537_dual__order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_rat
% 9.03/4.94      = ( ^ [B4: rat,A3: rat] :
% 9.03/4.94            ( ( ord_less_eq_rat @ B4 @ A3 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1538_dual__order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_num
% 9.03/4.94      = ( ^ [B4: num,A3: num] :
% 9.03/4.94            ( ( ord_less_eq_num @ B4 @ A3 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1539_dual__order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_nat
% 9.03/4.94      = ( ^ [B4: nat,A3: nat] :
% 9.03/4.94            ( ( ord_less_eq_nat @ B4 @ A3 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1540_dual__order_Ostrict__iff__order,axiom,
% 9.03/4.94      ( ord_less_int
% 9.03/4.94      = ( ^ [B4: int,A3: int] :
% 9.03/4.94            ( ( ord_less_eq_int @ B4 @ A3 )
% 9.03/4.94            & ( A3 != B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1541_dual__order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [B: real,A: real,C: real] :
% 9.03/4.94        ( ( ord_less_eq_real @ B @ A )
% 9.03/4.94       => ( ( ord_less_real @ C @ B )
% 9.03/4.94         => ( ord_less_real @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1542_dual__order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [B: set_int,A: set_int,C: set_int] :
% 9.03/4.94        ( ( ord_less_eq_set_int @ B @ A )
% 9.03/4.94       => ( ( ord_less_set_int @ C @ B )
% 9.03/4.94         => ( ord_less_set_int @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1543_dual__order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [B: rat,A: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.94       => ( ( ord_less_rat @ C @ B )
% 9.03/4.94         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1544_dual__order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [B: num,A: num,C: num] :
% 9.03/4.94        ( ( ord_less_eq_num @ B @ A )
% 9.03/4.94       => ( ( ord_less_num @ C @ B )
% 9.03/4.94         => ( ord_less_num @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1545_dual__order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [B: nat,A: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_eq_nat @ B @ A )
% 9.03/4.94       => ( ( ord_less_nat @ C @ B )
% 9.03/4.94         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1546_dual__order_Ostrict__trans1,axiom,
% 9.03/4.94      ! [B: int,A: int,C: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ B @ A )
% 9.03/4.94       => ( ( ord_less_int @ C @ B )
% 9.03/4.94         => ( ord_less_int @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1547_dual__order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [B: real,A: real,C: real] :
% 9.03/4.94        ( ( ord_less_real @ B @ A )
% 9.03/4.94       => ( ( ord_less_eq_real @ C @ B )
% 9.03/4.94         => ( ord_less_real @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1548_dual__order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [B: set_int,A: set_int,C: set_int] :
% 9.03/4.94        ( ( ord_less_set_int @ B @ A )
% 9.03/4.94       => ( ( ord_less_eq_set_int @ C @ B )
% 9.03/4.94         => ( ord_less_set_int @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1549_dual__order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [B: rat,A: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ B @ A )
% 9.03/4.94       => ( ( ord_less_eq_rat @ C @ B )
% 9.03/4.94         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1550_dual__order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [B: num,A: num,C: num] :
% 9.03/4.94        ( ( ord_less_num @ B @ A )
% 9.03/4.94       => ( ( ord_less_eq_num @ C @ B )
% 9.03/4.94         => ( ord_less_num @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1551_dual__order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [B: nat,A: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_nat @ B @ A )
% 9.03/4.94       => ( ( ord_less_eq_nat @ C @ B )
% 9.03/4.94         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1552_dual__order_Ostrict__trans2,axiom,
% 9.03/4.94      ! [B: int,A: int,C: int] :
% 9.03/4.94        ( ( ord_less_int @ B @ A )
% 9.03/4.94       => ( ( ord_less_eq_int @ C @ B )
% 9.03/4.94         => ( ord_less_int @ C @ A ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1553_dual__order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_real
% 9.03/4.94      = ( ^ [B4: real,A3: real] :
% 9.03/4.94            ( ( ord_less_eq_real @ B4 @ A3 )
% 9.03/4.94            & ~ ( ord_less_eq_real @ A3 @ B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1554_dual__order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_set_int
% 9.03/4.94      = ( ^ [B4: set_int,A3: set_int] :
% 9.03/4.94            ( ( ord_less_eq_set_int @ B4 @ A3 )
% 9.03/4.94            & ~ ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1555_dual__order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_rat
% 9.03/4.94      = ( ^ [B4: rat,A3: rat] :
% 9.03/4.94            ( ( ord_less_eq_rat @ B4 @ A3 )
% 9.03/4.94            & ~ ( ord_less_eq_rat @ A3 @ B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1556_dual__order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_num
% 9.03/4.94      = ( ^ [B4: num,A3: num] :
% 9.03/4.94            ( ( ord_less_eq_num @ B4 @ A3 )
% 9.03/4.94            & ~ ( ord_less_eq_num @ A3 @ B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1557_dual__order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_nat
% 9.03/4.94      = ( ^ [B4: nat,A3: nat] :
% 9.03/4.94            ( ( ord_less_eq_nat @ B4 @ A3 )
% 9.03/4.94            & ~ ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1558_dual__order_Ostrict__iff__not,axiom,
% 9.03/4.94      ( ord_less_int
% 9.03/4.94      = ( ^ [B4: int,A3: int] :
% 9.03/4.94            ( ( ord_less_eq_int @ B4 @ A3 )
% 9.03/4.94            & ~ ( ord_less_eq_int @ A3 @ B4 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1559_order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [A: real,B: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1560_order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [A: set_int,B: set_int] :
% 9.03/4.94        ( ( ord_less_set_int @ A @ B )
% 9.03/4.94       => ( ord_less_eq_set_int @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1561_order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [A: rat,B: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1562_order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [A: num,B: num] :
% 9.03/4.94        ( ( ord_less_num @ A @ B )
% 9.03/4.94       => ( ord_less_eq_num @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1563_order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [A: nat,B: nat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ( ord_less_eq_nat @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1564_order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [A: int,B: int] :
% 9.03/4.94        ( ( ord_less_int @ A @ B )
% 9.03/4.94       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1565_dual__order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [B: real,A: real] :
% 9.03/4.94        ( ( ord_less_real @ B @ A )
% 9.03/4.94       => ( ord_less_eq_real @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1566_dual__order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [B: set_int,A: set_int] :
% 9.03/4.94        ( ( ord_less_set_int @ B @ A )
% 9.03/4.94       => ( ord_less_eq_set_int @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1567_dual__order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [B: rat,A: rat] :
% 9.03/4.94        ( ( ord_less_rat @ B @ A )
% 9.03/4.94       => ( ord_less_eq_rat @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1568_dual__order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [B: num,A: num] :
% 9.03/4.94        ( ( ord_less_num @ B @ A )
% 9.03/4.94       => ( ord_less_eq_num @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1569_dual__order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [B: nat,A: nat] :
% 9.03/4.94        ( ( ord_less_nat @ B @ A )
% 9.03/4.94       => ( ord_less_eq_nat @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1570_dual__order_Ostrict__implies__order,axiom,
% 9.03/4.94      ! [B: int,A: int] :
% 9.03/4.94        ( ( ord_less_int @ B @ A )
% 9.03/4.94       => ( ord_less_eq_int @ B @ A ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1571_order__le__less,axiom,
% 9.03/4.94      ( ord_less_eq_real
% 9.03/4.94      = ( ^ [X4: real,Y6: real] :
% 9.03/4.94            ( ( ord_less_real @ X4 @ Y6 )
% 9.03/4.94            | ( X4 = Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1572_order__le__less,axiom,
% 9.03/4.94      ( ord_less_eq_set_int
% 9.03/4.94      = ( ^ [X4: set_int,Y6: set_int] :
% 9.03/4.94            ( ( ord_less_set_int @ X4 @ Y6 )
% 9.03/4.94            | ( X4 = Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1573_order__le__less,axiom,
% 9.03/4.94      ( ord_less_eq_rat
% 9.03/4.94      = ( ^ [X4: rat,Y6: rat] :
% 9.03/4.94            ( ( ord_less_rat @ X4 @ Y6 )
% 9.03/4.94            | ( X4 = Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1574_order__le__less,axiom,
% 9.03/4.94      ( ord_less_eq_num
% 9.03/4.94      = ( ^ [X4: num,Y6: num] :
% 9.03/4.94            ( ( ord_less_num @ X4 @ Y6 )
% 9.03/4.94            | ( X4 = Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1575_order__le__less,axiom,
% 9.03/4.94      ( ord_less_eq_nat
% 9.03/4.94      = ( ^ [X4: nat,Y6: nat] :
% 9.03/4.94            ( ( ord_less_nat @ X4 @ Y6 )
% 9.03/4.94            | ( X4 = Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1576_order__le__less,axiom,
% 9.03/4.94      ( ord_less_eq_int
% 9.03/4.94      = ( ^ [X4: int,Y6: int] :
% 9.03/4.94            ( ( ord_less_int @ X4 @ Y6 )
% 9.03/4.94            | ( X4 = Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1577_order__less__le,axiom,
% 9.03/4.94      ( ord_less_real
% 9.03/4.94      = ( ^ [X4: real,Y6: real] :
% 9.03/4.94            ( ( ord_less_eq_real @ X4 @ Y6 )
% 9.03/4.94            & ( X4 != Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1578_order__less__le,axiom,
% 9.03/4.94      ( ord_less_set_int
% 9.03/4.94      = ( ^ [X4: set_int,Y6: set_int] :
% 9.03/4.94            ( ( ord_less_eq_set_int @ X4 @ Y6 )
% 9.03/4.94            & ( X4 != Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1579_order__less__le,axiom,
% 9.03/4.94      ( ord_less_rat
% 9.03/4.94      = ( ^ [X4: rat,Y6: rat] :
% 9.03/4.94            ( ( ord_less_eq_rat @ X4 @ Y6 )
% 9.03/4.94            & ( X4 != Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1580_order__less__le,axiom,
% 9.03/4.94      ( ord_less_num
% 9.03/4.94      = ( ^ [X4: num,Y6: num] :
% 9.03/4.94            ( ( ord_less_eq_num @ X4 @ Y6 )
% 9.03/4.94            & ( X4 != Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1581_order__less__le,axiom,
% 9.03/4.94      ( ord_less_nat
% 9.03/4.94      = ( ^ [X4: nat,Y6: nat] :
% 9.03/4.94            ( ( ord_less_eq_nat @ X4 @ Y6 )
% 9.03/4.94            & ( X4 != Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1582_order__less__le,axiom,
% 9.03/4.94      ( ord_less_int
% 9.03/4.94      = ( ^ [X4: int,Y6: int] :
% 9.03/4.94            ( ( ord_less_eq_int @ X4 @ Y6 )
% 9.03/4.94            & ( X4 != Y6 ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1583_linorder__not__le,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ~ ( ord_less_eq_real @ X @ Y ) )
% 9.03/4.94        = ( ord_less_real @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1584_linorder__not__le,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
% 9.03/4.94        = ( ord_less_rat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1585_linorder__not__le,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ~ ( ord_less_eq_num @ X @ Y ) )
% 9.03/4.94        = ( ord_less_num @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1586_linorder__not__le,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
% 9.03/4.94        = ( ord_less_nat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1587_linorder__not__le,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ~ ( ord_less_eq_int @ X @ Y ) )
% 9.03/4.94        = ( ord_less_int @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1588_linorder__not__less,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ~ ( ord_less_real @ X @ Y ) )
% 9.03/4.94        = ( ord_less_eq_real @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1589_linorder__not__less,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ~ ( ord_less_rat @ X @ Y ) )
% 9.03/4.94        = ( ord_less_eq_rat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1590_linorder__not__less,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ~ ( ord_less_num @ X @ Y ) )
% 9.03/4.94        = ( ord_less_eq_num @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1591_linorder__not__less,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ~ ( ord_less_nat @ X @ Y ) )
% 9.03/4.94        = ( ord_less_eq_nat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1592_linorder__not__less,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ~ ( ord_less_int @ X @ Y ) )
% 9.03/4.94        = ( ord_less_eq_int @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1593_order__less__imp__le,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_real @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1594_order__less__imp__le,axiom,
% 9.03/4.94      ! [X: set_int,Y: set_int] :
% 9.03/4.94        ( ( ord_less_set_int @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_set_int @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1595_order__less__imp__le,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1596_order__less__imp__le,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_num @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1597_order__less__imp__le,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_nat @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1598_order__less__imp__le,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94       => ( ord_less_eq_int @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1599_order__le__neq__trans,axiom,
% 9.03/4.94      ! [A: real,B: real] :
% 9.03/4.94        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.94       => ( ( A != B )
% 9.03/4.94         => ( ord_less_real @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1600_order__le__neq__trans,axiom,
% 9.03/4.94      ! [A: set_int,B: set_int] :
% 9.03/4.94        ( ( ord_less_eq_set_int @ A @ B )
% 9.03/4.94       => ( ( A != B )
% 9.03/4.94         => ( ord_less_set_int @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1601_order__le__neq__trans,axiom,
% 9.03/4.94      ! [A: rat,B: rat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.94       => ( ( A != B )
% 9.03/4.94         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1602_order__le__neq__trans,axiom,
% 9.03/4.94      ! [A: num,B: num] :
% 9.03/4.94        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.94       => ( ( A != B )
% 9.03/4.94         => ( ord_less_num @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1603_order__le__neq__trans,axiom,
% 9.03/4.94      ! [A: nat,B: nat] :
% 9.03/4.94        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.94       => ( ( A != B )
% 9.03/4.94         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1604_order__le__neq__trans,axiom,
% 9.03/4.94      ! [A: int,B: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.94       => ( ( A != B )
% 9.03/4.94         => ( ord_less_int @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1605_order__neq__le__trans,axiom,
% 9.03/4.94      ! [A: real,B: real] :
% 9.03/4.94        ( ( A != B )
% 9.03/4.94       => ( ( ord_less_eq_real @ A @ B )
% 9.03/4.94         => ( ord_less_real @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1606_order__neq__le__trans,axiom,
% 9.03/4.94      ! [A: set_int,B: set_int] :
% 9.03/4.94        ( ( A != B )
% 9.03/4.94       => ( ( ord_less_eq_set_int @ A @ B )
% 9.03/4.94         => ( ord_less_set_int @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1607_order__neq__le__trans,axiom,
% 9.03/4.94      ! [A: rat,B: rat] :
% 9.03/4.94        ( ( A != B )
% 9.03/4.94       => ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.94         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1608_order__neq__le__trans,axiom,
% 9.03/4.94      ! [A: num,B: num] :
% 9.03/4.94        ( ( A != B )
% 9.03/4.94       => ( ( ord_less_eq_num @ A @ B )
% 9.03/4.94         => ( ord_less_num @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1609_order__neq__le__trans,axiom,
% 9.03/4.94      ! [A: nat,B: nat] :
% 9.03/4.94        ( ( A != B )
% 9.03/4.94       => ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.94         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1610_order__neq__le__trans,axiom,
% 9.03/4.94      ! [A: int,B: int] :
% 9.03/4.94        ( ( A != B )
% 9.03/4.94       => ( ( ord_less_eq_int @ A @ B )
% 9.03/4.94         => ( ord_less_int @ A @ B ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1611_order__le__less__trans,axiom,
% 9.03/4.94      ! [X: real,Y: real,Z: real] :
% 9.03/4.94        ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.94       => ( ( ord_less_real @ Y @ Z )
% 9.03/4.94         => ( ord_less_real @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1612_order__le__less__trans,axiom,
% 9.03/4.94      ! [X: set_int,Y: set_int,Z: set_int] :
% 9.03/4.94        ( ( ord_less_eq_set_int @ X @ Y )
% 9.03/4.94       => ( ( ord_less_set_int @ Y @ Z )
% 9.03/4.94         => ( ord_less_set_int @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1613_order__le__less__trans,axiom,
% 9.03/4.94      ! [X: rat,Y: rat,Z: rat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_rat @ Y @ Z )
% 9.03/4.94         => ( ord_less_rat @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1614_order__le__less__trans,axiom,
% 9.03/4.94      ! [X: num,Y: num,Z: num] :
% 9.03/4.94        ( ( ord_less_eq_num @ X @ Y )
% 9.03/4.94       => ( ( ord_less_num @ Y @ Z )
% 9.03/4.94         => ( ord_less_num @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1615_order__le__less__trans,axiom,
% 9.03/4.94      ! [X: nat,Y: nat,Z: nat] :
% 9.03/4.94        ( ( ord_less_eq_nat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_nat @ Y @ Z )
% 9.03/4.94         => ( ord_less_nat @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1616_order__le__less__trans,axiom,
% 9.03/4.94      ! [X: int,Y: int,Z: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ X @ Y )
% 9.03/4.94       => ( ( ord_less_int @ Y @ Z )
% 9.03/4.94         => ( ord_less_int @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1617_order__less__le__trans,axiom,
% 9.03/4.94      ! [X: real,Y: real,Z: real] :
% 9.03/4.94        ( ( ord_less_real @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_real @ Y @ Z )
% 9.03/4.94         => ( ord_less_real @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1618_order__less__le__trans,axiom,
% 9.03/4.94      ! [X: set_int,Y: set_int,Z: set_int] :
% 9.03/4.94        ( ( ord_less_set_int @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_set_int @ Y @ Z )
% 9.03/4.94         => ( ord_less_set_int @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1619_order__less__le__trans,axiom,
% 9.03/4.94      ! [X: rat,Y: rat,Z: rat] :
% 9.03/4.94        ( ( ord_less_rat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_rat @ Y @ Z )
% 9.03/4.94         => ( ord_less_rat @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1620_order__less__le__trans,axiom,
% 9.03/4.94      ! [X: num,Y: num,Z: num] :
% 9.03/4.94        ( ( ord_less_num @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_num @ Y @ Z )
% 9.03/4.94         => ( ord_less_num @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1621_order__less__le__trans,axiom,
% 9.03/4.94      ! [X: nat,Y: nat,Z: nat] :
% 9.03/4.94        ( ( ord_less_nat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_nat @ Y @ Z )
% 9.03/4.94         => ( ord_less_nat @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1622_order__less__le__trans,axiom,
% 9.03/4.94      ! [X: int,Y: int,Z: int] :
% 9.03/4.94        ( ( ord_less_int @ X @ Y )
% 9.03/4.94       => ( ( ord_less_eq_int @ Y @ Z )
% 9.03/4.94         => ( ord_less_int @ X @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1623_order__le__less__subst1,axiom,
% 9.03/4.94      ! [A: real,F: real > real,B: real,C: real] :
% 9.03/4.94        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1624_order__le__less__subst1,axiom,
% 9.03/4.94      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1625_order__le__less__subst1,axiom,
% 9.03/4.94      ! [A: real,F: num > real,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_num @ B @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1626_order__le__less__subst1,axiom,
% 9.03/4.94      ! [A: real,F: nat > real,B: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_nat @ B @ C )
% 9.03/4.94         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.94                ( ( ord_less_nat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1627_order__le__less__subst1,axiom,
% 9.03/4.94      ! [A: real,F: int > real,B: int,C: int] :
% 9.03/4.94        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_int @ B @ C )
% 9.03/4.94         => ( ! [X3: int,Y4: int] :
% 9.03/4.94                ( ( ord_less_int @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1628_order__le__less__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: real > rat,B: real,C: real] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_real @ B @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1629_order__le__less__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1630_order__le__less__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_num @ B @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1631_order__le__less__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_nat @ B @ C )
% 9.03/4.94         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.94                ( ( ord_less_nat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1632_order__le__less__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: int > rat,B: int,C: int] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_int @ B @ C )
% 9.03/4.94         => ( ! [X3: int,Y4: int] :
% 9.03/4.94                ( ( ord_less_int @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1633_order__le__less__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1634_order__le__less__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1635_order__le__less__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1636_order__le__less__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1637_order__le__less__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.03/4.94        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1638_order__le__less__subst2,axiom,
% 9.03/4.94      ! [A: num,B: num,F: num > real,C: real] :
% 9.03/4.94        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.94       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1639_order__le__less__subst2,axiom,
% 9.03/4.94      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.94       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1640_order__le__less__subst2,axiom,
% 9.03/4.94      ! [A: num,B: num,F: num > num,C: num] :
% 9.03/4.94        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.94       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1641_order__le__less__subst2,axiom,
% 9.03/4.94      ! [A: num,B: num,F: num > nat,C: nat] :
% 9.03/4.94        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.94       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1642_order__le__less__subst2,axiom,
% 9.03/4.94      ! [A: num,B: num,F: num > int,C: int] :
% 9.03/4.94        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.94       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1643_order__less__le__subst1,axiom,
% 9.03/4.94      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1644_order__less__le__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1645_order__less__le__subst1,axiom,
% 9.03/4.94      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_num @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1646_order__less__le__subst1,axiom,
% 9.03/4.94      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1647_order__less__le__subst1,axiom,
% 9.03/4.94      ! [A: int,F: rat > int,B: rat,C: rat] :
% 9.03/4.94        ( ( ord_less_int @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1648_order__less__le__subst1,axiom,
% 9.03/4.94      ! [A: real,F: num > real,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1649_order__less__le__subst1,axiom,
% 9.03/4.94      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1650_order__less__le__subst1,axiom,
% 9.03/4.94      ! [A: num,F: num > num,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_num @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1651_order__less__le__subst1,axiom,
% 9.03/4.94      ! [A: nat,F: num > nat,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_nat @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1652_order__less__le__subst1,axiom,
% 9.03/4.94      ! [A: int,F: num > int,B: num,C: num] :
% 9.03/4.94        ( ( ord_less_int @ A @ ( F @ B ) )
% 9.03/4.94       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1653_order__less__le__subst2,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > real,C: real] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1654_order__less__le__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1655_order__less__le__subst2,axiom,
% 9.03/4.94      ! [A: num,B: num,F: num > real,C: real] :
% 9.03/4.94        ( ( ord_less_num @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1656_order__less__le__subst2,axiom,
% 9.03/4.94      ! [A: nat,B: nat,F: nat > real,C: real] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.94                ( ( ord_less_nat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1657_order__less__le__subst2,axiom,
% 9.03/4.94      ! [A: int,B: int,F: int > real,C: real] :
% 9.03/4.94        ( ( ord_less_int @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: int,Y4: int] :
% 9.03/4.94                ( ( ord_less_int @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1658_order__less__le__subst2,axiom,
% 9.03/4.94      ! [A: real,B: real,F: real > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_real @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: real,Y4: real] :
% 9.03/4.94                ( ( ord_less_real @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1659_order__less__le__subst2,axiom,
% 9.03/4.94      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_rat @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.94                ( ( ord_less_rat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1660_order__less__le__subst2,axiom,
% 9.03/4.94      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_num @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: num,Y4: num] :
% 9.03/4.94                ( ( ord_less_num @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1661_order__less__le__subst2,axiom,
% 9.03/4.94      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_nat @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.94                ( ( ord_less_nat @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1662_order__less__le__subst2,axiom,
% 9.03/4.94      ! [A: int,B: int,F: int > rat,C: rat] :
% 9.03/4.94        ( ( ord_less_int @ A @ B )
% 9.03/4.94       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.03/4.94         => ( ! [X3: int,Y4: int] :
% 9.03/4.94                ( ( ord_less_int @ X3 @ Y4 )
% 9.03/4.94               => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.94           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1663_linorder__le__less__linear,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.94        | ( ord_less_real @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1664_linorder__le__less__linear,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.94        | ( ord_less_rat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1665_linorder__le__less__linear,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_eq_num @ X @ Y )
% 9.03/4.94        | ( ord_less_num @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1666_linorder__le__less__linear,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_eq_nat @ X @ Y )
% 9.03/4.94        | ( ord_less_nat @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1667_linorder__le__less__linear,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ X @ Y )
% 9.03/4.94        | ( ord_less_int @ Y @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1668_order__le__imp__less__or__eq,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.94       => ( ( ord_less_real @ X @ Y )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1669_order__le__imp__less__or__eq,axiom,
% 9.03/4.94      ! [X: set_int,Y: set_int] :
% 9.03/4.94        ( ( ord_less_eq_set_int @ X @ Y )
% 9.03/4.94       => ( ( ord_less_set_int @ X @ Y )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1670_order__le__imp__less__or__eq,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_rat @ X @ Y )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1671_order__le__imp__less__or__eq,axiom,
% 9.03/4.94      ! [X: num,Y: num] :
% 9.03/4.94        ( ( ord_less_eq_num @ X @ Y )
% 9.03/4.94       => ( ( ord_less_num @ X @ Y )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1672_order__le__imp__less__or__eq,axiom,
% 9.03/4.94      ! [X: nat,Y: nat] :
% 9.03/4.94        ( ( ord_less_eq_nat @ X @ Y )
% 9.03/4.94       => ( ( ord_less_nat @ X @ Y )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1673_order__le__imp__less__or__eq,axiom,
% 9.03/4.94      ! [X: int,Y: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ X @ Y )
% 9.03/4.94       => ( ( ord_less_int @ X @ Y )
% 9.03/4.94          | ( X = Y ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1674_ceiling__less__cancel,axiom,
% 9.03/4.94      ! [X: real,Y: real] :
% 9.03/4.94        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
% 9.03/4.94       => ( ord_less_real @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1675_ceiling__less__cancel,axiom,
% 9.03/4.94      ! [X: rat,Y: rat] :
% 9.03/4.94        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
% 9.03/4.94       => ( ord_less_rat @ X @ Y ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1676_nat__power__less__imp__less,axiom,
% 9.03/4.94      ! [I: nat,M: nat,N: nat] :
% 9.03/4.94        ( ( ord_less_nat @ zero_zero_nat @ I )
% 9.03/4.94       => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 9.03/4.94         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1677_of__nat__less__of__int__iff,axiom,
% 9.03/4.94      ! [N: nat,X: int] :
% 9.03/4.94        ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( ring_18347121197199848620nteger @ X ) )
% 9.03/4.94        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1678_of__nat__less__of__int__iff,axiom,
% 9.03/4.94      ! [N: nat,X: int] :
% 9.03/4.94        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
% 9.03/4.94        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1679_of__nat__less__of__int__iff,axiom,
% 9.03/4.94      ! [N: nat,X: int] :
% 9.03/4.94        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
% 9.03/4.94        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1680_of__nat__less__of__int__iff,axiom,
% 9.03/4.94      ! [N: nat,X: int] :
% 9.03/4.94        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
% 9.03/4.94        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1681_power__strict__decreasing,axiom,
% 9.03/4.94      ! [N: nat,N4: nat,A: real] :
% 9.03/4.94        ( ( ord_less_nat @ N @ N4 )
% 9.03/4.94       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.94         => ( ( ord_less_real @ A @ one_one_real )
% 9.03/4.94           => ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1682_power__strict__decreasing,axiom,
% 9.03/4.94      ! [N: nat,N4: nat,A: rat] :
% 9.03/4.94        ( ( ord_less_nat @ N @ N4 )
% 9.03/4.94       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.94         => ( ( ord_less_rat @ A @ one_one_rat )
% 9.03/4.94           => ( ord_less_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1683_power__strict__decreasing,axiom,
% 9.03/4.94      ! [N: nat,N4: nat,A: nat] :
% 9.03/4.94        ( ( ord_less_nat @ N @ N4 )
% 9.03/4.94       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.94         => ( ( ord_less_nat @ A @ one_one_nat )
% 9.03/4.94           => ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1684_power__strict__decreasing,axiom,
% 9.03/4.94      ! [N: nat,N4: nat,A: int] :
% 9.03/4.94        ( ( ord_less_nat @ N @ N4 )
% 9.03/4.94       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.94         => ( ( ord_less_int @ A @ one_one_int )
% 9.03/4.94           => ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1685_length__induct,axiom,
% 9.03/4.94      ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.03/4.94        ( ! [Xs2: list_VEBT_VEBT] :
% 9.03/4.94            ( ! [Ys: list_VEBT_VEBT] :
% 9.03/4.94                ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.03/4.94               => ( P @ Ys ) )
% 9.03/4.94           => ( P @ Xs2 ) )
% 9.03/4.94       => ( P @ Xs ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1686_length__induct,axiom,
% 9.03/4.94      ! [P: list_o > $o,Xs: list_o] :
% 9.03/4.94        ( ! [Xs2: list_o] :
% 9.03/4.94            ( ! [Ys: list_o] :
% 9.03/4.94                ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs2 ) )
% 9.03/4.94               => ( P @ Ys ) )
% 9.03/4.94           => ( P @ Xs2 ) )
% 9.03/4.94       => ( P @ Xs ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1687_length__induct,axiom,
% 9.03/4.94      ! [P: list_int > $o,Xs: list_int] :
% 9.03/4.94        ( ! [Xs2: list_int] :
% 9.03/4.94            ( ! [Ys: list_int] :
% 9.03/4.94                ( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs2 ) )
% 9.03/4.94               => ( P @ Ys ) )
% 9.03/4.94           => ( P @ Xs2 ) )
% 9.03/4.94       => ( P @ Xs ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1688_of__int__nonneg,axiom,
% 9.03/4.94      ! [Z: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.03/4.94       => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1689_of__int__nonneg,axiom,
% 9.03/4.94      ! [Z: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.03/4.94       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1690_of__int__nonneg,axiom,
% 9.03/4.94      ! [Z: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.03/4.94       => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1691_of__int__nonneg,axiom,
% 9.03/4.94      ! [Z: int] :
% 9.03/4.94        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.03/4.94       => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 9.03/4.94  
% 9.03/4.94  thf(fact_1692_one__less__power,axiom,
% 9.03/4.94      ! [A: real,N: nat] :
% 9.03/4.94        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.94       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.94         => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1693_one__less__power,axiom,
% 9.03/4.95      ! [A: rat,N: nat] :
% 9.03/4.95        ( ( ord_less_rat @ one_one_rat @ A )
% 9.03/4.95       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95         => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1694_one__less__power,axiom,
% 9.03/4.95      ! [A: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ one_one_nat @ A )
% 9.03/4.95       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95         => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1695_one__less__power,axiom,
% 9.03/4.95      ! [A: int,N: nat] :
% 9.03/4.95        ( ( ord_less_int @ one_one_int @ A )
% 9.03/4.95       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95         => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1696_power__eq__iff__eq__base,axiom,
% 9.03/4.95      ! [N: nat,A: real,B: real] :
% 9.03/4.95        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.95         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.95           => ( ( ( power_power_real @ A @ N )
% 9.03/4.95                = ( power_power_real @ B @ N ) )
% 9.03/4.95              = ( A = B ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1697_power__eq__iff__eq__base,axiom,
% 9.03/4.95      ! [N: nat,A: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.95         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.95           => ( ( ( power_power_rat @ A @ N )
% 9.03/4.95                = ( power_power_rat @ B @ N ) )
% 9.03/4.95              = ( A = B ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1698_power__eq__iff__eq__base,axiom,
% 9.03/4.95      ! [N: nat,A: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.95         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.95           => ( ( ( power_power_nat @ A @ N )
% 9.03/4.95                = ( power_power_nat @ B @ N ) )
% 9.03/4.95              = ( A = B ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1699_power__eq__iff__eq__base,axiom,
% 9.03/4.95      ! [N: nat,A: int,B: int] :
% 9.03/4.95        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.95         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.95           => ( ( ( power_power_int @ A @ N )
% 9.03/4.95                = ( power_power_int @ B @ N ) )
% 9.03/4.95              = ( A = B ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1700_power__eq__imp__eq__base,axiom,
% 9.03/4.95      ! [A: real,N: nat,B: real] :
% 9.03/4.95        ( ( ( power_power_real @ A @ N )
% 9.03/4.95          = ( power_power_real @ B @ N ) )
% 9.03/4.95       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.95         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.95           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95             => ( A = B ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1701_power__eq__imp__eq__base,axiom,
% 9.03/4.95      ! [A: rat,N: nat,B: rat] :
% 9.03/4.95        ( ( ( power_power_rat @ A @ N )
% 9.03/4.95          = ( power_power_rat @ B @ N ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.95         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.95           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95             => ( A = B ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1702_power__eq__imp__eq__base,axiom,
% 9.03/4.95      ! [A: nat,N: nat,B: nat] :
% 9.03/4.95        ( ( ( power_power_nat @ A @ N )
% 9.03/4.95          = ( power_power_nat @ B @ N ) )
% 9.03/4.95       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.95         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.95           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95             => ( A = B ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1703_power__eq__imp__eq__base,axiom,
% 9.03/4.95      ! [A: int,N: nat,B: int] :
% 9.03/4.95        ( ( ( power_power_int @ A @ N )
% 9.03/4.95          = ( power_power_int @ B @ N ) )
% 9.03/4.95       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.95         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.95           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95             => ( A = B ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1704_length__pos__if__in__set,axiom,
% 9.03/4.95      ! [X: complex,Xs: list_complex] :
% 9.03/4.95        ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 9.03/4.95       => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1705_length__pos__if__in__set,axiom,
% 9.03/4.95      ! [X: real,Xs: list_real] :
% 9.03/4.95        ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.03/4.95       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1706_length__pos__if__in__set,axiom,
% 9.03/4.95      ! [X: set_nat,Xs: list_set_nat] :
% 9.03/4.95        ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.03/4.95       => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1707_length__pos__if__in__set,axiom,
% 9.03/4.95      ! [X: nat,Xs: list_nat] :
% 9.03/4.95        ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.03/4.95       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1708_length__pos__if__in__set,axiom,
% 9.03/4.95      ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.03/4.95        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.03/4.95       => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1709_length__pos__if__in__set,axiom,
% 9.03/4.95      ! [X: $o,Xs: list_o] :
% 9.03/4.95        ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.03/4.95       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1710_length__pos__if__in__set,axiom,
% 9.03/4.95      ! [X: int,Xs: list_int] :
% 9.03/4.95        ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.03/4.95       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1711_half__gt__zero__iff,axiom,
% 9.03/4.95      ! [A: real] :
% 9.03/4.95        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.03/4.95        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1712_half__gt__zero__iff,axiom,
% 9.03/4.95      ! [A: rat] :
% 9.03/4.95        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.03/4.95        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1713_half__gt__zero,axiom,
% 9.03/4.95      ! [A: real] :
% 9.03/4.95        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.95       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1714_half__gt__zero,axiom,
% 9.03/4.95      ! [A: rat] :
% 9.03/4.95        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.95       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1715_nat__approx__posE,axiom,
% 9.03/4.95      ! [E2: real] :
% 9.03/4.95        ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.03/4.95       => ~ ! [N2: nat] :
% 9.03/4.95              ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1716_nat__approx__posE,axiom,
% 9.03/4.95      ! [E2: rat] :
% 9.03/4.95        ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 9.03/4.95       => ~ ! [N2: nat] :
% 9.03/4.95              ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1717_power__strict__increasing,axiom,
% 9.03/4.95      ! [N: nat,N4: nat,A: real] :
% 9.03/4.95        ( ( ord_less_nat @ N @ N4 )
% 9.03/4.95       => ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.95         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1718_power__strict__increasing,axiom,
% 9.03/4.95      ! [N: nat,N4: nat,A: rat] :
% 9.03/4.95        ( ( ord_less_nat @ N @ N4 )
% 9.03/4.95       => ( ( ord_less_rat @ one_one_rat @ A )
% 9.03/4.95         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1719_power__strict__increasing,axiom,
% 9.03/4.95      ! [N: nat,N4: nat,A: nat] :
% 9.03/4.95        ( ( ord_less_nat @ N @ N4 )
% 9.03/4.95       => ( ( ord_less_nat @ one_one_nat @ A )
% 9.03/4.95         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1720_power__strict__increasing,axiom,
% 9.03/4.95      ! [N: nat,N4: nat,A: int] :
% 9.03/4.95        ( ( ord_less_nat @ N @ N4 )
% 9.03/4.95       => ( ( ord_less_int @ one_one_int @ A )
% 9.03/4.95         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1721_power__less__imp__less__exp,axiom,
% 9.03/4.95      ! [A: real,M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.95       => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 9.03/4.95         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1722_power__less__imp__less__exp,axiom,
% 9.03/4.95      ! [A: rat,M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_rat @ one_one_rat @ A )
% 9.03/4.95       => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 9.03/4.95         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1723_power__less__imp__less__exp,axiom,
% 9.03/4.95      ! [A: nat,M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ one_one_nat @ A )
% 9.03/4.95       => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.03/4.95         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1724_power__less__imp__less__exp,axiom,
% 9.03/4.95      ! [A: int,M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_int @ one_one_int @ A )
% 9.03/4.95       => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.03/4.95         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1725_add__strict__increasing2,axiom,
% 9.03/4.95      ! [A: real,B: real,C: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.95       => ( ( ord_less_real @ B @ C )
% 9.03/4.95         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1726_add__strict__increasing2,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.95       => ( ( ord_less_rat @ B @ C )
% 9.03/4.95         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1727_add__strict__increasing2,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.95       => ( ( ord_less_nat @ B @ C )
% 9.03/4.95         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1728_add__strict__increasing2,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.95       => ( ( ord_less_int @ B @ C )
% 9.03/4.95         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1729_add__strict__increasing,axiom,
% 9.03/4.95      ! [A: real,B: real,C: real] :
% 9.03/4.95        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.95       => ( ( ord_less_eq_real @ B @ C )
% 9.03/4.95         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1730_add__strict__increasing,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1731_add__strict__increasing,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat] :
% 9.03/4.95        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ C )
% 9.03/4.95         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1732_add__strict__increasing,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int] :
% 9.03/4.95        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.95       => ( ( ord_less_eq_int @ B @ C )
% 9.03/4.95         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1733_add__pos__nonneg,axiom,
% 9.03/4.95      ! [A: real,B: real] :
% 9.03/4.95        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.95       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.95         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1734_add__pos__nonneg,axiom,
% 9.03/4.95      ! [A: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.95       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.95         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1735_add__pos__nonneg,axiom,
% 9.03/4.95      ! [A: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.95       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.95         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1736_add__pos__nonneg,axiom,
% 9.03/4.95      ! [A: int,B: int] :
% 9.03/4.95        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.95       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.95         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1737_add__nonpos__neg,axiom,
% 9.03/4.95      ! [A: real,B: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.95       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.03/4.95         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1738_add__nonpos__neg,axiom,
% 9.03/4.95      ! [A: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.95       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.03/4.95         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1739_add__nonpos__neg,axiom,
% 9.03/4.95      ! [A: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.03/4.95       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.03/4.95         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1740_add__nonpos__neg,axiom,
% 9.03/4.95      ! [A: int,B: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.95       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.03/4.95         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1741_add__nonneg__pos,axiom,
% 9.03/4.95      ! [A: real,B: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.95       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.95         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1742_add__nonneg__pos,axiom,
% 9.03/4.95      ! [A: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.95       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.03/4.95         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1743_add__nonneg__pos,axiom,
% 9.03/4.95      ! [A: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.95       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.03/4.95         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1744_add__nonneg__pos,axiom,
% 9.03/4.95      ! [A: int,B: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.95       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.95         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1745_add__neg__nonpos,axiom,
% 9.03/4.95      ! [A: real,B: real] :
% 9.03/4.95        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.95       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.03/4.95         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1746_add__neg__nonpos,axiom,
% 9.03/4.95      ! [A: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.03/4.95         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1747_add__neg__nonpos,axiom,
% 9.03/4.95      ! [A: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.03/4.95         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1748_add__neg__nonpos,axiom,
% 9.03/4.95      ! [A: int,B: int] :
% 9.03/4.95        ( ( ord_less_int @ A @ zero_zero_int )
% 9.03/4.95       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.03/4.95         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1749_power__less__imp__less__base,axiom,
% 9.03/4.95      ! [A: real,N: nat,B: real] :
% 9.03/4.95        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 9.03/4.95       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.95         => ( ord_less_real @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1750_power__less__imp__less__base,axiom,
% 9.03/4.95      ! [A: rat,N: nat,B: rat] :
% 9.03/4.95        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.95         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1751_power__less__imp__less__base,axiom,
% 9.03/4.95      ! [A: nat,N: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 9.03/4.95       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.95         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1752_power__less__imp__less__base,axiom,
% 9.03/4.95      ! [A: int,N: nat,B: int] :
% 9.03/4.95        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 9.03/4.95       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.95         => ( ord_less_int @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1753_ex__least__nat__less,axiom,
% 9.03/4.95      ! [P: nat > $o,N: nat] :
% 9.03/4.95        ( ( P @ N )
% 9.03/4.95       => ( ~ ( P @ zero_zero_nat )
% 9.03/4.95         => ? [K3: nat] :
% 9.03/4.95              ( ( ord_less_nat @ K3 @ N )
% 9.03/4.95              & ! [I4: nat] :
% 9.03/4.95                  ( ( ord_less_eq_nat @ I4 @ K3 )
% 9.03/4.95                 => ~ ( P @ I4 ) )
% 9.03/4.95              & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1754_power__0__left,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( ( N = zero_zero_nat )
% 9.03/4.95         => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.03/4.95            = one_one_rat ) )
% 9.03/4.95        & ( ( N != zero_zero_nat )
% 9.03/4.95         => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.03/4.95            = zero_zero_rat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1755_power__0__left,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( ( N = zero_zero_nat )
% 9.03/4.95         => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.03/4.95            = one_one_nat ) )
% 9.03/4.95        & ( ( N != zero_zero_nat )
% 9.03/4.95         => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.03/4.95            = zero_zero_nat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1756_power__0__left,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( ( N = zero_zero_nat )
% 9.03/4.95         => ( ( power_power_real @ zero_zero_real @ N )
% 9.03/4.95            = one_one_real ) )
% 9.03/4.95        & ( ( N != zero_zero_nat )
% 9.03/4.95         => ( ( power_power_real @ zero_zero_real @ N )
% 9.03/4.95            = zero_zero_real ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1757_power__0__left,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( ( N = zero_zero_nat )
% 9.03/4.95         => ( ( power_power_int @ zero_zero_int @ N )
% 9.03/4.95            = one_one_int ) )
% 9.03/4.95        & ( ( N != zero_zero_nat )
% 9.03/4.95         => ( ( power_power_int @ zero_zero_int @ N )
% 9.03/4.95            = zero_zero_int ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1758_power__0__left,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( ( N = zero_zero_nat )
% 9.03/4.95         => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.03/4.95            = one_one_complex ) )
% 9.03/4.95        & ( ( N != zero_zero_nat )
% 9.03/4.95         => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.03/4.95            = zero_zero_complex ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1759_ex__le__of__int,axiom,
% 9.03/4.95      ! [X: real] :
% 9.03/4.95      ? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1760_ex__le__of__int,axiom,
% 9.03/4.95      ! [X: rat] :
% 9.03/4.95      ? [Z3: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1761_nat__induct__non__zero,axiom,
% 9.03/4.95      ! [N: nat,P: nat > $o] :
% 9.03/4.95        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95       => ( ( P @ one_one_nat )
% 9.03/4.95         => ( ! [N2: nat] :
% 9.03/4.95                ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.03/4.95               => ( ( P @ N2 )
% 9.03/4.95                 => ( P @ ( suc @ N2 ) ) ) )
% 9.03/4.95           => ( P @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1762_power__gt__expt,axiom,
% 9.03/4.95      ! [N: nat,K: nat] :
% 9.03/4.95        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.03/4.95       => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1763_verit__comp__simplify1_I3_J,axiom,
% 9.03/4.95      ! [B7: real,A6: real] :
% 9.03/4.95        ( ( ~ ( ord_less_eq_real @ B7 @ A6 ) )
% 9.03/4.95        = ( ord_less_real @ A6 @ B7 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1764_verit__comp__simplify1_I3_J,axiom,
% 9.03/4.95      ! [B7: rat,A6: rat] :
% 9.03/4.95        ( ( ~ ( ord_less_eq_rat @ B7 @ A6 ) )
% 9.03/4.95        = ( ord_less_rat @ A6 @ B7 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1765_verit__comp__simplify1_I3_J,axiom,
% 9.03/4.95      ! [B7: num,A6: num] :
% 9.03/4.95        ( ( ~ ( ord_less_eq_num @ B7 @ A6 ) )
% 9.03/4.95        = ( ord_less_num @ A6 @ B7 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1766_verit__comp__simplify1_I3_J,axiom,
% 9.03/4.95      ! [B7: nat,A6: nat] :
% 9.03/4.95        ( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
% 9.03/4.95        = ( ord_less_nat @ A6 @ B7 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1767_verit__comp__simplify1_I3_J,axiom,
% 9.03/4.95      ! [B7: int,A6: int] :
% 9.03/4.95        ( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
% 9.03/4.95        = ( ord_less_int @ A6 @ B7 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1768_power__divide,axiom,
% 9.03/4.95      ! [A: complex,B: complex,N: nat] :
% 9.03/4.95        ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
% 9.03/4.95        = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1769_power__divide,axiom,
% 9.03/4.95      ! [A: real,B: real,N: nat] :
% 9.03/4.95        ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 9.03/4.95        = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1770_power__divide,axiom,
% 9.03/4.95      ! [A: rat,B: rat,N: nat] :
% 9.03/4.95        ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
% 9.03/4.95        = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1771_less__numeral__extra_I4_J,axiom,
% 9.03/4.95      ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1772_less__numeral__extra_I4_J,axiom,
% 9.03/4.95      ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1773_less__numeral__extra_I4_J,axiom,
% 9.03/4.95      ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1774_less__numeral__extra_I4_J,axiom,
% 9.03/4.95      ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1775_add__less__imp__less__right,axiom,
% 9.03/4.95      ! [A: real,C: real,B: real] :
% 9.03/4.95        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.03/4.95       => ( ord_less_real @ A @ B ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1776_add__less__imp__less__right,axiom,
% 9.03/4.95      ! [A: rat,C: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.03/4.95       => ( ord_less_rat @ A @ B ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1777_add__less__imp__less__right,axiom,
% 9.03/4.95      ! [A: nat,C: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.03/4.95       => ( ord_less_nat @ A @ B ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1778_add__less__imp__less__right,axiom,
% 9.03/4.95      ! [A: int,C: int,B: int] :
% 9.03/4.95        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.03/4.95       => ( ord_less_int @ A @ B ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1779_add__less__imp__less__left,axiom,
% 9.03/4.95      ! [C: real,A: real,B: real] :
% 9.03/4.95        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.03/4.95       => ( ord_less_real @ A @ B ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1780_add__less__imp__less__left,axiom,
% 9.03/4.95      ! [C: rat,A: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.03/4.95       => ( ord_less_rat @ A @ B ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1781_add__less__imp__less__left,axiom,
% 9.03/4.95      ! [C: nat,A: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.03/4.95       => ( ord_less_nat @ A @ B ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1782_add__less__imp__less__left,axiom,
% 9.03/4.95      ! [C: int,A: int,B: int] :
% 9.03/4.95        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.03/4.95       => ( ord_less_int @ A @ B ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1783_add__strict__right__mono,axiom,
% 9.03/4.95      ! [A: real,B: real,C: real] :
% 9.03/4.95        ( ( ord_less_real @ A @ B )
% 9.03/4.95       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1784_add__strict__right__mono,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_rat @ A @ B )
% 9.03/4.95       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1785_add__strict__right__mono,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat] :
% 9.03/4.95        ( ( ord_less_nat @ A @ B )
% 9.03/4.95       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1786_add__strict__right__mono,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int] :
% 9.03/4.95        ( ( ord_less_int @ A @ B )
% 9.03/4.95       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1787_add__strict__left__mono,axiom,
% 9.03/4.95      ! [A: real,B: real,C: real] :
% 9.03/4.95        ( ( ord_less_real @ A @ B )
% 9.03/4.95       => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1788_add__strict__left__mono,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_rat @ A @ B )
% 9.03/4.95       => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1789_add__strict__left__mono,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat] :
% 9.03/4.95        ( ( ord_less_nat @ A @ B )
% 9.03/4.95       => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1790_add__strict__left__mono,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int] :
% 9.03/4.95        ( ( ord_less_int @ A @ B )
% 9.03/4.95       => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1791_add__strict__mono,axiom,
% 9.03/4.95      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.95        ( ( ord_less_real @ A @ B )
% 9.03/4.95       => ( ( ord_less_real @ C @ D )
% 9.03/4.95         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1792_add__strict__mono,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.95        ( ( ord_less_rat @ A @ B )
% 9.03/4.95       => ( ( ord_less_rat @ C @ D )
% 9.03/4.95         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1793_add__strict__mono,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.03/4.95        ( ( ord_less_nat @ A @ B )
% 9.03/4.95       => ( ( ord_less_nat @ C @ D )
% 9.03/4.95         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1794_add__strict__mono,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.95        ( ( ord_less_int @ A @ B )
% 9.03/4.95       => ( ( ord_less_int @ C @ D )
% 9.03/4.95         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1795_add__mono__thms__linordered__field_I1_J,axiom,
% 9.03/4.95      ! [I: real,J: real,K: real,L: real] :
% 9.03/4.95        ( ( ( ord_less_real @ I @ J )
% 9.03/4.95          & ( K = L ) )
% 9.03/4.95       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1796_add__mono__thms__linordered__field_I1_J,axiom,
% 9.03/4.95      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.03/4.95        ( ( ( ord_less_rat @ I @ J )
% 9.03/4.95          & ( K = L ) )
% 9.03/4.95       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1797_add__mono__thms__linordered__field_I1_J,axiom,
% 9.03/4.95      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.95        ( ( ( ord_less_nat @ I @ J )
% 9.03/4.95          & ( K = L ) )
% 9.03/4.95       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1798_add__mono__thms__linordered__field_I1_J,axiom,
% 9.03/4.95      ! [I: int,J: int,K: int,L: int] :
% 9.03/4.95        ( ( ( ord_less_int @ I @ J )
% 9.03/4.95          & ( K = L ) )
% 9.03/4.95       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1799_add__mono__thms__linordered__field_I2_J,axiom,
% 9.03/4.95      ! [I: real,J: real,K: real,L: real] :
% 9.03/4.95        ( ( ( I = J )
% 9.03/4.95          & ( ord_less_real @ K @ L ) )
% 9.03/4.95       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1800_add__mono__thms__linordered__field_I2_J,axiom,
% 9.03/4.95      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.03/4.95        ( ( ( I = J )
% 9.03/4.95          & ( ord_less_rat @ K @ L ) )
% 9.03/4.95       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1801_add__mono__thms__linordered__field_I2_J,axiom,
% 9.03/4.95      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.95        ( ( ( I = J )
% 9.03/4.95          & ( ord_less_nat @ K @ L ) )
% 9.03/4.95       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1802_add__mono__thms__linordered__field_I2_J,axiom,
% 9.03/4.95      ! [I: int,J: int,K: int,L: int] :
% 9.03/4.95        ( ( ( I = J )
% 9.03/4.95          & ( ord_less_int @ K @ L ) )
% 9.03/4.95       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1803_add__mono__thms__linordered__field_I5_J,axiom,
% 9.03/4.95      ! [I: real,J: real,K: real,L: real] :
% 9.03/4.95        ( ( ( ord_less_real @ I @ J )
% 9.03/4.95          & ( ord_less_real @ K @ L ) )
% 9.03/4.95       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1804_add__mono__thms__linordered__field_I5_J,axiom,
% 9.03/4.95      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.03/4.95        ( ( ( ord_less_rat @ I @ J )
% 9.03/4.95          & ( ord_less_rat @ K @ L ) )
% 9.03/4.95       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1805_add__mono__thms__linordered__field_I5_J,axiom,
% 9.03/4.95      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.95        ( ( ( ord_less_nat @ I @ J )
% 9.03/4.95          & ( ord_less_nat @ K @ L ) )
% 9.03/4.95       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1806_add__mono__thms__linordered__field_I5_J,axiom,
% 9.03/4.95      ! [I: int,J: int,K: int,L: int] :
% 9.03/4.95        ( ( ( ord_less_int @ I @ J )
% 9.03/4.95          & ( ord_less_int @ K @ L ) )
% 9.03/4.95       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1807_reals__Archimedean2,axiom,
% 9.03/4.95      ! [X: real] :
% 9.03/4.95      ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1808_reals__Archimedean2,axiom,
% 9.03/4.95      ! [X: rat] :
% 9.03/4.95      ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1809_not__less__less__Suc__eq,axiom,
% 9.03/4.95      ! [N: nat,M: nat] :
% 9.03/4.95        ( ~ ( ord_less_nat @ N @ M )
% 9.03/4.95       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.03/4.95          = ( N = M ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1810_strict__inc__induct,axiom,
% 9.03/4.95      ! [I: nat,J: nat,P: nat > $o] :
% 9.03/4.95        ( ( ord_less_nat @ I @ J )
% 9.03/4.95       => ( ! [I2: nat] :
% 9.03/4.95              ( ( J
% 9.03/4.95                = ( suc @ I2 ) )
% 9.03/4.95             => ( P @ I2 ) )
% 9.03/4.95         => ( ! [I2: nat] :
% 9.03/4.95                ( ( ord_less_nat @ I2 @ J )
% 9.03/4.95               => ( ( P @ ( suc @ I2 ) )
% 9.03/4.95                 => ( P @ I2 ) ) )
% 9.03/4.95           => ( P @ I ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1811_less__Suc__induct,axiom,
% 9.03/4.95      ! [I: nat,J: nat,P: nat > nat > $o] :
% 9.03/4.95        ( ( ord_less_nat @ I @ J )
% 9.03/4.95       => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 9.03/4.95         => ( ! [I2: nat,J3: nat,K3: nat] :
% 9.03/4.95                ( ( ord_less_nat @ I2 @ J3 )
% 9.03/4.95               => ( ( ord_less_nat @ J3 @ K3 )
% 9.03/4.95                 => ( ( P @ I2 @ J3 )
% 9.03/4.95                   => ( ( P @ J3 @ K3 )
% 9.03/4.95                     => ( P @ I2 @ K3 ) ) ) ) )
% 9.03/4.95           => ( P @ I @ J ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1812_less__trans__Suc,axiom,
% 9.03/4.95      ! [I: nat,J: nat,K: nat] :
% 9.03/4.95        ( ( ord_less_nat @ I @ J )
% 9.03/4.95       => ( ( ord_less_nat @ J @ K )
% 9.03/4.95         => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1813_Suc__less__SucD,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.03/4.95       => ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1814_less__antisym,axiom,
% 9.03/4.95      ! [N: nat,M: nat] :
% 9.03/4.95        ( ~ ( ord_less_nat @ N @ M )
% 9.03/4.95       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.03/4.95         => ( M = N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1815_Suc__less__eq2,axiom,
% 9.03/4.95      ! [N: nat,M: nat] :
% 9.03/4.95        ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.03/4.95        = ( ? [M7: nat] :
% 9.03/4.95              ( ( M
% 9.03/4.95                = ( suc @ M7 ) )
% 9.03/4.95              & ( ord_less_nat @ N @ M7 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1816_All__less__Suc,axiom,
% 9.03/4.95      ! [N: nat,P: nat > $o] :
% 9.03/4.95        ( ( ! [I3: nat] :
% 9.03/4.95              ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 9.03/4.95             => ( P @ I3 ) ) )
% 9.03/4.95        = ( ( P @ N )
% 9.03/4.95          & ! [I3: nat] :
% 9.03/4.95              ( ( ord_less_nat @ I3 @ N )
% 9.03/4.95             => ( P @ I3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1817_not__less__eq,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ~ ( ord_less_nat @ M @ N ) )
% 9.03/4.95        = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1818_less__Suc__eq,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.03/4.95        = ( ( ord_less_nat @ M @ N )
% 9.03/4.95          | ( M = N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1819_Ex__less__Suc,axiom,
% 9.03/4.95      ! [N: nat,P: nat > $o] :
% 9.03/4.95        ( ( ? [I3: nat] :
% 9.03/4.95              ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 9.03/4.95              & ( P @ I3 ) ) )
% 9.03/4.95        = ( ( P @ N )
% 9.03/4.95          | ? [I3: nat] :
% 9.03/4.95              ( ( ord_less_nat @ I3 @ N )
% 9.03/4.95              & ( P @ I3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1820_less__SucI,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ M @ N )
% 9.03/4.95       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1821_less__SucE,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.03/4.95       => ( ~ ( ord_less_nat @ M @ N )
% 9.03/4.95         => ( M = N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1822_Suc__lessI,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ M @ N )
% 9.03/4.95       => ( ( ( suc @ M )
% 9.03/4.95           != N )
% 9.03/4.95         => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1823_Suc__lessE,axiom,
% 9.03/4.95      ! [I: nat,K: nat] :
% 9.03/4.95        ( ( ord_less_nat @ ( suc @ I ) @ K )
% 9.03/4.95       => ~ ! [J3: nat] :
% 9.03/4.95              ( ( ord_less_nat @ I @ J3 )
% 9.03/4.95             => ( K
% 9.03/4.95               != ( suc @ J3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1824_Suc__lessD,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ ( suc @ M ) @ N )
% 9.03/4.95       => ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1825_Nat_OlessE,axiom,
% 9.03/4.95      ! [I: nat,K: nat] :
% 9.03/4.95        ( ( ord_less_nat @ I @ K )
% 9.03/4.95       => ( ( K
% 9.03/4.95           != ( suc @ I ) )
% 9.03/4.95         => ~ ! [J3: nat] :
% 9.03/4.95                ( ( ord_less_nat @ I @ J3 )
% 9.03/4.95               => ( K
% 9.03/4.95                 != ( suc @ J3 ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1826_less__mono__imp__le__mono,axiom,
% 9.03/4.95      ! [F: nat > nat,I: nat,J: nat] :
% 9.03/4.95        ( ! [I2: nat,J3: nat] :
% 9.03/4.95            ( ( ord_less_nat @ I2 @ J3 )
% 9.03/4.95           => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
% 9.03/4.95       => ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.95         => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1827_le__neq__implies__less,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.95       => ( ( M != N )
% 9.03/4.95         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1828_less__or__eq__imp__le,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ( ord_less_nat @ M @ N )
% 9.03/4.95          | ( M = N ) )
% 9.03/4.95       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1829_le__eq__less__or__eq,axiom,
% 9.03/4.95      ( ord_less_eq_nat
% 9.03/4.95      = ( ^ [M6: nat,N3: nat] :
% 9.03/4.95            ( ( ord_less_nat @ M6 @ N3 )
% 9.03/4.95            | ( M6 = N3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1830_less__imp__le__nat,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ M @ N )
% 9.03/4.95       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1831_nat__less__le,axiom,
% 9.03/4.95      ( ord_less_nat
% 9.03/4.95      = ( ^ [M6: nat,N3: nat] :
% 9.03/4.95            ( ( ord_less_eq_nat @ M6 @ N3 )
% 9.03/4.95            & ( M6 != N3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1832_zero__le,axiom,
% 9.03/4.95      ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1833_le__numeral__extra_I3_J,axiom,
% 9.03/4.95      ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1834_le__numeral__extra_I3_J,axiom,
% 9.03/4.95      ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1835_le__numeral__extra_I3_J,axiom,
% 9.03/4.95      ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1836_le__numeral__extra_I3_J,axiom,
% 9.03/4.95      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1837_add__lessD1,axiom,
% 9.03/4.95      ! [I: nat,J: nat,K: nat] :
% 9.03/4.95        ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 9.03/4.95       => ( ord_less_nat @ I @ K ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1838_add__less__mono,axiom,
% 9.03/4.95      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.95        ( ( ord_less_nat @ I @ J )
% 9.03/4.95       => ( ( ord_less_nat @ K @ L )
% 9.03/4.95         => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1839_not__add__less1,axiom,
% 9.03/4.95      ! [I: nat,J: nat] :
% 9.03/4.95        ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1840_not__add__less2,axiom,
% 9.03/4.95      ! [J: nat,I: nat] :
% 9.03/4.95        ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1841_add__less__mono1,axiom,
% 9.03/4.95      ! [I: nat,J: nat,K: nat] :
% 9.03/4.95        ( ( ord_less_nat @ I @ J )
% 9.03/4.95       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1842_trans__less__add1,axiom,
% 9.03/4.95      ! [I: nat,J: nat,M: nat] :
% 9.03/4.95        ( ( ord_less_nat @ I @ J )
% 9.03/4.95       => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1843_trans__less__add2,axiom,
% 9.03/4.95      ! [I: nat,J: nat,M: nat] :
% 9.03/4.95        ( ( ord_less_nat @ I @ J )
% 9.03/4.95       => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1844_less__add__eq__less,axiom,
% 9.03/4.95      ! [K: nat,L: nat,M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ K @ L )
% 9.03/4.95       => ( ( ( plus_plus_nat @ M @ L )
% 9.03/4.95            = ( plus_plus_nat @ K @ N ) )
% 9.03/4.95         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1845_zero__neq__numeral,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ( zero_zero_complex
% 9.03/4.95       != ( numera6690914467698888265omplex @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1846_zero__neq__numeral,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ( zero_zero_real
% 9.03/4.95       != ( numeral_numeral_real @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1847_zero__neq__numeral,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ( zero_zero_rat
% 9.03/4.95       != ( numeral_numeral_rat @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1848_zero__neq__numeral,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ( zero_zero_nat
% 9.03/4.95       != ( numeral_numeral_nat @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1849_zero__neq__numeral,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ( zero_zero_int
% 9.03/4.95       != ( numeral_numeral_int @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1850_add_Ogroup__left__neutral,axiom,
% 9.03/4.95      ! [A: complex] :
% 9.03/4.95        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1851_add_Ogroup__left__neutral,axiom,
% 9.03/4.95      ! [A: real] :
% 9.03/4.95        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1852_add_Ogroup__left__neutral,axiom,
% 9.03/4.95      ! [A: rat] :
% 9.03/4.95        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1853_add_Ogroup__left__neutral,axiom,
% 9.03/4.95      ! [A: int] :
% 9.03/4.95        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1854_add_Ocomm__neutral,axiom,
% 9.03/4.95      ! [A: complex] :
% 9.03/4.95        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1855_add_Ocomm__neutral,axiom,
% 9.03/4.95      ! [A: real] :
% 9.03/4.95        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1856_add_Ocomm__neutral,axiom,
% 9.03/4.95      ! [A: rat] :
% 9.03/4.95        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1857_add_Ocomm__neutral,axiom,
% 9.03/4.95      ! [A: nat] :
% 9.03/4.95        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1858_add_Ocomm__neutral,axiom,
% 9.03/4.95      ! [A: int] :
% 9.03/4.95        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1859_comm__monoid__add__class_Oadd__0,axiom,
% 9.03/4.95      ! [A: complex] :
% 9.03/4.95        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1860_comm__monoid__add__class_Oadd__0,axiom,
% 9.03/4.95      ! [A: real] :
% 9.03/4.95        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1861_comm__monoid__add__class_Oadd__0,axiom,
% 9.03/4.95      ! [A: rat] :
% 9.03/4.95        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1862_comm__monoid__add__class_Oadd__0,axiom,
% 9.03/4.95      ! [A: nat] :
% 9.03/4.95        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1863_comm__monoid__add__class_Oadd__0,axiom,
% 9.03/4.95      ! [A: int] :
% 9.03/4.95        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1864_verit__sum__simplify,axiom,
% 9.03/4.95      ! [A: complex] :
% 9.03/4.95        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1865_verit__sum__simplify,axiom,
% 9.03/4.95      ! [A: real] :
% 9.03/4.95        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1866_verit__sum__simplify,axiom,
% 9.03/4.95      ! [A: rat] :
% 9.03/4.95        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1867_verit__sum__simplify,axiom,
% 9.03/4.95      ! [A: nat] :
% 9.03/4.95        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1868_verit__sum__simplify,axiom,
% 9.03/4.95      ! [A: int] :
% 9.03/4.95        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1869_power__not__zero,axiom,
% 9.03/4.95      ! [A: rat,N: nat] :
% 9.03/4.95        ( ( A != zero_zero_rat )
% 9.03/4.95       => ( ( power_power_rat @ A @ N )
% 9.03/4.95         != zero_zero_rat ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1870_power__not__zero,axiom,
% 9.03/4.95      ! [A: nat,N: nat] :
% 9.03/4.95        ( ( A != zero_zero_nat )
% 9.03/4.95       => ( ( power_power_nat @ A @ N )
% 9.03/4.95         != zero_zero_nat ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1871_power__not__zero,axiom,
% 9.03/4.95      ! [A: real,N: nat] :
% 9.03/4.95        ( ( A != zero_zero_real )
% 9.03/4.95       => ( ( power_power_real @ A @ N )
% 9.03/4.95         != zero_zero_real ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1872_power__not__zero,axiom,
% 9.03/4.95      ! [A: int,N: nat] :
% 9.03/4.95        ( ( A != zero_zero_int )
% 9.03/4.95       => ( ( power_power_int @ A @ N )
% 9.03/4.95         != zero_zero_int ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1873_power__not__zero,axiom,
% 9.03/4.95      ! [A: complex,N: nat] :
% 9.03/4.95        ( ( A != zero_zero_complex )
% 9.03/4.95       => ( ( power_power_complex @ A @ N )
% 9.03/4.95         != zero_zero_complex ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1874_num_Osize_I4_J,axiom,
% 9.03/4.95      ( ( size_size_num @ one )
% 9.03/4.95      = zero_zero_nat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1875_vebt__buildup_Ocases,axiom,
% 9.03/4.95      ! [X: nat] :
% 9.03/4.95        ( ( X != zero_zero_nat )
% 9.03/4.95       => ( ( X
% 9.03/4.95           != ( suc @ zero_zero_nat ) )
% 9.03/4.95         => ~ ! [Va: nat] :
% 9.03/4.95                ( X
% 9.03/4.95               != ( suc @ ( suc @ Va ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1876_exists__least__lemma,axiom,
% 9.03/4.95      ! [P: nat > $o] :
% 9.03/4.95        ( ~ ( P @ zero_zero_nat )
% 9.03/4.95       => ( ? [X_1: nat] : ( P @ X_1 )
% 9.03/4.95         => ? [N2: nat] :
% 9.03/4.95              ( ~ ( P @ N2 )
% 9.03/4.95              & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1877_not0__implies__Suc,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( N != zero_zero_nat )
% 9.03/4.95       => ? [M5: nat] :
% 9.03/4.95            ( N
% 9.03/4.95            = ( suc @ M5 ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1878_Zero__not__Suc,axiom,
% 9.03/4.95      ! [M: nat] :
% 9.03/4.95        ( zero_zero_nat
% 9.03/4.95       != ( suc @ M ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1879_Zero__neq__Suc,axiom,
% 9.03/4.95      ! [M: nat] :
% 9.03/4.95        ( zero_zero_nat
% 9.03/4.95       != ( suc @ M ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1880_Suc__neq__Zero,axiom,
% 9.03/4.95      ! [M: nat] :
% 9.03/4.95        ( ( suc @ M )
% 9.03/4.95       != zero_zero_nat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1881_zero__induct,axiom,
% 9.03/4.95      ! [P: nat > $o,K: nat] :
% 9.03/4.95        ( ( P @ K )
% 9.03/4.95       => ( ! [N2: nat] :
% 9.03/4.95              ( ( P @ ( suc @ N2 ) )
% 9.03/4.95             => ( P @ N2 ) )
% 9.03/4.95         => ( P @ zero_zero_nat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1882_diff__induct,axiom,
% 9.03/4.95      ! [P: nat > nat > $o,M: nat,N: nat] :
% 9.03/4.95        ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 9.03/4.95       => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 9.03/4.95         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.95                ( ( P @ X3 @ Y4 )
% 9.03/4.95               => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 9.03/4.95           => ( P @ M @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1883_nat__induct,axiom,
% 9.03/4.95      ! [P: nat > $o,N: nat] :
% 9.03/4.95        ( ( P @ zero_zero_nat )
% 9.03/4.95       => ( ! [N2: nat] :
% 9.03/4.95              ( ( P @ N2 )
% 9.03/4.95             => ( P @ ( suc @ N2 ) ) )
% 9.03/4.95         => ( P @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1884_old_Onat_Oexhaust,axiom,
% 9.03/4.95      ! [Y: nat] :
% 9.03/4.95        ( ( Y != zero_zero_nat )
% 9.03/4.95       => ~ ! [Nat3: nat] :
% 9.03/4.95              ( Y
% 9.03/4.95             != ( suc @ Nat3 ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1885_nat_OdiscI,axiom,
% 9.03/4.95      ! [Nat: nat,X22: nat] :
% 9.03/4.95        ( ( Nat
% 9.03/4.95          = ( suc @ X22 ) )
% 9.03/4.95       => ( Nat != zero_zero_nat ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1886_old_Onat_Odistinct_I1_J,axiom,
% 9.03/4.95      ! [Nat2: nat] :
% 9.03/4.95        ( zero_zero_nat
% 9.03/4.95       != ( suc @ Nat2 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1887_old_Onat_Odistinct_I2_J,axiom,
% 9.03/4.95      ! [Nat2: nat] :
% 9.03/4.95        ( ( suc @ Nat2 )
% 9.03/4.95       != zero_zero_nat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1888_nat_Odistinct_I1_J,axiom,
% 9.03/4.95      ! [X22: nat] :
% 9.03/4.95        ( zero_zero_nat
% 9.03/4.95       != ( suc @ X22 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1889_le__0__eq,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 9.03/4.95        = ( N = zero_zero_nat ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1890_bot__nat__0_Oextremum__uniqueI,axiom,
% 9.03/4.95      ! [A: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.03/4.95       => ( A = zero_zero_nat ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1891_bot__nat__0_Oextremum__unique,axiom,
% 9.03/4.95      ! [A: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.03/4.95        = ( A = zero_zero_nat ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1892_less__eq__nat_Osimps_I1_J,axiom,
% 9.03/4.95      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1893_plus__nat_Oadd__0,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( plus_plus_nat @ zero_zero_nat @ N )
% 9.03/4.95        = N ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1894_add__eq__self__zero,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ( plus_plus_nat @ M @ N )
% 9.03/4.95          = M )
% 9.03/4.95       => ( N = zero_zero_nat ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1895_floor__exists1,axiom,
% 9.03/4.95      ! [X: real] :
% 9.03/4.95      ? [X3: int] :
% 9.03/4.95        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
% 9.03/4.95        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 9.03/4.95        & ! [Y5: int] :
% 9.03/4.95            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
% 9.03/4.95              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 9.03/4.95           => ( Y5 = X3 ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1896_floor__exists1,axiom,
% 9.03/4.95      ! [X: rat] :
% 9.03/4.95      ? [X3: int] :
% 9.03/4.95        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
% 9.03/4.95        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 9.03/4.95        & ! [Y5: int] :
% 9.03/4.95            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
% 9.03/4.95              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 9.03/4.95           => ( Y5 = X3 ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1897_floor__exists,axiom,
% 9.03/4.95      ! [X: real] :
% 9.03/4.95      ? [Z3: int] :
% 9.03/4.95        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
% 9.03/4.95        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1898_floor__exists,axiom,
% 9.03/4.95      ! [X: rat] :
% 9.03/4.95      ? [Z3: int] :
% 9.03/4.95        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
% 9.03/4.95        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1899_power__Suc__less__one,axiom,
% 9.03/4.95      ! [A: real,N: nat] :
% 9.03/4.95        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.95       => ( ( ord_less_real @ A @ one_one_real )
% 9.03/4.95         => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1900_power__Suc__less__one,axiom,
% 9.03/4.95      ! [A: rat,N: nat] :
% 9.03/4.95        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.95       => ( ( ord_less_rat @ A @ one_one_rat )
% 9.03/4.95         => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1901_power__Suc__less__one,axiom,
% 9.03/4.95      ! [A: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.95       => ( ( ord_less_nat @ A @ one_one_nat )
% 9.03/4.95         => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1902_power__Suc__less__one,axiom,
% 9.03/4.95      ! [A: int,N: nat] :
% 9.03/4.95        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.95       => ( ( ord_less_int @ A @ one_one_int )
% 9.03/4.95         => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1903_self__le__power,axiom,
% 9.03/4.95      ! [A: real,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_real @ one_one_real @ A )
% 9.03/4.95       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95         => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1904_self__le__power,axiom,
% 9.03/4.95      ! [A: rat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.03/4.95       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95         => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1905_self__le__power,axiom,
% 9.03/4.95      ! [A: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.03/4.95       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95         => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1906_self__le__power,axiom,
% 9.03/4.95      ! [A: int,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_int @ one_one_int @ A )
% 9.03/4.95       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.95         => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1907_field__less__half__sum,axiom,
% 9.03/4.95      ! [X: real,Y: real] :
% 9.03/4.95        ( ( ord_less_real @ X @ Y )
% 9.03/4.95       => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1908_field__less__half__sum,axiom,
% 9.03/4.95      ! [X: rat,Y: rat] :
% 9.03/4.95        ( ( ord_less_rat @ X @ Y )
% 9.03/4.95       => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1909_inverse__of__nat__le,axiom,
% 9.03/4.95      ! [N: nat,M: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.95       => ( ( N != zero_zero_nat )
% 9.03/4.95         => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1910_inverse__of__nat__le,axiom,
% 9.03/4.95      ! [N: nat,M: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.95       => ( ( N != zero_zero_nat )
% 9.03/4.95         => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1911_power2__less__0,axiom,
% 9.03/4.95      ! [A: real] :
% 9.03/4.95        ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1912_power2__less__0,axiom,
% 9.03/4.95      ! [A: rat] :
% 9.03/4.95        ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1913_power2__less__0,axiom,
% 9.03/4.95      ! [A: int] :
% 9.03/4.95        ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1914_less__2__cases,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.95       => ( ( N = zero_zero_nat )
% 9.03/4.95          | ( N
% 9.03/4.95            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1915_less__2__cases__iff,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.95        = ( ( N = zero_zero_nat )
% 9.03/4.95          | ( N
% 9.03/4.95            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1916_divide__numeral__1,axiom,
% 9.03/4.95      ! [A: complex] :
% 9.03/4.95        ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1917_divide__numeral__1,axiom,
% 9.03/4.95      ! [A: real] :
% 9.03/4.95        ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1918_divide__numeral__1,axiom,
% 9.03/4.95      ! [A: rat] :
% 9.03/4.95        ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 9.03/4.95        = A ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1919_le__of__int__ceiling,axiom,
% 9.03/4.95      ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1920_le__of__int__ceiling,axiom,
% 9.03/4.95      ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1921_power__one__over,axiom,
% 9.03/4.95      ! [A: complex,N: nat] :
% 9.03/4.95        ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 9.03/4.95        = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1922_power__one__over,axiom,
% 9.03/4.95      ! [A: real,N: nat] :
% 9.03/4.95        ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 9.03/4.95        = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1923_power__one__over,axiom,
% 9.03/4.95      ! [A: rat,N: nat] :
% 9.03/4.95        ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 9.03/4.95        = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1924_add__less__le__mono,axiom,
% 9.03/4.95      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.95        ( ( ord_less_real @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_real @ C @ D )
% 9.03/4.95         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1925_add__less__le__mono,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.95        ( ( ord_less_rat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_rat @ C @ D )
% 9.03/4.95         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1926_add__less__le__mono,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.03/4.95        ( ( ord_less_nat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_nat @ C @ D )
% 9.03/4.95         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1927_add__less__le__mono,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.95        ( ( ord_less_int @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_int @ C @ D )
% 9.03/4.95         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1928_add__le__less__mono,axiom,
% 9.03/4.95      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.95       => ( ( ord_less_real @ C @ D )
% 9.03/4.95         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1929_add__le__less__mono,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ord_less_rat @ C @ D )
% 9.03/4.95         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1930_add__le__less__mono,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95       => ( ( ord_less_nat @ C @ D )
% 9.03/4.95         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1931_add__le__less__mono,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.95       => ( ( ord_less_int @ C @ D )
% 9.03/4.95         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1932_add__mono__thms__linordered__field_I3_J,axiom,
% 9.03/4.95      ! [I: real,J: real,K: real,L: real] :
% 9.03/4.95        ( ( ( ord_less_real @ I @ J )
% 9.03/4.95          & ( ord_less_eq_real @ K @ L ) )
% 9.03/4.95       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1933_add__mono__thms__linordered__field_I3_J,axiom,
% 9.03/4.95      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.03/4.95        ( ( ( ord_less_rat @ I @ J )
% 9.03/4.95          & ( ord_less_eq_rat @ K @ L ) )
% 9.03/4.95       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1934_add__mono__thms__linordered__field_I3_J,axiom,
% 9.03/4.95      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.95        ( ( ( ord_less_nat @ I @ J )
% 9.03/4.95          & ( ord_less_eq_nat @ K @ L ) )
% 9.03/4.95       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1935_add__mono__thms__linordered__field_I3_J,axiom,
% 9.03/4.95      ! [I: int,J: int,K: int,L: int] :
% 9.03/4.95        ( ( ( ord_less_int @ I @ J )
% 9.03/4.95          & ( ord_less_eq_int @ K @ L ) )
% 9.03/4.95       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1936_add__mono__thms__linordered__field_I4_J,axiom,
% 9.03/4.95      ! [I: real,J: real,K: real,L: real] :
% 9.03/4.95        ( ( ( ord_less_eq_real @ I @ J )
% 9.03/4.95          & ( ord_less_real @ K @ L ) )
% 9.03/4.95       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1937_add__mono__thms__linordered__field_I4_J,axiom,
% 9.03/4.95      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.03/4.95        ( ( ( ord_less_eq_rat @ I @ J )
% 9.03/4.95          & ( ord_less_rat @ K @ L ) )
% 9.03/4.95       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1938_add__mono__thms__linordered__field_I4_J,axiom,
% 9.03/4.95      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.95        ( ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.95          & ( ord_less_nat @ K @ L ) )
% 9.03/4.95       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1939_add__mono__thms__linordered__field_I4_J,axiom,
% 9.03/4.95      ! [I: int,J: int,K: int,L: int] :
% 9.03/4.95        ( ( ( ord_less_eq_int @ I @ J )
% 9.03/4.95          & ( ord_less_int @ K @ L ) )
% 9.03/4.95       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1940_not__numeral__less__one,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1941_not__numeral__less__one,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1942_not__numeral__less__one,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1943_not__numeral__less__one,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1944_le__imp__less__Suc,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.95       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1945_less__eq__Suc__le,axiom,
% 9.03/4.95      ( ord_less_nat
% 9.03/4.95      = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1946_less__Suc__eq__le,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.03/4.95        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1947_le__less__Suc__eq,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.95       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.03/4.95          = ( N = M ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1948_Suc__le__lessD,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.03/4.95       => ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1949_inc__induct,axiom,
% 9.03/4.95      ! [I: nat,J: nat,P: nat > $o] :
% 9.03/4.95        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.95       => ( ( P @ J )
% 9.03/4.95         => ( ! [N2: nat] :
% 9.03/4.95                ( ( ord_less_eq_nat @ I @ N2 )
% 9.03/4.95               => ( ( ord_less_nat @ N2 @ J )
% 9.03/4.95                 => ( ( P @ ( suc @ N2 ) )
% 9.03/4.95                   => ( P @ N2 ) ) ) )
% 9.03/4.95           => ( P @ I ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1950_dec__induct,axiom,
% 9.03/4.95      ! [I: nat,J: nat,P: nat > $o] :
% 9.03/4.95        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.95       => ( ( P @ I )
% 9.03/4.95         => ( ! [N2: nat] :
% 9.03/4.95                ( ( ord_less_eq_nat @ I @ N2 )
% 9.03/4.95               => ( ( ord_less_nat @ N2 @ J )
% 9.03/4.95                 => ( ( P @ N2 )
% 9.03/4.95                   => ( P @ ( suc @ N2 ) ) ) ) )
% 9.03/4.95           => ( P @ J ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1951_Suc__le__eq,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.03/4.95        = ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1952_Suc__leI,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ M @ N )
% 9.03/4.95       => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1953_less__natE,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ M @ N )
% 9.03/4.95       => ~ ! [Q2: nat] :
% 9.03/4.95              ( N
% 9.03/4.95             != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1954_less__add__Suc1,axiom,
% 9.03/4.95      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1955_less__add__Suc2,axiom,
% 9.03/4.95      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1956_less__iff__Suc__add,axiom,
% 9.03/4.95      ( ord_less_nat
% 9.03/4.95      = ( ^ [M6: nat,N3: nat] :
% 9.03/4.95          ? [K2: nat] :
% 9.03/4.95            ( N3
% 9.03/4.95            = ( suc @ ( plus_plus_nat @ M6 @ K2 ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1957_less__imp__Suc__add,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ M @ N )
% 9.03/4.95       => ? [K3: nat] :
% 9.03/4.95            ( N
% 9.03/4.95            = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1958_mono__nat__linear__lb,axiom,
% 9.03/4.95      ! [F: nat > nat,M: nat,K: nat] :
% 9.03/4.95        ( ! [M5: nat,N2: nat] :
% 9.03/4.95            ( ( ord_less_nat @ M5 @ N2 )
% 9.03/4.95           => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
% 9.03/4.95       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1959_zero__le__numeral,axiom,
% 9.03/4.95      ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1960_zero__le__numeral,axiom,
% 9.03/4.95      ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1961_zero__le__numeral,axiom,
% 9.03/4.95      ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1962_zero__le__numeral,axiom,
% 9.03/4.95      ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1963_not__numeral__le__zero,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1964_not__numeral__le__zero,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1965_not__numeral__le__zero,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1966_not__numeral__le__zero,axiom,
% 9.03/4.95      ! [N: num] :
% 9.03/4.95        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1967_add__nonpos__eq__0__iff,axiom,
% 9.03/4.95      ! [X: real,Y: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.03/4.95       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 9.03/4.95         => ( ( ( plus_plus_real @ X @ Y )
% 9.03/4.95              = zero_zero_real )
% 9.03/4.95            = ( ( X = zero_zero_real )
% 9.03/4.95              & ( Y = zero_zero_real ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1968_add__nonpos__eq__0__iff,axiom,
% 9.03/4.95      ! [X: rat,Y: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.03/4.95       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 9.03/4.95         => ( ( ( plus_plus_rat @ X @ Y )
% 9.03/4.95              = zero_zero_rat )
% 9.03/4.95            = ( ( X = zero_zero_rat )
% 9.03/4.95              & ( Y = zero_zero_rat ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1969_add__nonpos__eq__0__iff,axiom,
% 9.03/4.95      ! [X: nat,Y: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 9.03/4.95       => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 9.03/4.95         => ( ( ( plus_plus_nat @ X @ Y )
% 9.03/4.95              = zero_zero_nat )
% 9.03/4.95            = ( ( X = zero_zero_nat )
% 9.03/4.95              & ( Y = zero_zero_nat ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1970_add__nonpos__eq__0__iff,axiom,
% 9.03/4.95      ! [X: int,Y: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ X @ zero_zero_int )
% 9.03/4.95       => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 9.03/4.95         => ( ( ( plus_plus_int @ X @ Y )
% 9.03/4.95              = zero_zero_int )
% 9.03/4.95            = ( ( X = zero_zero_int )
% 9.03/4.95              & ( Y = zero_zero_int ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1971_add__nonneg__eq__0__iff,axiom,
% 9.03/4.95      ! [X: real,Y: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.95       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.95         => ( ( ( plus_plus_real @ X @ Y )
% 9.03/4.95              = zero_zero_real )
% 9.03/4.95            = ( ( X = zero_zero_real )
% 9.03/4.95              & ( Y = zero_zero_real ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1972_add__nonneg__eq__0__iff,axiom,
% 9.03/4.95      ! [X: rat,Y: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.95       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.95         => ( ( ( plus_plus_rat @ X @ Y )
% 9.03/4.95              = zero_zero_rat )
% 9.03/4.95            = ( ( X = zero_zero_rat )
% 9.03/4.95              & ( Y = zero_zero_rat ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1973_add__nonneg__eq__0__iff,axiom,
% 9.03/4.95      ! [X: nat,Y: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 9.03/4.95       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 9.03/4.95         => ( ( ( plus_plus_nat @ X @ Y )
% 9.03/4.95              = zero_zero_nat )
% 9.03/4.95            = ( ( X = zero_zero_nat )
% 9.03/4.95              & ( Y = zero_zero_nat ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1974_add__nonneg__eq__0__iff,axiom,
% 9.03/4.95      ! [X: int,Y: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.03/4.95       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.03/4.95         => ( ( ( plus_plus_int @ X @ Y )
% 9.03/4.95              = zero_zero_int )
% 9.03/4.95            = ( ( X = zero_zero_int )
% 9.03/4.95              & ( Y = zero_zero_int ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1975_add__nonpos__nonpos,axiom,
% 9.03/4.95      ! [A: real,B: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.95       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.03/4.95         => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1976_add__nonpos__nonpos,axiom,
% 9.03/4.95      ! [A: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.03/4.95         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1977_add__nonpos__nonpos,axiom,
% 9.03/4.95      ! [A: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.03/4.95         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1978_add__nonpos__nonpos,axiom,
% 9.03/4.95      ! [A: int,B: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.95       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.03/4.95         => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1979_add__nonneg__nonneg,axiom,
% 9.03/4.95      ! [A: real,B: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.95       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.95         => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1980_add__nonneg__nonneg,axiom,
% 9.03/4.95      ! [A: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.95       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.95         => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1981_add__nonneg__nonneg,axiom,
% 9.03/4.95      ! [A: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.95       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.95         => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1982_add__nonneg__nonneg,axiom,
% 9.03/4.95      ! [A: int,B: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.95       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.95         => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1983_add__increasing2,axiom,
% 9.03/4.95      ! [C: real,B: real,A: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.95       => ( ( ord_less_eq_real @ B @ A )
% 9.03/4.95         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1984_add__increasing2,axiom,
% 9.03/4.95      ! [C: rat,B: rat,A: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.95         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1985_add__increasing2,axiom,
% 9.03/4.95      ! [C: nat,B: nat,A: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ A )
% 9.03/4.95         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1986_add__increasing2,axiom,
% 9.03/4.95      ! [C: int,B: int,A: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.95       => ( ( ord_less_eq_int @ B @ A )
% 9.03/4.95         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1987_add__decreasing2,axiom,
% 9.03/4.95      ! [C: real,A: real,B: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.95       => ( ( ord_less_eq_real @ A @ B )
% 9.03/4.95         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1988_add__decreasing2,axiom,
% 9.03/4.95      ! [C: rat,A: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.95       => ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1989_add__decreasing2,axiom,
% 9.03/4.95      ! [C: nat,A: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 9.03/4.95       => ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1990_add__decreasing2,axiom,
% 9.03/4.95      ! [C: int,A: int,B: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.03/4.95       => ( ( ord_less_eq_int @ A @ B )
% 9.03/4.95         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1991_add__increasing,axiom,
% 9.03/4.95      ! [A: real,B: real,C: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.95       => ( ( ord_less_eq_real @ B @ C )
% 9.03/4.95         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1992_add__increasing,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1993_add__increasing,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ C )
% 9.03/4.95         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1994_add__increasing,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.95       => ( ( ord_less_eq_int @ B @ C )
% 9.03/4.95         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1995_add__decreasing,axiom,
% 9.03/4.95      ! [A: real,C: real,B: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.95       => ( ( ord_less_eq_real @ C @ B )
% 9.03/4.95         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1996_add__decreasing,axiom,
% 9.03/4.95      ! [A: rat,C: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.95       => ( ( ord_less_eq_rat @ C @ B )
% 9.03/4.95         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1997_add__decreasing,axiom,
% 9.03/4.95      ! [A: nat,C: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.03/4.95       => ( ( ord_less_eq_nat @ C @ B )
% 9.03/4.95         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1998_add__decreasing,axiom,
% 9.03/4.95      ! [A: int,C: int,B: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.95       => ( ( ord_less_eq_int @ C @ B )
% 9.03/4.95         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_1999_power__mono,axiom,
% 9.03/4.95      ! [A: real,B: real,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.95         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2000_power__mono,axiom,
% 9.03/4.95      ! [A: rat,B: rat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.95         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2001_power__mono,axiom,
% 9.03/4.95      ! [A: nat,B: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.95         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2002_power__mono,axiom,
% 9.03/4.95      ! [A: int,B: int,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.95         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2003_zero__le__power,axiom,
% 9.03/4.95      ! [A: real,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.95       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2004_zero__le__power,axiom,
% 9.03/4.95      ! [A: rat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.95       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2005_zero__le__power,axiom,
% 9.03/4.95      ! [A: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.95       => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2006_zero__le__power,axiom,
% 9.03/4.95      ! [A: int,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.95       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2007_of__nat__0__le__iff,axiom,
% 9.03/4.95      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2008_of__nat__0__le__iff,axiom,
% 9.03/4.95      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2009_of__nat__0__le__iff,axiom,
% 9.03/4.95      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2010_of__nat__0__le__iff,axiom,
% 9.03/4.95      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2011_nat__less__real__le,axiom,
% 9.03/4.95      ( ord_less_nat
% 9.03/4.95      = ( ^ [N3: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2012_power__0,axiom,
% 9.03/4.95      ! [A: rat] :
% 9.03/4.95        ( ( power_power_rat @ A @ zero_zero_nat )
% 9.03/4.95        = one_one_rat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2013_power__0,axiom,
% 9.03/4.95      ! [A: nat] :
% 9.03/4.95        ( ( power_power_nat @ A @ zero_zero_nat )
% 9.03/4.95        = one_one_nat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2014_power__0,axiom,
% 9.03/4.95      ! [A: real] :
% 9.03/4.95        ( ( power_power_real @ A @ zero_zero_nat )
% 9.03/4.95        = one_one_real ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2015_power__0,axiom,
% 9.03/4.95      ! [A: int] :
% 9.03/4.95        ( ( power_power_int @ A @ zero_zero_nat )
% 9.03/4.95        = one_one_int ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2016_power__0,axiom,
% 9.03/4.95      ! [A: complex] :
% 9.03/4.95        ( ( power_power_complex @ A @ zero_zero_nat )
% 9.03/4.95        = one_one_complex ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2017_of__nat__neq__0,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 9.03/4.95       != zero_zero_complex ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2018_of__nat__neq__0,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 9.03/4.95       != zero_zero_real ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2019_of__nat__neq__0,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 9.03/4.95       != zero_zero_rat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2020_of__nat__neq__0,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 9.03/4.95       != zero_zero_nat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2021_of__nat__neq__0,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95        ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 9.03/4.95       != zero_zero_int ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2022_add__is__1,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ( plus_plus_nat @ M @ N )
% 9.03/4.95          = ( suc @ zero_zero_nat ) )
% 9.03/4.95        = ( ( ( M
% 9.03/4.95              = ( suc @ zero_zero_nat ) )
% 9.03/4.95            & ( N = zero_zero_nat ) )
% 9.03/4.95          | ( ( M = zero_zero_nat )
% 9.03/4.95            & ( N
% 9.03/4.95              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2023_one__is__add,axiom,
% 9.03/4.95      ! [M: nat,N: nat] :
% 9.03/4.95        ( ( ( suc @ zero_zero_nat )
% 9.03/4.95          = ( plus_plus_nat @ M @ N ) )
% 9.03/4.95        = ( ( ( M
% 9.03/4.95              = ( suc @ zero_zero_nat ) )
% 9.03/4.95            & ( N = zero_zero_nat ) )
% 9.03/4.95          | ( ( M = zero_zero_nat )
% 9.03/4.95            & ( N
% 9.03/4.95              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2024_One__nat__def,axiom,
% 9.03/4.95      ( one_one_nat
% 9.03/4.95      = ( suc @ zero_zero_nat ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2025_pow_Osimps_I1_J,axiom,
% 9.03/4.95      ! [X: num] :
% 9.03/4.95        ( ( pow @ X @ one )
% 9.03/4.95        = X ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2026_power2__less__imp__less,axiom,
% 9.03/4.95      ! [X: real,Y: real] :
% 9.03/4.95        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.95       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.95         => ( ord_less_real @ X @ Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2027_power2__less__imp__less,axiom,
% 9.03/4.95      ! [X: rat,Y: rat] :
% 9.03/4.95        ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.95         => ( ord_less_rat @ X @ Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2028_power2__less__imp__less,axiom,
% 9.03/4.95      ! [X: nat,Y: nat] :
% 9.03/4.95        ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.95       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 9.03/4.95         => ( ord_less_nat @ X @ Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2029_power2__less__imp__less,axiom,
% 9.03/4.95      ! [X: int,Y: int] :
% 9.03/4.95        ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.95       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.03/4.95         => ( ord_less_int @ X @ Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2030_sum__power2__gt__zero__iff,axiom,
% 9.03/4.95      ! [X: real,Y: real] :
% 9.03/4.95        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.03/4.95        = ( ( X != zero_zero_real )
% 9.03/4.95          | ( Y != zero_zero_real ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2031_sum__power2__gt__zero__iff,axiom,
% 9.03/4.95      ! [X: rat,Y: rat] :
% 9.03/4.95        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.03/4.95        = ( ( X != zero_zero_rat )
% 9.03/4.95          | ( Y != zero_zero_rat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2032_sum__power2__gt__zero__iff,axiom,
% 9.03/4.95      ! [X: int,Y: int] :
% 9.03/4.95        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.03/4.95        = ( ( X != zero_zero_int )
% 9.03/4.95          | ( Y != zero_zero_int ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2033_not__sum__power2__lt__zero,axiom,
% 9.03/4.95      ! [X: real,Y: real] :
% 9.03/4.95        ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2034_not__sum__power2__lt__zero,axiom,
% 9.03/4.95      ! [X: rat,Y: rat] :
% 9.03/4.95        ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2035_not__sum__power2__lt__zero,axiom,
% 9.03/4.95      ! [X: int,Y: int] :
% 9.03/4.95        ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2036_ceiling__eq,axiom,
% 9.03/4.95      ! [N: int,X: real] :
% 9.03/4.95        ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 9.03/4.95       => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.03/4.95         => ( ( archim7802044766580827645g_real @ X )
% 9.03/4.95            = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2037_ceiling__eq,axiom,
% 9.03/4.95      ! [N: int,X: rat] :
% 9.03/4.95        ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
% 9.03/4.95       => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 9.03/4.95         => ( ( archim2889992004027027881ng_rat @ X )
% 9.03/4.95            = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2038_ceiling__le__iff,axiom,
% 9.03/4.95      ! [X: real,Z: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
% 9.03/4.95        = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2039_ceiling__le__iff,axiom,
% 9.03/4.95      ! [X: rat,Z: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
% 9.03/4.95        = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2040_ceiling__le,axiom,
% 9.03/4.95      ! [X: real,A: int] :
% 9.03/4.95        ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 9.03/4.95       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2041_ceiling__le,axiom,
% 9.03/4.95      ! [X: rat,A: int] :
% 9.03/4.95        ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
% 9.03/4.95       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2042_power__gt1,axiom,
% 9.03/4.95      ! [A: real,N: nat] :
% 9.03/4.95        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.95       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2043_power__gt1,axiom,
% 9.03/4.95      ! [A: rat,N: nat] :
% 9.03/4.95        ( ( ord_less_rat @ one_one_rat @ A )
% 9.03/4.95       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2044_power__gt1,axiom,
% 9.03/4.95      ! [A: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_nat @ one_one_nat @ A )
% 9.03/4.95       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2045_power__gt1,axiom,
% 9.03/4.95      ! [A: int,N: nat] :
% 9.03/4.95        ( ( ord_less_int @ one_one_int @ A )
% 9.03/4.95       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2046_nle__le,axiom,
% 9.03/4.95      ! [A: rat,B: rat] :
% 9.03/4.95        ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 9.03/4.95        = ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.95          & ( B != A ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2047_nle__le,axiom,
% 9.03/4.95      ! [A: num,B: num] :
% 9.03/4.95        ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 9.03/4.95        = ( ( ord_less_eq_num @ B @ A )
% 9.03/4.95          & ( B != A ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2048_nle__le,axiom,
% 9.03/4.95      ! [A: nat,B: nat] :
% 9.03/4.95        ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 9.03/4.95        = ( ( ord_less_eq_nat @ B @ A )
% 9.03/4.95          & ( B != A ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2049_nle__le,axiom,
% 9.03/4.95      ! [A: int,B: int] :
% 9.03/4.95        ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 9.03/4.95        = ( ( ord_less_eq_int @ B @ A )
% 9.03/4.95          & ( B != A ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2050_le__cases3,axiom,
% 9.03/4.95      ! [X: rat,Y: rat,Z: rat] :
% 9.03/4.95        ( ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.95         => ~ ( ord_less_eq_rat @ Y @ Z ) )
% 9.03/4.95       => ( ( ( ord_less_eq_rat @ Y @ X )
% 9.03/4.95           => ~ ( ord_less_eq_rat @ X @ Z ) )
% 9.03/4.95         => ( ( ( ord_less_eq_rat @ X @ Z )
% 9.03/4.95             => ~ ( ord_less_eq_rat @ Z @ Y ) )
% 9.03/4.95           => ( ( ( ord_less_eq_rat @ Z @ Y )
% 9.03/4.95               => ~ ( ord_less_eq_rat @ Y @ X ) )
% 9.03/4.95             => ( ( ( ord_less_eq_rat @ Y @ Z )
% 9.03/4.95                 => ~ ( ord_less_eq_rat @ Z @ X ) )
% 9.03/4.95               => ~ ( ( ord_less_eq_rat @ Z @ X )
% 9.03/4.95                   => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2051_le__cases3,axiom,
% 9.03/4.95      ! [X: num,Y: num,Z: num] :
% 9.03/4.95        ( ( ( ord_less_eq_num @ X @ Y )
% 9.03/4.95         => ~ ( ord_less_eq_num @ Y @ Z ) )
% 9.03/4.95       => ( ( ( ord_less_eq_num @ Y @ X )
% 9.03/4.95           => ~ ( ord_less_eq_num @ X @ Z ) )
% 9.03/4.95         => ( ( ( ord_less_eq_num @ X @ Z )
% 9.03/4.95             => ~ ( ord_less_eq_num @ Z @ Y ) )
% 9.03/4.95           => ( ( ( ord_less_eq_num @ Z @ Y )
% 9.03/4.95               => ~ ( ord_less_eq_num @ Y @ X ) )
% 9.03/4.95             => ( ( ( ord_less_eq_num @ Y @ Z )
% 9.03/4.95                 => ~ ( ord_less_eq_num @ Z @ X ) )
% 9.03/4.95               => ~ ( ( ord_less_eq_num @ Z @ X )
% 9.03/4.95                   => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2052_le__cases3,axiom,
% 9.03/4.95      ! [X: nat,Y: nat,Z: nat] :
% 9.03/4.95        ( ( ( ord_less_eq_nat @ X @ Y )
% 9.03/4.95         => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 9.03/4.95       => ( ( ( ord_less_eq_nat @ Y @ X )
% 9.03/4.95           => ~ ( ord_less_eq_nat @ X @ Z ) )
% 9.03/4.95         => ( ( ( ord_less_eq_nat @ X @ Z )
% 9.03/4.95             => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 9.03/4.95           => ( ( ( ord_less_eq_nat @ Z @ Y )
% 9.03/4.95               => ~ ( ord_less_eq_nat @ Y @ X ) )
% 9.03/4.95             => ( ( ( ord_less_eq_nat @ Y @ Z )
% 9.03/4.95                 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 9.03/4.95               => ~ ( ( ord_less_eq_nat @ Z @ X )
% 9.03/4.95                   => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2053_le__cases3,axiom,
% 9.03/4.95      ! [X: int,Y: int,Z: int] :
% 9.03/4.95        ( ( ( ord_less_eq_int @ X @ Y )
% 9.03/4.95         => ~ ( ord_less_eq_int @ Y @ Z ) )
% 9.03/4.95       => ( ( ( ord_less_eq_int @ Y @ X )
% 9.03/4.95           => ~ ( ord_less_eq_int @ X @ Z ) )
% 9.03/4.95         => ( ( ( ord_less_eq_int @ X @ Z )
% 9.03/4.95             => ~ ( ord_less_eq_int @ Z @ Y ) )
% 9.03/4.95           => ( ( ( ord_less_eq_int @ Z @ Y )
% 9.03/4.95               => ~ ( ord_less_eq_int @ Y @ X ) )
% 9.03/4.95             => ( ( ( ord_less_eq_int @ Y @ Z )
% 9.03/4.95                 => ~ ( ord_less_eq_int @ Z @ X ) )
% 9.03/4.95               => ~ ( ( ord_less_eq_int @ Z @ X )
% 9.03/4.95                   => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2054_order__class_Oorder__eq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: set_int,Z2: set_int] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [X4: set_int,Y6: set_int] :
% 9.03/4.95            ( ( ord_less_eq_set_int @ X4 @ Y6 )
% 9.03/4.95            & ( ord_less_eq_set_int @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2055_order__class_Oorder__eq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [X4: rat,Y6: rat] :
% 9.03/4.95            ( ( ord_less_eq_rat @ X4 @ Y6 )
% 9.03/4.95            & ( ord_less_eq_rat @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2056_order__class_Oorder__eq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [X4: num,Y6: num] :
% 9.03/4.95            ( ( ord_less_eq_num @ X4 @ Y6 )
% 9.03/4.95            & ( ord_less_eq_num @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2057_order__class_Oorder__eq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [X4: nat,Y6: nat] :
% 9.03/4.95            ( ( ord_less_eq_nat @ X4 @ Y6 )
% 9.03/4.95            & ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2058_order__class_Oorder__eq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [X4: int,Y6: int] :
% 9.03/4.95            ( ( ord_less_eq_int @ X4 @ Y6 )
% 9.03/4.95            & ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2059_ord__eq__le__trans,axiom,
% 9.03/4.95      ! [A: set_int,B: set_int,C: set_int] :
% 9.03/4.95        ( ( A = B )
% 9.03/4.95       => ( ( ord_less_eq_set_int @ B @ C )
% 9.03/4.95         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2060_ord__eq__le__trans,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat] :
% 9.03/4.95        ( ( A = B )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2061_ord__eq__le__trans,axiom,
% 9.03/4.95      ! [A: num,B: num,C: num] :
% 9.03/4.95        ( ( A = B )
% 9.03/4.95       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.95         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2062_ord__eq__le__trans,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat] :
% 9.03/4.95        ( ( A = B )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ C )
% 9.03/4.95         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2063_ord__eq__le__trans,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int] :
% 9.03/4.95        ( ( A = B )
% 9.03/4.95       => ( ( ord_less_eq_int @ B @ C )
% 9.03/4.95         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2064_ord__le__eq__trans,axiom,
% 9.03/4.95      ! [A: set_int,B: set_int,C: set_int] :
% 9.03/4.95        ( ( ord_less_eq_set_int @ A @ B )
% 9.03/4.95       => ( ( B = C )
% 9.03/4.95         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2065_ord__le__eq__trans,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( B = C )
% 9.03/4.95         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2066_ord__le__eq__trans,axiom,
% 9.03/4.95      ! [A: num,B: num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( B = C )
% 9.03/4.95         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2067_ord__le__eq__trans,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95       => ( ( B = C )
% 9.03/4.95         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2068_ord__le__eq__trans,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.95       => ( ( B = C )
% 9.03/4.95         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2069_order__antisym,axiom,
% 9.03/4.95      ! [X: set_int,Y: set_int] :
% 9.03/4.95        ( ( ord_less_eq_set_int @ X @ Y )
% 9.03/4.95       => ( ( ord_less_eq_set_int @ Y @ X )
% 9.03/4.95         => ( X = Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2070_order__antisym,axiom,
% 9.03/4.95      ! [X: rat,Y: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.95       => ( ( ord_less_eq_rat @ Y @ X )
% 9.03/4.95         => ( X = Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2071_order__antisym,axiom,
% 9.03/4.95      ! [X: num,Y: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ X @ Y )
% 9.03/4.95       => ( ( ord_less_eq_num @ Y @ X )
% 9.03/4.95         => ( X = Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2072_order__antisym,axiom,
% 9.03/4.95      ! [X: nat,Y: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ X @ Y )
% 9.03/4.95       => ( ( ord_less_eq_nat @ Y @ X )
% 9.03/4.95         => ( X = Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2073_order__antisym,axiom,
% 9.03/4.95      ! [X: int,Y: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ X @ Y )
% 9.03/4.95       => ( ( ord_less_eq_int @ Y @ X )
% 9.03/4.95         => ( X = Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2074_order_Otrans,axiom,
% 9.03/4.95      ! [A: set_int,B: set_int,C: set_int] :
% 9.03/4.95        ( ( ord_less_eq_set_int @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_set_int @ B @ C )
% 9.03/4.95         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2075_order_Otrans,axiom,
% 9.03/4.95      ! [A: rat,B: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2076_order_Otrans,axiom,
% 9.03/4.95      ! [A: num,B: num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.95         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2077_order_Otrans,axiom,
% 9.03/4.95      ! [A: nat,B: nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ C )
% 9.03/4.95         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2078_order_Otrans,axiom,
% 9.03/4.95      ! [A: int,B: int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_int @ B @ C )
% 9.03/4.95         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2079_order__trans,axiom,
% 9.03/4.95      ! [X: set_int,Y: set_int,Z: set_int] :
% 9.03/4.95        ( ( ord_less_eq_set_int @ X @ Y )
% 9.03/4.95       => ( ( ord_less_eq_set_int @ Y @ Z )
% 9.03/4.95         => ( ord_less_eq_set_int @ X @ Z ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2080_order__trans,axiom,
% 9.03/4.95      ! [X: rat,Y: rat,Z: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.95       => ( ( ord_less_eq_rat @ Y @ Z )
% 9.03/4.95         => ( ord_less_eq_rat @ X @ Z ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2081_order__trans,axiom,
% 9.03/4.95      ! [X: num,Y: num,Z: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ X @ Y )
% 9.03/4.95       => ( ( ord_less_eq_num @ Y @ Z )
% 9.03/4.95         => ( ord_less_eq_num @ X @ Z ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2082_order__trans,axiom,
% 9.03/4.95      ! [X: nat,Y: nat,Z: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ X @ Y )
% 9.03/4.95       => ( ( ord_less_eq_nat @ Y @ Z )
% 9.03/4.95         => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2083_order__trans,axiom,
% 9.03/4.95      ! [X: int,Y: int,Z: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ X @ Y )
% 9.03/4.95       => ( ( ord_less_eq_int @ Y @ Z )
% 9.03/4.95         => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2084_linorder__wlog,axiom,
% 9.03/4.95      ! [P: rat > rat > $o,A: rat,B: rat] :
% 9.03/4.95        ( ! [A5: rat,B6: rat] :
% 9.03/4.95            ( ( ord_less_eq_rat @ A5 @ B6 )
% 9.03/4.95           => ( P @ A5 @ B6 ) )
% 9.03/4.95       => ( ! [A5: rat,B6: rat] :
% 9.03/4.95              ( ( P @ B6 @ A5 )
% 9.03/4.95             => ( P @ A5 @ B6 ) )
% 9.03/4.95         => ( P @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2085_linorder__wlog,axiom,
% 9.03/4.95      ! [P: num > num > $o,A: num,B: num] :
% 9.03/4.95        ( ! [A5: num,B6: num] :
% 9.03/4.95            ( ( ord_less_eq_num @ A5 @ B6 )
% 9.03/4.95           => ( P @ A5 @ B6 ) )
% 9.03/4.95       => ( ! [A5: num,B6: num] :
% 9.03/4.95              ( ( P @ B6 @ A5 )
% 9.03/4.95             => ( P @ A5 @ B6 ) )
% 9.03/4.95         => ( P @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2086_linorder__wlog,axiom,
% 9.03/4.95      ! [P: nat > nat > $o,A: nat,B: nat] :
% 9.03/4.95        ( ! [A5: nat,B6: nat] :
% 9.03/4.95            ( ( ord_less_eq_nat @ A5 @ B6 )
% 9.03/4.95           => ( P @ A5 @ B6 ) )
% 9.03/4.95       => ( ! [A5: nat,B6: nat] :
% 9.03/4.95              ( ( P @ B6 @ A5 )
% 9.03/4.95             => ( P @ A5 @ B6 ) )
% 9.03/4.95         => ( P @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2087_linorder__wlog,axiom,
% 9.03/4.95      ! [P: int > int > $o,A: int,B: int] :
% 9.03/4.95        ( ! [A5: int,B6: int] :
% 9.03/4.95            ( ( ord_less_eq_int @ A5 @ B6 )
% 9.03/4.95           => ( P @ A5 @ B6 ) )
% 9.03/4.95       => ( ! [A5: int,B6: int] :
% 9.03/4.95              ( ( P @ B6 @ A5 )
% 9.03/4.95             => ( P @ A5 @ B6 ) )
% 9.03/4.95         => ( P @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2088_dual__order_Oeq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: set_int,Z2: set_int] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [A3: set_int,B4: set_int] :
% 9.03/4.95            ( ( ord_less_eq_set_int @ B4 @ A3 )
% 9.03/4.95            & ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2089_dual__order_Oeq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [A3: rat,B4: rat] :
% 9.03/4.95            ( ( ord_less_eq_rat @ B4 @ A3 )
% 9.03/4.95            & ( ord_less_eq_rat @ A3 @ B4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2090_dual__order_Oeq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [A3: num,B4: num] :
% 9.03/4.95            ( ( ord_less_eq_num @ B4 @ A3 )
% 9.03/4.95            & ( ord_less_eq_num @ A3 @ B4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2091_dual__order_Oeq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [A3: nat,B4: nat] :
% 9.03/4.95            ( ( ord_less_eq_nat @ B4 @ A3 )
% 9.03/4.95            & ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2092_dual__order_Oeq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [A3: int,B4: int] :
% 9.03/4.95            ( ( ord_less_eq_int @ B4 @ A3 )
% 9.03/4.95            & ( ord_less_eq_int @ A3 @ B4 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2093_dual__order_Oantisym,axiom,
% 9.03/4.95      ! [B: set_int,A: set_int] :
% 9.03/4.95        ( ( ord_less_eq_set_int @ B @ A )
% 9.03/4.95       => ( ( ord_less_eq_set_int @ A @ B )
% 9.03/4.95         => ( A = B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2094_dual__order_Oantisym,axiom,
% 9.03/4.95      ! [B: rat,A: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.95       => ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95         => ( A = B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2095_dual__order_Oantisym,axiom,
% 9.03/4.95      ! [B: num,A: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ B @ A )
% 9.03/4.95       => ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95         => ( A = B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2096_dual__order_Oantisym,axiom,
% 9.03/4.95      ! [B: nat,A: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ B @ A )
% 9.03/4.95       => ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95         => ( A = B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2097_dual__order_Oantisym,axiom,
% 9.03/4.95      ! [B: int,A: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ B @ A )
% 9.03/4.95       => ( ( ord_less_eq_int @ A @ B )
% 9.03/4.95         => ( A = B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2098_dual__order_Otrans,axiom,
% 9.03/4.95      ! [B: set_int,A: set_int,C: set_int] :
% 9.03/4.95        ( ( ord_less_eq_set_int @ B @ A )
% 9.03/4.95       => ( ( ord_less_eq_set_int @ C @ B )
% 9.03/4.95         => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2099_dual__order_Otrans,axiom,
% 9.03/4.95      ! [B: rat,A: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.95       => ( ( ord_less_eq_rat @ C @ B )
% 9.03/4.95         => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2100_dual__order_Otrans,axiom,
% 9.03/4.95      ! [B: num,A: num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ B @ A )
% 9.03/4.95       => ( ( ord_less_eq_num @ C @ B )
% 9.03/4.95         => ( ord_less_eq_num @ C @ A ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2101_dual__order_Otrans,axiom,
% 9.03/4.95      ! [B: nat,A: nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ B @ A )
% 9.03/4.95       => ( ( ord_less_eq_nat @ C @ B )
% 9.03/4.95         => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2102_dual__order_Otrans,axiom,
% 9.03/4.95      ! [B: int,A: int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ B @ A )
% 9.03/4.95       => ( ( ord_less_eq_int @ C @ B )
% 9.03/4.95         => ( ord_less_eq_int @ C @ A ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2103_antisym,axiom,
% 9.03/4.95      ! [A: set_int,B: set_int] :
% 9.03/4.95        ( ( ord_less_eq_set_int @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_set_int @ B @ A )
% 9.03/4.95         => ( A = B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2104_antisym,axiom,
% 9.03/4.95      ! [A: rat,B: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.95         => ( A = B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2105_antisym,axiom,
% 9.03/4.95      ! [A: num,B: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_num @ B @ A )
% 9.03/4.95         => ( A = B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2106_antisym,axiom,
% 9.03/4.95      ! [A: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ A )
% 9.03/4.95         => ( A = B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2107_antisym,axiom,
% 9.03/4.95      ! [A: int,B: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_int @ B @ A )
% 9.03/4.95         => ( A = B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2108_Orderings_Oorder__eq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: set_int,Z2: set_int] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [A3: set_int,B4: set_int] :
% 9.03/4.95            ( ( ord_less_eq_set_int @ A3 @ B4 )
% 9.03/4.95            & ( ord_less_eq_set_int @ B4 @ A3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2109_Orderings_Oorder__eq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [A3: rat,B4: rat] :
% 9.03/4.95            ( ( ord_less_eq_rat @ A3 @ B4 )
% 9.03/4.95            & ( ord_less_eq_rat @ B4 @ A3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2110_Orderings_Oorder__eq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [A3: num,B4: num] :
% 9.03/4.95            ( ( ord_less_eq_num @ A3 @ B4 )
% 9.03/4.95            & ( ord_less_eq_num @ B4 @ A3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2111_Orderings_Oorder__eq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [A3: nat,B4: nat] :
% 9.03/4.95            ( ( ord_less_eq_nat @ A3 @ B4 )
% 9.03/4.95            & ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2112_Orderings_Oorder__eq__iff,axiom,
% 9.03/4.95      ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 9.03/4.95      = ( ^ [A3: int,B4: int] :
% 9.03/4.95            ( ( ord_less_eq_int @ A3 @ B4 )
% 9.03/4.95            & ( ord_less_eq_int @ B4 @ A3 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2113_order__subst1,axiom,
% 9.03/4.95      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2114_order__subst1,axiom,
% 9.03/4.95      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2115_order__subst1,axiom,
% 9.03/4.95      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ C )
% 9.03/4.95         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.95                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2116_order__subst1,axiom,
% 9.03/4.95      ! [A: rat,F: int > rat,B: int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_int @ B @ C )
% 9.03/4.95         => ( ! [X3: int,Y4: int] :
% 9.03/4.95                ( ( ord_less_eq_int @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2117_order__subst1,axiom,
% 9.03/4.95      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2118_order__subst1,axiom,
% 9.03/4.95      ! [A: num,F: num > num,B: num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2119_order__subst1,axiom,
% 9.03/4.95      ! [A: num,F: nat > num,B: nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ C )
% 9.03/4.95         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.95                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2120_order__subst1,axiom,
% 9.03/4.95      ! [A: num,F: int > num,B: int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_int @ B @ C )
% 9.03/4.95         => ( ! [X3: int,Y4: int] :
% 9.03/4.95                ( ( ord_less_eq_int @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2121_order__subst1,axiom,
% 9.03/4.95      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2122_order__subst1,axiom,
% 9.03/4.95      ! [A: nat,F: num > nat,B: num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2123_order__subst2,axiom,
% 9.03/4.95      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2124_order__subst2,axiom,
% 9.03/4.95      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2125_order__subst2,axiom,
% 9.03/4.95      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2126_order__subst2,axiom,
% 9.03/4.95      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2127_order__subst2,axiom,
% 9.03/4.95      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2128_order__subst2,axiom,
% 9.03/4.95      ! [A: num,B: num,F: num > num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2129_order__subst2,axiom,
% 9.03/4.95      ! [A: num,B: num,F: num > nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2130_order__subst2,axiom,
% 9.03/4.95      ! [A: num,B: num,F: num > int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2131_order__subst2,axiom,
% 9.03/4.95      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.03/4.95         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.95                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2132_order__subst2,axiom,
% 9.03/4.95      ! [A: nat,B: nat,F: nat > num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 9.03/4.95         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.95                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2133_order__eq__refl,axiom,
% 9.03/4.95      ! [X: set_int,Y: set_int] :
% 9.03/4.95        ( ( X = Y )
% 9.03/4.95       => ( ord_less_eq_set_int @ X @ Y ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2134_order__eq__refl,axiom,
% 9.03/4.95      ! [X: rat,Y: rat] :
% 9.03/4.95        ( ( X = Y )
% 9.03/4.95       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2135_order__eq__refl,axiom,
% 9.03/4.95      ! [X: num,Y: num] :
% 9.03/4.95        ( ( X = Y )
% 9.03/4.95       => ( ord_less_eq_num @ X @ Y ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2136_order__eq__refl,axiom,
% 9.03/4.95      ! [X: nat,Y: nat] :
% 9.03/4.95        ( ( X = Y )
% 9.03/4.95       => ( ord_less_eq_nat @ X @ Y ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2137_order__eq__refl,axiom,
% 9.03/4.95      ! [X: int,Y: int] :
% 9.03/4.95        ( ( X = Y )
% 9.03/4.95       => ( ord_less_eq_int @ X @ Y ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2138_linorder__linear,axiom,
% 9.03/4.95      ! [X: rat,Y: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.95        | ( ord_less_eq_rat @ Y @ X ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2139_linorder__linear,axiom,
% 9.03/4.95      ! [X: num,Y: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ X @ Y )
% 9.03/4.95        | ( ord_less_eq_num @ Y @ X ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2140_linorder__linear,axiom,
% 9.03/4.95      ! [X: nat,Y: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ X @ Y )
% 9.03/4.95        | ( ord_less_eq_nat @ Y @ X ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2141_linorder__linear,axiom,
% 9.03/4.95      ! [X: int,Y: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ X @ Y )
% 9.03/4.95        | ( ord_less_eq_int @ Y @ X ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2142_ord__eq__le__subst,axiom,
% 9.03/4.95      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.03/4.95        ( ( A
% 9.03/4.95          = ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2143_ord__eq__le__subst,axiom,
% 9.03/4.95      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.03/4.95        ( ( A
% 9.03/4.95          = ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2144_ord__eq__le__subst,axiom,
% 9.03/4.95      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.03/4.95        ( ( A
% 9.03/4.95          = ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2145_ord__eq__le__subst,axiom,
% 9.03/4.95      ! [A: int,F: rat > int,B: rat,C: rat] :
% 9.03/4.95        ( ( A
% 9.03/4.95          = ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ B @ C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2146_ord__eq__le__subst,axiom,
% 9.03/4.95      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.03/4.95        ( ( A
% 9.03/4.95          = ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2147_ord__eq__le__subst,axiom,
% 9.03/4.95      ! [A: num,F: num > num,B: num,C: num] :
% 9.03/4.95        ( ( A
% 9.03/4.95          = ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2148_ord__eq__le__subst,axiom,
% 9.03/4.95      ! [A: nat,F: num > nat,B: num,C: num] :
% 9.03/4.95        ( ( A
% 9.03/4.95          = ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2149_ord__eq__le__subst,axiom,
% 9.03/4.95      ! [A: int,F: num > int,B: num,C: num] :
% 9.03/4.95        ( ( A
% 9.03/4.95          = ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_num @ B @ C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2150_ord__eq__le__subst,axiom,
% 9.03/4.95      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.03/4.95        ( ( A
% 9.03/4.95          = ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ C )
% 9.03/4.95         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.95                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2151_ord__eq__le__subst,axiom,
% 9.03/4.95      ! [A: num,F: nat > num,B: nat,C: nat] :
% 9.03/4.95        ( ( A
% 9.03/4.95          = ( F @ B ) )
% 9.03/4.95       => ( ( ord_less_eq_nat @ B @ C )
% 9.03/4.95         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.95                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2152_ord__le__eq__subst,axiom,
% 9.03/4.95      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ( F @ B )
% 9.03/4.95            = C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2153_ord__le__eq__subst,axiom,
% 9.03/4.95      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ( F @ B )
% 9.03/4.95            = C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2154_ord__le__eq__subst,axiom,
% 9.03/4.95      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ( F @ B )
% 9.03/4.95            = C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2155_ord__le__eq__subst,axiom,
% 9.03/4.95      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.95       => ( ( ( F @ B )
% 9.03/4.95            = C )
% 9.03/4.95         => ( ! [X3: rat,Y4: rat] :
% 9.03/4.95                ( ( ord_less_eq_rat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2156_ord__le__eq__subst,axiom,
% 9.03/4.95      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( ( F @ B )
% 9.03/4.95            = C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2157_ord__le__eq__subst,axiom,
% 9.03/4.95      ! [A: num,B: num,F: num > num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( ( F @ B )
% 9.03/4.95            = C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2158_ord__le__eq__subst,axiom,
% 9.03/4.95      ! [A: num,B: num,F: num > nat,C: nat] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( ( F @ B )
% 9.03/4.95            = C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2159_ord__le__eq__subst,axiom,
% 9.03/4.95      ! [A: num,B: num,F: num > int,C: int] :
% 9.03/4.95        ( ( ord_less_eq_num @ A @ B )
% 9.03/4.95       => ( ( ( F @ B )
% 9.03/4.95            = C )
% 9.03/4.95         => ( ! [X3: num,Y4: num] :
% 9.03/4.95                ( ( ord_less_eq_num @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2160_ord__le__eq__subst,axiom,
% 9.03/4.95      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95       => ( ( ( F @ B )
% 9.03/4.95            = C )
% 9.03/4.95         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.95                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2161_ord__le__eq__subst,axiom,
% 9.03/4.95      ! [A: nat,B: nat,F: nat > num,C: num] :
% 9.03/4.95        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.95       => ( ( ( F @ B )
% 9.03/4.95            = C )
% 9.03/4.95         => ( ! [X3: nat,Y4: nat] :
% 9.03/4.95                ( ( ord_less_eq_nat @ X3 @ Y4 )
% 9.03/4.95               => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.03/4.95           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2162_linorder__le__cases,axiom,
% 9.03/4.95      ! [X: rat,Y: rat] :
% 9.03/4.95        ( ~ ( ord_less_eq_rat @ X @ Y )
% 9.03/4.95       => ( ord_less_eq_rat @ Y @ X ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2163_linorder__le__cases,axiom,
% 9.03/4.95      ! [X: num,Y: num] :
% 9.03/4.95        ( ~ ( ord_less_eq_num @ X @ Y )
% 9.03/4.95       => ( ord_less_eq_num @ Y @ X ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2164_linorder__le__cases,axiom,
% 9.03/4.95      ! [X: nat,Y: nat] :
% 9.03/4.95        ( ~ ( ord_less_eq_nat @ X @ Y )
% 9.03/4.95       => ( ord_less_eq_nat @ Y @ X ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2165_linorder__le__cases,axiom,
% 9.03/4.95      ! [X: int,Y: int] :
% 9.03/4.95        ( ~ ( ord_less_eq_int @ X @ Y )
% 9.03/4.95       => ( ord_less_eq_int @ Y @ X ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2166_order__antisym__conv,axiom,
% 9.03/4.95      ! [Y: set_int,X: set_int] :
% 9.03/4.95        ( ( ord_less_eq_set_int @ Y @ X )
% 9.03/4.95       => ( ( ord_less_eq_set_int @ X @ Y )
% 9.03/4.95          = ( X = Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2167_order__antisym__conv,axiom,
% 9.03/4.95      ! [Y: rat,X: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ Y @ X )
% 9.03/4.95       => ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.95          = ( X = Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2168_order__antisym__conv,axiom,
% 9.03/4.95      ! [Y: num,X: num] :
% 9.03/4.95        ( ( ord_less_eq_num @ Y @ X )
% 9.03/4.95       => ( ( ord_less_eq_num @ X @ Y )
% 9.03/4.95          = ( X = Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2169_order__antisym__conv,axiom,
% 9.03/4.95      ! [Y: nat,X: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ Y @ X )
% 9.03/4.95       => ( ( ord_less_eq_nat @ X @ Y )
% 9.03/4.95          = ( X = Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2170_order__antisym__conv,axiom,
% 9.03/4.95      ! [Y: int,X: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ Y @ X )
% 9.03/4.95       => ( ( ord_less_eq_int @ X @ Y )
% 9.03/4.95          = ( X = Y ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2171_power__le__one,axiom,
% 9.03/4.95      ! [A: real,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.95       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.03/4.95         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2172_power__le__one,axiom,
% 9.03/4.95      ! [A: rat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.95       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.03/4.95         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2173_power__le__one,axiom,
% 9.03/4.95      ! [A: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.95       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.03/4.95         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2174_power__le__one,axiom,
% 9.03/4.95      ! [A: int,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.95       => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.03/4.95         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2175_power__le__imp__le__base,axiom,
% 9.03/4.95      ! [A: real,N: nat,B: real] :
% 9.03/4.95        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 9.03/4.95       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.95         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2176_power__le__imp__le__base,axiom,
% 9.03/4.95      ! [A: rat,N: nat,B: rat] :
% 9.03/4.95        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.95         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2177_power__le__imp__le__base,axiom,
% 9.03/4.95      ! [A: nat,N: nat,B: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 9.03/4.95       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.95         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2178_power__le__imp__le__base,axiom,
% 9.03/4.95      ! [A: int,N: nat,B: int] :
% 9.03/4.95        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 9.03/4.95       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.95         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2179_power__inject__base,axiom,
% 9.03/4.95      ! [A: real,N: nat,B: real] :
% 9.03/4.95        ( ( ( power_power_real @ A @ ( suc @ N ) )
% 9.03/4.95          = ( power_power_real @ B @ ( suc @ N ) ) )
% 9.03/4.95       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.95         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.95           => ( A = B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2180_power__inject__base,axiom,
% 9.03/4.95      ! [A: rat,N: nat,B: rat] :
% 9.03/4.95        ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 9.03/4.95          = ( power_power_rat @ B @ ( suc @ N ) ) )
% 9.03/4.95       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.95         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.95           => ( A = B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2181_power__inject__base,axiom,
% 9.03/4.95      ! [A: nat,N: nat,B: nat] :
% 9.03/4.95        ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 9.03/4.95          = ( power_power_nat @ B @ ( suc @ N ) ) )
% 9.03/4.95       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.95         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.95           => ( A = B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2182_power__inject__base,axiom,
% 9.03/4.95      ! [A: int,N: nat,B: int] :
% 9.03/4.95        ( ( ( power_power_int @ A @ ( suc @ N ) )
% 9.03/4.95          = ( power_power_int @ B @ ( suc @ N ) ) )
% 9.03/4.95       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.95         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.95           => ( A = B ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2183_numeral__1__eq__Suc__0,axiom,
% 9.03/4.95      ( ( numeral_numeral_nat @ one )
% 9.03/4.95      = ( suc @ zero_zero_nat ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2184_subset__code_I1_J,axiom,
% 9.03/4.95      ! [Xs: list_complex,B2: set_complex] :
% 9.03/4.95        ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B2 )
% 9.03/4.95        = ( ! [X4: complex] :
% 9.03/4.95              ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
% 9.03/4.95             => ( member_complex @ X4 @ B2 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2185_subset__code_I1_J,axiom,
% 9.03/4.95      ! [Xs: list_real,B2: set_real] :
% 9.03/4.95        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
% 9.03/4.95        = ( ! [X4: real] :
% 9.03/4.95              ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 9.03/4.95             => ( member_real @ X4 @ B2 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2186_subset__code_I1_J,axiom,
% 9.03/4.95      ! [Xs: list_set_nat,B2: set_set_nat] :
% 9.03/4.95        ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
% 9.03/4.95        = ( ! [X4: set_nat] :
% 9.03/4.95              ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 9.03/4.95             => ( member_set_nat @ X4 @ B2 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2187_subset__code_I1_J,axiom,
% 9.03/4.95      ! [Xs: list_nat,B2: set_nat] :
% 9.03/4.95        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
% 9.03/4.95        = ( ! [X4: nat] :
% 9.03/4.95              ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 9.03/4.95             => ( member_nat @ X4 @ B2 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2188_subset__code_I1_J,axiom,
% 9.03/4.95      ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.95        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
% 9.03/4.95        = ( ! [X4: vEBT_VEBT] :
% 9.03/4.95              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.03/4.95             => ( member_VEBT_VEBT @ X4 @ B2 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2189_subset__code_I1_J,axiom,
% 9.03/4.95      ! [Xs: list_int,B2: set_int] :
% 9.03/4.95        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
% 9.03/4.95        = ( ! [X4: int] :
% 9.03/4.95              ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 9.03/4.95             => ( member_int @ X4 @ B2 ) ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2190_num_Osize_I5_J,axiom,
% 9.03/4.95      ! [X22: num] :
% 9.03/4.95        ( ( size_size_num @ ( bit0 @ X22 ) )
% 9.03/4.95        = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2191_Ex__list__of__length,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95      ? [Xs2: list_VEBT_VEBT] :
% 9.03/4.95        ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.03/4.95        = N ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2192_Ex__list__of__length,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95      ? [Xs2: list_o] :
% 9.03/4.95        ( ( size_size_list_o @ Xs2 )
% 9.03/4.95        = N ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2193_Ex__list__of__length,axiom,
% 9.03/4.95      ! [N: nat] :
% 9.03/4.95      ? [Xs2: list_int] :
% 9.03/4.95        ( ( size_size_list_int @ Xs2 )
% 9.03/4.95        = N ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2194_neq__if__length__neq,axiom,
% 9.03/4.95      ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 9.03/4.95        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.03/4.95         != ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 9.03/4.95       => ( Xs != Ys2 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2195_neq__if__length__neq,axiom,
% 9.03/4.95      ! [Xs: list_o,Ys2: list_o] :
% 9.03/4.95        ( ( ( size_size_list_o @ Xs )
% 9.03/4.95         != ( size_size_list_o @ Ys2 ) )
% 9.03/4.95       => ( Xs != Ys2 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2196_neq__if__length__neq,axiom,
% 9.03/4.95      ! [Xs: list_int,Ys2: list_int] :
% 9.03/4.95        ( ( ( size_size_list_int @ Xs )
% 9.03/4.95         != ( size_size_list_int @ Ys2 ) )
% 9.03/4.95       => ( Xs != Ys2 ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2197_nat__one__le__power,axiom,
% 9.03/4.95      ! [I: nat,N: nat] :
% 9.03/4.95        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 9.03/4.95       => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2198_of__int__ceiling__le__add__one,axiom,
% 9.03/4.95      ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2199_of__int__ceiling__le__add__one,axiom,
% 9.03/4.95      ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2200_field__sum__of__halves,axiom,
% 9.03/4.95      ! [X: real] :
% 9.03/4.95        ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.03/4.95        = X ) ).
% 9.03/4.95  
% 9.03/4.95  thf(fact_2201_field__sum__of__halves,axiom,
% 9.03/4.95      ! [X: rat] :
% 9.03/4.95        ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.03/4.96        = X ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2202_power__le__imp__le__exp,axiom,
% 9.03/4.96      ! [A: real,M: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 9.03/4.96         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2203_power__le__imp__le__exp,axiom,
% 9.03/4.96      ! [A: rat,M: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_rat @ one_one_rat @ A )
% 9.03/4.96       => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 9.03/4.96         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2204_power__le__imp__le__exp,axiom,
% 9.03/4.96      ! [A: nat,M: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_nat @ one_one_nat @ A )
% 9.03/4.96       => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.03/4.96         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2205_power__le__imp__le__exp,axiom,
% 9.03/4.96      ! [A: int,M: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_int @ one_one_int @ A )
% 9.03/4.96       => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.03/4.96         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2206_less__exp,axiom,
% 9.03/4.96      ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2207_power__Suc__le__self,axiom,
% 9.03/4.96      ! [A: real,N: nat] :
% 9.03/4.96        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.96       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.03/4.96         => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2208_power__Suc__le__self,axiom,
% 9.03/4.96      ! [A: rat,N: nat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.96       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.03/4.96         => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2209_power__Suc__le__self,axiom,
% 9.03/4.96      ! [A: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.96       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.03/4.96         => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2210_power__Suc__le__self,axiom,
% 9.03/4.96      ! [A: int,N: nat] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.96       => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.03/4.96         => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2211_power__decreasing,axiom,
% 9.03/4.96      ! [N: nat,N4: nat,A: real] :
% 9.03/4.96        ( ( ord_less_eq_nat @ N @ N4 )
% 9.03/4.96       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.96         => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.03/4.96           => ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2212_power__decreasing,axiom,
% 9.03/4.96      ! [N: nat,N4: nat,A: rat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ N @ N4 )
% 9.03/4.96       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.96         => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.03/4.96           => ( ord_less_eq_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2213_power__decreasing,axiom,
% 9.03/4.96      ! [N: nat,N4: nat,A: nat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ N @ N4 )
% 9.03/4.96       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.96         => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.03/4.96           => ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2214_power__decreasing,axiom,
% 9.03/4.96      ! [N: nat,N4: nat,A: int] :
% 9.03/4.96        ( ( ord_less_eq_nat @ N @ N4 )
% 9.03/4.96       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.96         => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.03/4.96           => ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2215_zero__power2,axiom,
% 9.03/4.96      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96      = zero_zero_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2216_zero__power2,axiom,
% 9.03/4.96      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96      = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2217_zero__power2,axiom,
% 9.03/4.96      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96      = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2218_zero__power2,axiom,
% 9.03/4.96      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96      = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2219_zero__power2,axiom,
% 9.03/4.96      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96      = zero_zero_complex ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2220_numeral__2__eq__2,axiom,
% 9.03/4.96      ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 9.03/4.96      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2221_log2__of__power__le,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/4.96       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.96         => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2222_of__nat__less__two__power,axiom,
% 9.03/4.96      ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2223_of__nat__less__two__power,axiom,
% 9.03/4.96      ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2224_of__nat__less__two__power,axiom,
% 9.03/4.96      ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2225_zero__le__power2,axiom,
% 9.03/4.96      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2226_zero__le__power2,axiom,
% 9.03/4.96      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2227_zero__le__power2,axiom,
% 9.03/4.96      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2228_power2__eq__imp__eq,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96          = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.96       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.96           => ( X = Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2229_power2__eq__imp__eq,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96          = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.96       => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.96         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.96           => ( X = Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2230_power2__eq__imp__eq,axiom,
% 9.03/4.96      ! [X: nat,Y: nat] :
% 9.03/4.96        ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96          = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.96       => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 9.03/4.96         => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 9.03/4.96           => ( X = Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2231_power2__eq__imp__eq,axiom,
% 9.03/4.96      ! [X: int,Y: int] :
% 9.03/4.96        ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96          = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.96       => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.03/4.96         => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.03/4.96           => ( X = Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2232_power2__le__imp__le,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.96       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.96         => ( ord_less_eq_real @ X @ Y ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2233_power2__le__imp__le,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.96       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.96         => ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2234_power2__le__imp__le,axiom,
% 9.03/4.96      ! [X: nat,Y: nat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.96       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 9.03/4.96         => ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2235_power2__le__imp__le,axiom,
% 9.03/4.96      ! [X: int,Y: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.96       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.03/4.96         => ( ord_less_eq_int @ X @ Y ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2236_sum__power2__le__zero__iff,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 9.03/4.96        = ( ( X = zero_zero_real )
% 9.03/4.96          & ( Y = zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2237_sum__power2__le__zero__iff,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 9.03/4.96        = ( ( X = zero_zero_rat )
% 9.03/4.96          & ( Y = zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2238_sum__power2__le__zero__iff,axiom,
% 9.03/4.96      ! [X: int,Y: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 9.03/4.96        = ( ( X = zero_zero_int )
% 9.03/4.96          & ( Y = zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2239_sum__power2__ge__zero,axiom,
% 9.03/4.96      ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2240_sum__power2__ge__zero,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2241_sum__power2__ge__zero,axiom,
% 9.03/4.96      ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2242_one__div__two__eq__zero,axiom,
% 9.03/4.96      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96      = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2243_one__div__two__eq__zero,axiom,
% 9.03/4.96      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/4.96      = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2244_bits__1__div__2,axiom,
% 9.03/4.96      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96      = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2245_bits__1__div__2,axiom,
% 9.03/4.96      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/4.96      = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2246_add__self__div__2,axiom,
% 9.03/4.96      ! [M: nat] :
% 9.03/4.96        ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96        = M ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2247_div2__Suc__Suc,axiom,
% 9.03/4.96      ! [M: nat] :
% 9.03/4.96        ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96        = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2248_le__divide__eq__1__pos,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.03/4.96          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2249_le__divide__eq__1__pos,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.96       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.03/4.96          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2250_le__divide__eq__1__neg,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.03/4.96          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2251_le__divide__eq__1__neg,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.03/4.96          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2252_divide__le__eq__1__pos,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.03/4.96          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2253_divide__le__eq__1__pos,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.96       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.03/4.96          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2254_divide__le__eq__1__neg,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.03/4.96          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2255_divide__le__eq__1__neg,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.03/4.96          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2256_zero__less__divide__1__iff,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 9.03/4.96        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2257_zero__less__divide__1__iff,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 9.03/4.96        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2258_less__divide__eq__1__pos,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.03/4.96          = ( ord_less_real @ A @ B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2259_less__divide__eq__1__pos,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.96       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.03/4.96          = ( ord_less_rat @ A @ B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2260_less__divide__eq__1__neg,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.03/4.96          = ( ord_less_real @ B @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2261_less__divide__eq__1__neg,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.03/4.96          = ( ord_less_rat @ B @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2262_psubsetI,axiom,
% 9.03/4.96      ! [A2: set_int,B2: set_int] :
% 9.03/4.96        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.96       => ( ( A2 != B2 )
% 9.03/4.96         => ( ord_less_set_int @ A2 @ B2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2263_division__ring__divide__zero,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 9.03/4.96        = zero_zero_complex ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2264_division__ring__divide__zero,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( divide_divide_real @ A @ zero_zero_real )
% 9.03/4.96        = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2265_division__ring__divide__zero,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 9.03/4.96        = zero_zero_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2266_bits__div__by__0,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2267_bits__div__by__0,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( divide_divide_int @ A @ zero_zero_int )
% 9.03/4.96        = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2268_bits__div__0,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2269_bits__div__0,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( divide_divide_int @ zero_zero_int @ A )
% 9.03/4.96        = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2270_divide__cancel__right,axiom,
% 9.03/4.96      ! [A: complex,C: complex,B: complex] :
% 9.03/4.96        ( ( ( divide1717551699836669952omplex @ A @ C )
% 9.03/4.96          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_complex )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2271_divide__cancel__right,axiom,
% 9.03/4.96      ! [A: real,C: real,B: real] :
% 9.03/4.96        ( ( ( divide_divide_real @ A @ C )
% 9.03/4.96          = ( divide_divide_real @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_real )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2272_divide__cancel__right,axiom,
% 9.03/4.96      ! [A: rat,C: rat,B: rat] :
% 9.03/4.96        ( ( ( divide_divide_rat @ A @ C )
% 9.03/4.96          = ( divide_divide_rat @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_rat )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2273_divide__cancel__left,axiom,
% 9.03/4.96      ! [C: complex,A: complex,B: complex] :
% 9.03/4.96        ( ( ( divide1717551699836669952omplex @ C @ A )
% 9.03/4.96          = ( divide1717551699836669952omplex @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_complex )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2274_divide__cancel__left,axiom,
% 9.03/4.96      ! [C: real,A: real,B: real] :
% 9.03/4.96        ( ( ( divide_divide_real @ C @ A )
% 9.03/4.96          = ( divide_divide_real @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_real )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2275_divide__cancel__left,axiom,
% 9.03/4.96      ! [C: rat,A: rat,B: rat] :
% 9.03/4.96        ( ( ( divide_divide_rat @ C @ A )
% 9.03/4.96          = ( divide_divide_rat @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_rat )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2276_divide__eq__0__iff,axiom,
% 9.03/4.96      ! [A: complex,B: complex] :
% 9.03/4.96        ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.03/4.96          = zero_zero_complex )
% 9.03/4.96        = ( ( A = zero_zero_complex )
% 9.03/4.96          | ( B = zero_zero_complex ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2277_divide__eq__0__iff,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ( divide_divide_real @ A @ B )
% 9.03/4.96          = zero_zero_real )
% 9.03/4.96        = ( ( A = zero_zero_real )
% 9.03/4.96          | ( B = zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2278_divide__eq__0__iff,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ( divide_divide_rat @ A @ B )
% 9.03/4.96          = zero_zero_rat )
% 9.03/4.96        = ( ( A = zero_zero_rat )
% 9.03/4.96          | ( B = zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2279_bits__div__by__1,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( divide_divide_nat @ A @ one_one_nat )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2280_bits__div__by__1,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( divide_divide_int @ A @ one_one_int )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2281_log__eq__one,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96       => ( ( A != one_one_real )
% 9.03/4.96         => ( ( log @ A @ A )
% 9.03/4.96            = one_one_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2282_log__less__cancel__iff,axiom,
% 9.03/4.96      ! [A: real,X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.96           => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 9.03/4.96              = ( ord_less_real @ X @ Y ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2283_log__less__one__cancel__iff,axiom,
% 9.03/4.96      ! [A: real,X: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
% 9.03/4.96            = ( ord_less_real @ X @ A ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2284_one__less__log__cancel__iff,axiom,
% 9.03/4.96      ! [A: real,X: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
% 9.03/4.96            = ( ord_less_real @ A @ X ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2285_log__less__zero__cancel__iff,axiom,
% 9.03/4.96      ! [A: real,X: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
% 9.03/4.96            = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2286_zero__less__log__cancel__iff,axiom,
% 9.03/4.96      ! [A: real,X: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
% 9.03/4.96            = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2287_log__one,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( log @ A @ one_one_real )
% 9.03/4.96        = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2288_semiring__norm_I78_J,axiom,
% 9.03/4.96      ! [M: num,N: num] :
% 9.03/4.96        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.03/4.96        = ( ord_less_num @ M @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2289_semiring__norm_I75_J,axiom,
% 9.03/4.96      ! [M: num] :
% 9.03/4.96        ~ ( ord_less_num @ M @ one ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2290_divide__eq__1__iff,axiom,
% 9.03/4.96      ! [A: complex,B: complex] :
% 9.03/4.96        ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.03/4.96          = one_one_complex )
% 9.03/4.96        = ( ( B != zero_zero_complex )
% 9.03/4.96          & ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2291_divide__eq__1__iff,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ( divide_divide_real @ A @ B )
% 9.03/4.96          = one_one_real )
% 9.03/4.96        = ( ( B != zero_zero_real )
% 9.03/4.96          & ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2292_divide__eq__1__iff,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ( divide_divide_rat @ A @ B )
% 9.03/4.96          = one_one_rat )
% 9.03/4.96        = ( ( B != zero_zero_rat )
% 9.03/4.96          & ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2293_one__eq__divide__iff,axiom,
% 9.03/4.96      ! [A: complex,B: complex] :
% 9.03/4.96        ( ( one_one_complex
% 9.03/4.96          = ( divide1717551699836669952omplex @ A @ B ) )
% 9.03/4.96        = ( ( B != zero_zero_complex )
% 9.03/4.96          & ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2294_one__eq__divide__iff,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( one_one_real
% 9.03/4.96          = ( divide_divide_real @ A @ B ) )
% 9.03/4.96        = ( ( B != zero_zero_real )
% 9.03/4.96          & ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2295_one__eq__divide__iff,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( one_one_rat
% 9.03/4.96          = ( divide_divide_rat @ A @ B ) )
% 9.03/4.96        = ( ( B != zero_zero_rat )
% 9.03/4.96          & ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2296_divide__self,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( A != zero_zero_complex )
% 9.03/4.96       => ( ( divide1717551699836669952omplex @ A @ A )
% 9.03/4.96          = one_one_complex ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2297_divide__self,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( A != zero_zero_real )
% 9.03/4.96       => ( ( divide_divide_real @ A @ A )
% 9.03/4.96          = one_one_real ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2298_divide__self,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( A != zero_zero_rat )
% 9.03/4.96       => ( ( divide_divide_rat @ A @ A )
% 9.03/4.96          = one_one_rat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2299_divide__self__if,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( ( A = zero_zero_complex )
% 9.03/4.96         => ( ( divide1717551699836669952omplex @ A @ A )
% 9.03/4.96            = zero_zero_complex ) )
% 9.03/4.96        & ( ( A != zero_zero_complex )
% 9.03/4.96         => ( ( divide1717551699836669952omplex @ A @ A )
% 9.03/4.96            = one_one_complex ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2300_divide__self__if,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( ( A = zero_zero_real )
% 9.03/4.96         => ( ( divide_divide_real @ A @ A )
% 9.03/4.96            = zero_zero_real ) )
% 9.03/4.96        & ( ( A != zero_zero_real )
% 9.03/4.96         => ( ( divide_divide_real @ A @ A )
% 9.03/4.96            = one_one_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2301_divide__self__if,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( ( A = zero_zero_rat )
% 9.03/4.96         => ( ( divide_divide_rat @ A @ A )
% 9.03/4.96            = zero_zero_rat ) )
% 9.03/4.96        & ( ( A != zero_zero_rat )
% 9.03/4.96         => ( ( divide_divide_rat @ A @ A )
% 9.03/4.96            = one_one_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2302_divide__eq__eq__1,axiom,
% 9.03/4.96      ! [B: real,A: real] :
% 9.03/4.96        ( ( ( divide_divide_real @ B @ A )
% 9.03/4.96          = one_one_real )
% 9.03/4.96        = ( ( A != zero_zero_real )
% 9.03/4.96          & ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2303_divide__eq__eq__1,axiom,
% 9.03/4.96      ! [B: rat,A: rat] :
% 9.03/4.96        ( ( ( divide_divide_rat @ B @ A )
% 9.03/4.96          = one_one_rat )
% 9.03/4.96        = ( ( A != zero_zero_rat )
% 9.03/4.96          & ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2304_eq__divide__eq__1,axiom,
% 9.03/4.96      ! [B: real,A: real] :
% 9.03/4.96        ( ( one_one_real
% 9.03/4.96          = ( divide_divide_real @ B @ A ) )
% 9.03/4.96        = ( ( A != zero_zero_real )
% 9.03/4.96          & ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2305_eq__divide__eq__1,axiom,
% 9.03/4.96      ! [B: rat,A: rat] :
% 9.03/4.96        ( ( one_one_rat
% 9.03/4.96          = ( divide_divide_rat @ B @ A ) )
% 9.03/4.96        = ( ( A != zero_zero_rat )
% 9.03/4.96          & ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2306_one__divide__eq__0__iff,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( ( divide_divide_real @ one_one_real @ A )
% 9.03/4.96          = zero_zero_real )
% 9.03/4.96        = ( A = zero_zero_real ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2307_one__divide__eq__0__iff,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( ( divide_divide_rat @ one_one_rat @ A )
% 9.03/4.96          = zero_zero_rat )
% 9.03/4.96        = ( A = zero_zero_rat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2308_zero__eq__1__divide__iff,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( zero_zero_real
% 9.03/4.96          = ( divide_divide_real @ one_one_real @ A ) )
% 9.03/4.96        = ( A = zero_zero_real ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2309_zero__eq__1__divide__iff,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( zero_zero_rat
% 9.03/4.96          = ( divide_divide_rat @ one_one_rat @ A ) )
% 9.03/4.96        = ( A = zero_zero_rat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2310_div__by__Suc__0,axiom,
% 9.03/4.96      ! [M: nat] :
% 9.03/4.96        ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.03/4.96        = M ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2311_div__less,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_nat @ M @ N )
% 9.03/4.96       => ( ( divide_divide_nat @ M @ N )
% 9.03/4.96          = zero_zero_nat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2312_div__neg__neg__trivial,axiom,
% 9.03/4.96      ! [K: int,L: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.03/4.96       => ( ( ord_less_int @ L @ K )
% 9.03/4.96         => ( ( divide_divide_int @ K @ L )
% 9.03/4.96            = zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2313_div__pos__pos__trivial,axiom,
% 9.03/4.96      ! [K: int,L: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.03/4.96       => ( ( ord_less_int @ K @ L )
% 9.03/4.96         => ( ( divide_divide_int @ K @ L )
% 9.03/4.96            = zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2314_log__le__cancel__iff,axiom,
% 9.03/4.96      ! [A: real,X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.96           => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 9.03/4.96              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2315_log__le__one__cancel__iff,axiom,
% 9.03/4.96      ! [A: real,X: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
% 9.03/4.96            = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2316_one__le__log__cancel__iff,axiom,
% 9.03/4.96      ! [A: real,X: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
% 9.03/4.96            = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2317_log__le__zero__cancel__iff,axiom,
% 9.03/4.96      ! [A: real,X: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
% 9.03/4.96            = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2318_zero__le__log__cancel__iff,axiom,
% 9.03/4.96      ! [A: real,X: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96         => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
% 9.03/4.96            = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2319_semiring__norm_I76_J,axiom,
% 9.03/4.96      ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2320_enat__ord__number_I2_J,axiom,
% 9.03/4.96      ! [M: num,N: num] :
% 9.03/4.96        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.03/4.96        = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2321_divide__le__0__1__iff,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 9.03/4.96        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2322_divide__le__0__1__iff,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 9.03/4.96        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2323_zero__le__divide__1__iff,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 9.03/4.96        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2324_zero__le__divide__1__iff,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 9.03/4.96        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2325_divide__less__0__1__iff,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 9.03/4.96        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2326_divide__less__0__1__iff,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 9.03/4.96        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2327_divide__less__eq__1__neg,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.03/4.96          = ( ord_less_real @ A @ B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2328_divide__less__eq__1__neg,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.03/4.96          = ( ord_less_rat @ A @ B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2329_divide__less__eq__1__pos,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.03/4.96          = ( ord_less_real @ B @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2330_divide__less__eq__1__pos,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.96       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.03/4.96          = ( ord_less_rat @ B @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2331_numeral__less__real__of__nat__iff,axiom,
% 9.03/4.96      ! [W: num,N: nat] :
% 9.03/4.96        ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.96        = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2332_real__of__nat__less__numeral__iff,axiom,
% 9.03/4.96      ! [N: nat,W: num] :
% 9.03/4.96        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 9.03/4.96        = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2333_zle__add1__eq__le,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 9.03/4.96        = ( ord_less_eq_int @ W @ Z ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2334_log__pow__cancel,axiom,
% 9.03/4.96      ! [A: real,B: nat] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96       => ( ( A != one_one_real )
% 9.03/4.96         => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 9.03/4.96            = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2335_half__nonnegative__int__iff,axiom,
% 9.03/4.96      ! [K: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.03/4.96        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2336_half__negative__int__iff,axiom,
% 9.03/4.96      ! [K: int] :
% 9.03/4.96        ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 9.03/4.96        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2337_ceiling__less__one,axiom,
% 9.03/4.96      ! [X: real] :
% 9.03/4.96        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 9.03/4.96        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2338_ceiling__less__one,axiom,
% 9.03/4.96      ! [X: rat] :
% 9.03/4.96        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 9.03/4.96        = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2339_less__int__code_I1_J,axiom,
% 9.03/4.96      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2340_log__base__change,axiom,
% 9.03/4.96      ! [A: real,B: real,X: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96       => ( ( A != one_one_real )
% 9.03/4.96         => ( ( log @ B @ X )
% 9.03/4.96            = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2341_log__base__pow,axiom,
% 9.03/4.96      ! [A: real,N: nat,X: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96       => ( ( log @ ( power_power_real @ A @ N ) @ X )
% 9.03/4.96          = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2342_real__of__int__div4,axiom,
% 9.03/4.96      ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2343_int__one__le__iff__zero__less,axiom,
% 9.03/4.96      ! [Z: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ one_one_int @ Z )
% 9.03/4.96        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2344_real__arch__pow__inv,axiom,
% 9.03/4.96      ! [Y: real,X: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.96       => ( ( ord_less_real @ X @ one_one_real )
% 9.03/4.96         => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2345_odd__less__0__iff,axiom,
% 9.03/4.96      ! [Z: int] :
% 9.03/4.96        ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 9.03/4.96        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2346_psubsetE,axiom,
% 9.03/4.96      ! [A2: set_int,B2: set_int] :
% 9.03/4.96        ( ( ord_less_set_int @ A2 @ B2 )
% 9.03/4.96       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.96           => ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2347_psubset__eq,axiom,
% 9.03/4.96      ( ord_less_set_int
% 9.03/4.96      = ( ^ [A4: set_int,B5: set_int] :
% 9.03/4.96            ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.03/4.96            & ( A4 != B5 ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2348_psubset__imp__subset,axiom,
% 9.03/4.96      ! [A2: set_int,B2: set_int] :
% 9.03/4.96        ( ( ord_less_set_int @ A2 @ B2 )
% 9.03/4.96       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2349_psubset__subset__trans,axiom,
% 9.03/4.96      ! [A2: set_int,B2: set_int,C6: set_int] :
% 9.03/4.96        ( ( ord_less_set_int @ A2 @ B2 )
% 9.03/4.96       => ( ( ord_less_eq_set_int @ B2 @ C6 )
% 9.03/4.96         => ( ord_less_set_int @ A2 @ C6 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2350_subset__not__subset__eq,axiom,
% 9.03/4.96      ( ord_less_set_int
% 9.03/4.96      = ( ^ [A4: set_int,B5: set_int] :
% 9.03/4.96            ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.03/4.96            & ~ ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2351_subset__psubset__trans,axiom,
% 9.03/4.96      ! [A2: set_int,B2: set_int,C6: set_int] :
% 9.03/4.96        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.96       => ( ( ord_less_set_int @ B2 @ C6 )
% 9.03/4.96         => ( ord_less_set_int @ A2 @ C6 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2352_subset__iff__psubset__eq,axiom,
% 9.03/4.96      ( ord_less_eq_set_int
% 9.03/4.96      = ( ^ [A4: set_int,B5: set_int] :
% 9.03/4.96            ( ( ord_less_set_int @ A4 @ B5 )
% 9.03/4.96            | ( A4 = B5 ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2353_less__eq__real__def,axiom,
% 9.03/4.96      ( ord_less_eq_real
% 9.03/4.96      = ( ^ [X4: real,Y6: real] :
% 9.03/4.96            ( ( ord_less_real @ X4 @ Y6 )
% 9.03/4.96            | ( X4 = Y6 ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2354_less__eq__int__code_I1_J,axiom,
% 9.03/4.96      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2355_plus__int__code_I2_J,axiom,
% 9.03/4.96      ! [L: int] :
% 9.03/4.96        ( ( plus_plus_int @ zero_zero_int @ L )
% 9.03/4.96        = L ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2356_plus__int__code_I1_J,axiom,
% 9.03/4.96      ! [K: int] :
% 9.03/4.96        ( ( plus_plus_int @ K @ zero_zero_int )
% 9.03/4.96        = K ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2357_ile0__eq,axiom,
% 9.03/4.96      ! [N: extended_enat] :
% 9.03/4.96        ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 9.03/4.96        = ( N = zero_z5237406670263579293d_enat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2358_i0__lb,axiom,
% 9.03/4.96      ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2359_zero__less__imp__eq__int,axiom,
% 9.03/4.96      ! [K: int] :
% 9.03/4.96        ( ( ord_less_int @ zero_zero_int @ K )
% 9.03/4.96       => ? [N2: nat] :
% 9.03/4.96            ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.03/4.96            & ( K
% 9.03/4.96              = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2360_pos__int__cases,axiom,
% 9.03/4.96      ! [K: int] :
% 9.03/4.96        ( ( ord_less_int @ zero_zero_int @ K )
% 9.03/4.96       => ~ ! [N2: nat] :
% 9.03/4.96              ( ( K
% 9.03/4.96                = ( semiri1314217659103216013at_int @ N2 ) )
% 9.03/4.96             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2361_le__imp__0__less,axiom,
% 9.03/4.96      ! [Z: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.03/4.96       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2362_nat__int__comparison_I2_J,axiom,
% 9.03/4.96      ( ord_less_nat
% 9.03/4.96      = ( ^ [A3: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2363_real__arch__pow,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ X )
% 9.03/4.96       => ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2364_int__gr__induct,axiom,
% 9.03/4.96      ! [K: int,I: int,P: int > $o] :
% 9.03/4.96        ( ( ord_less_int @ K @ I )
% 9.03/4.96       => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 9.03/4.96         => ( ! [I2: int] :
% 9.03/4.96                ( ( ord_less_int @ K @ I2 )
% 9.03/4.96               => ( ( P @ I2 )
% 9.03/4.96                 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 9.03/4.96           => ( P @ I ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2365_zless__add1__eq,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 9.03/4.96        = ( ( ord_less_int @ W @ Z )
% 9.03/4.96          | ( W = Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2366_int__ops_I1_J,axiom,
% 9.03/4.96      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 9.03/4.96      = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2367_nonneg__int__cases,axiom,
% 9.03/4.96      ! [K: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.03/4.96       => ~ ! [N2: nat] :
% 9.03/4.96              ( K
% 9.03/4.96             != ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2368_zero__le__imp__eq__int,axiom,
% 9.03/4.96      ! [K: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.03/4.96       => ? [N2: nat] :
% 9.03/4.96            ( K
% 9.03/4.96            = ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2369_odd__nonzero,axiom,
% 9.03/4.96      ! [Z: int] :
% 9.03/4.96        ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 9.03/4.96       != zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2370_linordered__field__no__ub,axiom,
% 9.03/4.96      ! [X2: real] :
% 9.03/4.96      ? [X_12: real] : ( ord_less_real @ X2 @ X_12 ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2371_linordered__field__no__ub,axiom,
% 9.03/4.96      ! [X2: rat] :
% 9.03/4.96      ? [X_12: rat] : ( ord_less_rat @ X2 @ X_12 ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2372_linordered__field__no__lb,axiom,
% 9.03/4.96      ! [X2: real] :
% 9.03/4.96      ? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2373_linordered__field__no__lb,axiom,
% 9.03/4.96      ! [X2: rat] :
% 9.03/4.96      ? [Y4: rat] : ( ord_less_rat @ Y4 @ X2 ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2374_real__of__nat__div4,axiom,
% 9.03/4.96      ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2375_zless__iff__Suc__zadd,axiom,
% 9.03/4.96      ( ord_less_int
% 9.03/4.96      = ( ^ [W2: int,Z4: int] :
% 9.03/4.96          ? [N3: nat] :
% 9.03/4.96            ( Z4
% 9.03/4.96            = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2376_add1__zle__eq,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 9.03/4.96        = ( ord_less_int @ W @ Z ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2377_zless__imp__add1__zle,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ord_less_int @ W @ Z )
% 9.03/4.96       => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2378_nat__le__real__less,axiom,
% 9.03/4.96      ( ord_less_eq_nat
% 9.03/4.96      = ( ^ [N3: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2379_int__less__real__le,axiom,
% 9.03/4.96      ( ord_less_int
% 9.03/4.96      = ( ^ [N3: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2380_int__le__real__less,axiom,
% 9.03/4.96      ( ord_less_eq_int
% 9.03/4.96      = ( ^ [N3: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2381_not__exp__less__eq__0__int,axiom,
% 9.03/4.96      ! [N: nat] :
% 9.03/4.96        ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2382_log__of__power__eq,axiom,
% 9.03/4.96      ! [M: nat,B: real,N: nat] :
% 9.03/4.96        ( ( ( semiri5074537144036343181t_real @ M )
% 9.03/4.96          = ( power_power_real @ B @ N ) )
% 9.03/4.96       => ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.96         => ( ( semiri5074537144036343181t_real @ N )
% 9.03/4.96            = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2383_less__log__of__power,axiom,
% 9.03/4.96      ! [B: real,N: nat,M: real] :
% 9.03/4.96        ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 9.03/4.96       => ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.96         => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2384_le__log__of__power,axiom,
% 9.03/4.96      ! [B: real,N: nat,M: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 9.03/4.96       => ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.96         => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2385_log__of__power__less,axiom,
% 9.03/4.96      ! [M: nat,B: real,N: nat] :
% 9.03/4.96        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 9.03/4.96       => ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.96         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.96           => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2386_add__divide__distrib,axiom,
% 9.03/4.96      ! [A: complex,B: complex,C: complex] :
% 9.03/4.96        ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 9.03/4.96        = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2387_add__divide__distrib,axiom,
% 9.03/4.96      ! [A: real,B: real,C: real] :
% 9.03/4.96        ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.03/4.96        = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2388_add__divide__distrib,axiom,
% 9.03/4.96      ! [A: rat,B: rat,C: rat] :
% 9.03/4.96        ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.03/4.96        = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2389_div__le__dividend,axiom,
% 9.03/4.96      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2390_div__le__mono,axiom,
% 9.03/4.96      ! [M: nat,N: nat,K: nat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.96       => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2391_log__of__power__le,axiom,
% 9.03/4.96      ! [M: nat,B: real,N: nat] :
% 9.03/4.96        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 9.03/4.96       => ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.96         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.96           => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2392_less__log2__of__power,axiom,
% 9.03/4.96      ! [N: nat,M: nat] :
% 9.03/4.96        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 9.03/4.96       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2393_log2__of__power__less,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/4.96       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.96         => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2394_divide__right__mono__neg,axiom,
% 9.03/4.96      ! [A: real,B: real,C: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.96       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.96         => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2395_divide__right__mono__neg,axiom,
% 9.03/4.96      ! [A: rat,B: rat,C: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.96       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.96         => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2396_divide__nonpos__nonpos,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 9.03/4.96         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2397_divide__nonpos__nonpos,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 9.03/4.96         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2398_divide__nonpos__nonneg,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.96         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2399_divide__nonpos__nonneg,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.96         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2400_divide__nonneg__nonpos,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.96       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 9.03/4.96         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2401_divide__nonneg__nonpos,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.96       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 9.03/4.96         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2402_divide__nonneg__nonneg,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.96       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.96         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2403_divide__nonneg__nonneg,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.96       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.96         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2404_zero__le__divide__iff,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 9.03/4.96        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.96            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.03/4.96          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.96            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2405_zero__le__divide__iff,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 9.03/4.96        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.96            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 9.03/4.96          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.96            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2406_divide__right__mono,axiom,
% 9.03/4.96      ! [A: real,B: real,C: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.96       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.96         => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2407_divide__right__mono,axiom,
% 9.03/4.96      ! [A: rat,B: rat,C: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.96       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.96         => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2408_divide__le__0__iff,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 9.03/4.96        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.96            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.03/4.96          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.96            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2409_divide__le__0__iff,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 9.03/4.96        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.96            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 9.03/4.96          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.96            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2410_divide__strict__right__mono__neg,axiom,
% 9.03/4.96      ! [B: real,A: real,C: real] :
% 9.03/4.96        ( ( ord_less_real @ B @ A )
% 9.03/4.96       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.96         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2411_divide__strict__right__mono__neg,axiom,
% 9.03/4.96      ! [B: rat,A: rat,C: rat] :
% 9.03/4.96        ( ( ord_less_rat @ B @ A )
% 9.03/4.96       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.96         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2412_divide__strict__right__mono,axiom,
% 9.03/4.96      ! [A: real,B: real,C: real] :
% 9.03/4.96        ( ( ord_less_real @ A @ B )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.96         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2413_divide__strict__right__mono,axiom,
% 9.03/4.96      ! [A: rat,B: rat,C: rat] :
% 9.03/4.96        ( ( ord_less_rat @ A @ B )
% 9.03/4.96       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.96         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2414_zero__less__divide__iff,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 9.03/4.96        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96            & ( ord_less_real @ zero_zero_real @ B ) )
% 9.03/4.96          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.96            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2415_zero__less__divide__iff,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 9.03/4.96        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.96            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 9.03/4.96          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.96            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2416_divide__less__cancel,axiom,
% 9.03/4.96      ! [A: real,C: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 9.03/4.96        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.96           => ( ord_less_real @ A @ B ) )
% 9.03/4.96          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.96           => ( ord_less_real @ B @ A ) )
% 9.03/4.96          & ( C != zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2417_divide__less__cancel,axiom,
% 9.03/4.96      ! [A: rat,C: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.96        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.96           => ( ord_less_rat @ A @ B ) )
% 9.03/4.96          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.96           => ( ord_less_rat @ B @ A ) )
% 9.03/4.96          & ( C != zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2418_divide__less__0__iff,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 9.03/4.96        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96            & ( ord_less_real @ B @ zero_zero_real ) )
% 9.03/4.96          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.96            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2419_divide__less__0__iff,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 9.03/4.96        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.96            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 9.03/4.96          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.96            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2420_divide__pos__pos,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.96         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2421_divide__pos__pos,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.03/4.96       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.03/4.96         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2422_divide__pos__neg,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96       => ( ( ord_less_real @ Y @ zero_zero_real )
% 9.03/4.96         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2423_divide__pos__neg,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.03/4.96       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 9.03/4.96         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2424_divide__neg__pos,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_real @ X @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.96         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2425_divide__neg__pos,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_rat @ X @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.03/4.96         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2426_divide__neg__neg,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_real @ X @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_real @ Y @ zero_zero_real )
% 9.03/4.96         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2427_divide__neg__neg,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_rat @ X @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 9.03/4.96         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2428_right__inverse__eq,axiom,
% 9.03/4.96      ! [B: complex,A: complex] :
% 9.03/4.96        ( ( B != zero_zero_complex )
% 9.03/4.96       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.03/4.96            = one_one_complex )
% 9.03/4.96          = ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2429_right__inverse__eq,axiom,
% 9.03/4.96      ! [B: real,A: real] :
% 9.03/4.96        ( ( B != zero_zero_real )
% 9.03/4.96       => ( ( ( divide_divide_real @ A @ B )
% 9.03/4.96            = one_one_real )
% 9.03/4.96          = ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2430_right__inverse__eq,axiom,
% 9.03/4.96      ! [B: rat,A: rat] :
% 9.03/4.96        ( ( B != zero_zero_rat )
% 9.03/4.96       => ( ( ( divide_divide_rat @ A @ B )
% 9.03/4.96            = one_one_rat )
% 9.03/4.96          = ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2431_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 9.03/4.96        = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2432_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 9.03/4.96        = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2433_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ( divide_divide_nat @ M @ N )
% 9.03/4.96          = zero_zero_nat )
% 9.03/4.96        = ( ( ord_less_nat @ M @ N )
% 9.03/4.96          | ( N = zero_zero_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2434_Suc__div__le__mono,axiom,
% 9.03/4.96      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2435_field__le__epsilon,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ! [E: real] :
% 9.03/4.96            ( ( ord_less_real @ zero_zero_real @ E )
% 9.03/4.96           => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
% 9.03/4.96       => ( ord_less_eq_real @ X @ Y ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2436_field__le__epsilon,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ! [E: rat] :
% 9.03/4.96            ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.03/4.96           => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
% 9.03/4.96       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2437_divide__nonpos__pos,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.96         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2438_divide__nonpos__pos,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.03/4.96         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2439_divide__nonpos__neg,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_real @ Y @ zero_zero_real )
% 9.03/4.96         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2440_divide__nonpos__neg,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 9.03/4.96         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2441_divide__nonneg__pos,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.96         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2442_divide__nonneg__pos,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.96       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.03/4.96         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2443_divide__nonneg__neg,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.96       => ( ( ord_less_real @ Y @ zero_zero_real )
% 9.03/4.96         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2444_divide__nonneg__neg,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.96       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 9.03/4.96         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2445_divide__le__cancel,axiom,
% 9.03/4.96      ! [A: real,C: real,B: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 9.03/4.96        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.96           => ( ord_less_eq_real @ A @ B ) )
% 9.03/4.96          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.96           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2446_divide__le__cancel,axiom,
% 9.03/4.96      ! [A: rat,C: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.96        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.96           => ( ord_less_eq_rat @ A @ B ) )
% 9.03/4.96          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.96           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2447_frac__less2,axiom,
% 9.03/4.96      ! [X: real,Y: real,W: real,Z: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96       => ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.96         => ( ( ord_less_real @ zero_zero_real @ W )
% 9.03/4.96           => ( ( ord_less_real @ W @ Z )
% 9.03/4.96             => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2448_frac__less2,axiom,
% 9.03/4.96      ! [X: rat,Y: rat,W: rat,Z: rat] :
% 9.03/4.96        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.03/4.96       => ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.96         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 9.03/4.96           => ( ( ord_less_rat @ W @ Z )
% 9.03/4.96             => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2449_frac__less,axiom,
% 9.03/4.96      ! [X: real,Y: real,W: real,Z: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.96       => ( ( ord_less_real @ X @ Y )
% 9.03/4.96         => ( ( ord_less_real @ zero_zero_real @ W )
% 9.03/4.96           => ( ( ord_less_eq_real @ W @ Z )
% 9.03/4.96             => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2450_frac__less,axiom,
% 9.03/4.96      ! [X: rat,Y: rat,W: rat,Z: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.96       => ( ( ord_less_rat @ X @ Y )
% 9.03/4.96         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 9.03/4.96           => ( ( ord_less_eq_rat @ W @ Z )
% 9.03/4.96             => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2451_frac__le,axiom,
% 9.03/4.96      ! [Y: real,X: real,W: real,Z: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.96       => ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.96         => ( ( ord_less_real @ zero_zero_real @ W )
% 9.03/4.96           => ( ( ord_less_eq_real @ W @ Z )
% 9.03/4.96             => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2452_frac__le,axiom,
% 9.03/4.96      ! [Y: rat,X: rat,W: rat,Z: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.96       => ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.96         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 9.03/4.96           => ( ( ord_less_eq_rat @ W @ Z )
% 9.03/4.96             => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2453_divide__less__eq__1,axiom,
% 9.03/4.96      ! [B: real,A: real] :
% 9.03/4.96        ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.03/4.96        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96            & ( ord_less_real @ B @ A ) )
% 9.03/4.96          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.96            & ( ord_less_real @ A @ B ) )
% 9.03/4.96          | ( A = zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2454_divide__less__eq__1,axiom,
% 9.03/4.96      ! [B: rat,A: rat] :
% 9.03/4.96        ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.03/4.96        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.96            & ( ord_less_rat @ B @ A ) )
% 9.03/4.96          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.96            & ( ord_less_rat @ A @ B ) )
% 9.03/4.96          | ( A = zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2455_less__divide__eq__1,axiom,
% 9.03/4.96      ! [B: real,A: real] :
% 9.03/4.96        ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.03/4.96        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96            & ( ord_less_real @ A @ B ) )
% 9.03/4.96          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.96            & ( ord_less_real @ B @ A ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2456_less__divide__eq__1,axiom,
% 9.03/4.96      ! [B: rat,A: rat] :
% 9.03/4.96        ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.03/4.96        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.96            & ( ord_less_rat @ A @ B ) )
% 9.03/4.96          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.96            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2457_div__add__self1,axiom,
% 9.03/4.96      ! [B: nat,A: nat] :
% 9.03/4.96        ( ( B != zero_zero_nat )
% 9.03/4.96       => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.03/4.96          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2458_div__add__self1,axiom,
% 9.03/4.96      ! [B: int,A: int] :
% 9.03/4.96        ( ( B != zero_zero_int )
% 9.03/4.96       => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.03/4.96          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2459_div__add__self2,axiom,
% 9.03/4.96      ! [B: nat,A: nat] :
% 9.03/4.96        ( ( B != zero_zero_nat )
% 9.03/4.96       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.03/4.96          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2460_div__add__self2,axiom,
% 9.03/4.96      ! [B: int,A: int] :
% 9.03/4.96        ( ( B != zero_zero_int )
% 9.03/4.96       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.03/4.96          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2461_gt__half__sum,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ A @ B )
% 9.03/4.96       => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2462_gt__half__sum,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ A @ B )
% 9.03/4.96       => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2463_less__half__sum,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ A @ B )
% 9.03/4.96       => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2464_less__half__sum,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ A @ B )
% 9.03/4.96       => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2465_numeral__Bit0__div__2,axiom,
% 9.03/4.96      ! [N: num] :
% 9.03/4.96        ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.96        = ( numeral_numeral_nat @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2466_numeral__Bit0__div__2,axiom,
% 9.03/4.96      ! [N: num] :
% 9.03/4.96        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/4.96        = ( numeral_numeral_int @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2467_div__greater__zero__iff,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 9.03/4.96        = ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.96          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2468_div__le__mono2,axiom,
% 9.03/4.96      ! [M: nat,N: nat,K: nat] :
% 9.03/4.96        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.96       => ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.96         => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2469_div__less__dividend,axiom,
% 9.03/4.96      ! [N: nat,M: nat] :
% 9.03/4.96        ( ( ord_less_nat @ one_one_nat @ N )
% 9.03/4.96       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.96         => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2470_div__eq__dividend__iff,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.96       => ( ( ( divide_divide_nat @ M @ N )
% 9.03/4.96            = M )
% 9.03/4.96          = ( N = one_one_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2471_le__divide__eq__1,axiom,
% 9.03/4.96      ! [B: real,A: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.03/4.96        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96            & ( ord_less_eq_real @ A @ B ) )
% 9.03/4.96          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.96            & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2472_le__divide__eq__1,axiom,
% 9.03/4.96      ! [B: rat,A: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.03/4.96        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.96            & ( ord_less_eq_rat @ A @ B ) )
% 9.03/4.96          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.96            & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2473_divide__le__eq__1,axiom,
% 9.03/4.96      ! [B: real,A: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.03/4.96        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96            & ( ord_less_eq_real @ B @ A ) )
% 9.03/4.96          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.96            & ( ord_less_eq_real @ A @ B ) )
% 9.03/4.96          | ( A = zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2474_divide__le__eq__1,axiom,
% 9.03/4.96      ! [B: rat,A: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.03/4.96        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.96            & ( ord_less_eq_rat @ B @ A ) )
% 9.03/4.96          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.96            & ( ord_less_eq_rat @ A @ B ) )
% 9.03/4.96          | ( A = zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2475_exp__add__not__zero__imp__left,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.96         != zero_zero_nat )
% 9.03/4.96       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.03/4.96         != zero_zero_nat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2476_exp__add__not__zero__imp__left,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.96         != zero_zero_int )
% 9.03/4.96       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 9.03/4.96         != zero_zero_int ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2477_exp__add__not__zero__imp__right,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.96         != zero_zero_nat )
% 9.03/4.96       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.03/4.96         != zero_zero_nat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2478_exp__add__not__zero__imp__right,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.96         != zero_zero_int )
% 9.03/4.96       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.03/4.96         != zero_zero_int ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2479_div__exp__eq,axiom,
% 9.03/4.96      ! [A: nat,M: nat,N: nat] :
% 9.03/4.96        ( ( 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 ) )
% 9.03/4.96        = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2480_div__exp__eq,axiom,
% 9.03/4.96      ! [A: int,M: nat,N: nat] :
% 9.03/4.96        ( ( 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 ) )
% 9.03/4.96        = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2481_div__2__gt__zero,axiom,
% 9.03/4.96      ! [N: nat] :
% 9.03/4.96        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.03/4.96       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2482_Suc__n__div__2__gt__zero,axiom,
% 9.03/4.96      ! [N: nat] :
% 9.03/4.96        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.96       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2483_buildup__gives__valid,axiom,
% 9.03/4.96      ! [N: nat] :
% 9.03/4.96        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.96       => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2484_div__self,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( A != zero_zero_complex )
% 9.03/4.96       => ( ( divide1717551699836669952omplex @ A @ A )
% 9.03/4.96          = one_one_complex ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2485_div__self,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( A != zero_zero_real )
% 9.03/4.96       => ( ( divide_divide_real @ A @ A )
% 9.03/4.96          = one_one_real ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2486_div__self,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( A != zero_zero_rat )
% 9.03/4.96       => ( ( divide_divide_rat @ A @ A )
% 9.03/4.96          = one_one_rat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2487_div__self,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( A != zero_zero_nat )
% 9.03/4.96       => ( ( divide_divide_nat @ A @ A )
% 9.03/4.96          = one_one_nat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2488_div__self,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( A != zero_zero_int )
% 9.03/4.96       => ( ( divide_divide_int @ A @ A )
% 9.03/4.96          = one_one_int ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2489_arcosh__1,axiom,
% 9.03/4.96      ( ( arcosh_real @ one_one_real )
% 9.03/4.96      = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2490_nat__induct2,axiom,
% 9.03/4.96      ! [P: nat > $o,N: nat] :
% 9.03/4.96        ( ( P @ zero_zero_nat )
% 9.03/4.96       => ( ( P @ one_one_nat )
% 9.03/4.96         => ( ! [N2: nat] :
% 9.03/4.96                ( ( P @ N2 )
% 9.03/4.96               => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.03/4.96           => ( P @ N ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2491_high__bound__aux,axiom,
% 9.03/4.96      ! [Ma: nat,N: nat,M: nat] :
% 9.03/4.96        ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.03/4.96       => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2492_zdiv__numeral__Bit0,axiom,
% 9.03/4.96      ! [V: num,W: num] :
% 9.03/4.96        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 9.03/4.96        = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2493_pos2,axiom,
% 9.03/4.96      ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2494_div__by__1,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2495_div__by__1,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( divide_divide_real @ A @ one_one_real )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2496_div__by__1,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( divide_divide_rat @ A @ one_one_rat )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2497_div__by__1,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( divide_divide_nat @ A @ one_one_nat )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2498_div__by__1,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( divide_divide_int @ A @ one_one_int )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2499_div__by__0,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 9.03/4.96        = zero_zero_complex ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2500_div__by__0,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( divide_divide_real @ A @ zero_zero_real )
% 9.03/4.96        = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2501_div__by__0,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 9.03/4.96        = zero_zero_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2502_div__by__0,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2503_div__by__0,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( divide_divide_int @ A @ zero_zero_int )
% 9.03/4.96        = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2504_div__0,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 9.03/4.96        = zero_zero_complex ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2505_div__0,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( divide_divide_real @ zero_zero_real @ A )
% 9.03/4.96        = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2506_div__0,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( divide_divide_rat @ zero_zero_rat @ A )
% 9.03/4.96        = zero_zero_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2507_div__0,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2508_div__0,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( divide_divide_int @ zero_zero_int @ A )
% 9.03/4.96        = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2509_high__def,axiom,
% 9.03/4.96      ( vEBT_VEBT_high
% 9.03/4.96      = ( ^ [X4: nat,N3: nat] : ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2510_i0__less,axiom,
% 9.03/4.96      ! [N: extended_enat] :
% 9.03/4.96        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 9.03/4.96        = ( N != zero_z5237406670263579293d_enat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2511_psubsetD,axiom,
% 9.03/4.96      ! [A2: set_complex,B2: set_complex,C: complex] :
% 9.03/4.96        ( ( ord_less_set_complex @ A2 @ B2 )
% 9.03/4.96       => ( ( member_complex @ C @ A2 )
% 9.03/4.96         => ( member_complex @ C @ B2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2512_psubsetD,axiom,
% 9.03/4.96      ! [A2: set_real,B2: set_real,C: real] :
% 9.03/4.96        ( ( ord_less_set_real @ A2 @ B2 )
% 9.03/4.96       => ( ( member_real @ C @ A2 )
% 9.03/4.96         => ( member_real @ C @ B2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2513_psubsetD,axiom,
% 9.03/4.96      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 9.03/4.96        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 9.03/4.96       => ( ( member_set_nat @ C @ A2 )
% 9.03/4.96         => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2514_psubsetD,axiom,
% 9.03/4.96      ! [A2: set_nat,B2: set_nat,C: nat] :
% 9.03/4.96        ( ( ord_less_set_nat @ A2 @ B2 )
% 9.03/4.96       => ( ( member_nat @ C @ A2 )
% 9.03/4.96         => ( member_nat @ C @ B2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2515_psubsetD,axiom,
% 9.03/4.96      ! [A2: set_int,B2: set_int,C: int] :
% 9.03/4.96        ( ( ord_less_set_int @ A2 @ B2 )
% 9.03/4.96       => ( ( member_int @ C @ A2 )
% 9.03/4.96         => ( member_int @ C @ B2 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2516_not__iless0,axiom,
% 9.03/4.96      ! [N: extended_enat] :
% 9.03/4.96        ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2517_enat__less__induct,axiom,
% 9.03/4.96      ! [P: extended_enat > $o,N: extended_enat] :
% 9.03/4.96        ( ! [N2: extended_enat] :
% 9.03/4.96            ( ! [M3: extended_enat] :
% 9.03/4.96                ( ( ord_le72135733267957522d_enat @ M3 @ N2 )
% 9.03/4.96               => ( P @ M3 ) )
% 9.03/4.96           => ( P @ N2 ) )
% 9.03/4.96       => ( P @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2518_iadd__is__0,axiom,
% 9.03/4.96      ! [M: extended_enat,N: extended_enat] :
% 9.03/4.96        ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 9.03/4.96          = zero_z5237406670263579293d_enat )
% 9.03/4.96        = ( ( M = zero_z5237406670263579293d_enat )
% 9.03/4.96          & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2519_zero__one__enat__neq_I1_J,axiom,
% 9.03/4.96      zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2520_linorder__neqE__linordered__idom,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( X != Y )
% 9.03/4.96       => ( ~ ( ord_less_real @ X @ Y )
% 9.03/4.96         => ( ord_less_real @ Y @ X ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2521_linorder__neqE__linordered__idom,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( X != Y )
% 9.03/4.96       => ( ~ ( ord_less_rat @ X @ Y )
% 9.03/4.96         => ( ord_less_rat @ Y @ X ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2522_linorder__neqE__linordered__idom,axiom,
% 9.03/4.96      ! [X: int,Y: int] :
% 9.03/4.96        ( ( X != Y )
% 9.03/4.96       => ( ~ ( ord_less_int @ X @ Y )
% 9.03/4.96         => ( ord_less_int @ Y @ X ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2523_zero__neq__one,axiom,
% 9.03/4.96      zero_zero_complex != one_one_complex ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2524_zero__neq__one,axiom,
% 9.03/4.96      zero_zero_real != one_one_real ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2525_zero__neq__one,axiom,
% 9.03/4.96      zero_zero_rat != one_one_rat ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2526_zero__neq__one,axiom,
% 9.03/4.96      zero_zero_nat != one_one_nat ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2527_zero__neq__one,axiom,
% 9.03/4.96      zero_zero_int != one_one_int ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2528_realpow__pos__nth2,axiom,
% 9.03/4.96      ! [A: real,N: nat] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96       => ? [R3: real] :
% 9.03/4.96            ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.03/4.96            & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 9.03/4.96              = A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2529_pos__imp__zdiv__neg__iff,axiom,
% 9.03/4.96      ! [B: int,A: int] :
% 9.03/4.96        ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.96       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 9.03/4.96          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2530_neg__imp__zdiv__neg__iff,axiom,
% 9.03/4.96      ! [B: int,A: int] :
% 9.03/4.96        ( ( ord_less_int @ B @ zero_zero_int )
% 9.03/4.96       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 9.03/4.96          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2531_div__neg__pos__less0,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( ord_less_int @ A @ zero_zero_int )
% 9.03/4.96       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.96         => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2532_zdiv__int,axiom,
% 9.03/4.96      ! [A: nat,B: nat] :
% 9.03/4.96        ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 9.03/4.96        = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2533_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 9.03/4.96      ! [X: nat,N: nat,M: nat] :
% 9.03/4.96        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.03/4.96       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.96         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.96           => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2534_not__one__le__zero,axiom,
% 9.03/4.96      ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2535_not__one__le__zero,axiom,
% 9.03/4.96      ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2536_not__one__le__zero,axiom,
% 9.03/4.96      ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2537_not__one__le__zero,axiom,
% 9.03/4.96      ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2538_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.03/4.96      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2539_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.03/4.96      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2540_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.03/4.96      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2541_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.03/4.96      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2542_zero__less__one__class_Ozero__le__one,axiom,
% 9.03/4.96      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2543_zero__less__one__class_Ozero__le__one,axiom,
% 9.03/4.96      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2544_zero__less__one__class_Ozero__le__one,axiom,
% 9.03/4.96      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2545_zero__less__one__class_Ozero__le__one,axiom,
% 9.03/4.96      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2546_not__one__less__zero,axiom,
% 9.03/4.96      ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2547_not__one__less__zero,axiom,
% 9.03/4.96      ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2548_not__one__less__zero,axiom,
% 9.03/4.96      ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2549_not__one__less__zero,axiom,
% 9.03/4.96      ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2550_zero__less__one,axiom,
% 9.03/4.96      ord_less_real @ zero_zero_real @ one_one_real ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2551_zero__less__one,axiom,
% 9.03/4.96      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2552_zero__less__one,axiom,
% 9.03/4.96      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2553_zero__less__one,axiom,
% 9.03/4.96      ord_less_int @ zero_zero_int @ one_one_int ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2554_add__less__zeroD,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 9.03/4.96       => ( ( ord_less_real @ X @ zero_zero_real )
% 9.03/4.96          | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2555_add__less__zeroD,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
% 9.03/4.96       => ( ( ord_less_rat @ X @ zero_zero_rat )
% 9.03/4.96          | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2556_add__less__zeroD,axiom,
% 9.03/4.96      ! [X: int,Y: int] :
% 9.03/4.96        ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
% 9.03/4.96       => ( ( ord_less_int @ X @ zero_zero_int )
% 9.03/4.96          | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2557_add__mono1,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ A @ B )
% 9.03/4.96       => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2558_add__mono1,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ A @ B )
% 9.03/4.96       => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2559_add__mono1,axiom,
% 9.03/4.96      ! [A: nat,B: nat] :
% 9.03/4.96        ( ( ord_less_nat @ A @ B )
% 9.03/4.96       => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2560_add__mono1,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( ord_less_int @ A @ B )
% 9.03/4.96       => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2561_less__add__one,axiom,
% 9.03/4.96      ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2562_less__add__one,axiom,
% 9.03/4.96      ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2563_less__add__one,axiom,
% 9.03/4.96      ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2564_less__add__one,axiom,
% 9.03/4.96      ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2565_realpow__pos__nth,axiom,
% 9.03/4.96      ! [N: nat,A: real] :
% 9.03/4.96        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96         => ? [R3: real] :
% 9.03/4.96              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.03/4.96              & ( ( power_power_real @ R3 @ N )
% 9.03/4.96                = A ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2566_realpow__pos__nth__unique,axiom,
% 9.03/4.96      ! [N: nat,A: real] :
% 9.03/4.96        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.96       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.96         => ? [X3: real] :
% 9.03/4.96              ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.03/4.96              & ( ( power_power_real @ X3 @ N )
% 9.03/4.96                = A )
% 9.03/4.96              & ! [Y5: real] :
% 9.03/4.96                  ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 9.03/4.96                    & ( ( power_power_real @ Y5 @ N )
% 9.03/4.96                      = A ) )
% 9.03/4.96                 => ( Y5 = X3 ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2567_nonneg1__imp__zdiv__pos__iff,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.96       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 9.03/4.96          = ( ( ord_less_eq_int @ B @ A )
% 9.03/4.96            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2568_pos__imp__zdiv__nonneg__iff,axiom,
% 9.03/4.96      ! [B: int,A: int] :
% 9.03/4.96        ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.96       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 9.03/4.96          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2569_neg__imp__zdiv__nonneg__iff,axiom,
% 9.03/4.96      ! [B: int,A: int] :
% 9.03/4.96        ( ( ord_less_int @ B @ zero_zero_int )
% 9.03/4.96       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 9.03/4.96          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2570_pos__imp__zdiv__pos__iff,axiom,
% 9.03/4.96      ! [K: int,I: int] :
% 9.03/4.96        ( ( ord_less_int @ zero_zero_int @ K )
% 9.03/4.96       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 9.03/4.96          = ( ord_less_eq_int @ K @ I ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2571_div__nonpos__pos__le0,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.96       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.96         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2572_div__nonneg__neg__le0,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.96       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.03/4.96         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2573_div__positive__int,axiom,
% 9.03/4.96      ! [L: int,K: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ L @ K )
% 9.03/4.96       => ( ( ord_less_int @ zero_zero_int @ L )
% 9.03/4.96         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2574_div__int__pos__iff,axiom,
% 9.03/4.96      ! [K: int,L: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 9.03/4.96        = ( ( K = zero_zero_int )
% 9.03/4.96          | ( L = zero_zero_int )
% 9.03/4.96          | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.03/4.96            & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 9.03/4.96          | ( ( ord_less_int @ K @ zero_zero_int )
% 9.03/4.96            & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2575_zdiv__mono2__neg,axiom,
% 9.03/4.96      ! [A: int,B7: int,B: int] :
% 9.03/4.96        ( ( ord_less_int @ A @ zero_zero_int )
% 9.03/4.96       => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.03/4.96         => ( ( ord_less_eq_int @ B7 @ B )
% 9.03/4.96           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B7 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2576_zdiv__mono1__neg,axiom,
% 9.03/4.96      ! [A: int,A6: int,B: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ A @ A6 )
% 9.03/4.96       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.03/4.96         => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2577_zdiv__eq__0__iff,axiom,
% 9.03/4.96      ! [I: int,K: int] :
% 9.03/4.96        ( ( ( divide_divide_int @ I @ K )
% 9.03/4.96          = zero_zero_int )
% 9.03/4.96        = ( ( K = zero_zero_int )
% 9.03/4.96          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.03/4.96            & ( ord_less_int @ I @ K ) )
% 9.03/4.96          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 9.03/4.96            & ( ord_less_int @ K @ I ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2578_zdiv__mono2,axiom,
% 9.03/4.96      ! [A: int,B7: int,B: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.96       => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.03/4.96         => ( ( ord_less_eq_int @ B7 @ B )
% 9.03/4.96           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B7 ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2579_zdiv__mono1,axiom,
% 9.03/4.96      ! [A: int,A6: int,B: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ A @ A6 )
% 9.03/4.96       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.96         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2580_int__div__less__self,axiom,
% 9.03/4.96      ! [X: int,K: int] :
% 9.03/4.96        ( ( ord_less_int @ zero_zero_int @ X )
% 9.03/4.96       => ( ( ord_less_int @ one_one_int @ K )
% 9.03/4.96         => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2581_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.03/4.96      ! [A: nat,B: nat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.96       => ( ( ord_less_nat @ A @ B )
% 9.03/4.96         => ( ( divide_divide_nat @ A @ B )
% 9.03/4.96            = zero_zero_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2582_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.96       => ( ( ord_less_int @ A @ B )
% 9.03/4.96         => ( ( divide_divide_int @ A @ B )
% 9.03/4.96            = zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2583_div__positive,axiom,
% 9.03/4.96      ! [B: nat,A: nat] :
% 9.03/4.96        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.03/4.96       => ( ( ord_less_eq_nat @ B @ A )
% 9.03/4.96         => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2584_div__positive,axiom,
% 9.03/4.96      ! [B: int,A: int] :
% 9.03/4.96        ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.96       => ( ( ord_less_eq_int @ B @ A )
% 9.03/4.96         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2585_discrete,axiom,
% 9.03/4.96      ( ord_less_nat
% 9.03/4.96      = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2586_discrete,axiom,
% 9.03/4.96      ( ord_less_int
% 9.03/4.96      = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2587_zero__less__two,axiom,
% 9.03/4.96      ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2588_zero__less__two,axiom,
% 9.03/4.96      ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2589_zero__less__two,axiom,
% 9.03/4.96      ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2590_zero__less__two,axiom,
% 9.03/4.96      ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2591_high__inv,axiom,
% 9.03/4.96      ! [X: nat,N: nat,Y: nat] :
% 9.03/4.96        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/4.96       => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 9.03/4.96          = Y ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2592_buildup__nothing__in__leaf,axiom,
% 9.03/4.96      ! [N: nat,X: nat] :
% 9.03/4.96        ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2593_ceiling__log2__div2,axiom,
% 9.03/4.96      ! [N: nat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.03/4.96       => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.03/4.96          = ( 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 ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2594_artanh__0,axiom,
% 9.03/4.96      ( ( artanh_real @ zero_zero_real )
% 9.03/4.96      = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2595_arsinh__0,axiom,
% 9.03/4.96      ( ( arsinh_real @ zero_zero_real )
% 9.03/4.96      = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2596_ceiling__log__eq__powr__iff,axiom,
% 9.03/4.96      ! [X: real,B: real,K: nat] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.96       => ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.96         => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
% 9.03/4.96              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 9.03/4.96            = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 9.03/4.96              & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2597_floor__log__nat__eq__powr__iff,axiom,
% 9.03/4.96      ! [B: nat,K: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.03/4.96       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.96         => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.03/4.96              = ( semiri1314217659103216013at_int @ N ) )
% 9.03/4.96            = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 9.03/4.96              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2598_of__int__round__le,axiom,
% 9.03/4.96      ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2599_of__int__round__le,axiom,
% 9.03/4.96      ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2600_mult__cancel__right,axiom,
% 9.03/4.96      ! [A: complex,C: complex,B: complex] :
% 9.03/4.96        ( ( ( times_times_complex @ A @ C )
% 9.03/4.96          = ( times_times_complex @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_complex )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2601_mult__cancel__right,axiom,
% 9.03/4.96      ! [A: real,C: real,B: real] :
% 9.03/4.96        ( ( ( times_times_real @ A @ C )
% 9.03/4.96          = ( times_times_real @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_real )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2602_mult__cancel__right,axiom,
% 9.03/4.96      ! [A: rat,C: rat,B: rat] :
% 9.03/4.96        ( ( ( times_times_rat @ A @ C )
% 9.03/4.96          = ( times_times_rat @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_rat )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2603_mult__cancel__right,axiom,
% 9.03/4.96      ! [A: nat,C: nat,B: nat] :
% 9.03/4.96        ( ( ( times_times_nat @ A @ C )
% 9.03/4.96          = ( times_times_nat @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_nat )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2604_mult__cancel__right,axiom,
% 9.03/4.96      ! [A: int,C: int,B: int] :
% 9.03/4.96        ( ( ( times_times_int @ A @ C )
% 9.03/4.96          = ( times_times_int @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_int )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2605_mult__cancel__left,axiom,
% 9.03/4.96      ! [C: complex,A: complex,B: complex] :
% 9.03/4.96        ( ( ( times_times_complex @ C @ A )
% 9.03/4.96          = ( times_times_complex @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_complex )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2606_mult__cancel__left,axiom,
% 9.03/4.96      ! [C: real,A: real,B: real] :
% 9.03/4.96        ( ( ( times_times_real @ C @ A )
% 9.03/4.96          = ( times_times_real @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_real )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2607_mult__cancel__left,axiom,
% 9.03/4.96      ! [C: rat,A: rat,B: rat] :
% 9.03/4.96        ( ( ( times_times_rat @ C @ A )
% 9.03/4.96          = ( times_times_rat @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_rat )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2608_mult__cancel__left,axiom,
% 9.03/4.96      ! [C: nat,A: nat,B: nat] :
% 9.03/4.96        ( ( ( times_times_nat @ C @ A )
% 9.03/4.96          = ( times_times_nat @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_nat )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2609_mult__cancel__left,axiom,
% 9.03/4.96      ! [C: int,A: int,B: int] :
% 9.03/4.96        ( ( ( times_times_int @ C @ A )
% 9.03/4.96          = ( times_times_int @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_int )
% 9.03/4.96          | ( A = B ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2610_mult__eq__0__iff,axiom,
% 9.03/4.96      ! [A: complex,B: complex] :
% 9.03/4.96        ( ( ( times_times_complex @ A @ B )
% 9.03/4.96          = zero_zero_complex )
% 9.03/4.96        = ( ( A = zero_zero_complex )
% 9.03/4.96          | ( B = zero_zero_complex ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2611_mult__eq__0__iff,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ( times_times_real @ A @ B )
% 9.03/4.96          = zero_zero_real )
% 9.03/4.96        = ( ( A = zero_zero_real )
% 9.03/4.96          | ( B = zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2612_mult__eq__0__iff,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ( times_times_rat @ A @ B )
% 9.03/4.96          = zero_zero_rat )
% 9.03/4.96        = ( ( A = zero_zero_rat )
% 9.03/4.96          | ( B = zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2613_mult__eq__0__iff,axiom,
% 9.03/4.96      ! [A: nat,B: nat] :
% 9.03/4.96        ( ( ( times_times_nat @ A @ B )
% 9.03/4.96          = zero_zero_nat )
% 9.03/4.96        = ( ( A = zero_zero_nat )
% 9.03/4.96          | ( B = zero_zero_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2614_mult__eq__0__iff,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( ( times_times_int @ A @ B )
% 9.03/4.96          = zero_zero_int )
% 9.03/4.96        = ( ( A = zero_zero_int )
% 9.03/4.96          | ( B = zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2615_mult__zero__right,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( times_times_complex @ A @ zero_zero_complex )
% 9.03/4.96        = zero_zero_complex ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2616_mult__zero__right,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( times_times_real @ A @ zero_zero_real )
% 9.03/4.96        = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2617_mult__zero__right,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( times_times_rat @ A @ zero_zero_rat )
% 9.03/4.96        = zero_zero_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2618_mult__zero__right,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( times_times_nat @ A @ zero_zero_nat )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2619_mult__zero__right,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( times_times_int @ A @ zero_zero_int )
% 9.03/4.96        = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2620_mult__zero__left,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( times_times_complex @ zero_zero_complex @ A )
% 9.03/4.96        = zero_zero_complex ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2621_mult__zero__left,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( times_times_real @ zero_zero_real @ A )
% 9.03/4.96        = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2622_mult__zero__left,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( times_times_rat @ zero_zero_rat @ A )
% 9.03/4.96        = zero_zero_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2623_mult__zero__left,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( times_times_nat @ zero_zero_nat @ A )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2624_mult__zero__left,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( times_times_int @ zero_zero_int @ A )
% 9.03/4.96        = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2625_diff__self,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( minus_minus_complex @ A @ A )
% 9.03/4.96        = zero_zero_complex ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2626_diff__self,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( minus_minus_real @ A @ A )
% 9.03/4.96        = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2627_diff__self,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( minus_minus_rat @ A @ A )
% 9.03/4.96        = zero_zero_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2628_diff__self,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( minus_minus_int @ A @ A )
% 9.03/4.96        = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2629_diff__0__right,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2630_diff__0__right,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( minus_minus_real @ A @ zero_zero_real )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2631_diff__0__right,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2632_diff__0__right,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( minus_minus_int @ A @ zero_zero_int )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2633_zero__diff,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ zero_zero_nat @ A )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2634_diff__zero,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2635_diff__zero,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( minus_minus_real @ A @ zero_zero_real )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2636_diff__zero,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2637_diff__zero,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ A @ zero_zero_nat )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2638_diff__zero,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( minus_minus_int @ A @ zero_zero_int )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2639_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( minus_minus_complex @ A @ A )
% 9.03/4.96        = zero_zero_complex ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2640_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( minus_minus_real @ A @ A )
% 9.03/4.96        = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2641_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( minus_minus_rat @ A @ A )
% 9.03/4.96        = zero_zero_rat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2642_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ A @ A )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2643_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( minus_minus_int @ A @ A )
% 9.03/4.96        = zero_zero_int ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2644_numeral__times__numeral,axiom,
% 9.03/4.96      ! [M: num,N: num] :
% 9.03/4.96        ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 9.03/4.96        = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2645_numeral__times__numeral,axiom,
% 9.03/4.96      ! [M: num,N: num] :
% 9.03/4.96        ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.03/4.96        = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2646_numeral__times__numeral,axiom,
% 9.03/4.96      ! [M: num,N: num] :
% 9.03/4.96        ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.03/4.96        = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2647_numeral__times__numeral,axiom,
% 9.03/4.96      ! [M: num,N: num] :
% 9.03/4.96        ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.03/4.96        = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2648_numeral__times__numeral,axiom,
% 9.03/4.96      ! [M: num,N: num] :
% 9.03/4.96        ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.03/4.96        = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2649_mult__numeral__left__semiring__numeral,axiom,
% 9.03/4.96      ! [V: num,W: num,Z: complex] :
% 9.03/4.96        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 9.03/4.96        = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2650_mult__numeral__left__semiring__numeral,axiom,
% 9.03/4.96      ! [V: num,W: num,Z: real] :
% 9.03/4.96        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 9.03/4.96        = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2651_mult__numeral__left__semiring__numeral,axiom,
% 9.03/4.96      ! [V: num,W: num,Z: rat] :
% 9.03/4.96        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 9.03/4.96        = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2652_mult__numeral__left__semiring__numeral,axiom,
% 9.03/4.96      ! [V: num,W: num,Z: nat] :
% 9.03/4.96        ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 9.03/4.96        = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2653_mult__numeral__left__semiring__numeral,axiom,
% 9.03/4.96      ! [V: num,W: num,Z: int] :
% 9.03/4.96        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 9.03/4.96        = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2654_mult_Oright__neutral,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( times_times_complex @ A @ one_one_complex )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2655_mult_Oright__neutral,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( times_times_real @ A @ one_one_real )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2656_mult_Oright__neutral,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( times_times_rat @ A @ one_one_rat )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2657_mult_Oright__neutral,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( times_times_nat @ A @ one_one_nat )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2658_mult_Oright__neutral,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( times_times_int @ A @ one_one_int )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2659_mult__1,axiom,
% 9.03/4.96      ! [A: complex] :
% 9.03/4.96        ( ( times_times_complex @ one_one_complex @ A )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2660_mult__1,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( times_times_real @ one_one_real @ A )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2661_mult__1,axiom,
% 9.03/4.96      ! [A: rat] :
% 9.03/4.96        ( ( times_times_rat @ one_one_rat @ A )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2662_mult__1,axiom,
% 9.03/4.96      ! [A: nat] :
% 9.03/4.96        ( ( times_times_nat @ one_one_nat @ A )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2663_mult__1,axiom,
% 9.03/4.96      ! [A: int] :
% 9.03/4.96        ( ( times_times_int @ one_one_int @ A )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2664_add__diff__cancel,axiom,
% 9.03/4.96      ! [A: complex,B: complex] :
% 9.03/4.96        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2665_add__diff__cancel,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2666_add__diff__cancel,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2667_add__diff__cancel,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2668_diff__add__cancel,axiom,
% 9.03/4.96      ! [A: complex,B: complex] :
% 9.03/4.96        ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2669_diff__add__cancel,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2670_diff__add__cancel,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2671_diff__add__cancel,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2672_add__diff__cancel__left,axiom,
% 9.03/4.96      ! [C: complex,A: complex,B: complex] :
% 9.03/4.96        ( ( minus_minus_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
% 9.03/4.96        = ( minus_minus_complex @ A @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2673_add__diff__cancel__left,axiom,
% 9.03/4.96      ! [C: real,A: real,B: real] :
% 9.03/4.96        ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.03/4.96        = ( minus_minus_real @ A @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2674_add__diff__cancel__left,axiom,
% 9.03/4.96      ! [C: rat,A: rat,B: rat] :
% 9.03/4.96        ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.03/4.96        = ( minus_minus_rat @ A @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2675_add__diff__cancel__left,axiom,
% 9.03/4.96      ! [C: nat,A: nat,B: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.03/4.96        = ( minus_minus_nat @ A @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2676_add__diff__cancel__left,axiom,
% 9.03/4.96      ! [C: int,A: int,B: int] :
% 9.03/4.96        ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.03/4.96        = ( minus_minus_int @ A @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2677_add__diff__cancel__left_H,axiom,
% 9.03/4.96      ! [A: complex,B: complex] :
% 9.03/4.96        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ A )
% 9.03/4.96        = B ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2678_add__diff__cancel__left_H,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 9.03/4.96        = B ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2679_add__diff__cancel__left_H,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 9.03/4.96        = B ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2680_add__diff__cancel__left_H,axiom,
% 9.03/4.96      ! [A: nat,B: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 9.03/4.96        = B ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2681_add__diff__cancel__left_H,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 9.03/4.96        = B ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2682_add__diff__cancel__right,axiom,
% 9.03/4.96      ! [A: complex,C: complex,B: complex] :
% 9.03/4.96        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
% 9.03/4.96        = ( minus_minus_complex @ A @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2683_add__diff__cancel__right,axiom,
% 9.03/4.96      ! [A: real,C: real,B: real] :
% 9.03/4.96        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.03/4.96        = ( minus_minus_real @ A @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2684_add__diff__cancel__right,axiom,
% 9.03/4.96      ! [A: rat,C: rat,B: rat] :
% 9.03/4.96        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.03/4.96        = ( minus_minus_rat @ A @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2685_add__diff__cancel__right,axiom,
% 9.03/4.96      ! [A: nat,C: nat,B: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.03/4.96        = ( minus_minus_nat @ A @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2686_add__diff__cancel__right,axiom,
% 9.03/4.96      ! [A: int,C: int,B: int] :
% 9.03/4.96        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.03/4.96        = ( minus_minus_int @ A @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2687_add__diff__cancel__right_H,axiom,
% 9.03/4.96      ! [A: complex,B: complex] :
% 9.03/4.96        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2688_add__diff__cancel__right_H,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2689_add__diff__cancel__right_H,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2690_add__diff__cancel__right_H,axiom,
% 9.03/4.96      ! [A: nat,B: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2691_add__diff__cancel__right_H,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.03/4.96        = A ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2692_times__divide__eq__right,axiom,
% 9.03/4.96      ! [A: complex,B: complex,C: complex] :
% 9.03/4.96        ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 9.03/4.96        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2693_times__divide__eq__right,axiom,
% 9.03/4.96      ! [A: real,B: real,C: real] :
% 9.03/4.96        ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.03/4.96        = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2694_times__divide__eq__right,axiom,
% 9.03/4.96      ! [A: rat,B: rat,C: rat] :
% 9.03/4.96        ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.96        = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2695_divide__divide__eq__right,axiom,
% 9.03/4.96      ! [A: complex,B: complex,C: complex] :
% 9.03/4.96        ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 9.03/4.96        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2696_divide__divide__eq__right,axiom,
% 9.03/4.96      ! [A: real,B: real,C: real] :
% 9.03/4.96        ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.03/4.96        = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2697_divide__divide__eq__right,axiom,
% 9.03/4.96      ! [A: rat,B: rat,C: rat] :
% 9.03/4.96        ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.96        = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2698_divide__divide__eq__left,axiom,
% 9.03/4.96      ! [A: complex,B: complex,C: complex] :
% 9.03/4.96        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 9.03/4.96        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2699_divide__divide__eq__left,axiom,
% 9.03/4.96      ! [A: real,B: real,C: real] :
% 9.03/4.96        ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 9.03/4.96        = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2700_divide__divide__eq__left,axiom,
% 9.03/4.96      ! [A: rat,B: rat,C: rat] :
% 9.03/4.96        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 9.03/4.96        = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2701_times__divide__eq__left,axiom,
% 9.03/4.96      ! [B: complex,C: complex,A: complex] :
% 9.03/4.96        ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 9.03/4.96        = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2702_times__divide__eq__left,axiom,
% 9.03/4.96      ! [B: real,C: real,A: real] :
% 9.03/4.96        ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.03/4.96        = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2703_times__divide__eq__left,axiom,
% 9.03/4.96      ! [B: rat,C: rat,A: rat] :
% 9.03/4.96        ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.03/4.96        = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2704_of__int__mult,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ring_18347121197199848620nteger @ ( times_times_int @ W @ Z ) )
% 9.03/4.96        = ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2705_of__int__mult,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ring_17405671764205052669omplex @ ( times_times_int @ W @ Z ) )
% 9.03/4.96        = ( times_times_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2706_of__int__mult,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 9.03/4.96        = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2707_of__int__mult,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 9.03/4.96        = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2708_of__int__mult,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 9.03/4.96        = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2709_diff__Suc__Suc,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.03/4.96        = ( minus_minus_nat @ M @ N ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2710_Suc__diff__diff,axiom,
% 9.03/4.96      ! [M: nat,N: nat,K: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 9.03/4.96        = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2711_diff__0__eq__0,axiom,
% 9.03/4.96      ! [N: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ zero_zero_nat @ N )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2712_diff__self__eq__0,axiom,
% 9.03/4.96      ! [M: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ M @ M )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2713_of__int__diff,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ring_18347121197199848620nteger @ ( minus_minus_int @ W @ Z ) )
% 9.03/4.96        = ( minus_8373710615458151222nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2714_of__int__diff,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ring_17405671764205052669omplex @ ( minus_minus_int @ W @ Z ) )
% 9.03/4.96        = ( minus_minus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2715_of__int__diff,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 9.03/4.96        = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2716_of__int__diff,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 9.03/4.96        = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2717_of__int__diff,axiom,
% 9.03/4.96      ! [W: int,Z: int] :
% 9.03/4.96        ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 9.03/4.96        = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2718_mult__is__0,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ( times_times_nat @ M @ N )
% 9.03/4.96          = zero_zero_nat )
% 9.03/4.96        = ( ( M = zero_zero_nat )
% 9.03/4.96          | ( N = zero_zero_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2719_mult__0__right,axiom,
% 9.03/4.96      ! [M: nat] :
% 9.03/4.96        ( ( times_times_nat @ M @ zero_zero_nat )
% 9.03/4.96        = zero_zero_nat ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2720_mult__cancel1,axiom,
% 9.03/4.96      ! [K: nat,M: nat,N: nat] :
% 9.03/4.96        ( ( ( times_times_nat @ K @ M )
% 9.03/4.96          = ( times_times_nat @ K @ N ) )
% 9.03/4.96        = ( ( M = N )
% 9.03/4.96          | ( K = zero_zero_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2721_mult__cancel2,axiom,
% 9.03/4.96      ! [M: nat,K: nat,N: nat] :
% 9.03/4.96        ( ( ( times_times_nat @ M @ K )
% 9.03/4.96          = ( times_times_nat @ N @ K ) )
% 9.03/4.96        = ( ( M = N )
% 9.03/4.96          | ( K = zero_zero_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2722_diff__diff__cancel,axiom,
% 9.03/4.96      ! [I: nat,N: nat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ I @ N )
% 9.03/4.96       => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 9.03/4.96          = I ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2723_diff__diff__left,axiom,
% 9.03/4.96      ! [I: nat,J: nat,K: nat] :
% 9.03/4.96        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 9.03/4.96        = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2724_powr__0,axiom,
% 9.03/4.96      ! [Z: real] :
% 9.03/4.96        ( ( powr_real @ zero_zero_real @ Z )
% 9.03/4.96        = zero_zero_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2725_powr__eq__0__iff,axiom,
% 9.03/4.96      ! [W: real,Z: real] :
% 9.03/4.96        ( ( ( powr_real @ W @ Z )
% 9.03/4.96          = zero_zero_real )
% 9.03/4.96        = ( W = zero_zero_real ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2726_powr__one__eq__one,axiom,
% 9.03/4.96      ! [A: real] :
% 9.03/4.96        ( ( powr_real @ one_one_real @ A )
% 9.03/4.96        = one_one_real ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2727_nat__mult__eq__1__iff,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( ( times_times_nat @ M @ N )
% 9.03/4.96          = one_one_nat )
% 9.03/4.96        = ( ( M = one_one_nat )
% 9.03/4.96          & ( N = one_one_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2728_nat__1__eq__mult__iff,axiom,
% 9.03/4.96      ! [M: nat,N: nat] :
% 9.03/4.96        ( ( one_one_nat
% 9.03/4.96          = ( times_times_nat @ M @ N ) )
% 9.03/4.96        = ( ( M = one_one_nat )
% 9.03/4.96          & ( N = one_one_nat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2729_floor__of__int,axiom,
% 9.03/4.96      ! [Z: int] :
% 9.03/4.96        ( ( archim6058952711729229775r_real @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.96        = Z ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2730_floor__of__int,axiom,
% 9.03/4.96      ! [Z: int] :
% 9.03/4.96        ( ( archim3151403230148437115or_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.96        = Z ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2731_of__int__floor__cancel,axiom,
% 9.03/4.96      ! [X: real] :
% 9.03/4.96        ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
% 9.03/4.96          = X )
% 9.03/4.96        = ( ? [N3: int] :
% 9.03/4.96              ( X
% 9.03/4.96              = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2732_of__int__floor__cancel,axiom,
% 9.03/4.96      ! [X: rat] :
% 9.03/4.96        ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/4.96          = X )
% 9.03/4.96        = ( ? [N3: int] :
% 9.03/4.96              ( X
% 9.03/4.96              = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2733_round__of__int,axiom,
% 9.03/4.96      ! [N: int] :
% 9.03/4.96        ( ( archim8280529875227126926d_real @ ( ring_1_of_int_real @ N ) )
% 9.03/4.96        = N ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2734_round__of__int,axiom,
% 9.03/4.96      ! [N: int] :
% 9.03/4.96        ( ( archim7778729529865785530nd_rat @ ( ring_1_of_int_rat @ N ) )
% 9.03/4.96        = N ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2735_diff__ge__0__iff__ge,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 9.03/4.96        = ( ord_less_eq_real @ B @ A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2736_diff__ge__0__iff__ge,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 9.03/4.96        = ( ord_less_eq_rat @ B @ A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2737_diff__ge__0__iff__ge,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 9.03/4.96        = ( ord_less_eq_int @ B @ A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2738_diff__gt__0__iff__gt,axiom,
% 9.03/4.96      ! [A: real,B: real] :
% 9.03/4.96        ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 9.03/4.96        = ( ord_less_real @ B @ A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2739_diff__gt__0__iff__gt,axiom,
% 9.03/4.96      ! [A: rat,B: rat] :
% 9.03/4.96        ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 9.03/4.96        = ( ord_less_rat @ B @ A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2740_diff__gt__0__iff__gt,axiom,
% 9.03/4.96      ! [A: int,B: int] :
% 9.03/4.96        ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 9.03/4.96        = ( ord_less_int @ B @ A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2741_mult__cancel__left1,axiom,
% 9.03/4.96      ! [C: complex,B: complex] :
% 9.03/4.96        ( ( C
% 9.03/4.96          = ( times_times_complex @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_complex )
% 9.03/4.96          | ( B = one_one_complex ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2742_mult__cancel__left1,axiom,
% 9.03/4.96      ! [C: real,B: real] :
% 9.03/4.96        ( ( C
% 9.03/4.96          = ( times_times_real @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_real )
% 9.03/4.96          | ( B = one_one_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2743_mult__cancel__left1,axiom,
% 9.03/4.96      ! [C: rat,B: rat] :
% 9.03/4.96        ( ( C
% 9.03/4.96          = ( times_times_rat @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_rat )
% 9.03/4.96          | ( B = one_one_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2744_mult__cancel__left1,axiom,
% 9.03/4.96      ! [C: int,B: int] :
% 9.03/4.96        ( ( C
% 9.03/4.96          = ( times_times_int @ C @ B ) )
% 9.03/4.96        = ( ( C = zero_zero_int )
% 9.03/4.96          | ( B = one_one_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2745_mult__cancel__left2,axiom,
% 9.03/4.96      ! [C: complex,A: complex] :
% 9.03/4.96        ( ( ( times_times_complex @ C @ A )
% 9.03/4.96          = C )
% 9.03/4.96        = ( ( C = zero_zero_complex )
% 9.03/4.96          | ( A = one_one_complex ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2746_mult__cancel__left2,axiom,
% 9.03/4.96      ! [C: real,A: real] :
% 9.03/4.96        ( ( ( times_times_real @ C @ A )
% 9.03/4.96          = C )
% 9.03/4.96        = ( ( C = zero_zero_real )
% 9.03/4.96          | ( A = one_one_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2747_mult__cancel__left2,axiom,
% 9.03/4.96      ! [C: rat,A: rat] :
% 9.03/4.96        ( ( ( times_times_rat @ C @ A )
% 9.03/4.96          = C )
% 9.03/4.96        = ( ( C = zero_zero_rat )
% 9.03/4.96          | ( A = one_one_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2748_mult__cancel__left2,axiom,
% 9.03/4.96      ! [C: int,A: int] :
% 9.03/4.96        ( ( ( times_times_int @ C @ A )
% 9.03/4.96          = C )
% 9.03/4.96        = ( ( C = zero_zero_int )
% 9.03/4.96          | ( A = one_one_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2749_mult__cancel__right1,axiom,
% 9.03/4.96      ! [C: complex,B: complex] :
% 9.03/4.96        ( ( C
% 9.03/4.96          = ( times_times_complex @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_complex )
% 9.03/4.96          | ( B = one_one_complex ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2750_mult__cancel__right1,axiom,
% 9.03/4.96      ! [C: real,B: real] :
% 9.03/4.96        ( ( C
% 9.03/4.96          = ( times_times_real @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_real )
% 9.03/4.96          | ( B = one_one_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2751_mult__cancel__right1,axiom,
% 9.03/4.96      ! [C: rat,B: rat] :
% 9.03/4.96        ( ( C
% 9.03/4.96          = ( times_times_rat @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_rat )
% 9.03/4.96          | ( B = one_one_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2752_mult__cancel__right1,axiom,
% 9.03/4.96      ! [C: int,B: int] :
% 9.03/4.96        ( ( C
% 9.03/4.96          = ( times_times_int @ B @ C ) )
% 9.03/4.96        = ( ( C = zero_zero_int )
% 9.03/4.96          | ( B = one_one_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2753_mult__cancel__right2,axiom,
% 9.03/4.96      ! [A: complex,C: complex] :
% 9.03/4.96        ( ( ( times_times_complex @ A @ C )
% 9.03/4.96          = C )
% 9.03/4.96        = ( ( C = zero_zero_complex )
% 9.03/4.96          | ( A = one_one_complex ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2754_mult__cancel__right2,axiom,
% 9.03/4.96      ! [A: real,C: real] :
% 9.03/4.96        ( ( ( times_times_real @ A @ C )
% 9.03/4.96          = C )
% 9.03/4.96        = ( ( C = zero_zero_real )
% 9.03/4.96          | ( A = one_one_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2755_mult__cancel__right2,axiom,
% 9.03/4.96      ! [A: rat,C: rat] :
% 9.03/4.96        ( ( ( times_times_rat @ A @ C )
% 9.03/4.96          = C )
% 9.03/4.96        = ( ( C = zero_zero_rat )
% 9.03/4.96          | ( A = one_one_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2756_mult__cancel__right2,axiom,
% 9.03/4.96      ! [A: int,C: int] :
% 9.03/4.96        ( ( ( times_times_int @ A @ C )
% 9.03/4.96          = C )
% 9.03/4.96        = ( ( C = zero_zero_int )
% 9.03/4.96          | ( A = one_one_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2757_sum__squares__eq__zero__iff,axiom,
% 9.03/4.96      ! [X: real,Y: real] :
% 9.03/4.96        ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 9.03/4.96          = zero_zero_real )
% 9.03/4.96        = ( ( X = zero_zero_real )
% 9.03/4.96          & ( Y = zero_zero_real ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2758_sum__squares__eq__zero__iff,axiom,
% 9.03/4.96      ! [X: rat,Y: rat] :
% 9.03/4.96        ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 9.03/4.96          = zero_zero_rat )
% 9.03/4.96        = ( ( X = zero_zero_rat )
% 9.03/4.96          & ( Y = zero_zero_rat ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2759_sum__squares__eq__zero__iff,axiom,
% 9.03/4.96      ! [X: int,Y: int] :
% 9.03/4.96        ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 9.03/4.96          = zero_zero_int )
% 9.03/4.96        = ( ( X = zero_zero_int )
% 9.03/4.96          & ( Y = zero_zero_int ) ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2760_le__add__diff__inverse,axiom,
% 9.03/4.96      ! [B: real,A: real] :
% 9.03/4.96        ( ( ord_less_eq_real @ B @ A )
% 9.03/4.96       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 9.03/4.96          = A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2761_le__add__diff__inverse,axiom,
% 9.03/4.96      ! [B: rat,A: rat] :
% 9.03/4.96        ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.96       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 9.03/4.96          = A ) ) ).
% 9.03/4.96  
% 9.03/4.96  thf(fact_2762_le__add__diff__inverse,axiom,
% 9.03/4.96      ! [B: nat,A: nat] :
% 9.03/4.96        ( ( ord_less_eq_nat @ B @ A )
% 9.03/4.96       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2763_le__add__diff__inverse,axiom,
% 9.03/4.97      ! [B: int,A: int] :
% 9.03/4.97        ( ( ord_less_eq_int @ B @ A )
% 9.03/4.97       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2764_le__add__diff__inverse2,axiom,
% 9.03/4.97      ! [B: real,A: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ B @ A )
% 9.03/4.97       => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2765_le__add__diff__inverse2,axiom,
% 9.03/4.97      ! [B: rat,A: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.97       => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2766_le__add__diff__inverse2,axiom,
% 9.03/4.97      ! [B: nat,A: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ B @ A )
% 9.03/4.97       => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2767_le__add__diff__inverse2,axiom,
% 9.03/4.97      ! [B: int,A: int] :
% 9.03/4.97        ( ( ord_less_eq_int @ B @ A )
% 9.03/4.97       => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2768_diff__numeral__special_I9_J,axiom,
% 9.03/4.97      ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 9.03/4.97      = zero_zero_complex ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2769_diff__numeral__special_I9_J,axiom,
% 9.03/4.97      ( ( minus_minus_real @ one_one_real @ one_one_real )
% 9.03/4.97      = zero_zero_real ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2770_diff__numeral__special_I9_J,axiom,
% 9.03/4.97      ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 9.03/4.97      = zero_zero_rat ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2771_diff__numeral__special_I9_J,axiom,
% 9.03/4.97      ( ( minus_minus_int @ one_one_int @ one_one_int )
% 9.03/4.97      = zero_zero_int ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2772_diff__add__zero,axiom,
% 9.03/4.97      ! [A: nat,B: nat] :
% 9.03/4.97        ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.03/4.97        = zero_zero_nat ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2773_div__mult__mult1__if,axiom,
% 9.03/4.97      ! [C: nat,A: nat,B: nat] :
% 9.03/4.97        ( ( ( C = zero_zero_nat )
% 9.03/4.97         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.03/4.97            = zero_zero_nat ) )
% 9.03/4.97        & ( ( C != zero_zero_nat )
% 9.03/4.97         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.03/4.97            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2774_div__mult__mult1__if,axiom,
% 9.03/4.97      ! [C: int,A: int,B: int] :
% 9.03/4.97        ( ( ( C = zero_zero_int )
% 9.03/4.97         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.97            = zero_zero_int ) )
% 9.03/4.97        & ( ( C != zero_zero_int )
% 9.03/4.97         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.97            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2775_div__mult__mult2,axiom,
% 9.03/4.97      ! [C: nat,A: nat,B: nat] :
% 9.03/4.97        ( ( C != zero_zero_nat )
% 9.03/4.97       => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 9.03/4.97          = ( divide_divide_nat @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2776_div__mult__mult2,axiom,
% 9.03/4.97      ! [C: int,A: int,B: int] :
% 9.03/4.97        ( ( C != zero_zero_int )
% 9.03/4.97       => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.03/4.97          = ( divide_divide_int @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2777_div__mult__mult1,axiom,
% 9.03/4.97      ! [C: nat,A: nat,B: nat] :
% 9.03/4.97        ( ( C != zero_zero_nat )
% 9.03/4.97       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.03/4.97          = ( divide_divide_nat @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2778_div__mult__mult1,axiom,
% 9.03/4.97      ! [C: int,A: int,B: int] :
% 9.03/4.97        ( ( C != zero_zero_int )
% 9.03/4.97       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.97          = ( divide_divide_int @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2779_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.03/4.97      ! [C: complex,A: complex,B: complex] :
% 9.03/4.97        ( ( C != zero_zero_complex )
% 9.03/4.97       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 9.03/4.97          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2780_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.03/4.97      ! [C: real,A: real,B: real] :
% 9.03/4.97        ( ( C != zero_zero_real )
% 9.03/4.97       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 9.03/4.97          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2781_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.03/4.97      ! [C: rat,A: rat,B: rat] :
% 9.03/4.97        ( ( C != zero_zero_rat )
% 9.03/4.97       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.97          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2782_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.03/4.97      ! [C: complex,A: complex,B: complex] :
% 9.03/4.97        ( ( C != zero_zero_complex )
% 9.03/4.97       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 9.03/4.97          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2783_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.03/4.97      ! [C: real,A: real,B: real] :
% 9.03/4.97        ( ( C != zero_zero_real )
% 9.03/4.97       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.03/4.97          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2784_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.03/4.97      ! [C: rat,A: rat,B: rat] :
% 9.03/4.97        ( ( C != zero_zero_rat )
% 9.03/4.97       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.03/4.97          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2785_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.03/4.97      ! [C: complex,A: complex,B: complex] :
% 9.03/4.97        ( ( C != zero_zero_complex )
% 9.03/4.97       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 9.03/4.97          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2786_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.03/4.97      ! [C: real,A: real,B: real] :
% 9.03/4.97        ( ( C != zero_zero_real )
% 9.03/4.97       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 9.03/4.97          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2787_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.03/4.97      ! [C: rat,A: rat,B: rat] :
% 9.03/4.97        ( ( C != zero_zero_rat )
% 9.03/4.97       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 9.03/4.97          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2788_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.03/4.97      ! [C: complex,A: complex,B: complex] :
% 9.03/4.97        ( ( C != zero_zero_complex )
% 9.03/4.97       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.03/4.97          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2789_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.03/4.97      ! [C: real,A: real,B: real] :
% 9.03/4.97        ( ( C != zero_zero_real )
% 9.03/4.97       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.97          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2790_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.03/4.97      ! [C: rat,A: rat,B: rat] :
% 9.03/4.97        ( ( C != zero_zero_rat )
% 9.03/4.97       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.97          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2791_mult__divide__mult__cancel__left__if,axiom,
% 9.03/4.97      ! [C: complex,A: complex,B: complex] :
% 9.03/4.97        ( ( ( C = zero_zero_complex )
% 9.03/4.97         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.03/4.97            = zero_zero_complex ) )
% 9.03/4.97        & ( ( C != zero_zero_complex )
% 9.03/4.97         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.03/4.97            = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2792_mult__divide__mult__cancel__left__if,axiom,
% 9.03/4.97      ! [C: real,A: real,B: real] :
% 9.03/4.97        ( ( ( C = zero_zero_real )
% 9.03/4.97         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.97            = zero_zero_real ) )
% 9.03/4.97        & ( ( C != zero_zero_real )
% 9.03/4.97         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.97            = ( divide_divide_real @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2793_mult__divide__mult__cancel__left__if,axiom,
% 9.03/4.97      ! [C: rat,A: rat,B: rat] :
% 9.03/4.97        ( ( ( C = zero_zero_rat )
% 9.03/4.97         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.97            = zero_zero_rat ) )
% 9.03/4.97        & ( ( C != zero_zero_rat )
% 9.03/4.97         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.97            = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2794_nonzero__mult__div__cancel__right,axiom,
% 9.03/4.97      ! [B: complex,A: complex] :
% 9.03/4.97        ( ( B != zero_zero_complex )
% 9.03/4.97       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2795_nonzero__mult__div__cancel__right,axiom,
% 9.03/4.97      ! [B: real,A: real] :
% 9.03/4.97        ( ( B != zero_zero_real )
% 9.03/4.97       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2796_nonzero__mult__div__cancel__right,axiom,
% 9.03/4.97      ! [B: rat,A: rat] :
% 9.03/4.97        ( ( B != zero_zero_rat )
% 9.03/4.97       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2797_nonzero__mult__div__cancel__right,axiom,
% 9.03/4.97      ! [B: nat,A: nat] :
% 9.03/4.97        ( ( B != zero_zero_nat )
% 9.03/4.97       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2798_nonzero__mult__div__cancel__right,axiom,
% 9.03/4.97      ! [B: int,A: int] :
% 9.03/4.97        ( ( B != zero_zero_int )
% 9.03/4.97       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 9.03/4.97          = A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2799_nonzero__mult__div__cancel__left,axiom,
% 9.03/4.97      ! [A: complex,B: complex] :
% 9.03/4.97        ( ( A != zero_zero_complex )
% 9.03/4.97       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 9.03/4.97          = B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2800_nonzero__mult__div__cancel__left,axiom,
% 9.03/4.97      ! [A: real,B: real] :
% 9.03/4.97        ( ( A != zero_zero_real )
% 9.03/4.97       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 9.03/4.97          = B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2801_nonzero__mult__div__cancel__left,axiom,
% 9.03/4.97      ! [A: rat,B: rat] :
% 9.03/4.97        ( ( A != zero_zero_rat )
% 9.03/4.97       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 9.03/4.97          = B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2802_nonzero__mult__div__cancel__left,axiom,
% 9.03/4.97      ! [A: nat,B: nat] :
% 9.03/4.97        ( ( A != zero_zero_nat )
% 9.03/4.97       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 9.03/4.97          = B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2803_nonzero__mult__div__cancel__left,axiom,
% 9.03/4.97      ! [A: int,B: int] :
% 9.03/4.97        ( ( A != zero_zero_int )
% 9.03/4.97       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 9.03/4.97          = B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2804_distrib__left__numeral,axiom,
% 9.03/4.97      ! [V: num,B: complex,C: complex] :
% 9.03/4.97        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 9.03/4.97        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2805_distrib__left__numeral,axiom,
% 9.03/4.97      ! [V: num,B: real,C: real] :
% 9.03/4.97        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 9.03/4.97        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2806_distrib__left__numeral,axiom,
% 9.03/4.97      ! [V: num,B: rat,C: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 9.03/4.97        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2807_distrib__left__numeral,axiom,
% 9.03/4.97      ! [V: num,B: nat,C: nat] :
% 9.03/4.97        ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 9.03/4.97        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2808_distrib__left__numeral,axiom,
% 9.03/4.97      ! [V: num,B: int,C: int] :
% 9.03/4.97        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 9.03/4.97        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2809_distrib__right__numeral,axiom,
% 9.03/4.97      ! [A: complex,B: complex,V: num] :
% 9.03/4.97        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 9.03/4.97        = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2810_distrib__right__numeral,axiom,
% 9.03/4.97      ! [A: real,B: real,V: num] :
% 9.03/4.97        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 9.03/4.97        = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2811_distrib__right__numeral,axiom,
% 9.03/4.97      ! [A: rat,B: rat,V: num] :
% 9.03/4.97        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 9.03/4.97        = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2812_distrib__right__numeral,axiom,
% 9.03/4.97      ! [A: nat,B: nat,V: num] :
% 9.03/4.97        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 9.03/4.97        = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2813_distrib__right__numeral,axiom,
% 9.03/4.97      ! [A: int,B: int,V: num] :
% 9.03/4.97        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 9.03/4.97        = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2814_left__diff__distrib__numeral,axiom,
% 9.03/4.97      ! [A: complex,B: complex,V: num] :
% 9.03/4.97        ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 9.03/4.97        = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2815_left__diff__distrib__numeral,axiom,
% 9.03/4.97      ! [A: real,B: real,V: num] :
% 9.03/4.97        ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 9.03/4.97        = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2816_left__diff__distrib__numeral,axiom,
% 9.03/4.97      ! [A: rat,B: rat,V: num] :
% 9.03/4.97        ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 9.03/4.97        = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2817_left__diff__distrib__numeral,axiom,
% 9.03/4.97      ! [A: int,B: int,V: num] :
% 9.03/4.97        ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 9.03/4.97        = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2818_right__diff__distrib__numeral,axiom,
% 9.03/4.97      ! [V: num,B: complex,C: complex] :
% 9.03/4.97        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 9.03/4.97        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2819_right__diff__distrib__numeral,axiom,
% 9.03/4.97      ! [V: num,B: real,C: real] :
% 9.03/4.97        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 9.03/4.97        = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2820_right__diff__distrib__numeral,axiom,
% 9.03/4.97      ! [V: num,B: rat,C: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 9.03/4.97        = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2821_right__diff__distrib__numeral,axiom,
% 9.03/4.97      ! [V: num,B: int,C: int] :
% 9.03/4.97        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 9.03/4.97        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2822_of__nat__mult,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2823_of__nat__mult,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2824_of__nat__mult,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2825_of__nat__mult,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2826_of__nat__mult,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2827_mult__eq__1__iff,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( ( times_times_nat @ M @ N )
% 9.03/4.97          = ( suc @ zero_zero_nat ) )
% 9.03/4.97        = ( ( M
% 9.03/4.97            = ( suc @ zero_zero_nat ) )
% 9.03/4.97          & ( N
% 9.03/4.97            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2828_one__eq__mult__iff,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( ( suc @ zero_zero_nat )
% 9.03/4.97          = ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( ( M
% 9.03/4.97            = ( suc @ zero_zero_nat ) )
% 9.03/4.97          & ( N
% 9.03/4.97            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2829_zero__less__diff,axiom,
% 9.03/4.97      ! [N: nat,M: nat] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 9.03/4.97        = ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2830_powr__zero__eq__one,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ( X = zero_zero_real )
% 9.03/4.97         => ( ( powr_real @ X @ zero_zero_real )
% 9.03/4.97            = zero_zero_real ) )
% 9.03/4.97        & ( ( X != zero_zero_real )
% 9.03/4.97         => ( ( powr_real @ X @ zero_zero_real )
% 9.03/4.97            = one_one_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2831_mult__less__cancel2,axiom,
% 9.03/4.97      ! [M: nat,K: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 9.03/4.97        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.97          & ( ord_less_nat @ M @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2832_nat__0__less__mult__iff,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.97          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2833_nat__mult__less__cancel__disj,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.03/4.97        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.97          & ( ord_less_nat @ M @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2834_diff__is__0__eq,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( ( minus_minus_nat @ M @ N )
% 9.03/4.97          = zero_zero_nat )
% 9.03/4.97        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2835_diff__is__0__eq_H,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.97       => ( ( minus_minus_nat @ M @ N )
% 9.03/4.97          = zero_zero_nat ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2836_mult__Suc__right,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( times_times_nat @ M @ ( suc @ N ) )
% 9.03/4.97        = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2837_Nat_Odiff__diff__right,axiom,
% 9.03/4.97      ! [K: nat,J: nat,I: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ K @ J )
% 9.03/4.97       => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.03/4.97          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2838_Nat_Oadd__diff__assoc2,axiom,
% 9.03/4.97      ! [K: nat,J: nat,I: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ K @ J )
% 9.03/4.97       => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.03/4.97          = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2839_Nat_Oadd__diff__assoc,axiom,
% 9.03/4.97      ! [K: nat,J: nat,I: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ K @ J )
% 9.03/4.97       => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.03/4.97          = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2840_floor__zero,axiom,
% 9.03/4.97      ( ( archim6058952711729229775r_real @ zero_zero_real )
% 9.03/4.97      = zero_zero_int ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2841_floor__zero,axiom,
% 9.03/4.97      ( ( archim3151403230148437115or_rat @ zero_zero_rat )
% 9.03/4.97      = zero_zero_int ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2842_diff__Suc__1,axiom,
% 9.03/4.97      ! [N: nat] :
% 9.03/4.97        ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 9.03/4.97        = N ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2843_floor__numeral,axiom,
% 9.03/4.97      ! [V: num] :
% 9.03/4.97        ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
% 9.03/4.97        = ( numeral_numeral_int @ V ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2844_floor__numeral,axiom,
% 9.03/4.97      ! [V: num] :
% 9.03/4.97        ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
% 9.03/4.97        = ( numeral_numeral_int @ V ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2845_nat__mult__div__cancel__disj,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ( K = zero_zero_nat )
% 9.03/4.97         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.03/4.97            = zero_zero_nat ) )
% 9.03/4.97        & ( ( K != zero_zero_nat )
% 9.03/4.97         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.03/4.97            = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2846_powr__gt__zero,axiom,
% 9.03/4.97      ! [X: real,A: real] :
% 9.03/4.97        ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
% 9.03/4.97        = ( X != zero_zero_real ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2847_floor__one,axiom,
% 9.03/4.97      ( ( archim6058952711729229775r_real @ one_one_real )
% 9.03/4.97      = one_one_int ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2848_floor__one,axiom,
% 9.03/4.97      ( ( archim3151403230148437115or_rat @ one_one_rat )
% 9.03/4.97      = one_one_int ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2849_powr__nonneg__iff,axiom,
% 9.03/4.97      ! [A: real,X: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
% 9.03/4.97        = ( A = zero_zero_real ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2850_floor__diff__of__int,axiom,
% 9.03/4.97      ! [X: real,Z: int] :
% 9.03/4.97        ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2851_floor__diff__of__int,axiom,
% 9.03/4.97      ! [X: rat,Z: int] :
% 9.03/4.97        ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2852_powr__less__cancel__iff,axiom,
% 9.03/4.97      ! [X: real,A: real,B: real] :
% 9.03/4.97        ( ( ord_less_real @ one_one_real @ X )
% 9.03/4.97       => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 9.03/4.97          = ( ord_less_real @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2853_ceiling__diff__of__int,axiom,
% 9.03/4.97      ! [X: real,Z: int] :
% 9.03/4.97        ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2854_ceiling__diff__of__int,axiom,
% 9.03/4.97      ! [X: rat,Z: int] :
% 9.03/4.97        ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2855_floor__of__nat,axiom,
% 9.03/4.97      ! [N: nat] :
% 9.03/4.97        ( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.97        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2856_floor__of__nat,axiom,
% 9.03/4.97      ! [N: nat] :
% 9.03/4.97        ( ( archim3151403230148437115or_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.03/4.97        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2857_round__0,axiom,
% 9.03/4.97      ( ( archim8280529875227126926d_real @ zero_zero_real )
% 9.03/4.97      = zero_zero_int ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2858_round__0,axiom,
% 9.03/4.97      ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 9.03/4.97      = zero_zero_int ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2859_round__numeral,axiom,
% 9.03/4.97      ! [N: num] :
% 9.03/4.97        ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 9.03/4.97        = ( numeral_numeral_int @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2860_round__numeral,axiom,
% 9.03/4.97      ! [N: num] :
% 9.03/4.97        ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 9.03/4.97        = ( numeral_numeral_int @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2861_round__1,axiom,
% 9.03/4.97      ( ( archim8280529875227126926d_real @ one_one_real )
% 9.03/4.97      = one_one_int ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2862_round__1,axiom,
% 9.03/4.97      ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 9.03/4.97      = one_one_int ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2863_round__of__nat,axiom,
% 9.03/4.97      ! [N: nat] :
% 9.03/4.97        ( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.97        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2864_round__of__nat,axiom,
% 9.03/4.97      ! [N: nat] :
% 9.03/4.97        ( ( archim7778729529865785530nd_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.03/4.97        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2865_le__divide__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [A: real,B: real,W: num] :
% 9.03/4.97        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 9.03/4.97        = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2866_le__divide__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [A: rat,B: rat,W: num] :
% 9.03/4.97        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 9.03/4.97        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2867_divide__le__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [B: real,W: num,A: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 9.03/4.97        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2868_divide__le__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [B: rat,W: num,A: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 9.03/4.97        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2869_eq__divide__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [A: complex,B: complex,W: num] :
% 9.03/4.97        ( ( A
% 9.03/4.97          = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 9.03/4.97        = ( ( ( ( numera6690914467698888265omplex @ W )
% 9.03/4.97             != zero_zero_complex )
% 9.03/4.97           => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 9.03/4.97              = B ) )
% 9.03/4.97          & ( ( ( numera6690914467698888265omplex @ W )
% 9.03/4.97              = zero_zero_complex )
% 9.03/4.97           => ( A = zero_zero_complex ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2870_eq__divide__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [A: real,B: real,W: num] :
% 9.03/4.97        ( ( A
% 9.03/4.97          = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 9.03/4.97        = ( ( ( ( numeral_numeral_real @ W )
% 9.03/4.97             != zero_zero_real )
% 9.03/4.97           => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 9.03/4.97              = B ) )
% 9.03/4.97          & ( ( ( numeral_numeral_real @ W )
% 9.03/4.97              = zero_zero_real )
% 9.03/4.97           => ( A = zero_zero_real ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2871_eq__divide__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [A: rat,B: rat,W: num] :
% 9.03/4.97        ( ( A
% 9.03/4.97          = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 9.03/4.97        = ( ( ( ( numeral_numeral_rat @ W )
% 9.03/4.97             != zero_zero_rat )
% 9.03/4.97           => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 9.03/4.97              = B ) )
% 9.03/4.97          & ( ( ( numeral_numeral_rat @ W )
% 9.03/4.97              = zero_zero_rat )
% 9.03/4.97           => ( A = zero_zero_rat ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2872_divide__eq__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [B: complex,W: num,A: complex] :
% 9.03/4.97        ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 9.03/4.97          = A )
% 9.03/4.97        = ( ( ( ( numera6690914467698888265omplex @ W )
% 9.03/4.97             != zero_zero_complex )
% 9.03/4.97           => ( B
% 9.03/4.97              = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 9.03/4.97          & ( ( ( numera6690914467698888265omplex @ W )
% 9.03/4.97              = zero_zero_complex )
% 9.03/4.97           => ( A = zero_zero_complex ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2873_divide__eq__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [B: real,W: num,A: real] :
% 9.03/4.97        ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 9.03/4.97          = A )
% 9.03/4.97        = ( ( ( ( numeral_numeral_real @ W )
% 9.03/4.97             != zero_zero_real )
% 9.03/4.97           => ( B
% 9.03/4.97              = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 9.03/4.97          & ( ( ( numeral_numeral_real @ W )
% 9.03/4.97              = zero_zero_real )
% 9.03/4.97           => ( A = zero_zero_real ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2874_divide__eq__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [B: rat,W: num,A: rat] :
% 9.03/4.97        ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 9.03/4.97          = A )
% 9.03/4.97        = ( ( ( ( numeral_numeral_rat @ W )
% 9.03/4.97             != zero_zero_rat )
% 9.03/4.97           => ( B
% 9.03/4.97              = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 9.03/4.97          & ( ( ( numeral_numeral_rat @ W )
% 9.03/4.97              = zero_zero_rat )
% 9.03/4.97           => ( A = zero_zero_rat ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2875_less__divide__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [A: real,B: real,W: num] :
% 9.03/4.97        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 9.03/4.97        = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2876_less__divide__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [A: rat,B: rat,W: num] :
% 9.03/4.97        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 9.03/4.97        = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2877_divide__less__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [B: real,W: num,A: real] :
% 9.03/4.97        ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 9.03/4.97        = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2878_divide__less__eq__numeral1_I1_J,axiom,
% 9.03/4.97      ! [B: rat,W: num,A: rat] :
% 9.03/4.97        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 9.03/4.97        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2879_nonzero__divide__mult__cancel__left,axiom,
% 9.03/4.97      ! [A: complex,B: complex] :
% 9.03/4.97        ( ( A != zero_zero_complex )
% 9.03/4.97       => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 9.03/4.97          = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2880_nonzero__divide__mult__cancel__left,axiom,
% 9.03/4.97      ! [A: real,B: real] :
% 9.03/4.97        ( ( A != zero_zero_real )
% 9.03/4.97       => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 9.03/4.97          = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2881_nonzero__divide__mult__cancel__left,axiom,
% 9.03/4.97      ! [A: rat,B: rat] :
% 9.03/4.97        ( ( A != zero_zero_rat )
% 9.03/4.97       => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 9.03/4.97          = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2882_nonzero__divide__mult__cancel__right,axiom,
% 9.03/4.97      ! [B: complex,A: complex] :
% 9.03/4.97        ( ( B != zero_zero_complex )
% 9.03/4.97       => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 9.03/4.97          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2883_nonzero__divide__mult__cancel__right,axiom,
% 9.03/4.97      ! [B: real,A: real] :
% 9.03/4.97        ( ( B != zero_zero_real )
% 9.03/4.97       => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 9.03/4.97          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2884_nonzero__divide__mult__cancel__right,axiom,
% 9.03/4.97      ! [B: rat,A: rat] :
% 9.03/4.97        ( ( B != zero_zero_rat )
% 9.03/4.97       => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 9.03/4.97          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2885_div__mult__self1,axiom,
% 9.03/4.97      ! [B: nat,A: nat,C: nat] :
% 9.03/4.97        ( ( B != zero_zero_nat )
% 9.03/4.97       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 9.03/4.97          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2886_div__mult__self1,axiom,
% 9.03/4.97      ! [B: int,A: int,C: int] :
% 9.03/4.97        ( ( B != zero_zero_int )
% 9.03/4.97       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 9.03/4.97          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2887_div__mult__self2,axiom,
% 9.03/4.97      ! [B: nat,A: nat,C: nat] :
% 9.03/4.97        ( ( B != zero_zero_nat )
% 9.03/4.97       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 9.03/4.97          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2888_div__mult__self2,axiom,
% 9.03/4.97      ! [B: int,A: int,C: int] :
% 9.03/4.97        ( ( B != zero_zero_int )
% 9.03/4.97       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 9.03/4.97          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2889_div__mult__self3,axiom,
% 9.03/4.97      ! [B: nat,C: nat,A: nat] :
% 9.03/4.97        ( ( B != zero_zero_nat )
% 9.03/4.97       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 9.03/4.97          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2890_div__mult__self3,axiom,
% 9.03/4.97      ! [B: int,C: int,A: int] :
% 9.03/4.97        ( ( B != zero_zero_int )
% 9.03/4.97       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 9.03/4.97          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2891_div__mult__self4,axiom,
% 9.03/4.97      ! [B: nat,C: nat,A: nat] :
% 9.03/4.97        ( ( B != zero_zero_nat )
% 9.03/4.97       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 9.03/4.97          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2892_div__mult__self4,axiom,
% 9.03/4.97      ! [B: int,C: int,A: int] :
% 9.03/4.97        ( ( B != zero_zero_int )
% 9.03/4.97       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 9.03/4.97          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2893_Suc__pred,axiom,
% 9.03/4.97      ! [N: nat] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.97       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 9.03/4.97          = N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2894_one__le__mult__iff,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.03/4.97          & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2895_mult__le__cancel2,axiom,
% 9.03/4.97      ! [M: nat,K: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 9.03/4.97        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.97         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2896_nat__mult__le__cancel__disj,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.03/4.97        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.97         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2897_diff__Suc__diff__eq1,axiom,
% 9.03/4.97      ! [K: nat,J: nat,I: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ K @ J )
% 9.03/4.97       => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 9.03/4.97          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2898_diff__Suc__diff__eq2,axiom,
% 9.03/4.97      ! [K: nat,J: nat,I: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ K @ J )
% 9.03/4.97       => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 9.03/4.97          = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2899_floor__diff__numeral,axiom,
% 9.03/4.97      ! [X: real,V: num] :
% 9.03/4.97        ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2900_floor__diff__numeral,axiom,
% 9.03/4.97      ! [X: rat,V: num] :
% 9.03/4.97        ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2901_ceiling__diff__numeral,axiom,
% 9.03/4.97      ! [X: real,V: num] :
% 9.03/4.97        ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2902_ceiling__diff__numeral,axiom,
% 9.03/4.97      ! [X: rat,V: num] :
% 9.03/4.97        ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2903_div__mult__self1__is__m,axiom,
% 9.03/4.97      ! [N: nat,M: nat] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.97       => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 9.03/4.97          = M ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2904_div__mult__self__is__m,axiom,
% 9.03/4.97      ! [N: nat,M: nat] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.97       => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 9.03/4.97          = M ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2905_floor__diff__one,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2906_floor__diff__one,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2907_ceiling__diff__one,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2908_ceiling__diff__one,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 9.03/4.97        = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2909_power__add__numeral,axiom,
% 9.03/4.97      ! [A: complex,M: num,N: num] :
% 9.03/4.97        ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 9.03/4.97        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2910_power__add__numeral,axiom,
% 9.03/4.97      ! [A: real,M: num,N: num] :
% 9.03/4.97        ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 9.03/4.97        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2911_power__add__numeral,axiom,
% 9.03/4.97      ! [A: rat,M: num,N: num] :
% 9.03/4.97        ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 9.03/4.97        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2912_power__add__numeral,axiom,
% 9.03/4.97      ! [A: nat,M: num,N: num] :
% 9.03/4.97        ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 9.03/4.97        = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2913_power__add__numeral,axiom,
% 9.03/4.97      ! [A: int,M: num,N: num] :
% 9.03/4.97        ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 9.03/4.97        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2914_power__add__numeral2,axiom,
% 9.03/4.97      ! [A: complex,M: num,N: num,B: complex] :
% 9.03/4.97        ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 9.03/4.97        = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2915_power__add__numeral2,axiom,
% 9.03/4.97      ! [A: real,M: num,N: num,B: real] :
% 9.03/4.97        ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 9.03/4.97        = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2916_power__add__numeral2,axiom,
% 9.03/4.97      ! [A: rat,M: num,N: num,B: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 9.03/4.97        = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2917_power__add__numeral2,axiom,
% 9.03/4.97      ! [A: nat,M: num,N: num,B: nat] :
% 9.03/4.97        ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 9.03/4.97        = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2918_power__add__numeral2,axiom,
% 9.03/4.97      ! [A: int,M: num,N: num,B: int] :
% 9.03/4.97        ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 9.03/4.97        = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2919_powr__eq__one__iff,axiom,
% 9.03/4.97      ! [A: real,X: real] :
% 9.03/4.97        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.97       => ( ( ( powr_real @ A @ X )
% 9.03/4.97            = one_one_real )
% 9.03/4.97          = ( X = zero_zero_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2920_powr__one__gt__zero__iff,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ( powr_real @ X @ one_one_real )
% 9.03/4.97          = X )
% 9.03/4.97        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2921_powr__one,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.97       => ( ( powr_real @ X @ one_one_real )
% 9.03/4.97          = X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2922_powr__le__cancel__iff,axiom,
% 9.03/4.97      ! [X: real,A: real,B: real] :
% 9.03/4.97        ( ( ord_less_real @ one_one_real @ X )
% 9.03/4.97       => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 9.03/4.97          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2923_numeral__powr__numeral__real,axiom,
% 9.03/4.97      ! [M: num,N: num] :
% 9.03/4.97        ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.03/4.97        = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2924_zero__le__floor,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 9.03/4.97        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2925_zero__le__floor,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/4.97        = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2926_Suc__diff__1,axiom,
% 9.03/4.97      ! [N: nat] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.97       => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.03/4.97          = N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2927_floor__less__zero,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 9.03/4.97        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2928_floor__less__zero,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 9.03/4.97        = ( ord_less_rat @ X @ zero_zero_rat ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2929_numeral__le__floor,axiom,
% 9.03/4.97      ! [V: num,X: real] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 9.03/4.97        = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2930_numeral__le__floor,axiom,
% 9.03/4.97      ! [V: num,X: rat] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/4.97        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2931_zero__less__floor,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 9.03/4.97        = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2932_zero__less__floor,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/4.97        = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2933_floor__le__zero,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 9.03/4.97        = ( ord_less_real @ X @ one_one_real ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2934_floor__le__zero,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 9.03/4.97        = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2935_floor__less__numeral,axiom,
% 9.03/4.97      ! [X: real,V: num] :
% 9.03/4.97        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 9.03/4.97        = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2936_floor__less__numeral,axiom,
% 9.03/4.97      ! [X: rat,V: num] :
% 9.03/4.97        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 9.03/4.97        = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2937_one__le__floor,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 9.03/4.97        = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2938_one__le__floor,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/4.97        = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2939_floor__less__one,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 9.03/4.97        = ( ord_less_real @ X @ one_one_real ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2940_floor__less__one,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 9.03/4.97        = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2941_floor__numeral__power,axiom,
% 9.03/4.97      ! [X: num,N: nat] :
% 9.03/4.97        ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 9.03/4.97        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2942_floor__numeral__power,axiom,
% 9.03/4.97      ! [X: num,N: nat] :
% 9.03/4.97        ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 9.03/4.97        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2943_powr__log__cancel,axiom,
% 9.03/4.97      ! [A: real,X: real] :
% 9.03/4.97        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.97       => ( ( A != one_one_real )
% 9.03/4.97         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.97           => ( ( powr_real @ A @ ( log @ A @ X ) )
% 9.03/4.97              = X ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2944_log__powr__cancel,axiom,
% 9.03/4.97      ! [A: real,Y: real] :
% 9.03/4.97        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.97       => ( ( A != one_one_real )
% 9.03/4.97         => ( ( log @ A @ ( powr_real @ A @ Y ) )
% 9.03/4.97            = Y ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2945_floor__divide__eq__div__numeral,axiom,
% 9.03/4.97      ! [A: num,B: num] :
% 9.03/4.97        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 9.03/4.97        = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2946_powr__numeral,axiom,
% 9.03/4.97      ! [X: real,N: num] :
% 9.03/4.97        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.97       => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
% 9.03/4.97          = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2947_numeral__less__floor,axiom,
% 9.03/4.97      ! [V: num,X: real] :
% 9.03/4.97        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 9.03/4.97        = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2948_numeral__less__floor,axiom,
% 9.03/4.97      ! [V: num,X: rat] :
% 9.03/4.97        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/4.97        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2949_floor__le__numeral,axiom,
% 9.03/4.97      ! [X: real,V: num] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 9.03/4.97        = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2950_floor__le__numeral,axiom,
% 9.03/4.97      ! [X: rat,V: num] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 9.03/4.97        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2951_one__less__floor,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 9.03/4.97        = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2952_one__less__floor,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/4.97        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2953_floor__le__one,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 9.03/4.97        = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2954_floor__le__one,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 9.03/4.97        = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2955_ceiling__less__numeral,axiom,
% 9.03/4.97      ! [X: real,V: num] :
% 9.03/4.97        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 9.03/4.97        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2956_ceiling__less__numeral,axiom,
% 9.03/4.97      ! [X: rat,V: num] :
% 9.03/4.97        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 9.03/4.97        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2957_numeral__le__ceiling,axiom,
% 9.03/4.97      ! [V: num,X: real] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 9.03/4.97        = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2958_numeral__le__ceiling,axiom,
% 9.03/4.97      ! [V: num,X: rat] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/4.97        = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2959_floor__one__divide__eq__div__numeral,axiom,
% 9.03/4.97      ! [B: num] :
% 9.03/4.97        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 9.03/4.97        = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2960_eq__add__iff1,axiom,
% 9.03/4.97      ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
% 9.03/4.97        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
% 9.03/4.97          = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E2 ) @ C )
% 9.03/4.97          = D ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2961_eq__add__iff1,axiom,
% 9.03/4.97      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.03/4.97        ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 9.03/4.97          = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
% 9.03/4.97          = D ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2962_eq__add__iff1,axiom,
% 9.03/4.97      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.03/4.97        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 9.03/4.97          = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
% 9.03/4.97          = D ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2963_eq__add__iff1,axiom,
% 9.03/4.97      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.03/4.97        ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 9.03/4.97          = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
% 9.03/4.97          = D ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2964_eq__add__iff2,axiom,
% 9.03/4.97      ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
% 9.03/4.97        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
% 9.03/4.97          = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( C
% 9.03/4.97          = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2965_eq__add__iff2,axiom,
% 9.03/4.97      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.03/4.97        ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 9.03/4.97          = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( C
% 9.03/4.97          = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2966_eq__add__iff2,axiom,
% 9.03/4.97      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.03/4.97        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 9.03/4.97          = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( C
% 9.03/4.97          = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2967_eq__add__iff2,axiom,
% 9.03/4.97      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.03/4.97        ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 9.03/4.97          = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( C
% 9.03/4.97          = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2968_square__diff__square__factored,axiom,
% 9.03/4.97      ! [X: complex,Y: complex] :
% 9.03/4.97        ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ ( times_times_complex @ Y @ Y ) )
% 9.03/4.97        = ( times_times_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_complex @ X @ Y ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2969_square__diff__square__factored,axiom,
% 9.03/4.97      ! [X: real,Y: real] :
% 9.03/4.97        ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 9.03/4.97        = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2970_square__diff__square__factored,axiom,
% 9.03/4.97      ! [X: rat,Y: rat] :
% 9.03/4.97        ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 9.03/4.97        = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2971_square__diff__square__factored,axiom,
% 9.03/4.97      ! [X: int,Y: int] :
% 9.03/4.97        ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 9.03/4.97        = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2972_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 9.03/4.97        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2973_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 9.03/4.97        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2974_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.03/4.97        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2975_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 9.03/4.97        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2976_diff__eq__diff__eq,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.97        ( ( ( minus_minus_real @ A @ B )
% 9.03/4.97          = ( minus_minus_real @ C @ D ) )
% 9.03/4.97       => ( ( A = B )
% 9.03/4.97          = ( C = D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2977_diff__eq__diff__eq,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.97        ( ( ( minus_minus_rat @ A @ B )
% 9.03/4.97          = ( minus_minus_rat @ C @ D ) )
% 9.03/4.97       => ( ( A = B )
% 9.03/4.97          = ( C = D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2978_diff__eq__diff__eq,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.97        ( ( ( minus_minus_int @ A @ B )
% 9.03/4.97          = ( minus_minus_int @ C @ D ) )
% 9.03/4.97       => ( ( A = B )
% 9.03/4.97          = ( C = D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2979_mult_Oassoc,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 9.03/4.97        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2980_mult_Oassoc,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 9.03/4.97        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2981_mult_Oassoc,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.03/4.97        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2982_mult_Oassoc,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 9.03/4.97        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2983_mult_Ocommute,axiom,
% 9.03/4.97      ( times_times_real
% 9.03/4.97      = ( ^ [A3: real,B4: real] : ( times_times_real @ B4 @ A3 ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2984_mult_Ocommute,axiom,
% 9.03/4.97      ( times_times_rat
% 9.03/4.97      = ( ^ [A3: rat,B4: rat] : ( times_times_rat @ B4 @ A3 ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2985_mult_Ocommute,axiom,
% 9.03/4.97      ( times_times_nat
% 9.03/4.97      = ( ^ [A3: nat,B4: nat] : ( times_times_nat @ B4 @ A3 ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2986_mult_Ocommute,axiom,
% 9.03/4.97      ( times_times_int
% 9.03/4.97      = ( ^ [A3: int,B4: int] : ( times_times_int @ B4 @ A3 ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2987_mult_Oleft__commute,axiom,
% 9.03/4.97      ! [B: real,A: real,C: real] :
% 9.03/4.97        ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 9.03/4.97        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2988_mult_Oleft__commute,axiom,
% 9.03/4.97      ! [B: rat,A: rat,C: rat] :
% 9.03/4.97        ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 9.03/4.97        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2989_mult_Oleft__commute,axiom,
% 9.03/4.97      ! [B: nat,A: nat,C: nat] :
% 9.03/4.97        ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 9.03/4.97        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2990_mult_Oleft__commute,axiom,
% 9.03/4.97      ! [B: int,A: int,C: int] :
% 9.03/4.97        ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 9.03/4.97        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2991_diff__right__commute,axiom,
% 9.03/4.97      ! [A: real,C: real,B: real] :
% 9.03/4.97        ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 9.03/4.97        = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2992_diff__right__commute,axiom,
% 9.03/4.97      ! [A: rat,C: rat,B: rat] :
% 9.03/4.97        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 9.03/4.97        = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2993_diff__right__commute,axiom,
% 9.03/4.97      ! [A: nat,C: nat,B: nat] :
% 9.03/4.97        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 9.03/4.97        = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2994_diff__right__commute,axiom,
% 9.03/4.97      ! [A: int,C: int,B: int] :
% 9.03/4.97        ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 9.03/4.97        = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2995_diff__commute,axiom,
% 9.03/4.97      ! [I: nat,J: nat,K: nat] :
% 9.03/4.97        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 9.03/4.97        = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2996_diff__mult__distrib,axiom,
% 9.03/4.97      ! [M: nat,N: nat,K: nat] :
% 9.03/4.97        ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 9.03/4.97        = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2997_diff__mult__distrib2,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.97        = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2998_powr__powr__swap,axiom,
% 9.03/4.97      ! [X: real,A: real,B: real] :
% 9.03/4.97        ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 9.03/4.97        = ( powr_real @ ( powr_real @ X @ B ) @ A ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_2999_mult__diff__mult,axiom,
% 9.03/4.97      ! [X: complex,Y: complex,A: complex,B: complex] :
% 9.03/4.97        ( ( minus_minus_complex @ ( times_times_complex @ X @ Y ) @ ( times_times_complex @ A @ B ) )
% 9.03/4.97        = ( plus_plus_complex @ ( times_times_complex @ X @ ( minus_minus_complex @ Y @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X @ A ) @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3000_mult__diff__mult,axiom,
% 9.03/4.97      ! [X: real,Y: real,A: real,B: real] :
% 9.03/4.97        ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
% 9.03/4.97        = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3001_mult__diff__mult,axiom,
% 9.03/4.97      ! [X: rat,Y: rat,A: rat,B: rat] :
% 9.03/4.97        ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
% 9.03/4.97        = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3002_mult__diff__mult,axiom,
% 9.03/4.97      ! [X: int,Y: int,A: int,B: int] :
% 9.03/4.97        ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
% 9.03/4.97        = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3003_floor__le__round,axiom,
% 9.03/4.97      ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3004_floor__le__round,axiom,
% 9.03/4.97      ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3005_powr__diff,axiom,
% 9.03/4.97      ! [W: real,Z1: real,Z22: real] :
% 9.03/4.97        ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
% 9.03/4.97        = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3006_powr__add,axiom,
% 9.03/4.97      ! [X: real,A: real,B: real] :
% 9.03/4.97        ( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
% 9.03/4.97        = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3007_le__mult__floor,axiom,
% 9.03/4.97      ! [A: real,B: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.97       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.97         => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3008_le__mult__floor,axiom,
% 9.03/4.97      ! [A: rat,B: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.97       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.97         => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3009_ordered__ring__class_Ole__add__iff1,axiom,
% 9.03/4.97      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3010_ordered__ring__class_Ole__add__iff1,axiom,
% 9.03/4.97      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3011_ordered__ring__class_Ole__add__iff1,axiom,
% 9.03/4.97      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3012_ordered__ring__class_Ole__add__iff2,axiom,
% 9.03/4.97      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3013_ordered__ring__class_Ole__add__iff2,axiom,
% 9.03/4.97      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3014_ordered__ring__class_Ole__add__iff2,axiom,
% 9.03/4.97      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3015_less__add__iff1,axiom,
% 9.03/4.97      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.03/4.97        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3016_less__add__iff1,axiom,
% 9.03/4.97      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.03/4.97        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3017_less__add__iff1,axiom,
% 9.03/4.97      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.03/4.97        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3018_less__add__iff2,axiom,
% 9.03/4.97      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.03/4.97        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3019_less__add__iff2,axiom,
% 9.03/4.97      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.03/4.97        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3020_less__add__iff2,axiom,
% 9.03/4.97      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.03/4.97        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.03/4.97        = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3021_divide__diff__eq__iff,axiom,
% 9.03/4.97      ! [Z: complex,X: complex,Y: complex] :
% 9.03/4.97        ( ( Z != zero_zero_complex )
% 9.03/4.97       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
% 9.03/4.97          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3022_divide__diff__eq__iff,axiom,
% 9.03/4.97      ! [Z: real,X: real,Y: real] :
% 9.03/4.97        ( ( Z != zero_zero_real )
% 9.03/4.97       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
% 9.03/4.97          = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3023_divide__diff__eq__iff,axiom,
% 9.03/4.97      ! [Z: rat,X: rat,Y: rat] :
% 9.03/4.97        ( ( Z != zero_zero_rat )
% 9.03/4.97       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
% 9.03/4.97          = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3024_diff__divide__eq__iff,axiom,
% 9.03/4.97      ! [Z: complex,X: complex,Y: complex] :
% 9.03/4.97        ( ( Z != zero_zero_complex )
% 9.03/4.97       => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
% 9.03/4.97          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3025_diff__divide__eq__iff,axiom,
% 9.03/4.97      ! [Z: real,X: real,Y: real] :
% 9.03/4.97        ( ( Z != zero_zero_real )
% 9.03/4.97       => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
% 9.03/4.97          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3026_diff__divide__eq__iff,axiom,
% 9.03/4.97      ! [Z: rat,X: rat,Y: rat] :
% 9.03/4.97        ( ( Z != zero_zero_rat )
% 9.03/4.97       => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
% 9.03/4.97          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3027_diff__frac__eq,axiom,
% 9.03/4.97      ! [Y: complex,Z: complex,X: complex,W: complex] :
% 9.03/4.97        ( ( Y != zero_zero_complex )
% 9.03/4.97       => ( ( Z != zero_zero_complex )
% 9.03/4.97         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 9.03/4.97            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3028_diff__frac__eq,axiom,
% 9.03/4.97      ! [Y: real,Z: real,X: real,W: real] :
% 9.03/4.97        ( ( Y != zero_zero_real )
% 9.03/4.97       => ( ( Z != zero_zero_real )
% 9.03/4.97         => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 9.03/4.97            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3029_diff__frac__eq,axiom,
% 9.03/4.97      ! [Y: rat,Z: rat,X: rat,W: rat] :
% 9.03/4.97        ( ( Y != zero_zero_rat )
% 9.03/4.97       => ( ( Z != zero_zero_rat )
% 9.03/4.97         => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 9.03/4.97            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3030_add__divide__eq__if__simps_I4_J,axiom,
% 9.03/4.97      ! [Z: complex,A: complex,B: complex] :
% 9.03/4.97        ( ( ( Z = zero_zero_complex )
% 9.03/4.97         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 9.03/4.97            = A ) )
% 9.03/4.97        & ( ( Z != zero_zero_complex )
% 9.03/4.97         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 9.03/4.97            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3031_add__divide__eq__if__simps_I4_J,axiom,
% 9.03/4.97      ! [Z: real,A: real,B: real] :
% 9.03/4.97        ( ( ( Z = zero_zero_real )
% 9.03/4.97         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 9.03/4.97            = A ) )
% 9.03/4.97        & ( ( Z != zero_zero_real )
% 9.03/4.97         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 9.03/4.97            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3032_add__divide__eq__if__simps_I4_J,axiom,
% 9.03/4.97      ! [Z: rat,A: rat,B: rat] :
% 9.03/4.97        ( ( ( Z = zero_zero_rat )
% 9.03/4.97         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 9.03/4.97            = A ) )
% 9.03/4.97        & ( ( Z != zero_zero_rat )
% 9.03/4.97         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 9.03/4.97            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3033_square__diff__one__factored,axiom,
% 9.03/4.97      ! [X: complex] :
% 9.03/4.97        ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
% 9.03/4.97        = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3034_square__diff__one__factored,axiom,
% 9.03/4.97      ! [X: real] :
% 9.03/4.97        ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
% 9.03/4.97        = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3035_square__diff__one__factored,axiom,
% 9.03/4.97      ! [X: rat] :
% 9.03/4.97        ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 9.03/4.97        = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3036_square__diff__one__factored,axiom,
% 9.03/4.97      ! [X: int] :
% 9.03/4.97        ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 9.03/4.97        = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3037_of__nat__diff,axiom,
% 9.03/4.97      ! [N: nat,M: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.97       => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.97          = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3038_of__nat__diff,axiom,
% 9.03/4.97      ! [N: nat,M: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.97       => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.97          = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3039_of__nat__diff,axiom,
% 9.03/4.97      ! [N: nat,M: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.97       => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.97          = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3040_of__nat__diff,axiom,
% 9.03/4.97      ! [N: nat,M: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.97       => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.97          = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3041_of__nat__diff,axiom,
% 9.03/4.97      ! [N: nat,M: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.97       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.97          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3042_nat__diff__add__eq2,axiom,
% 9.03/4.97      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.97       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.03/4.97          = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3043_nat__diff__add__eq1,axiom,
% 9.03/4.97      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ J @ I )
% 9.03/4.97       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.03/4.97          = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3044_nat__le__add__iff2,axiom,
% 9.03/4.97      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.97       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.03/4.97          = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3045_nat__le__add__iff1,axiom,
% 9.03/4.97      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ J @ I )
% 9.03/4.97       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.03/4.97          = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3046_nat__eq__add__iff2,axiom,
% 9.03/4.97      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.97       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 9.03/4.97            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.03/4.97          = ( M
% 9.03/4.97            = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3047_nat__eq__add__iff1,axiom,
% 9.03/4.97      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ J @ I )
% 9.03/4.97       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 9.03/4.97            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.03/4.97          = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 9.03/4.97            = N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3048_diff__eq__diff__less__eq,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.97        ( ( ( minus_minus_real @ A @ B )
% 9.03/4.97          = ( minus_minus_real @ C @ D ) )
% 9.03/4.97       => ( ( ord_less_eq_real @ A @ B )
% 9.03/4.97          = ( ord_less_eq_real @ C @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3049_diff__eq__diff__less__eq,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.97        ( ( ( minus_minus_rat @ A @ B )
% 9.03/4.97          = ( minus_minus_rat @ C @ D ) )
% 9.03/4.97       => ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.97          = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3050_diff__eq__diff__less__eq,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.97        ( ( ( minus_minus_int @ A @ B )
% 9.03/4.97          = ( minus_minus_int @ C @ D ) )
% 9.03/4.97       => ( ( ord_less_eq_int @ A @ B )
% 9.03/4.97          = ( ord_less_eq_int @ C @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3051_diff__right__mono,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.97       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3052_diff__right__mono,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.97       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3053_diff__right__mono,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.97       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3054_diff__left__mono,axiom,
% 9.03/4.97      ! [B: real,A: real,C: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ B @ A )
% 9.03/4.97       => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3055_diff__left__mono,axiom,
% 9.03/4.97      ! [B: rat,A: rat,C: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.97       => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3056_diff__left__mono,axiom,
% 9.03/4.97      ! [B: int,A: int,C: int] :
% 9.03/4.97        ( ( ord_less_eq_int @ B @ A )
% 9.03/4.97       => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3057_diff__mono,axiom,
% 9.03/4.97      ! [A: real,B: real,D: real,C: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.97       => ( ( ord_less_eq_real @ D @ C )
% 9.03/4.97         => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3058_diff__mono,axiom,
% 9.03/4.97      ! [A: rat,B: rat,D: rat,C: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.97       => ( ( ord_less_eq_rat @ D @ C )
% 9.03/4.97         => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3059_diff__mono,axiom,
% 9.03/4.97      ! [A: int,B: int,D: int,C: int] :
% 9.03/4.97        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.97       => ( ( ord_less_eq_int @ D @ C )
% 9.03/4.97         => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3060_eq__iff__diff__eq__0,axiom,
% 9.03/4.97      ( ( ^ [Y3: complex,Z2: complex] : ( Y3 = Z2 ) )
% 9.03/4.97      = ( ^ [A3: complex,B4: complex] :
% 9.03/4.97            ( ( minus_minus_complex @ A3 @ B4 )
% 9.03/4.97            = zero_zero_complex ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3061_eq__iff__diff__eq__0,axiom,
% 9.03/4.97      ( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
% 9.03/4.97      = ( ^ [A3: real,B4: real] :
% 9.03/4.97            ( ( minus_minus_real @ A3 @ B4 )
% 9.03/4.97            = zero_zero_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3062_eq__iff__diff__eq__0,axiom,
% 9.03/4.97      ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 9.03/4.97      = ( ^ [A3: rat,B4: rat] :
% 9.03/4.97            ( ( minus_minus_rat @ A3 @ B4 )
% 9.03/4.97            = zero_zero_rat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3063_eq__iff__diff__eq__0,axiom,
% 9.03/4.97      ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 9.03/4.97      = ( ^ [A3: int,B4: int] :
% 9.03/4.97            ( ( minus_minus_int @ A3 @ B4 )
% 9.03/4.97            = zero_zero_int ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3064_diff__strict__mono,axiom,
% 9.03/4.97      ! [A: real,B: real,D: real,C: real] :
% 9.03/4.97        ( ( ord_less_real @ A @ B )
% 9.03/4.97       => ( ( ord_less_real @ D @ C )
% 9.03/4.97         => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3065_diff__strict__mono,axiom,
% 9.03/4.97      ! [A: rat,B: rat,D: rat,C: rat] :
% 9.03/4.97        ( ( ord_less_rat @ A @ B )
% 9.03/4.97       => ( ( ord_less_rat @ D @ C )
% 9.03/4.97         => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3066_diff__strict__mono,axiom,
% 9.03/4.97      ! [A: int,B: int,D: int,C: int] :
% 9.03/4.97        ( ( ord_less_int @ A @ B )
% 9.03/4.97       => ( ( ord_less_int @ D @ C )
% 9.03/4.97         => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3067_diff__eq__diff__less,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.97        ( ( ( minus_minus_real @ A @ B )
% 9.03/4.97          = ( minus_minus_real @ C @ D ) )
% 9.03/4.97       => ( ( ord_less_real @ A @ B )
% 9.03/4.97          = ( ord_less_real @ C @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3068_diff__eq__diff__less,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.97        ( ( ( minus_minus_rat @ A @ B )
% 9.03/4.97          = ( minus_minus_rat @ C @ D ) )
% 9.03/4.97       => ( ( ord_less_rat @ A @ B )
% 9.03/4.97          = ( ord_less_rat @ C @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3069_diff__eq__diff__less,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.97        ( ( ( minus_minus_int @ A @ B )
% 9.03/4.97          = ( minus_minus_int @ C @ D ) )
% 9.03/4.97       => ( ( ord_less_int @ A @ B )
% 9.03/4.97          = ( ord_less_int @ C @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3070_diff__strict__left__mono,axiom,
% 9.03/4.97      ! [B: real,A: real,C: real] :
% 9.03/4.97        ( ( ord_less_real @ B @ A )
% 9.03/4.97       => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3071_diff__strict__left__mono,axiom,
% 9.03/4.97      ! [B: rat,A: rat,C: rat] :
% 9.03/4.97        ( ( ord_less_rat @ B @ A )
% 9.03/4.97       => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3072_diff__strict__left__mono,axiom,
% 9.03/4.97      ! [B: int,A: int,C: int] :
% 9.03/4.97        ( ( ord_less_int @ B @ A )
% 9.03/4.97       => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3073_diff__strict__right__mono,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( ord_less_real @ A @ B )
% 9.03/4.97       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3074_diff__strict__right__mono,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( ord_less_rat @ A @ B )
% 9.03/4.97       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3075_diff__strict__right__mono,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( ord_less_int @ A @ B )
% 9.03/4.97       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3076_add__diff__add,axiom,
% 9.03/4.97      ! [A: complex,C: complex,B: complex,D: complex] :
% 9.03/4.97        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) )
% 9.03/4.97        = ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ ( minus_minus_complex @ C @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3077_add__diff__add,axiom,
% 9.03/4.97      ! [A: real,C: real,B: real,D: real] :
% 9.03/4.97        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 9.03/4.97        = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3078_add__diff__add,axiom,
% 9.03/4.97      ! [A: rat,C: rat,B: rat,D: rat] :
% 9.03/4.97        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 9.03/4.97        = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3079_add__diff__add,axiom,
% 9.03/4.97      ! [A: int,C: int,B: int,D: int] :
% 9.03/4.97        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 9.03/4.97        = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3080_group__cancel_Osub1,axiom,
% 9.03/4.97      ! [A2: complex,K: complex,A: complex,B: complex] :
% 9.03/4.97        ( ( A2
% 9.03/4.97          = ( plus_plus_complex @ K @ A ) )
% 9.03/4.97       => ( ( minus_minus_complex @ A2 @ B )
% 9.03/4.97          = ( plus_plus_complex @ K @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3081_group__cancel_Osub1,axiom,
% 9.03/4.97      ! [A2: real,K: real,A: real,B: real] :
% 9.03/4.97        ( ( A2
% 9.03/4.97          = ( plus_plus_real @ K @ A ) )
% 9.03/4.97       => ( ( minus_minus_real @ A2 @ B )
% 9.03/4.97          = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3082_group__cancel_Osub1,axiom,
% 9.03/4.97      ! [A2: rat,K: rat,A: rat,B: rat] :
% 9.03/4.97        ( ( A2
% 9.03/4.97          = ( plus_plus_rat @ K @ A ) )
% 9.03/4.97       => ( ( minus_minus_rat @ A2 @ B )
% 9.03/4.97          = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3083_group__cancel_Osub1,axiom,
% 9.03/4.97      ! [A2: int,K: int,A: int,B: int] :
% 9.03/4.97        ( ( A2
% 9.03/4.97          = ( plus_plus_int @ K @ A ) )
% 9.03/4.97       => ( ( minus_minus_int @ A2 @ B )
% 9.03/4.97          = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3084_diff__eq__eq,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( ( minus_minus_complex @ A @ B )
% 9.03/4.97          = C )
% 9.03/4.97        = ( A
% 9.03/4.97          = ( plus_plus_complex @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3085_diff__eq__eq,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( ( minus_minus_real @ A @ B )
% 9.03/4.97          = C )
% 9.03/4.97        = ( A
% 9.03/4.97          = ( plus_plus_real @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3086_diff__eq__eq,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( ( minus_minus_rat @ A @ B )
% 9.03/4.97          = C )
% 9.03/4.97        = ( A
% 9.03/4.97          = ( plus_plus_rat @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3087_diff__eq__eq,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( ( minus_minus_int @ A @ B )
% 9.03/4.97          = C )
% 9.03/4.97        = ( A
% 9.03/4.97          = ( plus_plus_int @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3088_eq__diff__eq,axiom,
% 9.03/4.97      ! [A: complex,C: complex,B: complex] :
% 9.03/4.97        ( ( A
% 9.03/4.97          = ( minus_minus_complex @ C @ B ) )
% 9.03/4.97        = ( ( plus_plus_complex @ A @ B )
% 9.03/4.97          = C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3089_eq__diff__eq,axiom,
% 9.03/4.97      ! [A: real,C: real,B: real] :
% 9.03/4.97        ( ( A
% 9.03/4.97          = ( minus_minus_real @ C @ B ) )
% 9.03/4.97        = ( ( plus_plus_real @ A @ B )
% 9.03/4.97          = C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3090_eq__diff__eq,axiom,
% 9.03/4.97      ! [A: rat,C: rat,B: rat] :
% 9.03/4.97        ( ( A
% 9.03/4.97          = ( minus_minus_rat @ C @ B ) )
% 9.03/4.97        = ( ( plus_plus_rat @ A @ B )
% 9.03/4.97          = C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3091_eq__diff__eq,axiom,
% 9.03/4.97      ! [A: int,C: int,B: int] :
% 9.03/4.97        ( ( A
% 9.03/4.97          = ( minus_minus_int @ C @ B ) )
% 9.03/4.97        = ( ( plus_plus_int @ A @ B )
% 9.03/4.97          = C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3092_add__diff__eq,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( plus_plus_complex @ A @ ( minus_minus_complex @ B @ C ) )
% 9.03/4.97        = ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3093_add__diff__eq,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 9.03/4.97        = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3094_add__diff__eq,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 9.03/4.97        = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3095_add__diff__eq,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 9.03/4.97        = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3096_diff__diff__eq2,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( minus_minus_complex @ A @ ( minus_minus_complex @ B @ C ) )
% 9.03/4.97        = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3097_diff__diff__eq2,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 9.03/4.97        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3098_diff__diff__eq2,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 9.03/4.97        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3099_diff__diff__eq2,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 9.03/4.97        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3100_diff__add__eq,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3101_diff__add__eq,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3102_diff__add__eq,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3103_diff__add__eq,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3104_diff__add__eq__diff__diff__swap,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( minus_minus_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 9.03/4.97        = ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3105_diff__add__eq__diff__diff__swap,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.03/4.97        = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3106_diff__add__eq__diff__diff__swap,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.03/4.97        = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3107_diff__add__eq__diff__diff__swap,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.03/4.97        = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3108_add__implies__diff,axiom,
% 9.03/4.97      ! [C: complex,B: complex,A: complex] :
% 9.03/4.97        ( ( ( plus_plus_complex @ C @ B )
% 9.03/4.97          = A )
% 9.03/4.97       => ( C
% 9.03/4.97          = ( minus_minus_complex @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3109_add__implies__diff,axiom,
% 9.03/4.97      ! [C: real,B: real,A: real] :
% 9.03/4.97        ( ( ( plus_plus_real @ C @ B )
% 9.03/4.97          = A )
% 9.03/4.97       => ( C
% 9.03/4.97          = ( minus_minus_real @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3110_add__implies__diff,axiom,
% 9.03/4.97      ! [C: rat,B: rat,A: rat] :
% 9.03/4.97        ( ( ( plus_plus_rat @ C @ B )
% 9.03/4.97          = A )
% 9.03/4.97       => ( C
% 9.03/4.97          = ( minus_minus_rat @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3111_add__implies__diff,axiom,
% 9.03/4.97      ! [C: nat,B: nat,A: nat] :
% 9.03/4.97        ( ( ( plus_plus_nat @ C @ B )
% 9.03/4.97          = A )
% 9.03/4.97       => ( C
% 9.03/4.97          = ( minus_minus_nat @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3112_add__implies__diff,axiom,
% 9.03/4.97      ! [C: int,B: int,A: int] :
% 9.03/4.97        ( ( ( plus_plus_int @ C @ B )
% 9.03/4.97          = A )
% 9.03/4.97       => ( C
% 9.03/4.97          = ( minus_minus_int @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3113_diff__diff__eq,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( minus_minus_complex @ ( minus_minus_complex @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3114_diff__diff__eq,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3115_diff__diff__eq,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3116_diff__diff__eq,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3117_diff__diff__eq,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3118_diff__divide__distrib,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3119_diff__divide__distrib,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3120_diff__divide__distrib,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.03/4.97        = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3121_mult__right__cancel,axiom,
% 9.03/4.97      ! [C: complex,A: complex,B: complex] :
% 9.03/4.97        ( ( C != zero_zero_complex )
% 9.03/4.97       => ( ( ( times_times_complex @ A @ C )
% 9.03/4.97            = ( times_times_complex @ B @ C ) )
% 9.03/4.97          = ( A = B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3122_mult__right__cancel,axiom,
% 9.03/4.97      ! [C: real,A: real,B: real] :
% 9.03/4.97        ( ( C != zero_zero_real )
% 9.03/4.97       => ( ( ( times_times_real @ A @ C )
% 9.03/4.97            = ( times_times_real @ B @ C ) )
% 9.03/4.97          = ( A = B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3123_mult__right__cancel,axiom,
% 9.03/4.97      ! [C: rat,A: rat,B: rat] :
% 9.03/4.97        ( ( C != zero_zero_rat )
% 9.03/4.97       => ( ( ( times_times_rat @ A @ C )
% 9.03/4.97            = ( times_times_rat @ B @ C ) )
% 9.03/4.97          = ( A = B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3124_mult__right__cancel,axiom,
% 9.03/4.97      ! [C: nat,A: nat,B: nat] :
% 9.03/4.97        ( ( C != zero_zero_nat )
% 9.03/4.97       => ( ( ( times_times_nat @ A @ C )
% 9.03/4.97            = ( times_times_nat @ B @ C ) )
% 9.03/4.97          = ( A = B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3125_mult__right__cancel,axiom,
% 9.03/4.97      ! [C: int,A: int,B: int] :
% 9.03/4.97        ( ( C != zero_zero_int )
% 9.03/4.97       => ( ( ( times_times_int @ A @ C )
% 9.03/4.97            = ( times_times_int @ B @ C ) )
% 9.03/4.97          = ( A = B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3126_mult__left__cancel,axiom,
% 9.03/4.97      ! [C: complex,A: complex,B: complex] :
% 9.03/4.97        ( ( C != zero_zero_complex )
% 9.03/4.97       => ( ( ( times_times_complex @ C @ A )
% 9.03/4.97            = ( times_times_complex @ C @ B ) )
% 9.03/4.97          = ( A = B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3127_mult__left__cancel,axiom,
% 9.03/4.97      ! [C: real,A: real,B: real] :
% 9.03/4.97        ( ( C != zero_zero_real )
% 9.03/4.97       => ( ( ( times_times_real @ C @ A )
% 9.03/4.97            = ( times_times_real @ C @ B ) )
% 9.03/4.97          = ( A = B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3128_mult__left__cancel,axiom,
% 9.03/4.97      ! [C: rat,A: rat,B: rat] :
% 9.03/4.97        ( ( C != zero_zero_rat )
% 9.03/4.97       => ( ( ( times_times_rat @ C @ A )
% 9.03/4.97            = ( times_times_rat @ C @ B ) )
% 9.03/4.97          = ( A = B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3129_mult__left__cancel,axiom,
% 9.03/4.97      ! [C: nat,A: nat,B: nat] :
% 9.03/4.97        ( ( C != zero_zero_nat )
% 9.03/4.97       => ( ( ( times_times_nat @ C @ A )
% 9.03/4.97            = ( times_times_nat @ C @ B ) )
% 9.03/4.97          = ( A = B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3130_mult__left__cancel,axiom,
% 9.03/4.97      ! [C: int,A: int,B: int] :
% 9.03/4.97        ( ( C != zero_zero_int )
% 9.03/4.97       => ( ( ( times_times_int @ C @ A )
% 9.03/4.97            = ( times_times_int @ C @ B ) )
% 9.03/4.97          = ( A = B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3131_no__zero__divisors,axiom,
% 9.03/4.97      ! [A: complex,B: complex] :
% 9.03/4.97        ( ( A != zero_zero_complex )
% 9.03/4.97       => ( ( B != zero_zero_complex )
% 9.03/4.97         => ( ( times_times_complex @ A @ B )
% 9.03/4.97           != zero_zero_complex ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3132_no__zero__divisors,axiom,
% 9.03/4.97      ! [A: real,B: real] :
% 9.03/4.97        ( ( A != zero_zero_real )
% 9.03/4.97       => ( ( B != zero_zero_real )
% 9.03/4.97         => ( ( times_times_real @ A @ B )
% 9.03/4.97           != zero_zero_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3133_no__zero__divisors,axiom,
% 9.03/4.97      ! [A: rat,B: rat] :
% 9.03/4.97        ( ( A != zero_zero_rat )
% 9.03/4.97       => ( ( B != zero_zero_rat )
% 9.03/4.97         => ( ( times_times_rat @ A @ B )
% 9.03/4.97           != zero_zero_rat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3134_no__zero__divisors,axiom,
% 9.03/4.97      ! [A: nat,B: nat] :
% 9.03/4.97        ( ( A != zero_zero_nat )
% 9.03/4.97       => ( ( B != zero_zero_nat )
% 9.03/4.97         => ( ( times_times_nat @ A @ B )
% 9.03/4.97           != zero_zero_nat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3135_no__zero__divisors,axiom,
% 9.03/4.97      ! [A: int,B: int] :
% 9.03/4.97        ( ( A != zero_zero_int )
% 9.03/4.97       => ( ( B != zero_zero_int )
% 9.03/4.97         => ( ( times_times_int @ A @ B )
% 9.03/4.97           != zero_zero_int ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3136_divisors__zero,axiom,
% 9.03/4.97      ! [A: complex,B: complex] :
% 9.03/4.97        ( ( ( times_times_complex @ A @ B )
% 9.03/4.97          = zero_zero_complex )
% 9.03/4.97       => ( ( A = zero_zero_complex )
% 9.03/4.97          | ( B = zero_zero_complex ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3137_divisors__zero,axiom,
% 9.03/4.97      ! [A: real,B: real] :
% 9.03/4.97        ( ( ( times_times_real @ A @ B )
% 9.03/4.97          = zero_zero_real )
% 9.03/4.97       => ( ( A = zero_zero_real )
% 9.03/4.97          | ( B = zero_zero_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3138_divisors__zero,axiom,
% 9.03/4.97      ! [A: rat,B: rat] :
% 9.03/4.97        ( ( ( times_times_rat @ A @ B )
% 9.03/4.97          = zero_zero_rat )
% 9.03/4.97       => ( ( A = zero_zero_rat )
% 9.03/4.97          | ( B = zero_zero_rat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3139_divisors__zero,axiom,
% 9.03/4.97      ! [A: nat,B: nat] :
% 9.03/4.97        ( ( ( times_times_nat @ A @ B )
% 9.03/4.97          = zero_zero_nat )
% 9.03/4.97       => ( ( A = zero_zero_nat )
% 9.03/4.97          | ( B = zero_zero_nat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3140_divisors__zero,axiom,
% 9.03/4.97      ! [A: int,B: int] :
% 9.03/4.97        ( ( ( times_times_int @ A @ B )
% 9.03/4.97          = zero_zero_int )
% 9.03/4.97       => ( ( A = zero_zero_int )
% 9.03/4.97          | ( B = zero_zero_int ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3141_mult__not__zero,axiom,
% 9.03/4.97      ! [A: complex,B: complex] :
% 9.03/4.97        ( ( ( times_times_complex @ A @ B )
% 9.03/4.97         != zero_zero_complex )
% 9.03/4.97       => ( ( A != zero_zero_complex )
% 9.03/4.97          & ( B != zero_zero_complex ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3142_mult__not__zero,axiom,
% 9.03/4.97      ! [A: real,B: real] :
% 9.03/4.97        ( ( ( times_times_real @ A @ B )
% 9.03/4.97         != zero_zero_real )
% 9.03/4.97       => ( ( A != zero_zero_real )
% 9.03/4.97          & ( B != zero_zero_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3143_mult__not__zero,axiom,
% 9.03/4.97      ! [A: rat,B: rat] :
% 9.03/4.97        ( ( ( times_times_rat @ A @ B )
% 9.03/4.97         != zero_zero_rat )
% 9.03/4.97       => ( ( A != zero_zero_rat )
% 9.03/4.97          & ( B != zero_zero_rat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3144_mult__not__zero,axiom,
% 9.03/4.97      ! [A: nat,B: nat] :
% 9.03/4.97        ( ( ( times_times_nat @ A @ B )
% 9.03/4.97         != zero_zero_nat )
% 9.03/4.97       => ( ( A != zero_zero_nat )
% 9.03/4.97          & ( B != zero_zero_nat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3145_mult__not__zero,axiom,
% 9.03/4.97      ! [A: int,B: int] :
% 9.03/4.97        ( ( ( times_times_int @ A @ B )
% 9.03/4.97         != zero_zero_int )
% 9.03/4.97       => ( ( A != zero_zero_int )
% 9.03/4.97          & ( B != zero_zero_int ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3146_comm__monoid__mult__class_Omult__1,axiom,
% 9.03/4.97      ! [A: complex] :
% 9.03/4.97        ( ( times_times_complex @ one_one_complex @ A )
% 9.03/4.97        = A ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3147_comm__monoid__mult__class_Omult__1,axiom,
% 9.03/4.97      ! [A: real] :
% 9.03/4.97        ( ( times_times_real @ one_one_real @ A )
% 9.03/4.97        = A ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3148_comm__monoid__mult__class_Omult__1,axiom,
% 9.03/4.97      ! [A: rat] :
% 9.03/4.97        ( ( times_times_rat @ one_one_rat @ A )
% 9.03/4.97        = A ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3149_comm__monoid__mult__class_Omult__1,axiom,
% 9.03/4.97      ! [A: nat] :
% 9.03/4.97        ( ( times_times_nat @ one_one_nat @ A )
% 9.03/4.97        = A ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3150_comm__monoid__mult__class_Omult__1,axiom,
% 9.03/4.97      ! [A: int] :
% 9.03/4.97        ( ( times_times_int @ one_one_int @ A )
% 9.03/4.97        = A ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3151_mult_Ocomm__neutral,axiom,
% 9.03/4.97      ! [A: complex] :
% 9.03/4.97        ( ( times_times_complex @ A @ one_one_complex )
% 9.03/4.97        = A ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3152_mult_Ocomm__neutral,axiom,
% 9.03/4.97      ! [A: real] :
% 9.03/4.97        ( ( times_times_real @ A @ one_one_real )
% 9.03/4.97        = A ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3153_mult_Ocomm__neutral,axiom,
% 9.03/4.97      ! [A: rat] :
% 9.03/4.97        ( ( times_times_rat @ A @ one_one_rat )
% 9.03/4.97        = A ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3154_mult_Ocomm__neutral,axiom,
% 9.03/4.97      ! [A: nat] :
% 9.03/4.97        ( ( times_times_nat @ A @ one_one_nat )
% 9.03/4.97        = A ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3155_mult_Ocomm__neutral,axiom,
% 9.03/4.97      ! [A: int] :
% 9.03/4.97        ( ( times_times_int @ A @ one_one_int )
% 9.03/4.97        = A ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3156_ring__class_Oring__distribs_I2_J,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3157_ring__class_Oring__distribs_I2_J,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3158_ring__class_Oring__distribs_I2_J,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3159_ring__class_Oring__distribs_I2_J,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3160_ring__class_Oring__distribs_I1_J,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 9.03/4.97        = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3161_ring__class_Oring__distribs_I1_J,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.03/4.97        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3162_ring__class_Oring__distribs_I1_J,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.03/4.97        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3163_ring__class_Oring__distribs_I1_J,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.03/4.97        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3164_comm__semiring__class_Odistrib,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3165_comm__semiring__class_Odistrib,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3166_comm__semiring__class_Odistrib,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3167_comm__semiring__class_Odistrib,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3168_comm__semiring__class_Odistrib,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3169_distrib__left,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 9.03/4.97        = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3170_distrib__left,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.03/4.97        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3171_distrib__left,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.03/4.97        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3172_distrib__left,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 9.03/4.97        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3173_distrib__left,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.03/4.97        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3174_distrib__right,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3175_distrib__right,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3176_distrib__right,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3177_distrib__right,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3178_distrib__right,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.03/4.97        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3179_combine__common__factor,axiom,
% 9.03/4.97      ! [A: complex,E2: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ C ) )
% 9.03/4.97        = ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E2 ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3180_combine__common__factor,axiom,
% 9.03/4.97      ! [A: real,E2: real,B: real,C: real] :
% 9.03/4.97        ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
% 9.03/4.97        = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3181_combine__common__factor,axiom,
% 9.03/4.97      ! [A: rat,E2: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
% 9.03/4.97        = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3182_combine__common__factor,axiom,
% 9.03/4.97      ! [A: nat,E2: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
% 9.03/4.97        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3183_combine__common__factor,axiom,
% 9.03/4.97      ! [A: int,E2: int,B: int,C: int] :
% 9.03/4.97        ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
% 9.03/4.97        = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3184_zero__induct__lemma,axiom,
% 9.03/4.97      ! [P: nat > $o,K: nat,I: nat] :
% 9.03/4.97        ( ( P @ K )
% 9.03/4.97       => ( ! [N2: nat] :
% 9.03/4.97              ( ( P @ ( suc @ N2 ) )
% 9.03/4.97             => ( P @ N2 ) )
% 9.03/4.97         => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3185_minus__nat_Odiff__0,axiom,
% 9.03/4.97      ! [M: nat] :
% 9.03/4.97        ( ( minus_minus_nat @ M @ zero_zero_nat )
% 9.03/4.97        = M ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3186_diffs0__imp__equal,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( ( minus_minus_nat @ M @ N )
% 9.03/4.97          = zero_zero_nat )
% 9.03/4.97       => ( ( ( minus_minus_nat @ N @ M )
% 9.03/4.97            = zero_zero_nat )
% 9.03/4.97         => ( M = N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3187_divide__divide__eq__left_H,axiom,
% 9.03/4.97      ! [A: complex,B: complex,C: complex] :
% 9.03/4.97        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 9.03/4.97        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3188_divide__divide__eq__left_H,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 9.03/4.97        = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3189_divide__divide__eq__left_H,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 9.03/4.97        = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3190_divide__divide__times__eq,axiom,
% 9.03/4.97      ! [X: complex,Y: complex,Z: complex,W: complex] :
% 9.03/4.97        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 9.03/4.97        = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3191_divide__divide__times__eq,axiom,
% 9.03/4.97      ! [X: real,Y: real,Z: real,W: real] :
% 9.03/4.97        ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
% 9.03/4.97        = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3192_divide__divide__times__eq,axiom,
% 9.03/4.97      ! [X: rat,Y: rat,Z: rat,W: rat] :
% 9.03/4.97        ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
% 9.03/4.97        = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3193_times__divide__times__eq,axiom,
% 9.03/4.97      ! [X: complex,Y: complex,Z: complex,W: complex] :
% 9.03/4.97        ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 9.03/4.97        = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3194_times__divide__times__eq,axiom,
% 9.03/4.97      ! [X: real,Y: real,Z: real,W: real] :
% 9.03/4.97        ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
% 9.03/4.97        = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3195_times__divide__times__eq,axiom,
% 9.03/4.97      ! [X: rat,Y: rat,Z: rat,W: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
% 9.03/4.97        = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3196_diff__less__mono2,axiom,
% 9.03/4.97      ! [M: nat,N: nat,L: nat] :
% 9.03/4.97        ( ( ord_less_nat @ M @ N )
% 9.03/4.97       => ( ( ord_less_nat @ M @ L )
% 9.03/4.97         => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3197_less__imp__diff__less,axiom,
% 9.03/4.97      ! [J: nat,K: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_nat @ J @ K )
% 9.03/4.97       => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3198_power__commuting__commutes,axiom,
% 9.03/4.97      ! [X: complex,Y: complex,N: nat] :
% 9.03/4.97        ( ( ( times_times_complex @ X @ Y )
% 9.03/4.97          = ( times_times_complex @ Y @ X ) )
% 9.03/4.97       => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
% 9.03/4.97          = ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3199_power__commuting__commutes,axiom,
% 9.03/4.97      ! [X: real,Y: real,N: nat] :
% 9.03/4.97        ( ( ( times_times_real @ X @ Y )
% 9.03/4.97          = ( times_times_real @ Y @ X ) )
% 9.03/4.97       => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
% 9.03/4.97          = ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3200_power__commuting__commutes,axiom,
% 9.03/4.97      ! [X: rat,Y: rat,N: nat] :
% 9.03/4.97        ( ( ( times_times_rat @ X @ Y )
% 9.03/4.97          = ( times_times_rat @ Y @ X ) )
% 9.03/4.97       => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
% 9.03/4.97          = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3201_power__commuting__commutes,axiom,
% 9.03/4.97      ! [X: nat,Y: nat,N: nat] :
% 9.03/4.97        ( ( ( times_times_nat @ X @ Y )
% 9.03/4.97          = ( times_times_nat @ Y @ X ) )
% 9.03/4.97       => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
% 9.03/4.97          = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3202_power__commuting__commutes,axiom,
% 9.03/4.97      ! [X: int,Y: int,N: nat] :
% 9.03/4.97        ( ( ( times_times_int @ X @ Y )
% 9.03/4.97          = ( times_times_int @ Y @ X ) )
% 9.03/4.97       => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
% 9.03/4.97          = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3203_power__mult__distrib,axiom,
% 9.03/4.97      ! [A: complex,B: complex,N: nat] :
% 9.03/4.97        ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
% 9.03/4.97        = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3204_power__mult__distrib,axiom,
% 9.03/4.97      ! [A: real,B: real,N: nat] :
% 9.03/4.97        ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
% 9.03/4.97        = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3205_power__mult__distrib,axiom,
% 9.03/4.97      ! [A: rat,B: rat,N: nat] :
% 9.03/4.97        ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
% 9.03/4.97        = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3206_power__mult__distrib,axiom,
% 9.03/4.97      ! [A: nat,B: nat,N: nat] :
% 9.03/4.97        ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
% 9.03/4.97        = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3207_power__mult__distrib,axiom,
% 9.03/4.97      ! [A: int,B: int,N: nat] :
% 9.03/4.97        ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
% 9.03/4.97        = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3208_power__commutes,axiom,
% 9.03/4.97      ! [A: complex,N: nat] :
% 9.03/4.97        ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 9.03/4.97        = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3209_power__commutes,axiom,
% 9.03/4.97      ! [A: real,N: nat] :
% 9.03/4.97        ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 9.03/4.97        = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3210_power__commutes,axiom,
% 9.03/4.97      ! [A: rat,N: nat] :
% 9.03/4.97        ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 9.03/4.97        = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3211_power__commutes,axiom,
% 9.03/4.97      ! [A: nat,N: nat] :
% 9.03/4.97        ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 9.03/4.97        = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3212_power__commutes,axiom,
% 9.03/4.97      ! [A: int,N: nat] :
% 9.03/4.97        ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 9.03/4.97        = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3213_eq__diff__iff,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ K @ M )
% 9.03/4.97       => ( ( ord_less_eq_nat @ K @ N )
% 9.03/4.97         => ( ( ( minus_minus_nat @ M @ K )
% 9.03/4.97              = ( minus_minus_nat @ N @ K ) )
% 9.03/4.97            = ( M = N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3214_le__diff__iff,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ K @ M )
% 9.03/4.97       => ( ( ord_less_eq_nat @ K @ N )
% 9.03/4.97         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.03/4.97            = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3215_Nat_Odiff__diff__eq,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ K @ M )
% 9.03/4.97       => ( ( ord_less_eq_nat @ K @ N )
% 9.03/4.97         => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.03/4.97            = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3216_diff__le__mono,axiom,
% 9.03/4.97      ! [M: nat,N: nat,L: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.97       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3217_diff__le__self,axiom,
% 9.03/4.97      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3218_le__diff__iff_H,axiom,
% 9.03/4.97      ! [A: nat,C: nat,B: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ C )
% 9.03/4.97       => ( ( ord_less_eq_nat @ B @ C )
% 9.03/4.97         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 9.03/4.97            = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3219_diff__le__mono2,axiom,
% 9.03/4.97      ! [M: nat,N: nat,L: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ M @ N )
% 9.03/4.97       => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3220_power__mult,axiom,
% 9.03/4.97      ! [A: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3221_power__mult,axiom,
% 9.03/4.97      ! [A: real,M: nat,N: nat] :
% 9.03/4.97        ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3222_power__mult,axiom,
% 9.03/4.97      ! [A: int,M: nat,N: nat] :
% 9.03/4.97        ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3223_power__mult,axiom,
% 9.03/4.97      ! [A: complex,M: nat,N: nat] :
% 9.03/4.97        ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 9.03/4.97        = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3224_Suc__mult__cancel1,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 9.03/4.97          = ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.03/4.97        = ( M = N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3225_mult__0,axiom,
% 9.03/4.97      ! [N: nat] :
% 9.03/4.97        ( ( times_times_nat @ zero_zero_nat @ N )
% 9.03/4.97        = zero_zero_nat ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3226_nat__mult__eq__cancel__disj,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ( times_times_nat @ K @ M )
% 9.03/4.97          = ( times_times_nat @ K @ N ) )
% 9.03/4.97        = ( ( K = zero_zero_nat )
% 9.03/4.97          | ( M = N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3227_diff__add__inverse2,axiom,
% 9.03/4.97      ! [M: nat,N: nat] :
% 9.03/4.97        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 9.03/4.97        = M ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3228_diff__add__inverse,axiom,
% 9.03/4.97      ! [N: nat,M: nat] :
% 9.03/4.97        ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 9.03/4.97        = M ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3229_diff__cancel2,axiom,
% 9.03/4.97      ! [M: nat,K: nat,N: nat] :
% 9.03/4.97        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 9.03/4.97        = ( minus_minus_nat @ M @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3230_Nat_Odiff__cancel,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.03/4.97        = ( minus_minus_nat @ M @ N ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3231_mult__of__nat__commute,axiom,
% 9.03/4.97      ! [X: nat,Y: complex] :
% 9.03/4.97        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
% 9.03/4.97        = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3232_mult__of__nat__commute,axiom,
% 9.03/4.97      ! [X: nat,Y: real] :
% 9.03/4.97        ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
% 9.03/4.97        = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3233_mult__of__nat__commute,axiom,
% 9.03/4.97      ! [X: nat,Y: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
% 9.03/4.97        = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3234_mult__of__nat__commute,axiom,
% 9.03/4.97      ! [X: nat,Y: nat] :
% 9.03/4.97        ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
% 9.03/4.97        = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3235_mult__of__nat__commute,axiom,
% 9.03/4.97      ! [X: nat,Y: int] :
% 9.03/4.97        ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
% 9.03/4.97        = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3236_le__cube,axiom,
% 9.03/4.97      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3237_le__square,axiom,
% 9.03/4.97      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3238_mult__le__mono,axiom,
% 9.03/4.97      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.97       => ( ( ord_less_eq_nat @ K @ L )
% 9.03/4.97         => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3239_mult__le__mono1,axiom,
% 9.03/4.97      ! [I: nat,J: nat,K: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.97       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3240_mult__le__mono2,axiom,
% 9.03/4.97      ! [I: nat,J: nat,K: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.97       => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3241_left__add__mult__distrib,axiom,
% 9.03/4.97      ! [I: nat,U: nat,J: nat,K: nat] :
% 9.03/4.97        ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 9.03/4.97        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3242_add__mult__distrib2,axiom,
% 9.03/4.97      ! [K: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.97        = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3243_add__mult__distrib,axiom,
% 9.03/4.97      ! [M: nat,N: nat,K: nat] :
% 9.03/4.97        ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 9.03/4.97        = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3244_mult__of__int__commute,axiom,
% 9.03/4.97      ! [X: int,Y: code_integer] :
% 9.03/4.97        ( ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ X ) @ Y )
% 9.03/4.97        = ( times_3573771949741848930nteger @ Y @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3245_mult__of__int__commute,axiom,
% 9.03/4.97      ! [X: int,Y: complex] :
% 9.03/4.97        ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y )
% 9.03/4.97        = ( times_times_complex @ Y @ ( ring_17405671764205052669omplex @ X ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3246_mult__of__int__commute,axiom,
% 9.03/4.97      ! [X: int,Y: real] :
% 9.03/4.97        ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
% 9.03/4.97        = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3247_mult__of__int__commute,axiom,
% 9.03/4.97      ! [X: int,Y: rat] :
% 9.03/4.97        ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
% 9.03/4.97        = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3248_mult__of__int__commute,axiom,
% 9.03/4.97      ! [X: int,Y: int] :
% 9.03/4.97        ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
% 9.03/4.97        = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3249_nat__mult__1__right,axiom,
% 9.03/4.97      ! [N: nat] :
% 9.03/4.97        ( ( times_times_nat @ N @ one_one_nat )
% 9.03/4.97        = N ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3250_nat__mult__1,axiom,
% 9.03/4.97      ! [N: nat] :
% 9.03/4.97        ( ( times_times_nat @ one_one_nat @ N )
% 9.03/4.97        = N ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3251_div__mult2__eq,axiom,
% 9.03/4.97      ! [M: nat,N: nat,Q3: nat] :
% 9.03/4.97        ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q3 ) )
% 9.03/4.97        = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3252_frac__le__eq,axiom,
% 9.03/4.97      ! [Y: real,Z: real,X: real,W: real] :
% 9.03/4.97        ( ( Y != zero_zero_real )
% 9.03/4.97       => ( ( Z != zero_zero_real )
% 9.03/4.97         => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 9.03/4.97            = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3253_frac__le__eq,axiom,
% 9.03/4.97      ! [Y: rat,Z: rat,X: rat,W: rat] :
% 9.03/4.97        ( ( Y != zero_zero_rat )
% 9.03/4.97       => ( ( Z != zero_zero_rat )
% 9.03/4.97         => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 9.03/4.97            = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3254_frac__less__eq,axiom,
% 9.03/4.97      ! [Y: real,Z: real,X: real,W: real] :
% 9.03/4.97        ( ( Y != zero_zero_real )
% 9.03/4.97       => ( ( Z != zero_zero_real )
% 9.03/4.97         => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 9.03/4.97            = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3255_frac__less__eq,axiom,
% 9.03/4.97      ! [Y: rat,Z: rat,X: rat,W: rat] :
% 9.03/4.97        ( ( Y != zero_zero_rat )
% 9.03/4.97       => ( ( Z != zero_zero_rat )
% 9.03/4.97         => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 9.03/4.97            = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3256_nat__less__add__iff2,axiom,
% 9.03/4.97      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.97       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.03/4.97          = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3257_nat__less__add__iff1,axiom,
% 9.03/4.97      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ J @ I )
% 9.03/4.97       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.03/4.97          = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3258_mult__eq__if,axiom,
% 9.03/4.97      ( times_times_nat
% 9.03/4.97      = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3259_scaling__mono,axiom,
% 9.03/4.97      ! [U: real,V: real,R2: real,S: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ U @ V )
% 9.03/4.97       => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 9.03/4.97         => ( ( ord_less_eq_real @ R2 @ S )
% 9.03/4.97           => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3260_scaling__mono,axiom,
% 9.03/4.97      ! [U: rat,V: rat,R2: rat,S: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ U @ V )
% 9.03/4.97       => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 9.03/4.97         => ( ( ord_less_eq_rat @ R2 @ S )
% 9.03/4.97           => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3261_floor__mono,axiom,
% 9.03/4.97      ! [X: real,Y: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.97       => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3262_floor__mono,axiom,
% 9.03/4.97      ! [X: rat,Y: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.97       => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3263_of__int__floor__le,axiom,
% 9.03/4.97      ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3264_of__int__floor__le,axiom,
% 9.03/4.97      ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3265_floor__less__cancel,axiom,
% 9.03/4.97      ! [X: real,Y: real] :
% 9.03/4.97        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
% 9.03/4.97       => ( ord_less_real @ X @ Y ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3266_floor__less__cancel,axiom,
% 9.03/4.97      ! [X: rat,Y: rat] :
% 9.03/4.97        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
% 9.03/4.97       => ( ord_less_rat @ X @ Y ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3267_powr__non__neg,axiom,
% 9.03/4.97      ! [A: real,X: real] :
% 9.03/4.97        ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3268_powr__less__mono2__neg,axiom,
% 9.03/4.97      ! [A: real,X: real,Y: real] :
% 9.03/4.97        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.97       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.97         => ( ( ord_less_real @ X @ Y )
% 9.03/4.97           => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3269_powr__mono2,axiom,
% 9.03/4.97      ! [A: real,X: real,Y: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.97       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.97         => ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.97           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3270_powr__ge__pzero,axiom,
% 9.03/4.97      ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3271_powr__less__cancel,axiom,
% 9.03/4.97      ! [X: real,A: real,B: real] :
% 9.03/4.97        ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 9.03/4.97       => ( ( ord_less_real @ one_one_real @ X )
% 9.03/4.97         => ( ord_less_real @ A @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3272_powr__less__mono,axiom,
% 9.03/4.97      ! [A: real,B: real,X: real] :
% 9.03/4.97        ( ( ord_less_real @ A @ B )
% 9.03/4.97       => ( ( ord_less_real @ one_one_real @ X )
% 9.03/4.97         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3273_powr__mono,axiom,
% 9.03/4.97      ! [A: real,B: real,X: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.97       => ( ( ord_less_eq_real @ one_one_real @ X )
% 9.03/4.97         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3274_power__eq__if,axiom,
% 9.03/4.97      ( power_power_complex
% 9.03/4.97      = ( ^ [P4: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P4 @ ( power_power_complex @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3275_power__eq__if,axiom,
% 9.03/4.97      ( power_power_real
% 9.03/4.97      = ( ^ [P4: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P4 @ ( power_power_real @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3276_power__eq__if,axiom,
% 9.03/4.97      ( power_power_rat
% 9.03/4.97      = ( ^ [P4: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P4 @ ( power_power_rat @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3277_power__eq__if,axiom,
% 9.03/4.97      ( power_power_nat
% 9.03/4.97      = ( ^ [P4: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3278_power__eq__if,axiom,
% 9.03/4.97      ( power_power_int
% 9.03/4.97      = ( ^ [P4: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P4 @ ( power_power_int @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3279_power__minus__mult,axiom,
% 9.03/4.97      ! [N: nat,A: complex] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.97       => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.03/4.97          = ( power_power_complex @ A @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3280_power__minus__mult,axiom,
% 9.03/4.97      ! [N: nat,A: real] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.97       => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.03/4.97          = ( power_power_real @ A @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3281_power__minus__mult,axiom,
% 9.03/4.97      ! [N: nat,A: rat] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.97       => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.03/4.97          = ( power_power_rat @ A @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3282_power__minus__mult,axiom,
% 9.03/4.97      ! [N: nat,A: nat] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.97       => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.03/4.97          = ( power_power_nat @ A @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3283_power__minus__mult,axiom,
% 9.03/4.97      ! [N: nat,A: int] :
% 9.03/4.97        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.97       => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.03/4.97          = ( power_power_int @ A @ N ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3284_mult__ceiling__le,axiom,
% 9.03/4.97      ! [A: real,B: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.97       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.97         => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3285_mult__ceiling__le,axiom,
% 9.03/4.97      ! [A: rat,B: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.97       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.97         => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3286_le__iff__diff__le__0,axiom,
% 9.03/4.97      ( ord_less_eq_real
% 9.03/4.97      = ( ^ [A3: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3287_le__iff__diff__le__0,axiom,
% 9.03/4.97      ( ord_less_eq_rat
% 9.03/4.97      = ( ^ [A3: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B4 ) @ zero_zero_rat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3288_le__iff__diff__le__0,axiom,
% 9.03/4.97      ( ord_less_eq_int
% 9.03/4.97      = ( ^ [A3: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3289_less__iff__diff__less__0,axiom,
% 9.03/4.97      ( ord_less_real
% 9.03/4.97      = ( ^ [A3: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3290_less__iff__diff__less__0,axiom,
% 9.03/4.97      ( ord_less_rat
% 9.03/4.97      = ( ^ [A3: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B4 ) @ zero_zero_rat ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3291_less__iff__diff__less__0,axiom,
% 9.03/4.97      ( ord_less_int
% 9.03/4.97      = ( ^ [A3: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3292_floor__le__ceiling,axiom,
% 9.03/4.97      ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3293_floor__le__ceiling,axiom,
% 9.03/4.97      ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3294_diff__le__eq,axiom,
% 9.03/4.97      ! [A: real,B: real,C: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.03/4.97        = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3295_diff__le__eq,axiom,
% 9.03/4.97      ! [A: rat,B: rat,C: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.03/4.97        = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3296_diff__le__eq,axiom,
% 9.03/4.97      ! [A: int,B: int,C: int] :
% 9.03/4.97        ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.03/4.97        = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3297_le__diff__eq,axiom,
% 9.03/4.97      ! [A: real,C: real,B: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 9.03/4.97        = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3298_le__diff__eq,axiom,
% 9.03/4.97      ! [A: rat,C: rat,B: rat] :
% 9.03/4.97        ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 9.03/4.97        = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3299_le__diff__eq,axiom,
% 9.03/4.97      ! [A: int,C: int,B: int] :
% 9.03/4.97        ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 9.03/4.97        = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3300_diff__add,axiom,
% 9.03/4.97      ! [A: nat,B: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 9.03/4.97          = B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3301_le__add__diff,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97       => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3302_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97       => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 9.03/4.97          = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3303_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97       => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 9.03/4.97          = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3304_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97       => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 9.03/4.97          = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3305_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 9.03/4.97          = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3306_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97       => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 9.03/4.97          = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3307_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97       => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 9.03/4.97          = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3308_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 9.03/4.97      ! [A: nat,B: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97       => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 9.03/4.97          = B ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3309_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 9.03/4.97      ! [A: nat,B: nat,C: nat] :
% 9.03/4.97        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97       => ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.97         => ( ( ( minus_minus_nat @ B @ A )
% 9.03/4.97              = C )
% 9.03/4.97            = ( B
% 9.03/4.97              = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 9.03/4.97  
% 9.03/4.97  thf(fact_3310_add__le__imp__le__diff,axiom,
% 9.03/4.97      ! [I: real,K: real,N: real] :
% 9.03/4.97        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.03/4.97       => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3311_add__le__imp__le__diff,axiom,
% 9.03/4.98      ! [I: rat,K: rat,N: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.03/4.98       => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3312_add__le__imp__le__diff,axiom,
% 9.03/4.98      ! [I: nat,K: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.03/4.98       => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3313_add__le__imp__le__diff,axiom,
% 9.03/4.98      ! [I: int,K: int,N: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.03/4.98       => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3314_add__le__add__imp__diff__le,axiom,
% 9.03/4.98      ! [I: real,K: real,N: real,J: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.03/4.98       => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 9.03/4.98         => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.03/4.98           => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 9.03/4.98             => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3315_add__le__add__imp__diff__le,axiom,
% 9.03/4.98      ! [I: rat,K: rat,N: rat,J: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.03/4.98       => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 9.03/4.98         => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.03/4.98           => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 9.03/4.98             => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3316_add__le__add__imp__diff__le,axiom,
% 9.03/4.98      ! [I: nat,K: nat,N: nat,J: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.03/4.98       => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 9.03/4.98         => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.03/4.98           => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 9.03/4.98             => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3317_add__le__add__imp__diff__le,axiom,
% 9.03/4.98      ! [I: int,K: int,N: int,J: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.03/4.98       => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 9.03/4.98         => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.03/4.98           => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 9.03/4.98             => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3318_diff__less__eq,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.03/4.98        = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3319_diff__less__eq,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.03/4.98        = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3320_diff__less__eq,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int] :
% 9.03/4.98        ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.03/4.98        = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3321_less__diff__eq,axiom,
% 9.03/4.98      ! [A: real,C: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 9.03/4.98        = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3322_less__diff__eq,axiom,
% 9.03/4.98      ! [A: rat,C: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 9.03/4.98        = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3323_less__diff__eq,axiom,
% 9.03/4.98      ! [A: int,C: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 9.03/4.98        = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3324_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ~ ( ord_less_real @ A @ B )
% 9.03/4.98       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 9.03/4.98          = A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3325_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ~ ( ord_less_rat @ A @ B )
% 9.03/4.98       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 9.03/4.98          = A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3326_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ~ ( ord_less_nat @ A @ B )
% 9.03/4.98       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 9.03/4.98          = A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3327_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ~ ( ord_less_int @ A @ B )
% 9.03/4.98       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 9.03/4.98          = A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3328_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3329_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3330_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3331_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3332_zero__le__mult__iff,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.03/4.98        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.03/4.98          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.98            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3333_zero__le__mult__iff,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.03/4.98        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 9.03/4.98          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.98            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3334_zero__le__mult__iff,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 9.03/4.98        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 9.03/4.98          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.98            & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3335_mult__nonneg__nonpos2,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.03/4.98         => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3336_mult__nonneg__nonpos2,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.03/4.98         => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3337_mult__nonneg__nonpos2,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.98       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.03/4.98         => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3338_mult__nonneg__nonpos2,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.03/4.98         => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3339_mult__nonpos__nonneg,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.98         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3340_mult__nonpos__nonneg,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.98         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3341_mult__nonpos__nonneg,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.03/4.98       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.98         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3342_mult__nonpos__nonneg,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.98         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3343_mult__nonneg__nonpos,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.03/4.98         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3344_mult__nonneg__nonpos,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.03/4.98         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3345_mult__nonneg__nonpos,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.98       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.03/4.98         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3346_mult__nonneg__nonpos,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.03/4.98         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3347_mult__nonneg__nonneg,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.98         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3348_mult__nonneg__nonneg,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.98         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3349_mult__nonneg__nonneg,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.98       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.98         => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3350_mult__nonneg__nonneg,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.98         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3351_split__mult__neg__le,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.03/4.98          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.98            & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 9.03/4.98       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3352_split__mult__neg__le,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 9.03/4.98          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.98            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 9.03/4.98       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3353_split__mult__neg__le,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.98            & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 9.03/4.98          | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.03/4.98            & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 9.03/4.98       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3354_split__mult__neg__le,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 9.03/4.98          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.98            & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 9.03/4.98       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3355_mult__le__0__iff,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 9.03/4.98        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.03/4.98          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.98            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3356_mult__le__0__iff,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 9.03/4.98        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 9.03/4.98          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.98            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3357_mult__le__0__iff,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 9.03/4.98        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 9.03/4.98          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.98            & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3358_mult__right__mono,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3359_mult__right__mono,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3360_mult__right__mono,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3361_mult__right__mono,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3362_mult__right__mono__neg,axiom,
% 9.03/4.98      ! [B: real,A: real,C: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ B @ A )
% 9.03/4.98       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.98         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3363_mult__right__mono__neg,axiom,
% 9.03/4.98      ! [B: rat,A: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.98       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.98         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3364_mult__right__mono__neg,axiom,
% 9.03/4.98      ! [B: int,A: int,C: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ B @ A )
% 9.03/4.98       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.03/4.98         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3365_mult__left__mono,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3366_mult__left__mono,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3367_mult__left__mono,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3368_mult__left__mono,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3369_mult__nonpos__nonpos,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.03/4.98         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3370_mult__nonpos__nonpos,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.98       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.03/4.98         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3371_mult__nonpos__nonpos,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.98       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.03/4.98         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3372_mult__left__mono__neg,axiom,
% 9.03/4.98      ! [B: real,A: real,C: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ B @ A )
% 9.03/4.98       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.98         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3373_mult__left__mono__neg,axiom,
% 9.03/4.98      ! [B: rat,A: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.98       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.98         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3374_mult__left__mono__neg,axiom,
% 9.03/4.98      ! [B: int,A: int,C: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ B @ A )
% 9.03/4.98       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.03/4.98         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3375_split__mult__pos__le,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.03/4.98          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.98            & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 9.03/4.98       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3376_split__mult__pos__le,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 9.03/4.98          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/4.98            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 9.03/4.98       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3377_split__mult__pos__le,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 9.03/4.98          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.98            & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 9.03/4.98       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3378_zero__le__square,axiom,
% 9.03/4.98      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3379_zero__le__square,axiom,
% 9.03/4.98      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3380_zero__le__square,axiom,
% 9.03/4.98      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3381_mult__mono_H,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_real @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3382_mult__mono_H,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_rat @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3383_mult__mono_H,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_nat @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.98           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3384_mult__mono_H,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_int @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3385_mult__mono,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_real @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.98           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3386_mult__mono,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_rat @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.98           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3387_mult__mono,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_nat @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.98           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3388_mult__mono,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_int @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.98           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3389_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3390_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3391_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat] :
% 9.03/4.98        ( ( ord_less_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.03/4.98         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3392_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int] :
% 9.03/4.98        ( ( ord_less_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3393_mult__less__cancel__right__disj,axiom,
% 9.03/4.98      ! [A: real,C: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98            & ( ord_less_real @ A @ B ) )
% 9.03/4.98          | ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98            & ( ord_less_real @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3394_mult__less__cancel__right__disj,axiom,
% 9.03/4.98      ! [A: rat,C: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98            & ( ord_less_rat @ A @ B ) )
% 9.03/4.98          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3395_mult__less__cancel__right__disj,axiom,
% 9.03/4.98      ! [A: int,C: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98            & ( ord_less_int @ A @ B ) )
% 9.03/4.98          | ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98            & ( ord_less_int @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3396_mult__strict__right__mono,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3397_mult__strict__right__mono,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3398_mult__strict__right__mono,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat] :
% 9.03/4.98        ( ( ord_less_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.03/4.98         => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3399_mult__strict__right__mono,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int] :
% 9.03/4.98        ( ( ord_less_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3400_mult__strict__right__mono__neg,axiom,
% 9.03/4.98      ! [B: real,A: real,C: real] :
% 9.03/4.98        ( ( ord_less_real @ B @ A )
% 9.03/4.98       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3401_mult__strict__right__mono__neg,axiom,
% 9.03/4.98      ! [B: rat,A: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_rat @ B @ A )
% 9.03/4.98       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3402_mult__strict__right__mono__neg,axiom,
% 9.03/4.98      ! [B: int,A: int,C: int] :
% 9.03/4.98        ( ( ord_less_int @ B @ A )
% 9.03/4.98       => ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3403_mult__less__cancel__left__disj,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98            & ( ord_less_real @ A @ B ) )
% 9.03/4.98          | ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98            & ( ord_less_real @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3404_mult__less__cancel__left__disj,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98            & ( ord_less_rat @ A @ B ) )
% 9.03/4.98          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3405_mult__less__cancel__left__disj,axiom,
% 9.03/4.98      ! [C: int,A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98            & ( ord_less_int @ A @ B ) )
% 9.03/4.98          | ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98            & ( ord_less_int @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3406_mult__strict__left__mono,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3407_mult__strict__left__mono,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3408_mult__strict__left__mono,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat] :
% 9.03/4.98        ( ( ord_less_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.03/4.98         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3409_mult__strict__left__mono,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int] :
% 9.03/4.98        ( ( ord_less_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3410_mult__strict__left__mono__neg,axiom,
% 9.03/4.98      ! [B: real,A: real,C: real] :
% 9.03/4.98        ( ( ord_less_real @ B @ A )
% 9.03/4.98       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3411_mult__strict__left__mono__neg,axiom,
% 9.03/4.98      ! [B: rat,A: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_rat @ B @ A )
% 9.03/4.98       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3412_mult__strict__left__mono__neg,axiom,
% 9.03/4.98      ! [B: int,A: int,C: int] :
% 9.03/4.98        ( ( ord_less_int @ B @ A )
% 9.03/4.98       => ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3413_mult__less__cancel__left__pos,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.98          = ( ord_less_real @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3414_mult__less__cancel__left__pos,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.98          = ( ord_less_rat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3415_mult__less__cancel__left__pos,axiom,
% 9.03/4.98      ! [C: int,A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.98          = ( ord_less_int @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3416_mult__less__cancel__left__neg,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.98          = ( ord_less_real @ B @ A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3417_mult__less__cancel__left__neg,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.98          = ( ord_less_rat @ B @ A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3418_mult__less__cancel__left__neg,axiom,
% 9.03/4.98      ! [C: int,A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.98          = ( ord_less_int @ B @ A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3419_zero__less__mult__pos2,axiom,
% 9.03/4.98      ! [B: real,A: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.98         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3420_zero__less__mult__pos2,axiom,
% 9.03/4.98      ! [B: rat,A: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 9.03/4.98       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.98         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3421_zero__less__mult__pos2,axiom,
% 9.03/4.98      ! [B: nat,A: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.98         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3422_zero__less__mult__pos2,axiom,
% 9.03/4.98      ! [B: int,A: int] :
% 9.03/4.98        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 9.03/4.98       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.98         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3423_zero__less__mult__pos,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.98         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3424_zero__less__mult__pos,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.03/4.98       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.98         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3425_zero__less__mult__pos,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.98         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3426_zero__less__mult__pos,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 9.03/4.98       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.98         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3427_zero__less__mult__iff,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.98            & ( ord_less_real @ zero_zero_real @ B ) )
% 9.03/4.98          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.98            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3428_zero__less__mult__iff,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.98            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 9.03/4.98          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.98            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3429_zero__less__mult__iff,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 9.03/4.98        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.98            & ( ord_less_int @ zero_zero_int @ B ) )
% 9.03/4.98          | ( ( ord_less_int @ A @ zero_zero_int )
% 9.03/4.98            & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3430_mult__pos__neg2,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.98       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.03/4.98         => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3431_mult__pos__neg2,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.98       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.03/4.98         => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3432_mult__pos__neg2,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.98       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.03/4.98         => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3433_mult__pos__neg2,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.98       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.03/4.98         => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3434_mult__pos__pos,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.98         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3435_mult__pos__pos,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.98       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.03/4.98         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3436_mult__pos__pos,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.03/4.98         => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3437_mult__pos__pos,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.98       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.98         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3438_mult__pos__neg,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.98       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.03/4.98         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3439_mult__pos__neg,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.98       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.03/4.98         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3440_mult__pos__neg,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.98       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.03/4.98         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3441_mult__pos__neg,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.98       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.03/4.98         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3442_mult__neg__pos,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.98         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3443_mult__neg__pos,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.98       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.03/4.98         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3444_mult__neg__pos,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.03/4.98         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3445_mult__neg__pos,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ A @ zero_zero_int )
% 9.03/4.98       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.98         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3446_mult__less__0__iff,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.98            & ( ord_less_real @ B @ zero_zero_real ) )
% 9.03/4.98          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.98            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3447_mult__less__0__iff,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.98            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 9.03/4.98          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.98            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3448_mult__less__0__iff,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 9.03/4.98        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.98            & ( ord_less_int @ B @ zero_zero_int ) )
% 9.03/4.98          | ( ( ord_less_int @ A @ zero_zero_int )
% 9.03/4.98            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3449_not__square__less__zero,axiom,
% 9.03/4.98      ! [A: real] :
% 9.03/4.98        ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3450_not__square__less__zero,axiom,
% 9.03/4.98      ! [A: rat] :
% 9.03/4.98        ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3451_not__square__less__zero,axiom,
% 9.03/4.98      ! [A: int] :
% 9.03/4.98        ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3452_mult__neg__neg,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.03/4.98         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3453_mult__neg__neg,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.98       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.03/4.98         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3454_mult__neg__neg,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ A @ zero_zero_int )
% 9.03/4.98       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.03/4.98         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3455_less__1__mult,axiom,
% 9.03/4.98      ! [M: real,N: real] :
% 9.03/4.98        ( ( ord_less_real @ one_one_real @ M )
% 9.03/4.98       => ( ( ord_less_real @ one_one_real @ N )
% 9.03/4.98         => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3456_less__1__mult,axiom,
% 9.03/4.98      ! [M: rat,N: rat] :
% 9.03/4.98        ( ( ord_less_rat @ one_one_rat @ M )
% 9.03/4.98       => ( ( ord_less_rat @ one_one_rat @ N )
% 9.03/4.98         => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3457_less__1__mult,axiom,
% 9.03/4.98      ! [M: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_nat @ one_one_nat @ M )
% 9.03/4.98       => ( ( ord_less_nat @ one_one_nat @ N )
% 9.03/4.98         => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3458_less__1__mult,axiom,
% 9.03/4.98      ! [M: int,N: int] :
% 9.03/4.98        ( ( ord_less_int @ one_one_int @ M )
% 9.03/4.98       => ( ( ord_less_int @ one_one_int @ N )
% 9.03/4.98         => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3459_nonzero__eq__divide__eq,axiom,
% 9.03/4.98      ! [C: complex,A: complex,B: complex] :
% 9.03/4.98        ( ( C != zero_zero_complex )
% 9.03/4.98       => ( ( A
% 9.03/4.98            = ( divide1717551699836669952omplex @ B @ C ) )
% 9.03/4.98          = ( ( times_times_complex @ A @ C )
% 9.03/4.98            = B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3460_nonzero__eq__divide__eq,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( C != zero_zero_real )
% 9.03/4.98       => ( ( A
% 9.03/4.98            = ( divide_divide_real @ B @ C ) )
% 9.03/4.98          = ( ( times_times_real @ A @ C )
% 9.03/4.98            = B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3461_nonzero__eq__divide__eq,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( C != zero_zero_rat )
% 9.03/4.98       => ( ( A
% 9.03/4.98            = ( divide_divide_rat @ B @ C ) )
% 9.03/4.98          = ( ( times_times_rat @ A @ C )
% 9.03/4.98            = B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3462_nonzero__divide__eq__eq,axiom,
% 9.03/4.98      ! [C: complex,B: complex,A: complex] :
% 9.03/4.98        ( ( C != zero_zero_complex )
% 9.03/4.98       => ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.03/4.98            = A )
% 9.03/4.98          = ( B
% 9.03/4.98            = ( times_times_complex @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3463_nonzero__divide__eq__eq,axiom,
% 9.03/4.98      ! [C: real,B: real,A: real] :
% 9.03/4.98        ( ( C != zero_zero_real )
% 9.03/4.98       => ( ( ( divide_divide_real @ B @ C )
% 9.03/4.98            = A )
% 9.03/4.98          = ( B
% 9.03/4.98            = ( times_times_real @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3464_nonzero__divide__eq__eq,axiom,
% 9.03/4.98      ! [C: rat,B: rat,A: rat] :
% 9.03/4.98        ( ( C != zero_zero_rat )
% 9.03/4.98       => ( ( ( divide_divide_rat @ B @ C )
% 9.03/4.98            = A )
% 9.03/4.98          = ( B
% 9.03/4.98            = ( times_times_rat @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3465_eq__divide__imp,axiom,
% 9.03/4.98      ! [C: complex,A: complex,B: complex] :
% 9.03/4.98        ( ( C != zero_zero_complex )
% 9.03/4.98       => ( ( ( times_times_complex @ A @ C )
% 9.03/4.98            = B )
% 9.03/4.98         => ( A
% 9.03/4.98            = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3466_eq__divide__imp,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( C != zero_zero_real )
% 9.03/4.98       => ( ( ( times_times_real @ A @ C )
% 9.03/4.98            = B )
% 9.03/4.98         => ( A
% 9.03/4.98            = ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3467_eq__divide__imp,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( C != zero_zero_rat )
% 9.03/4.98       => ( ( ( times_times_rat @ A @ C )
% 9.03/4.98            = B )
% 9.03/4.98         => ( A
% 9.03/4.98            = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3468_divide__eq__imp,axiom,
% 9.03/4.98      ! [C: complex,B: complex,A: complex] :
% 9.03/4.98        ( ( C != zero_zero_complex )
% 9.03/4.98       => ( ( B
% 9.03/4.98            = ( times_times_complex @ A @ C ) )
% 9.03/4.98         => ( ( divide1717551699836669952omplex @ B @ C )
% 9.03/4.98            = A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3469_divide__eq__imp,axiom,
% 9.03/4.98      ! [C: real,B: real,A: real] :
% 9.03/4.98        ( ( C != zero_zero_real )
% 9.03/4.98       => ( ( B
% 9.03/4.98            = ( times_times_real @ A @ C ) )
% 9.03/4.98         => ( ( divide_divide_real @ B @ C )
% 9.03/4.98            = A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3470_divide__eq__imp,axiom,
% 9.03/4.98      ! [C: rat,B: rat,A: rat] :
% 9.03/4.98        ( ( C != zero_zero_rat )
% 9.03/4.98       => ( ( B
% 9.03/4.98            = ( times_times_rat @ A @ C ) )
% 9.03/4.98         => ( ( divide_divide_rat @ B @ C )
% 9.03/4.98            = A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3471_eq__divide__eq,axiom,
% 9.03/4.98      ! [A: complex,B: complex,C: complex] :
% 9.03/4.98        ( ( A
% 9.03/4.98          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.03/4.98        = ( ( ( C != zero_zero_complex )
% 9.03/4.98           => ( ( times_times_complex @ A @ C )
% 9.03/4.98              = B ) )
% 9.03/4.98          & ( ( C = zero_zero_complex )
% 9.03/4.98           => ( A = zero_zero_complex ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3472_eq__divide__eq,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( A
% 9.03/4.98          = ( divide_divide_real @ B @ C ) )
% 9.03/4.98        = ( ( ( C != zero_zero_real )
% 9.03/4.98           => ( ( times_times_real @ A @ C )
% 9.03/4.98              = B ) )
% 9.03/4.98          & ( ( C = zero_zero_real )
% 9.03/4.98           => ( A = zero_zero_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3473_eq__divide__eq,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( A
% 9.03/4.98          = ( divide_divide_rat @ B @ C ) )
% 9.03/4.98        = ( ( ( C != zero_zero_rat )
% 9.03/4.98           => ( ( times_times_rat @ A @ C )
% 9.03/4.98              = B ) )
% 9.03/4.98          & ( ( C = zero_zero_rat )
% 9.03/4.98           => ( A = zero_zero_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3474_divide__eq__eq,axiom,
% 9.03/4.98      ! [B: complex,C: complex,A: complex] :
% 9.03/4.98        ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.03/4.98          = A )
% 9.03/4.98        = ( ( ( C != zero_zero_complex )
% 9.03/4.98           => ( B
% 9.03/4.98              = ( times_times_complex @ A @ C ) ) )
% 9.03/4.98          & ( ( C = zero_zero_complex )
% 9.03/4.98           => ( A = zero_zero_complex ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3475_divide__eq__eq,axiom,
% 9.03/4.98      ! [B: real,C: real,A: real] :
% 9.03/4.98        ( ( ( divide_divide_real @ B @ C )
% 9.03/4.98          = A )
% 9.03/4.98        = ( ( ( C != zero_zero_real )
% 9.03/4.98           => ( B
% 9.03/4.98              = ( times_times_real @ A @ C ) ) )
% 9.03/4.98          & ( ( C = zero_zero_real )
% 9.03/4.98           => ( A = zero_zero_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3476_divide__eq__eq,axiom,
% 9.03/4.98      ! [B: rat,C: rat,A: rat] :
% 9.03/4.98        ( ( ( divide_divide_rat @ B @ C )
% 9.03/4.98          = A )
% 9.03/4.98        = ( ( ( C != zero_zero_rat )
% 9.03/4.98           => ( B
% 9.03/4.98              = ( times_times_rat @ A @ C ) ) )
% 9.03/4.98          & ( ( C = zero_zero_rat )
% 9.03/4.98           => ( A = zero_zero_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3477_frac__eq__eq,axiom,
% 9.03/4.98      ! [Y: complex,Z: complex,X: complex,W: complex] :
% 9.03/4.98        ( ( Y != zero_zero_complex )
% 9.03/4.98       => ( ( Z != zero_zero_complex )
% 9.03/4.98         => ( ( ( divide1717551699836669952omplex @ X @ Y )
% 9.03/4.98              = ( divide1717551699836669952omplex @ W @ Z ) )
% 9.03/4.98            = ( ( times_times_complex @ X @ Z )
% 9.03/4.98              = ( times_times_complex @ W @ Y ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3478_frac__eq__eq,axiom,
% 9.03/4.98      ! [Y: real,Z: real,X: real,W: real] :
% 9.03/4.98        ( ( Y != zero_zero_real )
% 9.03/4.98       => ( ( Z != zero_zero_real )
% 9.03/4.98         => ( ( ( divide_divide_real @ X @ Y )
% 9.03/4.98              = ( divide_divide_real @ W @ Z ) )
% 9.03/4.98            = ( ( times_times_real @ X @ Z )
% 9.03/4.98              = ( times_times_real @ W @ Y ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3479_frac__eq__eq,axiom,
% 9.03/4.98      ! [Y: rat,Z: rat,X: rat,W: rat] :
% 9.03/4.98        ( ( Y != zero_zero_rat )
% 9.03/4.98       => ( ( Z != zero_zero_rat )
% 9.03/4.98         => ( ( ( divide_divide_rat @ X @ Y )
% 9.03/4.98              = ( divide_divide_rat @ W @ Z ) )
% 9.03/4.98            = ( ( times_times_rat @ X @ Z )
% 9.03/4.98              = ( times_times_rat @ W @ Y ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3480_mult__numeral__1,axiom,
% 9.03/4.98      ! [A: complex] :
% 9.03/4.98        ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 9.03/4.98        = A ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3481_mult__numeral__1,axiom,
% 9.03/4.98      ! [A: real] :
% 9.03/4.98        ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 9.03/4.98        = A ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3482_mult__numeral__1,axiom,
% 9.03/4.98      ! [A: rat] :
% 9.03/4.98        ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 9.03/4.98        = A ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3483_mult__numeral__1,axiom,
% 9.03/4.98      ! [A: nat] :
% 9.03/4.98        ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 9.03/4.98        = A ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3484_mult__numeral__1,axiom,
% 9.03/4.98      ! [A: int] :
% 9.03/4.98        ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 9.03/4.98        = A ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3485_mult__numeral__1__right,axiom,
% 9.03/4.98      ! [A: complex] :
% 9.03/4.98        ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 9.03/4.98        = A ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3486_mult__numeral__1__right,axiom,
% 9.03/4.98      ! [A: real] :
% 9.03/4.98        ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 9.03/4.98        = A ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3487_mult__numeral__1__right,axiom,
% 9.03/4.98      ! [A: rat] :
% 9.03/4.98        ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 9.03/4.98        = A ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3488_mult__numeral__1__right,axiom,
% 9.03/4.98      ! [A: nat] :
% 9.03/4.98        ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 9.03/4.98        = A ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3489_mult__numeral__1__right,axiom,
% 9.03/4.98      ! [A: int] :
% 9.03/4.98        ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 9.03/4.98        = A ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3490_left__right__inverse__power,axiom,
% 9.03/4.98      ! [X: complex,Y: complex,N: nat] :
% 9.03/4.98        ( ( ( times_times_complex @ X @ Y )
% 9.03/4.98          = one_one_complex )
% 9.03/4.98       => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
% 9.03/4.98          = one_one_complex ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3491_left__right__inverse__power,axiom,
% 9.03/4.98      ! [X: real,Y: real,N: nat] :
% 9.03/4.98        ( ( ( times_times_real @ X @ Y )
% 9.03/4.98          = one_one_real )
% 9.03/4.98       => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
% 9.03/4.98          = one_one_real ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3492_left__right__inverse__power,axiom,
% 9.03/4.98      ! [X: rat,Y: rat,N: nat] :
% 9.03/4.98        ( ( ( times_times_rat @ X @ Y )
% 9.03/4.98          = one_one_rat )
% 9.03/4.98       => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
% 9.03/4.98          = one_one_rat ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3493_left__right__inverse__power,axiom,
% 9.03/4.98      ! [X: nat,Y: nat,N: nat] :
% 9.03/4.98        ( ( ( times_times_nat @ X @ Y )
% 9.03/4.98          = one_one_nat )
% 9.03/4.98       => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
% 9.03/4.98          = one_one_nat ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3494_left__right__inverse__power,axiom,
% 9.03/4.98      ! [X: int,Y: int,N: nat] :
% 9.03/4.98        ( ( ( times_times_int @ X @ Y )
% 9.03/4.98          = one_one_int )
% 9.03/4.98       => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
% 9.03/4.98          = one_one_int ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3495_Suc__diff__Suc,axiom,
% 9.03/4.98      ! [N: nat,M: nat] :
% 9.03/4.98        ( ( ord_less_nat @ N @ M )
% 9.03/4.98       => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 9.03/4.98          = ( minus_minus_nat @ M @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3496_diff__less__Suc,axiom,
% 9.03/4.98      ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3497_diff__less,axiom,
% 9.03/4.98      ! [N: nat,M: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.98         => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3498_Suc__diff__le,axiom,
% 9.03/4.98      ! [N: nat,M: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.98       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 9.03/4.98          = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3499_power__Suc,axiom,
% 9.03/4.98      ! [A: complex,N: nat] :
% 9.03/4.98        ( ( power_power_complex @ A @ ( suc @ N ) )
% 9.03/4.98        = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3500_power__Suc,axiom,
% 9.03/4.98      ! [A: real,N: nat] :
% 9.03/4.98        ( ( power_power_real @ A @ ( suc @ N ) )
% 9.03/4.98        = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3501_power__Suc,axiom,
% 9.03/4.98      ! [A: rat,N: nat] :
% 9.03/4.98        ( ( power_power_rat @ A @ ( suc @ N ) )
% 9.03/4.98        = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3502_power__Suc,axiom,
% 9.03/4.98      ! [A: nat,N: nat] :
% 9.03/4.98        ( ( power_power_nat @ A @ ( suc @ N ) )
% 9.03/4.98        = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3503_power__Suc,axiom,
% 9.03/4.98      ! [A: int,N: nat] :
% 9.03/4.98        ( ( power_power_int @ A @ ( suc @ N ) )
% 9.03/4.98        = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3504_power__Suc2,axiom,
% 9.03/4.98      ! [A: complex,N: nat] :
% 9.03/4.98        ( ( power_power_complex @ A @ ( suc @ N ) )
% 9.03/4.98        = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3505_power__Suc2,axiom,
% 9.03/4.98      ! [A: real,N: nat] :
% 9.03/4.98        ( ( power_power_real @ A @ ( suc @ N ) )
% 9.03/4.98        = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3506_power__Suc2,axiom,
% 9.03/4.98      ! [A: rat,N: nat] :
% 9.03/4.98        ( ( power_power_rat @ A @ ( suc @ N ) )
% 9.03/4.98        = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3507_power__Suc2,axiom,
% 9.03/4.98      ! [A: nat,N: nat] :
% 9.03/4.98        ( ( power_power_nat @ A @ ( suc @ N ) )
% 9.03/4.98        = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3508_power__Suc2,axiom,
% 9.03/4.98      ! [A: int,N: nat] :
% 9.03/4.98        ( ( power_power_int @ A @ ( suc @ N ) )
% 9.03/4.98        = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3509_less__diff__iff,axiom,
% 9.03/4.98      ! [K: nat,M: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ K @ M )
% 9.03/4.98       => ( ( ord_less_eq_nat @ K @ N )
% 9.03/4.98         => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.03/4.98            = ( ord_less_nat @ M @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3510_diff__less__mono,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat] :
% 9.03/4.98        ( ( ord_less_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_nat @ C @ A )
% 9.03/4.98         => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3511_diff__add__0,axiom,
% 9.03/4.98      ! [N: nat,M: nat] :
% 9.03/4.98        ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 9.03/4.98        = zero_zero_nat ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3512_div__mult2__eq_H,axiom,
% 9.03/4.98      ! [A: nat,M: nat,N: nat] :
% 9.03/4.98        ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 9.03/4.98        = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3513_div__mult2__eq_H,axiom,
% 9.03/4.98      ! [A: int,M: nat,N: nat] :
% 9.03/4.98        ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.03/4.98        = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3514_add__diff__inverse__nat,axiom,
% 9.03/4.98      ! [M: nat,N: nat] :
% 9.03/4.98        ( ~ ( ord_less_nat @ M @ N )
% 9.03/4.98       => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.98          = M ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3515_less__diff__conv,axiom,
% 9.03/4.98      ! [I: nat,J: nat,K: nat] :
% 9.03/4.98        ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.03/4.98        = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3516_Nat_Ole__imp__diff__is__add,axiom,
% 9.03/4.98      ! [I: nat,J: nat,K: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ I @ J )
% 9.03/4.98       => ( ( ( minus_minus_nat @ J @ I )
% 9.03/4.98            = K )
% 9.03/4.98          = ( J
% 9.03/4.98            = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3517_Nat_Odiff__add__assoc2,axiom,
% 9.03/4.98      ! [K: nat,J: nat,I: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ K @ J )
% 9.03/4.98       => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 9.03/4.98          = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3518_Nat_Odiff__add__assoc,axiom,
% 9.03/4.98      ! [K: nat,J: nat,I: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ K @ J )
% 9.03/4.98       => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 9.03/4.98          = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3519_Nat_Ole__diff__conv2,axiom,
% 9.03/4.98      ! [K: nat,J: nat,I: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ K @ J )
% 9.03/4.98       => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.03/4.98          = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3520_le__diff__conv,axiom,
% 9.03/4.98      ! [J: nat,K: nat,I: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.03/4.98        = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3521_power__add,axiom,
% 9.03/4.98      ! [A: complex,M: nat,N: nat] :
% 9.03/4.98        ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.98        = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3522_power__add,axiom,
% 9.03/4.98      ! [A: real,M: nat,N: nat] :
% 9.03/4.98        ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.98        = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3523_power__add,axiom,
% 9.03/4.98      ! [A: rat,M: nat,N: nat] :
% 9.03/4.98        ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.98        = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3524_power__add,axiom,
% 9.03/4.98      ! [A: nat,M: nat,N: nat] :
% 9.03/4.98        ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.98        = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3525_power__add,axiom,
% 9.03/4.98      ! [A: int,M: nat,N: nat] :
% 9.03/4.98        ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 9.03/4.98        = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3526_Suc__mult__less__cancel1,axiom,
% 9.03/4.98      ! [K: nat,M: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.03/4.98        = ( ord_less_nat @ M @ N ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3527_mult__less__mono1,axiom,
% 9.03/4.98      ! [I: nat,J: nat,K: nat] :
% 9.03/4.98        ( ( ord_less_nat @ I @ J )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.98         => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3528_mult__less__mono2,axiom,
% 9.03/4.98      ! [I: nat,J: nat,K: nat] :
% 9.03/4.98        ( ( ord_less_nat @ I @ J )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.98         => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3529_nat__mult__eq__cancel1,axiom,
% 9.03/4.98      ! [K: nat,M: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.98       => ( ( ( times_times_nat @ K @ M )
% 9.03/4.98            = ( times_times_nat @ K @ N ) )
% 9.03/4.98          = ( M = N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3530_nat__mult__less__cancel1,axiom,
% 9.03/4.98      ! [K: nat,M: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.98       => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.03/4.98          = ( ord_less_nat @ M @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3531_Suc__mult__le__cancel1,axiom,
% 9.03/4.98      ! [K: nat,M: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.03/4.98        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3532_diff__Suc__eq__diff__pred,axiom,
% 9.03/4.98      ! [M: nat,N: nat] :
% 9.03/4.98        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 9.03/4.98        = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3533_mult__Suc,axiom,
% 9.03/4.98      ! [M: nat,N: nat] :
% 9.03/4.98        ( ( times_times_nat @ ( suc @ M ) @ N )
% 9.03/4.98        = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3534_mult__eq__self__implies__10,axiom,
% 9.03/4.98      ! [M: nat,N: nat] :
% 9.03/4.98        ( ( M
% 9.03/4.98          = ( times_times_nat @ M @ N ) )
% 9.03/4.98       => ( ( N = one_one_nat )
% 9.03/4.98          | ( M = zero_zero_nat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3535_less__mult__imp__div__less,axiom,
% 9.03/4.98      ! [M: nat,I: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 9.03/4.98       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3536_times__div__less__eq__dividend,axiom,
% 9.03/4.98      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3537_div__times__less__eq__dividend,axiom,
% 9.03/4.98      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3538_floor__divide__lower,axiom,
% 9.03/4.98      ! [Q3: real,P5: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ Q3 )
% 9.03/4.98       => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ Q3 ) @ P5 ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3539_floor__divide__lower,axiom,
% 9.03/4.98      ! [Q3: rat,P5: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 9.03/4.98       => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q3 ) ) ) @ Q3 ) @ P5 ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3540_power__odd__eq,axiom,
% 9.03/4.98      ! [A: complex,N: nat] :
% 9.03/4.98        ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.03/4.98        = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3541_power__odd__eq,axiom,
% 9.03/4.98      ! [A: real,N: nat] :
% 9.03/4.98        ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.03/4.98        = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3542_power__odd__eq,axiom,
% 9.03/4.98      ! [A: rat,N: nat] :
% 9.03/4.98        ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.03/4.98        = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3543_power__odd__eq,axiom,
% 9.03/4.98      ! [A: nat,N: nat] :
% 9.03/4.98        ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.03/4.98        = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3544_power__odd__eq,axiom,
% 9.03/4.98      ! [A: int,N: nat] :
% 9.03/4.98        ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.03/4.98        = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3545_floor__divide__upper,axiom,
% 9.03/4.98      ! [Q3: real,P5: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ Q3 )
% 9.03/4.98       => ( ord_less_real @ P5 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3546_floor__divide__upper,axiom,
% 9.03/4.98      ! [Q3: rat,P5: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 9.03/4.98       => ( ord_less_rat @ P5 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3547_le__floor__iff,axiom,
% 9.03/4.98      ! [Z: int,X: real] :
% 9.03/4.98        ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
% 9.03/4.98        = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3548_le__floor__iff,axiom,
% 9.03/4.98      ! [Z: int,X: rat] :
% 9.03/4.98        ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/4.98        = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3549_floor__less__iff,axiom,
% 9.03/4.98      ! [X: real,Z: int] :
% 9.03/4.98        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z )
% 9.03/4.98        = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3550_floor__less__iff,axiom,
% 9.03/4.98      ! [X: rat,Z: int] :
% 9.03/4.98        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
% 9.03/4.98        = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3551_powr__mono2_H,axiom,
% 9.03/4.98      ! [A: real,X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.98         => ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.98           => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3552_powr__less__mono2,axiom,
% 9.03/4.98      ! [A: real,X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.98         => ( ( ord_less_real @ X @ Y )
% 9.03/4.98           => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3553_le__floor__add,axiom,
% 9.03/4.98      ! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3554_le__floor__add,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3555_powr__inj,axiom,
% 9.03/4.98      ! [A: real,X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.98       => ( ( A != one_one_real )
% 9.03/4.98         => ( ( ( powr_real @ A @ X )
% 9.03/4.98              = ( powr_real @ A @ Y ) )
% 9.03/4.98            = ( X = Y ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3556_gr__one__powr,axiom,
% 9.03/4.98      ! [X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_real @ one_one_real @ X )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.98         => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3557_floor__add__int,axiom,
% 9.03/4.98      ! [X: real,Z: int] :
% 9.03/4.98        ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z )
% 9.03/4.98        = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3558_floor__add__int,axiom,
% 9.03/4.98      ! [X: rat,Z: int] :
% 9.03/4.98        ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
% 9.03/4.98        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3559_int__add__floor,axiom,
% 9.03/4.98      ! [Z: int,X: real] :
% 9.03/4.98        ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X ) )
% 9.03/4.98        = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3560_int__add__floor,axiom,
% 9.03/4.98      ! [Z: int,X: rat] :
% 9.03/4.98        ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/4.98        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3561_ge__one__powr__ge__zero,axiom,
% 9.03/4.98      ! [X: real,A: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ one_one_real @ X )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98         => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3562_powr__mono__both,axiom,
% 9.03/4.98      ! [A: real,B: real,X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98       => ( ( ord_less_eq_real @ A @ B )
% 9.03/4.98         => ( ( ord_less_eq_real @ one_one_real @ X )
% 9.03/4.98           => ( ( ord_less_eq_real @ X @ Y )
% 9.03/4.98             => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3563_powr__le1,axiom,
% 9.03/4.98      ! [A: real,X: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.98         => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.03/4.98           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3564_powr__divide,axiom,
% 9.03/4.98      ! [X: real,Y: real,A: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.98         => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
% 9.03/4.98            = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3565_power2__diff,axiom,
% 9.03/4.98      ! [X: complex,Y: complex] :
% 9.03/4.98        ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.98        = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3566_power2__diff,axiom,
% 9.03/4.98      ! [X: real,Y: real] :
% 9.03/4.98        ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.98        = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3567_power2__diff,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] :
% 9.03/4.98        ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.98        = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3568_power2__diff,axiom,
% 9.03/4.98      ! [X: int,Y: int] :
% 9.03/4.98        ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.98        = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3569_floor__divide__of__int__eq,axiom,
% 9.03/4.98      ! [K: int,L: int] :
% 9.03/4.98        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
% 9.03/4.98        = ( divide_divide_int @ K @ L ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3570_floor__divide__of__int__eq,axiom,
% 9.03/4.98      ! [K: int,L: int] :
% 9.03/4.98        ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
% 9.03/4.98        = ( divide_divide_int @ K @ L ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3571_floor__power,axiom,
% 9.03/4.98      ! [X: real,N: nat] :
% 9.03/4.98        ( ( X
% 9.03/4.98          = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
% 9.03/4.98       => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N ) )
% 9.03/4.98          = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3572_floor__power,axiom,
% 9.03/4.98      ! [X: rat,N: nat] :
% 9.03/4.98        ( ( X
% 9.03/4.98          = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
% 9.03/4.98       => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N ) )
% 9.03/4.98          = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3573_ceiling__divide__lower,axiom,
% 9.03/4.98      ! [Q3: real,P5: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ Q3 )
% 9.03/4.98       => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P5 ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3574_ceiling__divide__lower,axiom,
% 9.03/4.98      ! [Q3: rat,P5: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 9.03/4.98       => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P5 ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3575_log__base__powr,axiom,
% 9.03/4.98      ! [A: real,B: real,X: real] :
% 9.03/4.98        ( ( A != zero_zero_real )
% 9.03/4.98       => ( ( log @ ( powr_real @ A @ B ) @ X )
% 9.03/4.98          = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3576_round__mono,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ X @ Y )
% 9.03/4.98       => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3577_mult__less__le__imp__less,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_real @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98           => ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3578_mult__less__le__imp__less,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_rat @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98           => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3579_mult__less__le__imp__less,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.03/4.98        ( ( ord_less_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_nat @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.98           => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.03/4.98             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3580_mult__less__le__imp__less,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.98        ( ( ord_less_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_int @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98           => ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3581_mult__le__less__imp__less,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_real @ C @ D )
% 9.03/4.98         => ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.98           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3582_mult__le__less__imp__less,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_rat @ C @ D )
% 9.03/4.98         => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.98           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3583_mult__le__less__imp__less,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_nat @ C @ D )
% 9.03/4.98         => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.98           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3584_mult__le__less__imp__less,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_int @ C @ D )
% 9.03/4.98         => ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.98           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3585_mult__right__le__imp__le,axiom,
% 9.03/4.98      ! [A: real,C: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3586_mult__right__le__imp__le,axiom,
% 9.03/4.98      ! [A: rat,C: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.03/4.98       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3587_mult__right__le__imp__le,axiom,
% 9.03/4.98      ! [A: nat,C: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.03/4.98         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3588_mult__right__le__imp__le,axiom,
% 9.03/4.98      ! [A: int,C: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.03/4.98       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3589_mult__left__le__imp__le,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3590_mult__left__le__imp__le,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.98       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3591_mult__left__le__imp__le,axiom,
% 9.03/4.98      ! [C: nat,A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.03/4.98       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.03/4.98         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3592_mult__left__le__imp__le,axiom,
% 9.03/4.98      ! [C: int,A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.98       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3593_mult__le__cancel__left__pos,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.98          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3594_mult__le__cancel__left__pos,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.98          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3595_mult__le__cancel__left__pos,axiom,
% 9.03/4.98      ! [C: int,A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.98          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3596_mult__le__cancel__left__neg,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.98          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3597_mult__le__cancel__left__neg,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.98          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3598_mult__le__cancel__left__neg,axiom,
% 9.03/4.98      ! [C: int,A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.98          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3599_mult__less__cancel__right,axiom,
% 9.03/4.98      ! [A: real,C: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_real @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_real @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3600_mult__less__cancel__right,axiom,
% 9.03/4.98      ! [A: rat,C: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_rat @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3601_mult__less__cancel__right,axiom,
% 9.03/4.98      ! [A: int,C: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_int @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_int @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3602_mult__strict__mono_H,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_real @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3603_mult__strict__mono_H,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_rat @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3604_mult__strict__mono_H,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.03/4.98        ( ( ord_less_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_nat @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.98           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3605_mult__strict__mono_H,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.98        ( ( ord_less_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_int @ C @ D )
% 9.03/4.98         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3606_mult__right__less__imp__less,axiom,
% 9.03/4.98      ! [A: real,C: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ord_less_real @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3607_mult__right__less__imp__less,axiom,
% 9.03/4.98      ! [A: rat,C: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3608_mult__right__less__imp__less,axiom,
% 9.03/4.98      ! [A: nat,C: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 9.03/4.98       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3609_mult__right__less__imp__less,axiom,
% 9.03/4.98      ! [A: int,C: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98         => ( ord_less_int @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3610_mult__less__cancel__left,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_real @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_real @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3611_mult__less__cancel__left,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_rat @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3612_mult__less__cancel__left,axiom,
% 9.03/4.98      ! [C: int,A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_int @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_int @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3613_mult__strict__mono,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real,D: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_real @ C @ D )
% 9.03/4.98         => ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.98           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3614_mult__strict__mono,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_rat @ C @ D )
% 9.03/4.98         => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.03/4.98           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3615_mult__strict__mono,axiom,
% 9.03/4.98      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.03/4.98        ( ( ord_less_nat @ A @ B )
% 9.03/4.98       => ( ( ord_less_nat @ C @ D )
% 9.03/4.98         => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.03/4.98           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3616_mult__strict__mono,axiom,
% 9.03/4.98      ! [A: int,B: int,C: int,D: int] :
% 9.03/4.98        ( ( ord_less_int @ A @ B )
% 9.03/4.98       => ( ( ord_less_int @ C @ D )
% 9.03/4.98         => ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/4.98           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3617_mult__left__less__imp__less,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ord_less_real @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3618_mult__left__less__imp__less,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3619_mult__left__less__imp__less,axiom,
% 9.03/4.98      ! [C: nat,A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.03/4.98       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3620_mult__left__less__imp__less,axiom,
% 9.03/4.98      ! [C: int,A: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98         => ( ord_less_int @ A @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3621_mult__le__cancel__right,axiom,
% 9.03/4.98      ! [A: real,C: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_eq_real @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3622_mult__le__cancel__right,axiom,
% 9.03/4.98      ! [A: rat,C: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_eq_rat @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3623_mult__le__cancel__right,axiom,
% 9.03/4.98      ! [A: int,C: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_eq_int @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3624_mult__le__cancel__left,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_eq_real @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3625_mult__le__cancel__left,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_eq_rat @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3626_mult__le__cancel__left,axiom,
% 9.03/4.98      ! [C: int,A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_eq_int @ A @ B ) )
% 9.03/4.98          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3627_mult__left__le__one__le,axiom,
% 9.03/4.98      ! [X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.98         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.03/4.98           => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3628_mult__left__le__one__le,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.98         => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 9.03/4.98           => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3629_mult__left__le__one__le,axiom,
% 9.03/4.98      ! [X: int,Y: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.03/4.98         => ( ( ord_less_eq_int @ Y @ one_one_int )
% 9.03/4.98           => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3630_mult__right__le__one__le,axiom,
% 9.03/4.98      ! [X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.98         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.03/4.98           => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3631_mult__right__le__one__le,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.98         => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 9.03/4.98           => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3632_mult__right__le__one__le,axiom,
% 9.03/4.98      ! [X: int,Y: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.03/4.98         => ( ( ord_less_eq_int @ Y @ one_one_int )
% 9.03/4.98           => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3633_mult__le__one,axiom,
% 9.03/4.98      ! [A: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ one_one_real )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/4.98         => ( ( ord_less_eq_real @ B @ one_one_real )
% 9.03/4.98           => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3634_mult__le__one,axiom,
% 9.03/4.98      ! [A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/4.98         => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 9.03/4.98           => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3635_mult__le__one,axiom,
% 9.03/4.98      ! [A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.03/4.98       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.03/4.98         => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 9.03/4.98           => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3636_mult__le__one,axiom,
% 9.03/4.98      ! [A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ A @ one_one_int )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.98         => ( ( ord_less_eq_int @ B @ one_one_int )
% 9.03/4.98           => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3637_mult__left__le,axiom,
% 9.03/4.98      ! [C: real,A: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ C @ one_one_real )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/4.98         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3638_mult__left__le,axiom,
% 9.03/4.98      ! [C: rat,A: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ C @ one_one_rat )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/4.98         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3639_mult__left__le,axiom,
% 9.03/4.98      ! [C: nat,A: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ C @ one_one_nat )
% 9.03/4.98       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/4.98         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3640_mult__left__le,axiom,
% 9.03/4.98      ! [C: int,A: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ C @ one_one_int )
% 9.03/4.98       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.98         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3641_sum__squares__le__zero__iff,axiom,
% 9.03/4.98      ! [X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
% 9.03/4.98        = ( ( X = zero_zero_real )
% 9.03/4.98          & ( Y = zero_zero_real ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3642_sum__squares__le__zero__iff,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
% 9.03/4.98        = ( ( X = zero_zero_rat )
% 9.03/4.98          & ( Y = zero_zero_rat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3643_sum__squares__le__zero__iff,axiom,
% 9.03/4.98      ! [X: int,Y: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
% 9.03/4.98        = ( ( X = zero_zero_int )
% 9.03/4.98          & ( Y = zero_zero_int ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3644_sum__squares__ge__zero,axiom,
% 9.03/4.98      ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3645_sum__squares__ge__zero,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3646_sum__squares__ge__zero,axiom,
% 9.03/4.98      ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3647_sum__squares__gt__zero__iff,axiom,
% 9.03/4.98      ! [X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
% 9.03/4.98        = ( ( X != zero_zero_real )
% 9.03/4.98          | ( Y != zero_zero_real ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3648_sum__squares__gt__zero__iff,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
% 9.03/4.98        = ( ( X != zero_zero_rat )
% 9.03/4.98          | ( Y != zero_zero_rat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3649_sum__squares__gt__zero__iff,axiom,
% 9.03/4.98      ! [X: int,Y: int] :
% 9.03/4.98        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
% 9.03/4.98        = ( ( X != zero_zero_int )
% 9.03/4.98          | ( Y != zero_zero_int ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3650_not__sum__squares__lt__zero,axiom,
% 9.03/4.98      ! [X: real,Y: real] :
% 9.03/4.98        ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3651_not__sum__squares__lt__zero,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] :
% 9.03/4.98        ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3652_not__sum__squares__lt__zero,axiom,
% 9.03/4.98      ! [X: int,Y: int] :
% 9.03/4.98        ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3653_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.03/4.98      ! [C: nat,A: nat,B: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/4.98       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.03/4.98          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3654_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.03/4.98      ! [C: int,A: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 9.03/4.98          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3655_divide__strict__left__mono__neg,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.03/4.98           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3656_divide__strict__left__mono__neg,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.03/4.98           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3657_divide__strict__left__mono,axiom,
% 9.03/4.98      ! [B: real,A: real,C: real] :
% 9.03/4.98        ( ( ord_less_real @ B @ A )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.03/4.98           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3658_divide__strict__left__mono,axiom,
% 9.03/4.98      ! [B: rat,A: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_rat @ B @ A )
% 9.03/4.98       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.03/4.98           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3659_mult__imp__less__div__pos,axiom,
% 9.03/4.98      ! [Y: real,Z: real,X: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.98       => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
% 9.03/4.98         => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3660_mult__imp__less__div__pos,axiom,
% 9.03/4.98      ! [Y: rat,Z: rat,X: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.03/4.98       => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
% 9.03/4.98         => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3661_mult__imp__div__pos__less,axiom,
% 9.03/4.98      ! [Y: real,X: real,Z: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.98       => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
% 9.03/4.98         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3662_mult__imp__div__pos__less,axiom,
% 9.03/4.98      ! [Y: rat,X: rat,Z: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.03/4.98       => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
% 9.03/4.98         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3663_pos__less__divide__eq,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.03/4.98          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3664_pos__less__divide__eq,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.98          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3665_pos__divide__less__eq,axiom,
% 9.03/4.98      ! [C: real,B: real,A: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.03/4.98          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3666_pos__divide__less__eq,axiom,
% 9.03/4.98      ! [C: rat,B: rat,A: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.03/4.98          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3667_neg__less__divide__eq,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.03/4.98          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3668_neg__less__divide__eq,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.98          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3669_neg__divide__less__eq,axiom,
% 9.03/4.98      ! [C: real,B: real,A: real] :
% 9.03/4.98        ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.03/4.98          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3670_neg__divide__less__eq,axiom,
% 9.03/4.98      ! [C: rat,B: rat,A: rat] :
% 9.03/4.98        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.03/4.98          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3671_less__divide__eq,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.03/4.98          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98               => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.03/4.98              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3672_less__divide__eq,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.03/4.98          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98               => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.03/4.98              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3673_divide__less__eq,axiom,
% 9.03/4.98      ! [B: real,C: real,A: real] :
% 9.03/4.98        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.03/4.98          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98               => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.03/4.98              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3674_divide__less__eq,axiom,
% 9.03/4.98      ! [B: rat,C: rat,A: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.03/4.98          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.03/4.98              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3675_eq__divide__eq__numeral_I1_J,axiom,
% 9.03/4.98      ! [W: num,B: complex,C: complex] :
% 9.03/4.98        ( ( ( numera6690914467698888265omplex @ W )
% 9.03/4.98          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.03/4.98        = ( ( ( C != zero_zero_complex )
% 9.03/4.98           => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 9.03/4.98              = B ) )
% 9.03/4.98          & ( ( C = zero_zero_complex )
% 9.03/4.98           => ( ( numera6690914467698888265omplex @ W )
% 9.03/4.98              = zero_zero_complex ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3676_eq__divide__eq__numeral_I1_J,axiom,
% 9.03/4.98      ! [W: num,B: real,C: real] :
% 9.03/4.98        ( ( ( numeral_numeral_real @ W )
% 9.03/4.98          = ( divide_divide_real @ B @ C ) )
% 9.03/4.98        = ( ( ( C != zero_zero_real )
% 9.03/4.98           => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 9.03/4.98              = B ) )
% 9.03/4.98          & ( ( C = zero_zero_real )
% 9.03/4.98           => ( ( numeral_numeral_real @ W )
% 9.03/4.98              = zero_zero_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3677_eq__divide__eq__numeral_I1_J,axiom,
% 9.03/4.98      ! [W: num,B: rat,C: rat] :
% 9.03/4.98        ( ( ( numeral_numeral_rat @ W )
% 9.03/4.98          = ( divide_divide_rat @ B @ C ) )
% 9.03/4.98        = ( ( ( C != zero_zero_rat )
% 9.03/4.98           => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 9.03/4.98              = B ) )
% 9.03/4.98          & ( ( C = zero_zero_rat )
% 9.03/4.98           => ( ( numeral_numeral_rat @ W )
% 9.03/4.98              = zero_zero_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3678_divide__eq__eq__numeral_I1_J,axiom,
% 9.03/4.98      ! [B: complex,C: complex,W: num] :
% 9.03/4.98        ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.03/4.98          = ( numera6690914467698888265omplex @ W ) )
% 9.03/4.98        = ( ( ( C != zero_zero_complex )
% 9.03/4.98           => ( B
% 9.03/4.98              = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 9.03/4.98          & ( ( C = zero_zero_complex )
% 9.03/4.98           => ( ( numera6690914467698888265omplex @ W )
% 9.03/4.98              = zero_zero_complex ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3679_divide__eq__eq__numeral_I1_J,axiom,
% 9.03/4.98      ! [B: real,C: real,W: num] :
% 9.03/4.98        ( ( ( divide_divide_real @ B @ C )
% 9.03/4.98          = ( numeral_numeral_real @ W ) )
% 9.03/4.98        = ( ( ( C != zero_zero_real )
% 9.03/4.98           => ( B
% 9.03/4.98              = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 9.03/4.98          & ( ( C = zero_zero_real )
% 9.03/4.98           => ( ( numeral_numeral_real @ W )
% 9.03/4.98              = zero_zero_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3680_divide__eq__eq__numeral_I1_J,axiom,
% 9.03/4.98      ! [B: rat,C: rat,W: num] :
% 9.03/4.98        ( ( ( divide_divide_rat @ B @ C )
% 9.03/4.98          = ( numeral_numeral_rat @ W ) )
% 9.03/4.98        = ( ( ( C != zero_zero_rat )
% 9.03/4.98           => ( B
% 9.03/4.98              = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 9.03/4.98          & ( ( C = zero_zero_rat )
% 9.03/4.98           => ( ( numeral_numeral_rat @ W )
% 9.03/4.98              = zero_zero_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3681_add__divide__eq__if__simps_I2_J,axiom,
% 9.03/4.98      ! [Z: complex,A: complex,B: complex] :
% 9.03/4.98        ( ( ( Z = zero_zero_complex )
% 9.03/4.98         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 9.03/4.98            = B ) )
% 9.03/4.98        & ( ( Z != zero_zero_complex )
% 9.03/4.98         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 9.03/4.98            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3682_add__divide__eq__if__simps_I2_J,axiom,
% 9.03/4.98      ! [Z: real,A: real,B: real] :
% 9.03/4.98        ( ( ( Z = zero_zero_real )
% 9.03/4.98         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 9.03/4.98            = B ) )
% 9.03/4.98        & ( ( Z != zero_zero_real )
% 9.03/4.98         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 9.03/4.98            = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3683_add__divide__eq__if__simps_I2_J,axiom,
% 9.03/4.98      ! [Z: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ( Z = zero_zero_rat )
% 9.03/4.98         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 9.03/4.98            = B ) )
% 9.03/4.98        & ( ( Z != zero_zero_rat )
% 9.03/4.98         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 9.03/4.98            = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3684_add__divide__eq__if__simps_I1_J,axiom,
% 9.03/4.98      ! [Z: complex,A: complex,B: complex] :
% 9.03/4.98        ( ( ( Z = zero_zero_complex )
% 9.03/4.98         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 9.03/4.98            = A ) )
% 9.03/4.98        & ( ( Z != zero_zero_complex )
% 9.03/4.98         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 9.03/4.98            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3685_add__divide__eq__if__simps_I1_J,axiom,
% 9.03/4.98      ! [Z: real,A: real,B: real] :
% 9.03/4.98        ( ( ( Z = zero_zero_real )
% 9.03/4.98         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 9.03/4.98            = A ) )
% 9.03/4.98        & ( ( Z != zero_zero_real )
% 9.03/4.98         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 9.03/4.98            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3686_add__divide__eq__if__simps_I1_J,axiom,
% 9.03/4.98      ! [Z: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ( Z = zero_zero_rat )
% 9.03/4.98         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 9.03/4.98            = A ) )
% 9.03/4.98        & ( ( Z != zero_zero_rat )
% 9.03/4.98         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 9.03/4.98            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3687_add__frac__eq,axiom,
% 9.03/4.98      ! [Y: complex,Z: complex,X: complex,W: complex] :
% 9.03/4.98        ( ( Y != zero_zero_complex )
% 9.03/4.98       => ( ( Z != zero_zero_complex )
% 9.03/4.98         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 9.03/4.98            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3688_add__frac__eq,axiom,
% 9.03/4.98      ! [Y: real,Z: real,X: real,W: real] :
% 9.03/4.98        ( ( Y != zero_zero_real )
% 9.03/4.98       => ( ( Z != zero_zero_real )
% 9.03/4.98         => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 9.03/4.98            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3689_add__frac__eq,axiom,
% 9.03/4.98      ! [Y: rat,Z: rat,X: rat,W: rat] :
% 9.03/4.98        ( ( Y != zero_zero_rat )
% 9.03/4.98       => ( ( Z != zero_zero_rat )
% 9.03/4.98         => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 9.03/4.98            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3690_add__frac__num,axiom,
% 9.03/4.98      ! [Y: complex,X: complex,Z: complex] :
% 9.03/4.98        ( ( Y != zero_zero_complex )
% 9.03/4.98       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
% 9.03/4.98          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3691_add__frac__num,axiom,
% 9.03/4.98      ! [Y: real,X: real,Z: real] :
% 9.03/4.98        ( ( Y != zero_zero_real )
% 9.03/4.98       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
% 9.03/4.98          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3692_add__frac__num,axiom,
% 9.03/4.98      ! [Y: rat,X: rat,Z: rat] :
% 9.03/4.98        ( ( Y != zero_zero_rat )
% 9.03/4.98       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
% 9.03/4.98          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3693_add__num__frac,axiom,
% 9.03/4.98      ! [Y: complex,Z: complex,X: complex] :
% 9.03/4.98        ( ( Y != zero_zero_complex )
% 9.03/4.98       => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
% 9.03/4.98          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3694_add__num__frac,axiom,
% 9.03/4.98      ! [Y: real,Z: real,X: real] :
% 9.03/4.98        ( ( Y != zero_zero_real )
% 9.03/4.98       => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
% 9.03/4.98          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3695_add__num__frac,axiom,
% 9.03/4.98      ! [Y: rat,Z: rat,X: rat] :
% 9.03/4.98        ( ( Y != zero_zero_rat )
% 9.03/4.98       => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
% 9.03/4.98          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3696_add__divide__eq__iff,axiom,
% 9.03/4.98      ! [Z: complex,X: complex,Y: complex] :
% 9.03/4.98        ( ( Z != zero_zero_complex )
% 9.03/4.98       => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
% 9.03/4.98          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3697_add__divide__eq__iff,axiom,
% 9.03/4.98      ! [Z: real,X: real,Y: real] :
% 9.03/4.98        ( ( Z != zero_zero_real )
% 9.03/4.98       => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
% 9.03/4.98          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3698_add__divide__eq__iff,axiom,
% 9.03/4.98      ! [Z: rat,X: rat,Y: rat] :
% 9.03/4.98        ( ( Z != zero_zero_rat )
% 9.03/4.98       => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
% 9.03/4.98          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3699_divide__add__eq__iff,axiom,
% 9.03/4.98      ! [Z: complex,X: complex,Y: complex] :
% 9.03/4.98        ( ( Z != zero_zero_complex )
% 9.03/4.98       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
% 9.03/4.98          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3700_divide__add__eq__iff,axiom,
% 9.03/4.98      ! [Z: real,X: real,Y: real] :
% 9.03/4.98        ( ( Z != zero_zero_real )
% 9.03/4.98       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
% 9.03/4.98          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3701_divide__add__eq__iff,axiom,
% 9.03/4.98      ! [Z: rat,X: rat,Y: rat] :
% 9.03/4.98        ( ( Z != zero_zero_rat )
% 9.03/4.98       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
% 9.03/4.98          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3702_ex__less__of__nat__mult,axiom,
% 9.03/4.98      ! [X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.98       => ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3703_ex__less__of__nat__mult,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.03/4.98       => ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3704_power__gt1__lemma,axiom,
% 9.03/4.98      ! [A: real,N: nat] :
% 9.03/4.98        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.98       => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3705_power__gt1__lemma,axiom,
% 9.03/4.98      ! [A: rat,N: nat] :
% 9.03/4.98        ( ( ord_less_rat @ one_one_rat @ A )
% 9.03/4.98       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3706_power__gt1__lemma,axiom,
% 9.03/4.98      ! [A: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_nat @ one_one_nat @ A )
% 9.03/4.98       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3707_power__gt1__lemma,axiom,
% 9.03/4.98      ! [A: int,N: nat] :
% 9.03/4.98        ( ( ord_less_int @ one_one_int @ A )
% 9.03/4.98       => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3708_power__less__power__Suc,axiom,
% 9.03/4.98      ! [A: real,N: nat] :
% 9.03/4.98        ( ( ord_less_real @ one_one_real @ A )
% 9.03/4.98       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3709_power__less__power__Suc,axiom,
% 9.03/4.98      ! [A: rat,N: nat] :
% 9.03/4.98        ( ( ord_less_rat @ one_one_rat @ A )
% 9.03/4.98       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3710_power__less__power__Suc,axiom,
% 9.03/4.98      ! [A: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_nat @ one_one_nat @ A )
% 9.03/4.98       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3711_power__less__power__Suc,axiom,
% 9.03/4.98      ! [A: int,N: nat] :
% 9.03/4.98        ( ( ord_less_int @ one_one_int @ A )
% 9.03/4.98       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3712_diff__Suc__less,axiom,
% 9.03/4.98      ! [N: nat,I: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.98       => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3713_round__def,axiom,
% 9.03/4.98      ( archim8280529875227126926d_real
% 9.03/4.98      = ( ^ [X4: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3714_round__def,axiom,
% 9.03/4.98      ( archim7778729529865785530nd_rat
% 9.03/4.98      = ( ^ [X4: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3715_nat__diff__split,axiom,
% 9.03/4.98      ! [P: nat > $o,A: nat,B: nat] :
% 9.03/4.98        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 9.03/4.98        = ( ( ( ord_less_nat @ A @ B )
% 9.03/4.98           => ( P @ zero_zero_nat ) )
% 9.03/4.98          & ! [D3: nat] :
% 9.03/4.98              ( ( A
% 9.03/4.98                = ( plus_plus_nat @ B @ D3 ) )
% 9.03/4.98             => ( P @ D3 ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3716_nat__diff__split__asm,axiom,
% 9.03/4.98      ! [P: nat > $o,A: nat,B: nat] :
% 9.03/4.98        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 9.03/4.98        = ( ~ ( ( ( ord_less_nat @ A @ B )
% 9.03/4.98                & ~ ( P @ zero_zero_nat ) )
% 9.03/4.98              | ? [D3: nat] :
% 9.03/4.98                  ( ( A
% 9.03/4.98                    = ( plus_plus_nat @ B @ D3 ) )
% 9.03/4.98                  & ~ ( P @ D3 ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3717_less__diff__conv2,axiom,
% 9.03/4.98      ! [K: nat,J: nat,I: nat] :
% 9.03/4.98        ( ( ord_less_eq_nat @ K @ J )
% 9.03/4.98       => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.03/4.98          = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3718_n__less__n__mult__m,axiom,
% 9.03/4.98      ! [N: nat,M: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.98       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.03/4.98         => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3719_n__less__m__mult__n,axiom,
% 9.03/4.98      ! [N: nat,M: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.98       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.03/4.98         => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3720_one__less__mult,axiom,
% 9.03/4.98      ! [N: nat,M: nat] :
% 9.03/4.98        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.03/4.98       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.03/4.98         => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3721_nat__mult__le__cancel1,axiom,
% 9.03/4.98      ! [K: nat,M: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.98       => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.03/4.98          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3722_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 9.03/4.98      ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 9.03/4.98        ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3723_ceiling__ge__round,axiom,
% 9.03/4.98      ! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3724_ceiling__ge__round,axiom,
% 9.03/4.98      ! [X: rat] : ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3725_div__less__iff__less__mult,axiom,
% 9.03/4.98      ! [Q3: nat,M: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 9.03/4.98       => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
% 9.03/4.98          = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3726_nat__mult__div__cancel1,axiom,
% 9.03/4.98      ! [K: nat,M: nat,N: nat] :
% 9.03/4.98        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.98       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.03/4.98          = ( divide_divide_nat @ M @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3727_one__add__floor,axiom,
% 9.03/4.98      ! [X: real] :
% 9.03/4.98        ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 9.03/4.98        = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3728_one__add__floor,axiom,
% 9.03/4.98      ! [X: rat] :
% 9.03/4.98        ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 9.03/4.98        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3729_floor__log__eq__powr__iff,axiom,
% 9.03/4.98      ! [X: real,B: real,K: int] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.98       => ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.98         => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
% 9.03/4.98              = K )
% 9.03/4.98            = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 9.03/4.98              & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3730_floor__divide__of__nat__eq,axiom,
% 9.03/4.98      ! [M: nat,N: nat] :
% 9.03/4.98        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.03/4.98        = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3731_floor__divide__of__nat__eq,axiom,
% 9.03/4.98      ! [M: nat,N: nat] :
% 9.03/4.98        ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
% 9.03/4.98        = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3732_powr__realpow,axiom,
% 9.03/4.98      ! [X: real,N: nat] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.98       => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.98          = ( power_power_real @ X @ N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3733_powr__less__iff,axiom,
% 9.03/4.98      ! [B: real,X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.98         => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
% 9.03/4.98            = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3734_less__powr__iff,axiom,
% 9.03/4.98      ! [B: real,X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.98         => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
% 9.03/4.98            = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3735_log__less__iff,axiom,
% 9.03/4.98      ! [B: real,X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.98         => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
% 9.03/4.98            = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3736_less__log__iff,axiom,
% 9.03/4.98      ! [B: real,X: real,Y: real] :
% 9.03/4.98        ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.98       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.98         => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
% 9.03/4.98            = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3737_ceiling__altdef,axiom,
% 9.03/4.98      ( archim7802044766580827645g_real
% 9.03/4.98      = ( ^ [X4: real] :
% 9.03/4.98            ( if_int
% 9.03/4.98            @ ( X4
% 9.03/4.98              = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) )
% 9.03/4.98            @ ( archim6058952711729229775r_real @ X4 )
% 9.03/4.98            @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3738_ceiling__altdef,axiom,
% 9.03/4.98      ( archim2889992004027027881ng_rat
% 9.03/4.98      = ( ^ [X4: rat] :
% 9.03/4.98            ( if_int
% 9.03/4.98            @ ( X4
% 9.03/4.98              = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) )
% 9.03/4.98            @ ( archim3151403230148437115or_rat @ X4 )
% 9.03/4.98            @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3739_of__int__round__ge,axiom,
% 9.03/4.98      ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3740_of__int__round__ge,axiom,
% 9.03/4.98      ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3741_of__int__round__gt,axiom,
% 9.03/4.98      ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3742_of__int__round__gt,axiom,
% 9.03/4.98      ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3743_real__of__int__floor__add__one__gt,axiom,
% 9.03/4.98      ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3744_floor__eq,axiom,
% 9.03/4.98      ! [N: int,X: real] :
% 9.03/4.98        ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 9.03/4.98       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.03/4.98         => ( ( archim6058952711729229775r_real @ X )
% 9.03/4.98            = N ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3745_real__of__int__floor__add__one__ge,axiom,
% 9.03/4.98      ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3746_power2__commute,axiom,
% 9.03/4.98      ! [X: complex,Y: complex] :
% 9.03/4.98        ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.98        = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3747_power2__commute,axiom,
% 9.03/4.98      ! [X: real,Y: real] :
% 9.03/4.98        ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.98        = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3748_power2__commute,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] :
% 9.03/4.98        ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.98        = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3749_power2__commute,axiom,
% 9.03/4.98      ! [X: int,Y: int] :
% 9.03/4.98        ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.98        = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3750_field__le__mult__one__interval,axiom,
% 9.03/4.98      ! [X: real,Y: real] :
% 9.03/4.98        ( ! [Z3: real] :
% 9.03/4.98            ( ( ord_less_real @ zero_zero_real @ Z3 )
% 9.03/4.98           => ( ( ord_less_real @ Z3 @ one_one_real )
% 9.03/4.98             => ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ Y ) ) )
% 9.03/4.98       => ( ord_less_eq_real @ X @ Y ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3751_field__le__mult__one__interval,axiom,
% 9.03/4.98      ! [X: rat,Y: rat] :
% 9.03/4.98        ( ! [Z3: rat] :
% 9.03/4.98            ( ( ord_less_rat @ zero_zero_rat @ Z3 )
% 9.03/4.98           => ( ( ord_less_rat @ Z3 @ one_one_rat )
% 9.03/4.98             => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y ) ) )
% 9.03/4.98       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3752_mult__less__cancel__right2,axiom,
% 9.03/4.98      ! [A: real,C: real] :
% 9.03/4.98        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 9.03/4.98        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_real @ A @ one_one_real ) )
% 9.03/4.98          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3753_mult__less__cancel__right2,axiom,
% 9.03/4.98      ! [A: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 9.03/4.98        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_rat @ A @ one_one_rat ) )
% 9.03/4.98          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3754_mult__less__cancel__right2,axiom,
% 9.03/4.98      ! [A: int,C: int] :
% 9.03/4.98        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 9.03/4.98        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_int @ A @ one_one_int ) )
% 9.03/4.98          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3755_mult__less__cancel__right1,axiom,
% 9.03/4.98      ! [C: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_real @ one_one_real @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3756_mult__less__cancel__right1,axiom,
% 9.03/4.98      ! [C: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_rat @ one_one_rat @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3757_mult__less__cancel__right1,axiom,
% 9.03/4.98      ! [C: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_int @ one_one_int @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3758_mult__less__cancel__left2,axiom,
% 9.03/4.98      ! [C: real,A: real] :
% 9.03/4.98        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 9.03/4.98        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_real @ A @ one_one_real ) )
% 9.03/4.98          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3759_mult__less__cancel__left2,axiom,
% 9.03/4.98      ! [C: rat,A: rat] :
% 9.03/4.98        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 9.03/4.98        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_rat @ A @ one_one_rat ) )
% 9.03/4.98          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3760_mult__less__cancel__left2,axiom,
% 9.03/4.98      ! [C: int,A: int] :
% 9.03/4.98        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 9.03/4.98        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_int @ A @ one_one_int ) )
% 9.03/4.98          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3761_mult__less__cancel__left1,axiom,
% 9.03/4.98      ! [C: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_real @ one_one_real @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3762_mult__less__cancel__left1,axiom,
% 9.03/4.98      ! [C: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_rat @ one_one_rat @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3763_mult__less__cancel__left1,axiom,
% 9.03/4.98      ! [C: int,B: int] :
% 9.03/4.98        ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_int @ one_one_int @ B ) )
% 9.03/4.98          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3764_mult__le__cancel__right2,axiom,
% 9.03/4.98      ! [A: real,C: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_eq_real @ A @ one_one_real ) )
% 9.03/4.98          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3765_mult__le__cancel__right2,axiom,
% 9.03/4.98      ! [A: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 9.03/4.98          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3766_mult__le__cancel__right2,axiom,
% 9.03/4.98      ! [A: int,C: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 9.03/4.98        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_eq_int @ A @ one_one_int ) )
% 9.03/4.98          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3767_mult__le__cancel__right1,axiom,
% 9.03/4.98      ! [C: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_eq_real @ one_one_real @ B ) )
% 9.03/4.98          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3768_mult__le__cancel__right1,axiom,
% 9.03/4.98      ! [C: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 9.03/4.98          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3769_mult__le__cancel__right1,axiom,
% 9.03/4.98      ! [C: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_eq_int @ one_one_int @ B ) )
% 9.03/4.98          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3770_mult__le__cancel__left2,axiom,
% 9.03/4.98      ! [C: real,A: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_eq_real @ A @ one_one_real ) )
% 9.03/4.98          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3771_mult__le__cancel__left2,axiom,
% 9.03/4.98      ! [C: rat,A: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 9.03/4.98          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3772_mult__le__cancel__left2,axiom,
% 9.03/4.98      ! [C: int,A: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 9.03/4.98        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_eq_int @ A @ one_one_int ) )
% 9.03/4.98          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3773_mult__le__cancel__left1,axiom,
% 9.03/4.98      ! [C: real,B: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_eq_real @ one_one_real @ B ) )
% 9.03/4.98          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3774_mult__le__cancel__left1,axiom,
% 9.03/4.98      ! [C: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 9.03/4.98          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3775_mult__le__cancel__left1,axiom,
% 9.03/4.98      ! [C: int,B: int] :
% 9.03/4.98        ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 9.03/4.98        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.03/4.98           => ( ord_less_eq_int @ one_one_int @ B ) )
% 9.03/4.98          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.03/4.98           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3776_divide__left__mono__neg,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.03/4.98         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.03/4.98           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3777_divide__left__mono__neg,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ B )
% 9.03/4.98       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.03/4.98         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.03/4.98           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3778_mult__imp__le__div__pos,axiom,
% 9.03/4.98      ! [Y: real,Z: real,X: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.98       => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
% 9.03/4.98         => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3779_mult__imp__le__div__pos,axiom,
% 9.03/4.98      ! [Y: rat,Z: rat,X: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.03/4.98       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
% 9.03/4.98         => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3780_mult__imp__div__pos__le,axiom,
% 9.03/4.98      ! [Y: real,X: real,Z: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.98       => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
% 9.03/4.98         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3781_mult__imp__div__pos__le,axiom,
% 9.03/4.98      ! [Y: rat,X: rat,Z: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.03/4.98       => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
% 9.03/4.98         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3782_pos__le__divide__eq,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.03/4.98          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3783_pos__le__divide__eq,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.98          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3784_pos__divide__le__eq,axiom,
% 9.03/4.98      ! [C: real,B: real,A: real] :
% 9.03/4.98        ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.03/4.98          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3785_pos__divide__le__eq,axiom,
% 9.03/4.98      ! [C: rat,B: rat,A: rat] :
% 9.03/4.98        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.03/4.98          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3786_neg__le__divide__eq,axiom,
% 9.03/4.98      ! [C: real,A: real,B: real] :
% 9.03/4.98        ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.03/4.98          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3787_neg__le__divide__eq,axiom,
% 9.03/4.98      ! [C: rat,A: rat,B: rat] :
% 9.03/4.98        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.98          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3788_neg__divide__le__eq,axiom,
% 9.03/4.98      ! [C: real,B: real,A: real] :
% 9.03/4.98        ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.03/4.98          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3789_neg__divide__le__eq,axiom,
% 9.03/4.98      ! [C: rat,B: rat,A: rat] :
% 9.03/4.98        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.03/4.98          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3790_divide__left__mono,axiom,
% 9.03/4.98      ! [B: real,A: real,C: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ B @ A )
% 9.03/4.98       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.98         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.03/4.98           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3791_divide__left__mono,axiom,
% 9.03/4.98      ! [B: rat,A: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ B @ A )
% 9.03/4.98       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.03/4.98         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.03/4.98           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3792_le__divide__eq,axiom,
% 9.03/4.98      ! [A: real,B: real,C: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.03/4.98          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98               => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.03/4.98              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3793_le__divide__eq,axiom,
% 9.03/4.98      ! [A: rat,B: rat,C: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.98        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.03/4.98          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.98           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98               => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.03/4.98              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.98               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3794_divide__le__eq,axiom,
% 9.03/4.98      ! [B: real,C: real,A: real] :
% 9.03/4.98        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.03/4.98        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.03/4.98          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.98           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.03/4.98              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.98               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.03/4.98  
% 9.03/4.98  thf(fact_3795_divide__le__eq,axiom,
% 9.03/4.98      ! [B: rat,C: rat,A: rat] :
% 9.03/4.98        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.03/4.99        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.99           => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.03/4.99          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.99           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.99               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.03/4.99              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.99               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3796_convex__bound__le,axiom,
% 9.03/4.99      ! [X: real,A: real,Y: real,U: real,V: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ X @ A )
% 9.03/4.99       => ( ( ord_less_eq_real @ Y @ A )
% 9.03/4.99         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 9.03/4.99           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 9.03/4.99             => ( ( ( plus_plus_real @ U @ V )
% 9.03/4.99                  = one_one_real )
% 9.03/4.99               => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3797_convex__bound__le,axiom,
% 9.03/4.99      ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 9.03/4.99        ( ( ord_less_eq_rat @ X @ A )
% 9.03/4.99       => ( ( ord_less_eq_rat @ Y @ A )
% 9.03/4.99         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 9.03/4.99           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 9.03/4.99             => ( ( ( plus_plus_rat @ U @ V )
% 9.03/4.99                  = one_one_rat )
% 9.03/4.99               => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3798_convex__bound__le,axiom,
% 9.03/4.99      ! [X: int,A: int,Y: int,U: int,V: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ X @ A )
% 9.03/4.99       => ( ( ord_less_eq_int @ Y @ A )
% 9.03/4.99         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.03/4.99           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 9.03/4.99             => ( ( ( plus_plus_int @ U @ V )
% 9.03/4.99                  = one_one_int )
% 9.03/4.99               => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3799_less__divide__eq__numeral_I1_J,axiom,
% 9.03/4.99      ! [W: num,B: real,C: real] :
% 9.03/4.99        ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 9.03/4.99        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.99           => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 9.03/4.99          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.99           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.99               => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 9.03/4.99              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.99               => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3800_less__divide__eq__numeral_I1_J,axiom,
% 9.03/4.99      ! [W: num,B: rat,C: rat] :
% 9.03/4.99        ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.99        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.99           => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 9.03/4.99          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.99           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.99               => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 9.03/4.99              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.99               => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3801_divide__less__eq__numeral_I1_J,axiom,
% 9.03/4.99      ! [B: real,C: real,W: num] :
% 9.03/4.99        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 9.03/4.99        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.99           => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 9.03/4.99          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.99           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.99               => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 9.03/4.99              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.99               => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3802_divide__less__eq__numeral_I1_J,axiom,
% 9.03/4.99      ! [B: rat,C: rat,W: num] :
% 9.03/4.99        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 9.03/4.99        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.99           => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 9.03/4.99          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.99           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.99               => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 9.03/4.99              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.99               => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3803_power__Suc__less,axiom,
% 9.03/4.99      ! [A: real,N: nat] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.99       => ( ( ord_less_real @ A @ one_one_real )
% 9.03/4.99         => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3804_power__Suc__less,axiom,
% 9.03/4.99      ! [A: rat,N: nat] :
% 9.03/4.99        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/4.99       => ( ( ord_less_rat @ A @ one_one_rat )
% 9.03/4.99         => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3805_power__Suc__less,axiom,
% 9.03/4.99      ! [A: nat,N: nat] :
% 9.03/4.99        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.03/4.99       => ( ( ord_less_nat @ A @ one_one_nat )
% 9.03/4.99         => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3806_power__Suc__less,axiom,
% 9.03/4.99      ! [A: int,N: nat] :
% 9.03/4.99        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/4.99       => ( ( ord_less_int @ A @ one_one_int )
% 9.03/4.99         => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3807_mult__2,axiom,
% 9.03/4.99      ! [Z: complex] :
% 9.03/4.99        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 9.03/4.99        = ( plus_plus_complex @ Z @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3808_mult__2,axiom,
% 9.03/4.99      ! [Z: real] :
% 9.03/4.99        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 9.03/4.99        = ( plus_plus_real @ Z @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3809_mult__2,axiom,
% 9.03/4.99      ! [Z: rat] :
% 9.03/4.99        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 9.03/4.99        = ( plus_plus_rat @ Z @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3810_mult__2,axiom,
% 9.03/4.99      ! [Z: nat] :
% 9.03/4.99        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 9.03/4.99        = ( plus_plus_nat @ Z @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3811_mult__2,axiom,
% 9.03/4.99      ! [Z: int] :
% 9.03/4.99        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 9.03/4.99        = ( plus_plus_int @ Z @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3812_mult__2__right,axiom,
% 9.03/4.99      ! [Z: complex] :
% 9.03/4.99        ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( plus_plus_complex @ Z @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3813_mult__2__right,axiom,
% 9.03/4.99      ! [Z: real] :
% 9.03/4.99        ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( plus_plus_real @ Z @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3814_mult__2__right,axiom,
% 9.03/4.99      ! [Z: rat] :
% 9.03/4.99        ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( plus_plus_rat @ Z @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3815_mult__2__right,axiom,
% 9.03/4.99      ! [Z: nat] :
% 9.03/4.99        ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( plus_plus_nat @ Z @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3816_mult__2__right,axiom,
% 9.03/4.99      ! [Z: int] :
% 9.03/4.99        ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( plus_plus_int @ Z @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3817_left__add__twice,axiom,
% 9.03/4.99      ! [A: complex,B: complex] :
% 9.03/4.99        ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 9.03/4.99        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3818_left__add__twice,axiom,
% 9.03/4.99      ! [A: real,B: real] :
% 9.03/4.99        ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 9.03/4.99        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3819_left__add__twice,axiom,
% 9.03/4.99      ! [A: rat,B: rat] :
% 9.03/4.99        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 9.03/4.99        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3820_left__add__twice,axiom,
% 9.03/4.99      ! [A: nat,B: nat] :
% 9.03/4.99        ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.03/4.99        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3821_left__add__twice,axiom,
% 9.03/4.99      ! [A: int,B: int] :
% 9.03/4.99        ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 9.03/4.99        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3822_power__diff,axiom,
% 9.03/4.99      ! [A: complex,N: nat,M: nat] :
% 9.03/4.99        ( ( A != zero_zero_complex )
% 9.03/4.99       => ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.99         => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.99            = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3823_power__diff,axiom,
% 9.03/4.99      ! [A: real,N: nat,M: nat] :
% 9.03/4.99        ( ( A != zero_zero_real )
% 9.03/4.99       => ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.99         => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.99            = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3824_power__diff,axiom,
% 9.03/4.99      ! [A: rat,N: nat,M: nat] :
% 9.03/4.99        ( ( A != zero_zero_rat )
% 9.03/4.99       => ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.99         => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.99            = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3825_power__diff,axiom,
% 9.03/4.99      ! [A: nat,N: nat,M: nat] :
% 9.03/4.99        ( ( A != zero_zero_nat )
% 9.03/4.99       => ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.99         => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.99            = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3826_power__diff,axiom,
% 9.03/4.99      ! [A: int,N: nat,M: nat] :
% 9.03/4.99        ( ( A != zero_zero_int )
% 9.03/4.99       => ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.99         => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 9.03/4.99            = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3827_power4__eq__xxxx,axiom,
% 9.03/4.99      ! [X: complex] :
% 9.03/4.99        ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.03/4.99        = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3828_power4__eq__xxxx,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.03/4.99        = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3829_power4__eq__xxxx,axiom,
% 9.03/4.99      ! [X: rat] :
% 9.03/4.99        ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.03/4.99        = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3830_power4__eq__xxxx,axiom,
% 9.03/4.99      ! [X: nat] :
% 9.03/4.99        ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.03/4.99        = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3831_power4__eq__xxxx,axiom,
% 9.03/4.99      ! [X: int] :
% 9.03/4.99        ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.03/4.99        = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3832_power2__eq__square,axiom,
% 9.03/4.99      ! [A: complex] :
% 9.03/4.99        ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( times_times_complex @ A @ A ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3833_power2__eq__square,axiom,
% 9.03/4.99      ! [A: real] :
% 9.03/4.99        ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( times_times_real @ A @ A ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3834_power2__eq__square,axiom,
% 9.03/4.99      ! [A: rat] :
% 9.03/4.99        ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( times_times_rat @ A @ A ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3835_power2__eq__square,axiom,
% 9.03/4.99      ! [A: nat] :
% 9.03/4.99        ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( times_times_nat @ A @ A ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3836_power2__eq__square,axiom,
% 9.03/4.99      ! [A: int] :
% 9.03/4.99        ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( times_times_int @ A @ A ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3837_of__int__ceiling__diff__one__le,axiom,
% 9.03/4.99      ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3838_of__int__ceiling__diff__one__le,axiom,
% 9.03/4.99      ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3839_power__even__eq,axiom,
% 9.03/4.99      ! [A: nat,N: nat] :
% 9.03/4.99        ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/4.99        = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3840_power__even__eq,axiom,
% 9.03/4.99      ! [A: real,N: nat] :
% 9.03/4.99        ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/4.99        = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3841_power__even__eq,axiom,
% 9.03/4.99      ! [A: int,N: nat] :
% 9.03/4.99        ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/4.99        = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3842_power__even__eq,axiom,
% 9.03/4.99      ! [A: complex,N: nat] :
% 9.03/4.99        ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/4.99        = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3843_Suc__double__not__eq__double,axiom,
% 9.03/4.99      ! [M: nat,N: nat] :
% 9.03/4.99        ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.03/4.99       != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3844_double__not__eq__Suc__double,axiom,
% 9.03/4.99      ! [M: nat,N: nat] :
% 9.03/4.99        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.03/4.99       != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3845_Suc__pred_H,axiom,
% 9.03/4.99      ! [N: nat] :
% 9.03/4.99        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.99       => ( N
% 9.03/4.99          = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3846_Suc__diff__eq__diff__pred,axiom,
% 9.03/4.99      ! [N: nat,M: nat] :
% 9.03/4.99        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.99       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 9.03/4.99          = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3847_div__if,axiom,
% 9.03/4.99      ( divide_divide_nat
% 9.03/4.99      = ( ^ [M6: nat,N3: nat] :
% 9.03/4.99            ( if_nat
% 9.03/4.99            @ ( ( ord_less_nat @ M6 @ N3 )
% 9.03/4.99              | ( N3 = zero_zero_nat ) )
% 9.03/4.99            @ zero_zero_nat
% 9.03/4.99            @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3848_div__geq,axiom,
% 9.03/4.99      ! [N: nat,M: nat] :
% 9.03/4.99        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.99       => ( ~ ( ord_less_nat @ M @ N )
% 9.03/4.99         => ( ( divide_divide_nat @ M @ N )
% 9.03/4.99            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3849_add__eq__if,axiom,
% 9.03/4.99      ( plus_plus_nat
% 9.03/4.99      = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3850_div__nat__eqI,axiom,
% 9.03/4.99      ! [N: nat,Q3: nat,M: nat] :
% 9.03/4.99        ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
% 9.03/4.99       => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
% 9.03/4.99         => ( ( divide_divide_nat @ M @ N )
% 9.03/4.99            = Q3 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3851_less__eq__div__iff__mult__less__eq,axiom,
% 9.03/4.99      ! [Q3: nat,M: nat,N: nat] :
% 9.03/4.99        ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 9.03/4.99       => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
% 9.03/4.99          = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3852_split__div,axiom,
% 9.03/4.99      ! [P: nat > $o,M: nat,N: nat] :
% 9.03/4.99        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 9.03/4.99        = ( ( ( N = zero_zero_nat )
% 9.03/4.99           => ( P @ zero_zero_nat ) )
% 9.03/4.99          & ( ( N != zero_zero_nat )
% 9.03/4.99           => ! [I3: nat,J2: nat] :
% 9.03/4.99                ( ( ord_less_nat @ J2 @ N )
% 9.03/4.99               => ( ( M
% 9.03/4.99                    = ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J2 ) )
% 9.03/4.99                 => ( P @ I3 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3853_dividend__less__div__times,axiom,
% 9.03/4.99      ! [N: nat,M: nat] :
% 9.03/4.99        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.99       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3854_dividend__less__times__div,axiom,
% 9.03/4.99      ! [N: nat,M: nat] :
% 9.03/4.99        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.99       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3855_floor__split,axiom,
% 9.03/4.99      ! [P: int > $o,T: real] :
% 9.03/4.99        ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 9.03/4.99        = ( ! [I3: int] :
% 9.03/4.99              ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I3 ) @ T )
% 9.03/4.99                & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) ) )
% 9.03/4.99             => ( P @ I3 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3856_floor__split,axiom,
% 9.03/4.99      ! [P: int > $o,T: rat] :
% 9.03/4.99        ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 9.03/4.99        = ( ! [I3: int] :
% 9.03/4.99              ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I3 ) @ T )
% 9.03/4.99                & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) ) )
% 9.03/4.99             => ( P @ I3 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3857_floor__eq__iff,axiom,
% 9.03/4.99      ! [X: real,A: int] :
% 9.03/4.99        ( ( ( archim6058952711729229775r_real @ X )
% 9.03/4.99          = A )
% 9.03/4.99        = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
% 9.03/4.99          & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3858_floor__eq__iff,axiom,
% 9.03/4.99      ! [X: rat,A: int] :
% 9.03/4.99        ( ( ( archim3151403230148437115or_rat @ X )
% 9.03/4.99          = A )
% 9.03/4.99        = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
% 9.03/4.99          & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3859_floor__unique,axiom,
% 9.03/4.99      ! [Z: int,X: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
% 9.03/4.99       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
% 9.03/4.99         => ( ( archim6058952711729229775r_real @ X )
% 9.03/4.99            = Z ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3860_floor__unique,axiom,
% 9.03/4.99      ! [Z: int,X: rat] :
% 9.03/4.99        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
% 9.03/4.99       => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
% 9.03/4.99         => ( ( archim3151403230148437115or_rat @ X )
% 9.03/4.99            = Z ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3861_less__floor__iff,axiom,
% 9.03/4.99      ! [Z: int,X: real] :
% 9.03/4.99        ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
% 9.03/4.99        = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3862_less__floor__iff,axiom,
% 9.03/4.99      ! [Z: int,X: rat] :
% 9.03/4.99        ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/4.99        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3863_floor__le__iff,axiom,
% 9.03/4.99      ! [X: real,Z: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
% 9.03/4.99        = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3864_floor__le__iff,axiom,
% 9.03/4.99      ! [X: rat,Z: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
% 9.03/4.99        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3865_floor__correct,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
% 9.03/4.99        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3866_floor__correct,axiom,
% 9.03/4.99      ! [X: rat] :
% 9.03/4.99        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
% 9.03/4.99        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3867_le__log__iff,axiom,
% 9.03/4.99      ! [B: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99         => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
% 9.03/4.99            = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3868_log__le__iff,axiom,
% 9.03/4.99      ! [B: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99         => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
% 9.03/4.99            = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3869_le__powr__iff,axiom,
% 9.03/4.99      ! [B: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99         => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
% 9.03/4.99            = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3870_powr__le__iff,axiom,
% 9.03/4.99      ! [B: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ one_one_real @ B )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99         => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
% 9.03/4.99            = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3871_floor__eq2,axiom,
% 9.03/4.99      ! [N: int,X: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
% 9.03/4.99       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.03/4.99         => ( ( archim6058952711729229775r_real @ X )
% 9.03/4.99            = N ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3872_floor__divide__real__eq__div,axiom,
% 9.03/4.99      ! [B: int,A: real] :
% 9.03/4.99        ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/4.99       => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 9.03/4.99          = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3873_convex__bound__lt,axiom,
% 9.03/4.99      ! [X: real,A: real,Y: real,U: real,V: real] :
% 9.03/4.99        ( ( ord_less_real @ X @ A )
% 9.03/4.99       => ( ( ord_less_real @ Y @ A )
% 9.03/4.99         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 9.03/4.99           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 9.03/4.99             => ( ( ( plus_plus_real @ U @ V )
% 9.03/4.99                  = one_one_real )
% 9.03/4.99               => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3874_convex__bound__lt,axiom,
% 9.03/4.99      ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 9.03/4.99        ( ( ord_less_rat @ X @ A )
% 9.03/4.99       => ( ( ord_less_rat @ Y @ A )
% 9.03/4.99         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 9.03/4.99           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 9.03/4.99             => ( ( ( plus_plus_rat @ U @ V )
% 9.03/4.99                  = one_one_rat )
% 9.03/4.99               => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3875_convex__bound__lt,axiom,
% 9.03/4.99      ! [X: int,A: int,Y: int,U: int,V: int] :
% 9.03/4.99        ( ( ord_less_int @ X @ A )
% 9.03/4.99       => ( ( ord_less_int @ Y @ A )
% 9.03/4.99         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.03/4.99           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 9.03/4.99             => ( ( ( plus_plus_int @ U @ V )
% 9.03/4.99                  = one_one_int )
% 9.03/4.99               => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3876_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.03/4.99      ! [N: nat,M: nat] :
% 9.03/4.99        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.03/4.99         != zero_zero_nat )
% 9.03/4.99       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 9.03/4.99         != zero_zero_nat ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3877_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.03/4.99      ! [N: nat,M: nat] :
% 9.03/4.99        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.03/4.99         != zero_zero_int )
% 9.03/4.99       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 9.03/4.99         != zero_zero_int ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3878_le__divide__eq__numeral_I1_J,axiom,
% 9.03/4.99      ! [W: num,B: real,C: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 9.03/4.99        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.99           => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 9.03/4.99          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.99           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.99               => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 9.03/4.99              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.99               => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3879_le__divide__eq__numeral_I1_J,axiom,
% 9.03/4.99      ! [W: num,B: rat,C: rat] :
% 9.03/4.99        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 9.03/4.99        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.99           => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 9.03/4.99          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.99           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.99               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 9.03/4.99              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.99               => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3880_divide__le__eq__numeral_I1_J,axiom,
% 9.03/4.99      ! [B: real,C: real,W: num] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 9.03/4.99        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.99           => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 9.03/4.99          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/4.99           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.99               => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 9.03/4.99              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/4.99               => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3881_divide__le__eq__numeral_I1_J,axiom,
% 9.03/4.99      ! [B: rat,C: rat,W: num] :
% 9.03/4.99        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 9.03/4.99        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.99           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 9.03/4.99          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/4.99           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.99               => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 9.03/4.99              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/4.99               => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3882_round__unique,axiom,
% 9.03/4.99      ! [X: real,Y: int] :
% 9.03/4.99        ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.03/4.99         => ( ( archim8280529875227126926d_real @ X )
% 9.03/4.99            = Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3883_round__unique,axiom,
% 9.03/4.99      ! [X: rat,Y: int] :
% 9.03/4.99        ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
% 9.03/4.99       => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 9.03/4.99         => ( ( archim7778729529865785530nd_rat @ X )
% 9.03/4.99            = Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3884_power__diff__power__eq,axiom,
% 9.03/4.99      ! [A: nat,N: nat,M: nat] :
% 9.03/4.99        ( ( A != zero_zero_nat )
% 9.03/4.99       => ( ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.99           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.03/4.99              = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 9.03/4.99          & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.03/4.99           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.03/4.99              = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3885_power__diff__power__eq,axiom,
% 9.03/4.99      ! [A: int,N: nat,M: nat] :
% 9.03/4.99        ( ( A != zero_zero_int )
% 9.03/4.99       => ( ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.99           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.03/4.99              = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 9.03/4.99          & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.03/4.99           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.03/4.99              = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3886_ceiling__correct,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
% 9.03/4.99        & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3887_ceiling__correct,axiom,
% 9.03/4.99      ! [X: rat] :
% 9.03/4.99        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
% 9.03/4.99        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3888_ceiling__unique,axiom,
% 9.03/4.99      ! [Z: int,X: real] :
% 9.03/4.99        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
% 9.03/4.99       => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
% 9.03/4.99         => ( ( archim7802044766580827645g_real @ X )
% 9.03/4.99            = Z ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3889_ceiling__unique,axiom,
% 9.03/4.99      ! [Z: int,X: rat] :
% 9.03/4.99        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
% 9.03/4.99       => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
% 9.03/4.99         => ( ( archim2889992004027027881ng_rat @ X )
% 9.03/4.99            = Z ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3890_ceiling__eq__iff,axiom,
% 9.03/4.99      ! [X: real,A: int] :
% 9.03/4.99        ( ( ( archim7802044766580827645g_real @ X )
% 9.03/4.99          = A )
% 9.03/4.99        = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
% 9.03/4.99          & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3891_ceiling__eq__iff,axiom,
% 9.03/4.99      ! [X: rat,A: int] :
% 9.03/4.99        ( ( ( archim2889992004027027881ng_rat @ X )
% 9.03/4.99          = A )
% 9.03/4.99        = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
% 9.03/4.99          & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3892_ceiling__split,axiom,
% 9.03/4.99      ! [P: int > $o,T: real] :
% 9.03/4.99        ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 9.03/4.99        = ( ! [I3: int] :
% 9.03/4.99              ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) @ T )
% 9.03/4.99                & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I3 ) ) )
% 9.03/4.99             => ( P @ I3 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3893_ceiling__split,axiom,
% 9.03/4.99      ! [P: int > $o,T: rat] :
% 9.03/4.99        ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 9.03/4.99        = ( ! [I3: int] :
% 9.03/4.99              ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T )
% 9.03/4.99                & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I3 ) ) )
% 9.03/4.99             => ( P @ I3 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3894_ceiling__less__iff,axiom,
% 9.03/4.99      ! [X: real,Z: int] :
% 9.03/4.99        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
% 9.03/4.99        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3895_ceiling__less__iff,axiom,
% 9.03/4.99      ! [X: rat,Z: int] :
% 9.03/4.99        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
% 9.03/4.99        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3896_diff__le__diff__pow,axiom,
% 9.03/4.99      ! [K: nat,M: nat,N: nat] :
% 9.03/4.99        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.03/4.99       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3897_le__ceiling__iff,axiom,
% 9.03/4.99      ! [Z: int,X: real] :
% 9.03/4.99        ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
% 9.03/4.99        = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3898_le__ceiling__iff,axiom,
% 9.03/4.99      ! [Z: int,X: rat] :
% 9.03/4.99        ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/4.99        = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3899_le__div__geq,axiom,
% 9.03/4.99      ! [N: nat,M: nat] :
% 9.03/4.99        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.99       => ( ( ord_less_eq_nat @ N @ M )
% 9.03/4.99         => ( ( divide_divide_nat @ M @ N )
% 9.03/4.99            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3900_split__div_H,axiom,
% 9.03/4.99      ! [P: nat > $o,M: nat,N: nat] :
% 9.03/4.99        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 9.03/4.99        = ( ( ( N = zero_zero_nat )
% 9.03/4.99            & ( P @ zero_zero_nat ) )
% 9.03/4.99          | ? [Q4: nat] :
% 9.03/4.99              ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
% 9.03/4.99              & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
% 9.03/4.99              & ( P @ Q4 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3901_power2__sum,axiom,
% 9.03/4.99      ! [X: complex,Y: complex] :
% 9.03/4.99        ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3902_power2__sum,axiom,
% 9.03/4.99      ! [X: real,Y: real] :
% 9.03/4.99        ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3903_power2__sum,axiom,
% 9.03/4.99      ! [X: rat,Y: rat] :
% 9.03/4.99        ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3904_power2__sum,axiom,
% 9.03/4.99      ! [X: nat,Y: nat] :
% 9.03/4.99        ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3905_power2__sum,axiom,
% 9.03/4.99      ! [X: int,Y: int] :
% 9.03/4.99        ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99        = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3906_zero__le__even__power_H,axiom,
% 9.03/4.99      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3907_zero__le__even__power_H,axiom,
% 9.03/4.99      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3908_zero__le__even__power_H,axiom,
% 9.03/4.99      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3909_nat__bit__induct,axiom,
% 9.03/4.99      ! [P: nat > $o,N: nat] :
% 9.03/4.99        ( ( P @ zero_zero_nat )
% 9.03/4.99       => ( ! [N2: nat] :
% 9.03/4.99              ( ( P @ N2 )
% 9.03/4.99             => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.03/4.99               => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 9.03/4.99         => ( ! [N2: nat] :
% 9.03/4.99                ( ( P @ N2 )
% 9.03/4.99               => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 9.03/4.99           => ( P @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3910_ceiling__divide__upper,axiom,
% 9.03/4.99      ! [Q3: real,P5: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ Q3 )
% 9.03/4.99       => ( ord_less_eq_real @ P5 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3911_ceiling__divide__upper,axiom,
% 9.03/4.99      ! [Q3: rat,P5: rat] :
% 9.03/4.99        ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 9.03/4.99       => ( ord_less_eq_rat @ P5 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3912_sum__squares__bound,axiom,
% 9.03/4.99      ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3913_sum__squares__bound,axiom,
% 9.03/4.99      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3914_odd__0__le__power__imp__0__le,axiom,
% 9.03/4.99      ! [A: real,N: nat] :
% 9.03/4.99        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.03/4.99       => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3915_odd__0__le__power__imp__0__le,axiom,
% 9.03/4.99      ! [A: rat,N: nat] :
% 9.03/4.99        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.03/4.99       => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3916_odd__0__le__power__imp__0__le,axiom,
% 9.03/4.99      ! [A: int,N: nat] :
% 9.03/4.99        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.03/4.99       => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3917_odd__power__less__zero,axiom,
% 9.03/4.99      ! [A: real,N: nat] :
% 9.03/4.99        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/4.99       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3918_odd__power__less__zero,axiom,
% 9.03/4.99      ! [A: rat,N: nat] :
% 9.03/4.99        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/4.99       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3919_odd__power__less__zero,axiom,
% 9.03/4.99      ! [A: int,N: nat] :
% 9.03/4.99        ( ( ord_less_int @ A @ zero_zero_int )
% 9.03/4.99       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3920_int__power__div__base,axiom,
% 9.03/4.99      ! [M: nat,K: int] :
% 9.03/4.99        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.99       => ( ( ord_less_int @ zero_zero_int @ K )
% 9.03/4.99         => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 9.03/4.99            = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3921_arith__geo__mean,axiom,
% 9.03/4.99      ! [U: real,X: real,Y: real] :
% 9.03/4.99        ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99          = ( times_times_real @ X @ Y ) )
% 9.03/4.99       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.99         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.99           => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3922_arith__geo__mean,axiom,
% 9.03/4.99      ! [U: rat,X: rat,Y: rat] :
% 9.03/4.99        ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99          = ( times_times_rat @ X @ Y ) )
% 9.03/4.99       => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/4.99         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.03/4.99           => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3923_floor__log2__div2,axiom,
% 9.03/4.99      ! [N: nat] :
% 9.03/4.99        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.03/4.99       => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.03/4.99          = ( 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 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3924_floor__log__nat__eq__if,axiom,
% 9.03/4.99      ! [B: nat,N: nat,K: nat] :
% 9.03/4.99        ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 9.03/4.99       => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 9.03/4.99         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.03/4.99           => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.03/4.99              = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3925_low__inv,axiom,
% 9.03/4.99      ! [X: nat,N: nat,Y: nat] :
% 9.03/4.99        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/4.99       => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 9.03/4.99          = X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3926_set__bit__0,axiom,
% 9.03/4.99      ! [A: int] :
% 9.03/4.99        ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 9.03/4.99        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3927_set__bit__0,axiom,
% 9.03/4.99      ! [A: nat] :
% 9.03/4.99        ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 9.03/4.99        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3928_artanh__def,axiom,
% 9.03/4.99      ( artanh_real
% 9.03/4.99      = ( ^ [X4: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3929_unset__bit__0,axiom,
% 9.03/4.99      ! [A: int] :
% 9.03/4.99        ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 9.03/4.99        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3930_unset__bit__0,axiom,
% 9.03/4.99      ! [A: nat] :
% 9.03/4.99        ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 9.03/4.99        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3931_lemma__termdiff3,axiom,
% 9.03/4.99      ! [H: real,Z: real,K4: real,N: nat] :
% 9.03/4.99        ( ( H != zero_zero_real )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K4 )
% 9.03/4.99         => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H ) ) @ K4 )
% 9.03/4.99           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H ) @ ( 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 @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3932_lemma__termdiff3,axiom,
% 9.03/4.99      ! [H: complex,Z: complex,K4: real,N: nat] :
% 9.03/4.99        ( ( H != zero_zero_complex )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K4 )
% 9.03/4.99         => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H ) ) @ K4 )
% 9.03/4.99           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H ) @ ( 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 @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3933_both__member__options__def,axiom,
% 9.03/4.99      ( vEBT_V8194947554948674370ptions
% 9.03/4.99      = ( ^ [T2: vEBT_VEBT,X4: nat] :
% 9.03/4.99            ( ( vEBT_V5719532721284313246member @ T2 @ X4 )
% 9.03/4.99            | ( vEBT_VEBT_membermima @ T2 @ X4 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3934_mult__le__cancel__iff2,axiom,
% 9.03/4.99      ! [Z: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ Z )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
% 9.03/4.99          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3935_mult__le__cancel__iff2,axiom,
% 9.03/4.99      ! [Z: rat,X: rat,Y: rat] :
% 9.03/4.99        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 9.03/4.99       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
% 9.03/4.99          = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3936_mult__le__cancel__iff2,axiom,
% 9.03/4.99      ! [Z: int,X: int,Y: int] :
% 9.03/4.99        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.03/4.99       => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
% 9.03/4.99          = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3937_mult__le__cancel__iff1,axiom,
% 9.03/4.99      ! [Z: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ Z )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
% 9.03/4.99          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3938_mult__le__cancel__iff1,axiom,
% 9.03/4.99      ! [Z: rat,X: rat,Y: rat] :
% 9.03/4.99        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 9.03/4.99       => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 9.03/4.99          = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3939_mult__le__cancel__iff1,axiom,
% 9.03/4.99      ! [Z: int,X: int,Y: int] :
% 9.03/4.99        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.03/4.99       => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 9.03/4.99          = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3940_buildup__nothing__in__min__max,axiom,
% 9.03/4.99      ! [N: nat,X: nat] :
% 9.03/4.99        ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3941_Diff__insert0,axiom,
% 9.03/4.99      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.03/4.99        ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 9.03/4.99       => ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B2 ) )
% 9.03/4.99          = ( minus_5127226145743854075T_VEBT @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3942_Diff__insert0,axiom,
% 9.03/4.99      ! [X: $o,A2: set_o,B2: set_o] :
% 9.03/4.99        ( ~ ( member_o @ X @ A2 )
% 9.03/4.99       => ( ( minus_minus_set_o @ A2 @ ( insert_o @ X @ B2 ) )
% 9.03/4.99          = ( minus_minus_set_o @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3943_Diff__insert0,axiom,
% 9.03/4.99      ! [X: complex,A2: set_complex,B2: set_complex] :
% 9.03/4.99        ( ~ ( member_complex @ X @ A2 )
% 9.03/4.99       => ( ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ B2 ) )
% 9.03/4.99          = ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3944_Diff__insert0,axiom,
% 9.03/4.99      ! [X: real,A2: set_real,B2: set_real] :
% 9.03/4.99        ( ~ ( member_real @ X @ A2 )
% 9.03/4.99       => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 9.03/4.99          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3945_Diff__insert0,axiom,
% 9.03/4.99      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.99        ( ~ ( member_set_nat @ X @ A2 )
% 9.03/4.99       => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 9.03/4.99          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3946_Diff__insert0,axiom,
% 9.03/4.99      ! [X: int,A2: set_int,B2: set_int] :
% 9.03/4.99        ( ~ ( member_int @ X @ A2 )
% 9.03/4.99       => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 9.03/4.99          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3947_Diff__insert0,axiom,
% 9.03/4.99      ! [X: nat,A2: set_nat,B2: set_nat] :
% 9.03/4.99        ( ~ ( member_nat @ X @ A2 )
% 9.03/4.99       => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 9.03/4.99          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3948_insert__Diff1,axiom,
% 9.03/4.99      ! [X: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.03/4.99        ( ( member_VEBT_VEBT @ X @ B2 )
% 9.03/4.99       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B2 )
% 9.03/4.99          = ( minus_5127226145743854075T_VEBT @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3949_insert__Diff1,axiom,
% 9.03/4.99      ! [X: $o,B2: set_o,A2: set_o] :
% 9.03/4.99        ( ( member_o @ X @ B2 )
% 9.03/4.99       => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
% 9.03/4.99          = ( minus_minus_set_o @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3950_insert__Diff1,axiom,
% 9.03/4.99      ! [X: complex,B2: set_complex,A2: set_complex] :
% 9.03/4.99        ( ( member_complex @ X @ B2 )
% 9.03/4.99       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
% 9.03/4.99          = ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3951_insert__Diff1,axiom,
% 9.03/4.99      ! [X: real,B2: set_real,A2: set_real] :
% 9.03/4.99        ( ( member_real @ X @ B2 )
% 9.03/4.99       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 9.03/4.99          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3952_insert__Diff1,axiom,
% 9.03/4.99      ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 9.03/4.99        ( ( member_set_nat @ X @ B2 )
% 9.03/4.99       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 9.03/4.99          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3953_insert__Diff1,axiom,
% 9.03/4.99      ! [X: int,B2: set_int,A2: set_int] :
% 9.03/4.99        ( ( member_int @ X @ B2 )
% 9.03/4.99       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 9.03/4.99          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3954_insert__Diff1,axiom,
% 9.03/4.99      ! [X: nat,B2: set_nat,A2: set_nat] :
% 9.03/4.99        ( ( member_nat @ X @ B2 )
% 9.03/4.99       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 9.03/4.99          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3955_idiff__0,axiom,
% 9.03/4.99      ! [N: extended_enat] :
% 9.03/4.99        ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 9.03/4.99        = zero_z5237406670263579293d_enat ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3956_idiff__0__right,axiom,
% 9.03/4.99      ! [N: extended_enat] :
% 9.03/4.99        ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 9.03/4.99        = N ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3957_ln__less__cancel__iff,axiom,
% 9.03/4.99      ! [X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.99         => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 9.03/4.99            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3958_ln__inj__iff,axiom,
% 9.03/4.99      ! [X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.99         => ( ( ( ln_ln_real @ X )
% 9.03/4.99              = ( ln_ln_real @ Y ) )
% 9.03/4.99            = ( X = Y ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3959_unset__bit__nonnegative__int__iff,axiom,
% 9.03/4.99      ! [N: nat,K: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 9.03/4.99        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3960_set__bit__nonnegative__int__iff,axiom,
% 9.03/4.99      ! [N: nat,K: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 9.03/4.99        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3961_unset__bit__negative__int__iff,axiom,
% 9.03/4.99      ! [N: nat,K: int] :
% 9.03/4.99        ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 9.03/4.99        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3962_set__bit__negative__int__iff,axiom,
% 9.03/4.99      ! [N: nat,K: int] :
% 9.03/4.99        ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 9.03/4.99        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3963_real__divide__square__eq,axiom,
% 9.03/4.99      ! [R2: real,A: real] :
% 9.03/4.99        ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
% 9.03/4.99        = ( divide_divide_real @ A @ R2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3964_semiring__norm_I13_J,axiom,
% 9.03/4.99      ! [M: num,N: num] :
% 9.03/4.99        ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.03/4.99        = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3965_semiring__norm_I11_J,axiom,
% 9.03/4.99      ! [M: num] :
% 9.03/4.99        ( ( times_times_num @ M @ one )
% 9.03/4.99        = M ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3966_semiring__norm_I12_J,axiom,
% 9.03/4.99      ! [N: num] :
% 9.03/4.99        ( ( times_times_num @ one @ N )
% 9.03/4.99        = N ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3967_ln__le__cancel__iff,axiom,
% 9.03/4.99      ! [X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.99         => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 9.03/4.99            = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3968_ln__less__zero__iff,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 9.03/4.99          = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3969_ln__gt__zero__iff,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 9.03/4.99          = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3970_ln__eq__zero__iff,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ( ln_ln_real @ X )
% 9.03/4.99            = zero_zero_real )
% 9.03/4.99          = ( X = one_one_real ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3971_not__real__square__gt__zero,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 9.03/4.99        = ( X = zero_zero_real ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3972_ln__one,axiom,
% 9.03/4.99      ( ( ln_ln_real @ one_one_real )
% 9.03/4.99      = zero_zero_real ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3973_num__double,axiom,
% 9.03/4.99      ! [N: num] :
% 9.03/4.99        ( ( times_times_num @ ( bit0 @ one ) @ N )
% 9.03/4.99        = ( bit0 @ N ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3974_power__mult__numeral,axiom,
% 9.03/4.99      ! [A: nat,M: num,N: num] :
% 9.03/4.99        ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 9.03/4.99        = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3975_power__mult__numeral,axiom,
% 9.03/4.99      ! [A: real,M: num,N: num] :
% 9.03/4.99        ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 9.03/4.99        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3976_power__mult__numeral,axiom,
% 9.03/4.99      ! [A: int,M: num,N: num] :
% 9.03/4.99        ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 9.03/4.99        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3977_power__mult__numeral,axiom,
% 9.03/4.99      ! [A: complex,M: num,N: num] :
% 9.03/4.99        ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 9.03/4.99        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3978_ln__le__zero__iff,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 9.03/4.99          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3979_ln__ge__zero__iff,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 9.03/4.99          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3980_zle__diff1__eq,axiom,
% 9.03/4.99      ! [W: int,Z: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 9.03/4.99        = ( ord_less_int @ W @ Z ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3981_int__distrib_I4_J,axiom,
% 9.03/4.99      ! [W: int,Z1: int,Z22: int] :
% 9.03/4.99        ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 9.03/4.99        = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3982_int__distrib_I3_J,axiom,
% 9.03/4.99      ! [Z1: int,Z22: int,W: int] :
% 9.03/4.99        ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 9.03/4.99        = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3983_ln__powr,axiom,
% 9.03/4.99      ! [X: real,Y: real] :
% 9.03/4.99        ( ( X != zero_zero_real )
% 9.03/4.99       => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
% 9.03/4.99          = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3984_double__diff,axiom,
% 9.03/4.99      ! [A2: set_nat,B2: set_nat,C6: set_nat] :
% 9.03/4.99        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.03/4.99       => ( ( ord_less_eq_set_nat @ B2 @ C6 )
% 9.03/4.99         => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C6 @ A2 ) )
% 9.03/4.99            = A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3985_double__diff,axiom,
% 9.03/4.99      ! [A2: set_int,B2: set_int,C6: set_int] :
% 9.03/4.99        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.03/4.99       => ( ( ord_less_eq_set_int @ B2 @ C6 )
% 9.03/4.99         => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C6 @ A2 ) )
% 9.03/4.99            = A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3986_Diff__subset,axiom,
% 9.03/4.99      ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3987_Diff__subset,axiom,
% 9.03/4.99      ! [A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ A2 ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3988_Diff__mono,axiom,
% 9.03/4.99      ! [A2: set_nat,C6: set_nat,D2: set_nat,B2: set_nat] :
% 9.03/4.99        ( ( ord_less_eq_set_nat @ A2 @ C6 )
% 9.03/4.99       => ( ( ord_less_eq_set_nat @ D2 @ B2 )
% 9.03/4.99         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C6 @ D2 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3989_Diff__mono,axiom,
% 9.03/4.99      ! [A2: set_int,C6: set_int,D2: set_int,B2: set_int] :
% 9.03/4.99        ( ( ord_less_eq_set_int @ A2 @ C6 )
% 9.03/4.99       => ( ( ord_less_eq_set_int @ D2 @ B2 )
% 9.03/4.99         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( minus_minus_set_int @ C6 @ D2 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3990_insert__Diff__if,axiom,
% 9.03/4.99      ! [X: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.03/4.99        ( ( ( member_VEBT_VEBT @ X @ B2 )
% 9.03/4.99         => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( minus_5127226145743854075T_VEBT @ A2 @ B2 ) ) )
% 9.03/4.99        & ( ~ ( member_VEBT_VEBT @ X @ B2 )
% 9.03/4.99         => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( insert_VEBT_VEBT @ X @ ( minus_5127226145743854075T_VEBT @ A2 @ B2 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3991_insert__Diff__if,axiom,
% 9.03/4.99      ! [X: $o,B2: set_o,A2: set_o] :
% 9.03/4.99        ( ( ( member_o @ X @ B2 )
% 9.03/4.99         => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( minus_minus_set_o @ A2 @ B2 ) ) )
% 9.03/4.99        & ( ~ ( member_o @ X @ B2 )
% 9.03/4.99         => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( insert_o @ X @ ( minus_minus_set_o @ A2 @ B2 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3992_insert__Diff__if,axiom,
% 9.03/4.99      ! [X: complex,B2: set_complex,A2: set_complex] :
% 9.03/4.99        ( ( ( member_complex @ X @ B2 )
% 9.03/4.99         => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( minus_811609699411566653omplex @ A2 @ B2 ) ) )
% 9.03/4.99        & ( ~ ( member_complex @ X @ B2 )
% 9.03/4.99         => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( insert_complex @ X @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3993_insert__Diff__if,axiom,
% 9.03/4.99      ! [X: real,B2: set_real,A2: set_real] :
% 9.03/4.99        ( ( ( member_real @ X @ B2 )
% 9.03/4.99         => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( minus_minus_set_real @ A2 @ B2 ) ) )
% 9.03/4.99        & ( ~ ( member_real @ X @ B2 )
% 9.03/4.99         => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3994_insert__Diff__if,axiom,
% 9.03/4.99      ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 9.03/4.99        ( ( ( member_set_nat @ X @ B2 )
% 9.03/4.99         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
% 9.03/4.99        & ( ~ ( member_set_nat @ X @ B2 )
% 9.03/4.99         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3995_insert__Diff__if,axiom,
% 9.03/4.99      ! [X: int,B2: set_int,A2: set_int] :
% 9.03/4.99        ( ( ( member_int @ X @ B2 )
% 9.03/4.99         => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( minus_minus_set_int @ A2 @ B2 ) ) )
% 9.03/4.99        & ( ~ ( member_int @ X @ B2 )
% 9.03/4.99         => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3996_insert__Diff__if,axiom,
% 9.03/4.99      ! [X: nat,B2: set_nat,A2: set_nat] :
% 9.03/4.99        ( ( ( member_nat @ X @ B2 )
% 9.03/4.99         => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( minus_minus_set_nat @ A2 @ B2 ) ) )
% 9.03/4.99        & ( ~ ( member_nat @ X @ B2 )
% 9.03/4.99         => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 9.03/4.99            = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3997_minus__int__code_I1_J,axiom,
% 9.03/4.99      ! [K: int] :
% 9.03/4.99        ( ( minus_minus_int @ K @ zero_zero_int )
% 9.03/4.99        = K ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3998_times__int__code_I2_J,axiom,
% 9.03/4.99      ! [L: int] :
% 9.03/4.99        ( ( times_times_int @ zero_zero_int @ L )
% 9.03/4.99        = zero_zero_int ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_3999_times__int__code_I1_J,axiom,
% 9.03/4.99      ! [K: int] :
% 9.03/4.99        ( ( times_times_int @ K @ zero_zero_int )
% 9.03/4.99        = zero_zero_int ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4000_int__diff__cases,axiom,
% 9.03/4.99      ! [Z: int] :
% 9.03/4.99        ~ ! [M5: nat,N2: nat] :
% 9.03/4.99            ( Z
% 9.03/4.99           != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4001_psubset__imp__ex__mem,axiom,
% 9.03/4.99      ! [A2: set_complex,B2: set_complex] :
% 9.03/4.99        ( ( ord_less_set_complex @ A2 @ B2 )
% 9.03/4.99       => ? [B6: complex] : ( member_complex @ B6 @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4002_psubset__imp__ex__mem,axiom,
% 9.03/4.99      ! [A2: set_real,B2: set_real] :
% 9.03/4.99        ( ( ord_less_set_real @ A2 @ B2 )
% 9.03/4.99       => ? [B6: real] : ( member_real @ B6 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4003_psubset__imp__ex__mem,axiom,
% 9.03/4.99      ! [A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.99        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 9.03/4.99       => ? [B6: set_nat] : ( member_set_nat @ B6 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4004_psubset__imp__ex__mem,axiom,
% 9.03/4.99      ! [A2: set_int,B2: set_int] :
% 9.03/4.99        ( ( ord_less_set_int @ A2 @ B2 )
% 9.03/4.99       => ? [B6: int] : ( member_int @ B6 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4005_psubset__imp__ex__mem,axiom,
% 9.03/4.99      ! [A2: set_nat,B2: set_nat] :
% 9.03/4.99        ( ( ord_less_set_nat @ A2 @ B2 )
% 9.03/4.99       => ? [B6: nat] : ( member_nat @ B6 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4006_int__distrib_I1_J,axiom,
% 9.03/4.99      ! [Z1: int,Z22: int,W: int] :
% 9.03/4.99        ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 9.03/4.99        = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4007_int__distrib_I2_J,axiom,
% 9.03/4.99      ! [W: int,Z1: int,Z22: int] :
% 9.03/4.99        ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 9.03/4.99        = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4008_powr__powr,axiom,
% 9.03/4.99      ! [X: real,A: real,B: real] :
% 9.03/4.99        ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 9.03/4.99        = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4009_imult__is__0,axiom,
% 9.03/4.99      ! [M: extended_enat,N: extended_enat] :
% 9.03/4.99        ( ( ( times_7803423173614009249d_enat @ M @ N )
% 9.03/4.99          = zero_z5237406670263579293d_enat )
% 9.03/4.99        = ( ( M = zero_z5237406670263579293d_enat )
% 9.03/4.99          | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4010_ln__eq__minus__one,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ( ln_ln_real @ X )
% 9.03/4.99            = ( minus_minus_real @ X @ one_one_real ) )
% 9.03/4.99         => ( X = one_one_real ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4011_ln__div,axiom,
% 9.03/4.99      ! [X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.99         => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
% 9.03/4.99            = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4012_ln__mult,axiom,
% 9.03/4.99      ! [X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.99         => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
% 9.03/4.99            = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4013_unset__bit__less__eq,axiom,
% 9.03/4.99      ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4014_set__bit__greater__eq,axiom,
% 9.03/4.99      ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4015_ln__le__minus__one,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4016_ln__diff__le,axiom,
% 9.03/4.99      ! [X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.99         => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4017_ln__less__self,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4018_ln__realpow,axiom,
% 9.03/4.99      ! [X: real,N: nat] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
% 9.03/4.99          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4019_log__def,axiom,
% 9.03/4.99      ( log
% 9.03/4.99      = ( ^ [A3: real,X4: real] : ( divide_divide_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ A3 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4020_image__diff__subset,axiom,
% 9.03/4.99      ! [F: nat > set_nat,A2: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4021_image__diff__subset,axiom,
% 9.03/4.99      ! [F: int > nat,A2: set_int,B2: set_int] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_int_nat @ F @ A2 ) @ ( image_int_nat @ F @ B2 ) ) @ ( image_int_nat @ F @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4022_image__diff__subset,axiom,
% 9.03/4.99      ! [F: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4023_image__diff__subset,axiom,
% 9.03/4.99      ! [F: int > int,A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_int_int @ F @ A2 ) @ ( image_int_int @ F @ B2 ) ) @ ( image_int_int @ F @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4024_image__diff__subset,axiom,
% 9.03/4.99      ! [F: nat > int,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B2 ) ) @ ( image_nat_int @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4025_div__mult2__numeral__eq,axiom,
% 9.03/4.99      ! [A: nat,K: num,L: num] :
% 9.03/4.99        ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
% 9.03/4.99        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4026_div__mult2__numeral__eq,axiom,
% 9.03/4.99      ! [A: int,K: num,L: num] :
% 9.03/4.99        ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
% 9.03/4.99        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4027_subset__Diff__insert,axiom,
% 9.03/4.99      ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,X: vEBT_VEBT,C6: set_VEBT_VEBT] :
% 9.03/4.99        ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( minus_5127226145743854075T_VEBT @ B2 @ ( insert_VEBT_VEBT @ X @ C6 ) ) )
% 9.03/4.99        = ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( minus_5127226145743854075T_VEBT @ B2 @ C6 ) )
% 9.03/4.99          & ~ ( member_VEBT_VEBT @ X @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4028_subset__Diff__insert,axiom,
% 9.03/4.99      ! [A2: set_o,B2: set_o,X: $o,C6: set_o] :
% 9.03/4.99        ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ ( insert_o @ X @ C6 ) ) )
% 9.03/4.99        = ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ C6 ) )
% 9.03/4.99          & ~ ( member_o @ X @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4029_subset__Diff__insert,axiom,
% 9.03/4.99      ! [A2: set_complex,B2: set_complex,X: complex,C6: set_complex] :
% 9.03/4.99        ( ( ord_le211207098394363844omplex @ A2 @ ( minus_811609699411566653omplex @ B2 @ ( insert_complex @ X @ C6 ) ) )
% 9.03/4.99        = ( ( ord_le211207098394363844omplex @ A2 @ ( minus_811609699411566653omplex @ B2 @ C6 ) )
% 9.03/4.99          & ~ ( member_complex @ X @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4030_subset__Diff__insert,axiom,
% 9.03/4.99      ! [A2: set_real,B2: set_real,X: real,C6: set_real] :
% 9.03/4.99        ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X @ C6 ) ) )
% 9.03/4.99        = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C6 ) )
% 9.03/4.99          & ~ ( member_real @ X @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4031_subset__Diff__insert,axiom,
% 9.03/4.99      ! [A2: set_set_nat,B2: set_set_nat,X: set_nat,C6: set_set_nat] :
% 9.03/4.99        ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X @ C6 ) ) )
% 9.03/4.99        = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C6 ) )
% 9.03/4.99          & ~ ( member_set_nat @ X @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4032_subset__Diff__insert,axiom,
% 9.03/4.99      ! [A2: set_nat,B2: set_nat,X: nat,C6: set_nat] :
% 9.03/4.99        ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C6 ) ) )
% 9.03/4.99        = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C6 ) )
% 9.03/4.99          & ~ ( member_nat @ X @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4033_subset__Diff__insert,axiom,
% 9.03/4.99      ! [A2: set_int,B2: set_int,X: int,C6: set_int] :
% 9.03/4.99        ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X @ C6 ) ) )
% 9.03/4.99        = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C6 ) )
% 9.03/4.99          & ~ ( member_int @ X @ A2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4034_int__le__induct,axiom,
% 9.03/4.99      ! [I: int,K: int,P: int > $o] :
% 9.03/4.99        ( ( ord_less_eq_int @ I @ K )
% 9.03/4.99       => ( ( P @ K )
% 9.03/4.99         => ( ! [I2: int] :
% 9.03/4.99                ( ( ord_less_eq_int @ I2 @ K )
% 9.03/4.99               => ( ( P @ I2 )
% 9.03/4.99                 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 9.03/4.99           => ( P @ I ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4035_int__less__induct,axiom,
% 9.03/4.99      ! [I: int,K: int,P: int > $o] :
% 9.03/4.99        ( ( ord_less_int @ I @ K )
% 9.03/4.99       => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 9.03/4.99         => ( ! [I2: int] :
% 9.03/4.99                ( ( ord_less_int @ I2 @ K )
% 9.03/4.99               => ( ( P @ I2 )
% 9.03/4.99                 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 9.03/4.99           => ( P @ I ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4036_zmult__zless__mono2,axiom,
% 9.03/4.99      ! [I: int,J: int,K: int] :
% 9.03/4.99        ( ( ord_less_int @ I @ J )
% 9.03/4.99       => ( ( ord_less_int @ zero_zero_int @ K )
% 9.03/4.99         => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4037_int__ops_I7_J,axiom,
% 9.03/4.99      ! [A: nat,B: nat] :
% 9.03/4.99        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 9.03/4.99        = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4038_ln__powr__bound2,axiom,
% 9.03/4.99      ! [X: real,A: real] :
% 9.03/4.99        ( ( ord_less_real @ one_one_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.99         => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4039_log__eq__div__ln__mult__log,axiom,
% 9.03/4.99      ! [A: real,B: real,X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.99       => ( ( A != one_one_real )
% 9.03/4.99         => ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.99           => ( ( B != one_one_real )
% 9.03/4.99             => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99               => ( ( log @ A @ X )
% 9.03/4.99                  = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4040_enat__0__less__mult__iff,axiom,
% 9.03/4.99      ! [M: extended_enat,N: extended_enat] :
% 9.03/4.99        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 9.03/4.99        = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 9.03/4.99          & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4041_add__diff__assoc__enat,axiom,
% 9.03/4.99      ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
% 9.03/4.99        ( ( ord_le2932123472753598470d_enat @ Z @ Y )
% 9.03/4.99       => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
% 9.03/4.99          = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4042_ln__bound,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4043_ln__gt__zero__imp__gt__one,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99         => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4044_ln__less__zero,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ X @ one_one_real )
% 9.03/4.99         => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4045_ln__gt__zero,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ one_one_real @ X )
% 9.03/4.99       => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4046_ln__ge__zero,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ one_one_real @ X )
% 9.03/4.99       => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4047_int__induct,axiom,
% 9.03/4.99      ! [P: int > $o,K: int,I: int] :
% 9.03/4.99        ( ( P @ K )
% 9.03/4.99       => ( ! [I2: int] :
% 9.03/4.99              ( ( ord_less_eq_int @ K @ I2 )
% 9.03/4.99             => ( ( P @ I2 )
% 9.03/4.99               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 9.03/4.99         => ( ! [I2: int] :
% 9.03/4.99                ( ( ord_less_eq_int @ I2 @ K )
% 9.03/4.99               => ( ( P @ I2 )
% 9.03/4.99                 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 9.03/4.99           => ( P @ I ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4048_reals__Archimedean3,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ! [Y5: real] :
% 9.03/4.99          ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4049_pos__zmult__eq__1__iff,axiom,
% 9.03/4.99      ! [M: int,N: int] :
% 9.03/4.99        ( ( ord_less_int @ zero_zero_int @ M )
% 9.03/4.99       => ( ( ( times_times_int @ M @ N )
% 9.03/4.99            = one_one_int )
% 9.03/4.99          = ( ( M = one_one_int )
% 9.03/4.99            & ( N = one_one_int ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4050_powr__mult,axiom,
% 9.03/4.99      ! [X: real,Y: real,A: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.03/4.99         => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
% 9.03/4.99            = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4051_zdiv__zmult2__eq,axiom,
% 9.03/4.99      ! [C: int,A: int,B: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/4.99       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 9.03/4.99          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4052_log__powr,axiom,
% 9.03/4.99      ! [X: real,B: real,Y: real] :
% 9.03/4.99        ( ( X != zero_zero_real )
% 9.03/4.99       => ( ( log @ B @ ( powr_real @ X @ Y ) )
% 9.03/4.99          = ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4053_ln__ge__zero__imp__ge__one,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99         => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4054_ln__one__plus__pos__lower__bound,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.03/4.99         => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4055_ln__add__one__self__le__self,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4056_int__ops_I6_J,axiom,
% 9.03/4.99      ! [A: nat,B: nat] :
% 9.03/4.99        ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 9.03/4.99         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 9.03/4.99            = zero_zero_int ) )
% 9.03/4.99        & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 9.03/4.99         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 9.03/4.99            = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4057_zmult__zless__mono2__lemma,axiom,
% 9.03/4.99      ! [I: int,J: int,K: nat] :
% 9.03/4.99        ( ( ord_less_int @ I @ J )
% 9.03/4.99       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.03/4.99         => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4058_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 9.03/4.99      ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 9.03/4.99        ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4059_unique__quotient__lemma__neg,axiom,
% 9.03/4.99      ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 9.03/4.99       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.03/4.99         => ( ( ord_less_int @ B @ R2 )
% 9.03/4.99           => ( ( ord_less_int @ B @ R4 )
% 9.03/4.99             => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4060_unique__quotient__lemma,axiom,
% 9.03/4.99      ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 9.03/4.99       => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 9.03/4.99         => ( ( ord_less_int @ R4 @ B )
% 9.03/4.99           => ( ( ord_less_int @ R2 @ B )
% 9.03/4.99             => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4061_zdiv__mono2__neg__lemma,axiom,
% 9.03/4.99      ! [B: int,Q3: int,R2: int,B7: int,Q5: int,R4: int] :
% 9.03/4.99        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
% 9.03/4.99          = ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 9.03/4.99       => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) @ zero_zero_int )
% 9.03/4.99         => ( ( ord_less_int @ R2 @ B )
% 9.03/4.99           => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 9.03/4.99             => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.03/4.99               => ( ( ord_less_eq_int @ B7 @ B )
% 9.03/4.99                 => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4062_zdiv__mono2__lemma,axiom,
% 9.03/4.99      ! [B: int,Q3: int,R2: int,B7: int,Q5: int,R4: int] :
% 9.03/4.99        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
% 9.03/4.99          = ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 9.03/4.99       => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 9.03/4.99         => ( ( ord_less_int @ R4 @ B7 )
% 9.03/4.99           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.03/4.99             => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.03/4.99               => ( ( ord_less_eq_int @ B7 @ B )
% 9.03/4.99                 => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4063_q__pos__lemma,axiom,
% 9.03/4.99      ! [B7: int,Q5: int,R4: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 9.03/4.99       => ( ( ord_less_int @ R4 @ B7 )
% 9.03/4.99         => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.03/4.99           => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4064_ceiling__diff__floor__le__1,axiom,
% 9.03/4.99      ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4065_ceiling__diff__floor__le__1,axiom,
% 9.03/4.99      ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4066_real__of__int__floor__gt__diff__one,axiom,
% 9.03/4.99      ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4067_real__of__int__floor__ge__diff__one,axiom,
% 9.03/4.99      ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4068_real__of__nat__div2,axiom,
% 9.03/4.99      ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4069_log__divide,axiom,
% 9.03/4.99      ! [A: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.99       => ( ( A != one_one_real )
% 9.03/4.99         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99           => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.99             => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
% 9.03/4.99                = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4070_four__x__squared,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/4.99        = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4071_real__of__nat__div3,axiom,
% 9.03/4.99      ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4072_real__archimedian__rdiv__eq__0,axiom,
% 9.03/4.99      ! [X: real,C: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.03/4.99         => ( ! [M5: nat] :
% 9.03/4.99                ( ( ord_less_nat @ zero_zero_nat @ M5 )
% 9.03/4.99               => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
% 9.03/4.99           => ( X = zero_zero_real ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4073_real__of__int__div2,axiom,
% 9.03/4.99      ! [N: int,X: 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 @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4074_real__of__int__div3,axiom,
% 9.03/4.99      ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4075_powr__mult__base,axiom,
% 9.03/4.99      ! [X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
% 9.03/4.99          = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4076_log__mult,axiom,
% 9.03/4.99      ! [A: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.99       => ( ( A != one_one_real )
% 9.03/4.99         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99           => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/4.99             => ( ( log @ A @ ( times_times_real @ X @ Y ) )
% 9.03/4.99                = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4077_split__zdiv,axiom,
% 9.03/4.99      ! [P: int > $o,N: int,K: int] :
% 9.03/4.99        ( ( P @ ( divide_divide_int @ N @ K ) )
% 9.03/4.99        = ( ( ( K = zero_zero_int )
% 9.03/4.99           => ( P @ zero_zero_int ) )
% 9.03/4.99          & ( ( ord_less_int @ zero_zero_int @ K )
% 9.03/4.99           => ! [I3: int,J2: int] :
% 9.03/4.99                ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
% 9.03/4.99                  & ( ord_less_int @ J2 @ K )
% 9.03/4.99                  & ( N
% 9.03/4.99                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 9.03/4.99               => ( P @ I3 ) ) )
% 9.03/4.99          & ( ( ord_less_int @ K @ zero_zero_int )
% 9.03/4.99           => ! [I3: int,J2: int] :
% 9.03/4.99                ( ( ( ord_less_int @ K @ J2 )
% 9.03/4.99                  & ( ord_less_eq_int @ J2 @ zero_zero_int )
% 9.03/4.99                  & ( N
% 9.03/4.99                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 9.03/4.99               => ( P @ I3 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4078_int__div__neg__eq,axiom,
% 9.03/4.99      ! [A: int,B: int,Q3: int,R2: int] :
% 9.03/4.99        ( ( A
% 9.03/4.99          = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 9.03/4.99       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.03/4.99         => ( ( ord_less_int @ B @ R2 )
% 9.03/4.99           => ( ( divide_divide_int @ A @ B )
% 9.03/4.99              = Q3 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4079_int__div__pos__eq,axiom,
% 9.03/4.99      ! [A: int,B: int,Q3: int,R2: int] :
% 9.03/4.99        ( ( A
% 9.03/4.99          = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 9.03/4.99       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.03/4.99         => ( ( ord_less_int @ R2 @ B )
% 9.03/4.99           => ( ( divide_divide_int @ A @ B )
% 9.03/4.99              = Q3 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4080_log__nat__power,axiom,
% 9.03/4.99      ! [X: real,B: real,N: nat] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ( log @ B @ ( power_power_real @ X @ N ) )
% 9.03/4.99          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4081_ln__powr__bound,axiom,
% 9.03/4.99      ! [X: real,A: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ one_one_real @ X )
% 9.03/4.99       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/4.99         => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4082_minus__log__eq__powr,axiom,
% 9.03/4.99      ! [B: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.99       => ( ( B != one_one_real )
% 9.03/4.99         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99           => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
% 9.03/4.99              = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4083_div__pos__geq,axiom,
% 9.03/4.99      ! [L: int,K: int] :
% 9.03/4.99        ( ( ord_less_int @ zero_zero_int @ L )
% 9.03/4.99       => ( ( ord_less_eq_int @ L @ K )
% 9.03/4.99         => ( ( divide_divide_int @ K @ L )
% 9.03/4.99            = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4084_add__log__eq__powr,axiom,
% 9.03/4.99      ! [B: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.99       => ( ( B != one_one_real )
% 9.03/4.99         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99           => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
% 9.03/4.99              = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4085_log__add__eq__powr,axiom,
% 9.03/4.99      ! [B: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ B )
% 9.03/4.99       => ( ( B != one_one_real )
% 9.03/4.99         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/4.99           => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
% 9.03/4.99              = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4086_L2__set__mult__ineq__lemma,axiom,
% 9.03/4.99      ! [A: real,C: real,B: 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 @ B @ 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 @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4087_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 9.03/4.99      ! [X: nat,N: nat,M: nat] :
% 9.03/4.99        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.03/4.99       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.99         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.03/4.99           => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4088_pos__zdiv__mult__2,axiom,
% 9.03/4.99      ! [A: int,B: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/4.99       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.03/4.99          = ( divide_divide_int @ B @ A ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4089_neg__zdiv__mult__2,axiom,
% 9.03/4.99      ! [A: int,B: int] :
% 9.03/4.99        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/4.99       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.03/4.99          = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4090_mult__less__iff1,axiom,
% 9.03/4.99      ! [Z: real,X: real,Y: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ Z )
% 9.03/4.99       => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
% 9.03/4.99          = ( ord_less_real @ X @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4091_mult__less__iff1,axiom,
% 9.03/4.99      ! [Z: rat,X: rat,Y: rat] :
% 9.03/4.99        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 9.03/4.99       => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 9.03/4.99          = ( ord_less_rat @ X @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4092_mult__less__iff1,axiom,
% 9.03/4.99      ! [Z: int,X: int,Y: int] :
% 9.03/4.99        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.03/4.99       => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 9.03/4.99          = ( ord_less_int @ X @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4093_real__average__minus__second,axiom,
% 9.03/4.99      ! [B: real,A: real] :
% 9.03/4.99        ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 9.03/4.99        = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4094_real__average__minus__first,axiom,
% 9.03/4.99      ! [A: real,B: real] :
% 9.03/4.99        ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 9.03/4.99        = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4095_norm__divide__numeral,axiom,
% 9.03/4.99      ! [A: real,W: num] :
% 9.03/4.99        ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 9.03/4.99        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4096_norm__divide__numeral,axiom,
% 9.03/4.99      ! [A: complex,W: num] :
% 9.03/4.99        ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 9.03/4.99        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4097_norm__mult__numeral1,axiom,
% 9.03/4.99      ! [W: num,A: real] :
% 9.03/4.99        ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 9.03/4.99        = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4098_norm__mult__numeral1,axiom,
% 9.03/4.99      ! [W: num,A: complex] :
% 9.03/4.99        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 9.03/4.99        = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4099_norm__mult__numeral2,axiom,
% 9.03/4.99      ! [A: real,W: num] :
% 9.03/4.99        ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 9.03/4.99        = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4100_norm__mult__numeral2,axiom,
% 9.03/4.99      ! [A: complex,W: num] :
% 9.03/4.99        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 9.03/4.99        = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4101_norm__le__zero__iff,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 9.03/4.99        = ( X = zero_zero_real ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4102_norm__le__zero__iff,axiom,
% 9.03/4.99      ! [X: complex] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 9.03/4.99        = ( X = zero_zero_complex ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4103_zero__less__norm__iff,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 9.03/4.99        = ( X != zero_zero_real ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4104_zero__less__norm__iff,axiom,
% 9.03/4.99      ! [X: complex] :
% 9.03/4.99        ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 9.03/4.99        = ( X != zero_zero_complex ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4105_norm__of__nat,axiom,
% 9.03/4.99      ! [N: nat] :
% 9.03/4.99        ( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/4.99        = ( semiri5074537144036343181t_real @ N ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4106_norm__of__nat,axiom,
% 9.03/4.99      ! [N: nat] :
% 9.03/4.99        ( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N ) )
% 9.03/4.99        = ( semiri5074537144036343181t_real @ N ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4107_DiffI,axiom,
% 9.03/4.99      ! [C: complex,A2: set_complex,B2: set_complex] :
% 9.03/4.99        ( ( member_complex @ C @ A2 )
% 9.03/4.99       => ( ~ ( member_complex @ C @ B2 )
% 9.03/4.99         => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4108_DiffI,axiom,
% 9.03/4.99      ! [C: real,A2: set_real,B2: set_real] :
% 9.03/4.99        ( ( member_real @ C @ A2 )
% 9.03/4.99       => ( ~ ( member_real @ C @ B2 )
% 9.03/4.99         => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4109_DiffI,axiom,
% 9.03/4.99      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.99        ( ( member_set_nat @ C @ A2 )
% 9.03/4.99       => ( ~ ( member_set_nat @ C @ B2 )
% 9.03/4.99         => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4110_DiffI,axiom,
% 9.03/4.99      ! [C: int,A2: set_int,B2: set_int] :
% 9.03/4.99        ( ( member_int @ C @ A2 )
% 9.03/4.99       => ( ~ ( member_int @ C @ B2 )
% 9.03/4.99         => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4111_DiffI,axiom,
% 9.03/4.99      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.03/4.99        ( ( member_nat @ C @ A2 )
% 9.03/4.99       => ( ~ ( member_nat @ C @ B2 )
% 9.03/4.99         => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4112_Diff__iff,axiom,
% 9.03/4.99      ! [C: complex,A2: set_complex,B2: set_complex] :
% 9.03/4.99        ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.03/4.99        = ( ( member_complex @ C @ A2 )
% 9.03/4.99          & ~ ( member_complex @ C @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4113_Diff__iff,axiom,
% 9.03/4.99      ! [C: real,A2: set_real,B2: set_real] :
% 9.03/4.99        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.03/4.99        = ( ( member_real @ C @ A2 )
% 9.03/4.99          & ~ ( member_real @ C @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4114_Diff__iff,axiom,
% 9.03/4.99      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.99        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.03/4.99        = ( ( member_set_nat @ C @ A2 )
% 9.03/4.99          & ~ ( member_set_nat @ C @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4115_Diff__iff,axiom,
% 9.03/4.99      ! [C: int,A2: set_int,B2: set_int] :
% 9.03/4.99        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.03/4.99        = ( ( member_int @ C @ A2 )
% 9.03/4.99          & ~ ( member_int @ C @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4116_Diff__iff,axiom,
% 9.03/4.99      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.03/4.99        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.03/4.99        = ( ( member_nat @ C @ A2 )
% 9.03/4.99          & ~ ( member_nat @ C @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4117_Diff__idemp,axiom,
% 9.03/4.99      ! [A2: set_nat,B2: set_nat] :
% 9.03/4.99        ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
% 9.03/4.99        = ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4118_norm__eq__zero,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ( real_V7735802525324610683m_real @ X )
% 9.03/4.99          = zero_zero_real )
% 9.03/4.99        = ( X = zero_zero_real ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4119_norm__eq__zero,axiom,
% 9.03/4.99      ! [X: complex] :
% 9.03/4.99        ( ( ( real_V1022390504157884413omplex @ X )
% 9.03/4.99          = zero_zero_real )
% 9.03/4.99        = ( X = zero_zero_complex ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4120_norm__zero,axiom,
% 9.03/4.99      ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 9.03/4.99      = zero_zero_real ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4121_norm__zero,axiom,
% 9.03/4.99      ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 9.03/4.99      = zero_zero_real ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4122_norm__one,axiom,
% 9.03/4.99      ( ( real_V7735802525324610683m_real @ one_one_real )
% 9.03/4.99      = one_one_real ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4123_norm__one,axiom,
% 9.03/4.99      ( ( real_V1022390504157884413omplex @ one_one_complex )
% 9.03/4.99      = one_one_real ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4124_norm__numeral,axiom,
% 9.03/4.99      ! [W: num] :
% 9.03/4.99        ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 9.03/4.99        = ( numeral_numeral_real @ W ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4125_norm__numeral,axiom,
% 9.03/4.99      ! [W: num] :
% 9.03/4.99        ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.03/4.99        = ( numeral_numeral_real @ W ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4126_DiffE,axiom,
% 9.03/4.99      ! [C: complex,A2: set_complex,B2: set_complex] :
% 9.03/4.99        ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.03/4.99       => ~ ( ( member_complex @ C @ A2 )
% 9.03/4.99           => ( member_complex @ C @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4127_DiffE,axiom,
% 9.03/4.99      ! [C: real,A2: set_real,B2: set_real] :
% 9.03/4.99        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.03/4.99       => ~ ( ( member_real @ C @ A2 )
% 9.03/4.99           => ( member_real @ C @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4128_DiffE,axiom,
% 9.03/4.99      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.99        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.03/4.99       => ~ ( ( member_set_nat @ C @ A2 )
% 9.03/4.99           => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4129_DiffE,axiom,
% 9.03/4.99      ! [C: int,A2: set_int,B2: set_int] :
% 9.03/4.99        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.03/4.99       => ~ ( ( member_int @ C @ A2 )
% 9.03/4.99           => ( member_int @ C @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4130_DiffE,axiom,
% 9.03/4.99      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.03/4.99        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.03/4.99       => ~ ( ( member_nat @ C @ A2 )
% 9.03/4.99           => ( member_nat @ C @ B2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4131_DiffD1,axiom,
% 9.03/4.99      ! [C: complex,A2: set_complex,B2: set_complex] :
% 9.03/4.99        ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.03/4.99       => ( member_complex @ C @ A2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4132_DiffD1,axiom,
% 9.03/4.99      ! [C: real,A2: set_real,B2: set_real] :
% 9.03/4.99        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.03/4.99       => ( member_real @ C @ A2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4133_DiffD1,axiom,
% 9.03/4.99      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.99        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.03/4.99       => ( member_set_nat @ C @ A2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4134_DiffD1,axiom,
% 9.03/4.99      ! [C: int,A2: set_int,B2: set_int] :
% 9.03/4.99        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.03/4.99       => ( member_int @ C @ A2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4135_DiffD1,axiom,
% 9.03/4.99      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.03/4.99        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.03/4.99       => ( member_nat @ C @ A2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4136_DiffD2,axiom,
% 9.03/4.99      ! [C: complex,A2: set_complex,B2: set_complex] :
% 9.03/4.99        ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.03/4.99       => ~ ( member_complex @ C @ B2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4137_DiffD2,axiom,
% 9.03/4.99      ! [C: real,A2: set_real,B2: set_real] :
% 9.03/4.99        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.03/4.99       => ~ ( member_real @ C @ B2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4138_DiffD2,axiom,
% 9.03/4.99      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.03/4.99        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.03/4.99       => ~ ( member_set_nat @ C @ B2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4139_DiffD2,axiom,
% 9.03/4.99      ! [C: int,A2: set_int,B2: set_int] :
% 9.03/4.99        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.03/4.99       => ~ ( member_int @ C @ B2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4140_DiffD2,axiom,
% 9.03/4.99      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.03/4.99        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.03/4.99       => ~ ( member_nat @ C @ B2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4141_norm__diff__ineq,axiom,
% 9.03/4.99      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4142_norm__diff__ineq,axiom,
% 9.03/4.99      ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4143_norm__triangle__ineq2,axiom,
% 9.03/4.99      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4144_norm__triangle__ineq2,axiom,
% 9.03/4.99      ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4145_norm__not__less__zero,axiom,
% 9.03/4.99      ! [X: complex] :
% 9.03/4.99        ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4146_norm__ge__zero,axiom,
% 9.03/4.99      ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4147_norm__divide,axiom,
% 9.03/4.99      ! [A: real,B: real] :
% 9.03/4.99        ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 9.03/4.99        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4148_norm__divide,axiom,
% 9.03/4.99      ! [A: complex,B: complex] :
% 9.03/4.99        ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.03/4.99        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4149_norm__power,axiom,
% 9.03/4.99      ! [X: real,N: nat] :
% 9.03/4.99        ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
% 9.03/4.99        = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4150_norm__power,axiom,
% 9.03/4.99      ! [X: complex,N: nat] :
% 9.03/4.99        ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
% 9.03/4.99        = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4151_nonzero__norm__divide,axiom,
% 9.03/4.99      ! [B: real,A: real] :
% 9.03/4.99        ( ( B != zero_zero_real )
% 9.03/4.99       => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 9.03/4.99          = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4152_nonzero__norm__divide,axiom,
% 9.03/4.99      ! [B: complex,A: complex] :
% 9.03/4.99        ( ( B != zero_zero_complex )
% 9.03/4.99       => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.03/4.99          = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4153_power__eq__imp__eq__norm,axiom,
% 9.03/4.99      ! [W: real,N: nat,Z: real] :
% 9.03/4.99        ( ( ( power_power_real @ W @ N )
% 9.03/4.99          = ( power_power_real @ Z @ N ) )
% 9.03/4.99       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.99         => ( ( real_V7735802525324610683m_real @ W )
% 9.03/4.99            = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4154_power__eq__imp__eq__norm,axiom,
% 9.03/4.99      ! [W: complex,N: nat,Z: complex] :
% 9.03/4.99        ( ( ( power_power_complex @ W @ N )
% 9.03/4.99          = ( power_power_complex @ Z @ N ) )
% 9.03/4.99       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/4.99         => ( ( real_V1022390504157884413omplex @ W )
% 9.03/4.99            = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4155_norm__mult__less,axiom,
% 9.03/4.99      ! [X: real,R2: real,Y: real,S: real] :
% 9.03/4.99        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
% 9.03/4.99       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 9.03/4.99         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4156_norm__mult__less,axiom,
% 9.03/4.99      ! [X: complex,R2: real,Y: complex,S: real] :
% 9.03/4.99        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
% 9.03/4.99       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 9.03/4.99         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4157_norm__mult__ineq,axiom,
% 9.03/4.99      ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4158_norm__mult__ineq,axiom,
% 9.03/4.99      ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4159_norm__triangle__lt,axiom,
% 9.03/4.99      ! [X: real,Y: real,E2: real] :
% 9.03/4.99        ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
% 9.03/4.99       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4160_norm__triangle__lt,axiom,
% 9.03/4.99      ! [X: complex,Y: complex,E2: real] :
% 9.03/4.99        ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
% 9.03/4.99       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4161_norm__add__less,axiom,
% 9.03/4.99      ! [X: real,R2: real,Y: real,S: real] :
% 9.03/4.99        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
% 9.03/4.99       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 9.03/4.99         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4162_norm__add__less,axiom,
% 9.03/4.99      ! [X: complex,R2: real,Y: complex,S: real] :
% 9.03/4.99        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
% 9.03/4.99       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 9.03/4.99         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4163_norm__add__leD,axiom,
% 9.03/4.99      ! [A: real,B: real,C: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 9.03/4.99       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4164_norm__add__leD,axiom,
% 9.03/4.99      ! [A: complex,B: complex,C: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 9.03/4.99       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4165_norm__triangle__le,axiom,
% 9.03/4.99      ! [X: real,Y: real,E2: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
% 9.03/4.99       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4166_norm__triangle__le,axiom,
% 9.03/4.99      ! [X: complex,Y: complex,E2: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
% 9.03/4.99       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4167_norm__triangle__ineq,axiom,
% 9.03/4.99      ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4168_norm__triangle__ineq,axiom,
% 9.03/4.99      ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4169_norm__triangle__mono,axiom,
% 9.03/4.99      ! [A: real,R2: real,B: real,S: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
% 9.03/4.99         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4170_norm__triangle__mono,axiom,
% 9.03/4.99      ! [A: complex,R2: real,B: complex,S: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
% 9.03/4.99         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4171_norm__power__ineq,axiom,
% 9.03/4.99      ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4172_norm__power__ineq,axiom,
% 9.03/4.99      ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4173_norm__diff__triangle__less,axiom,
% 9.03/4.99      ! [X: real,Y: real,E1: real,Z: real,E22: real] :
% 9.03/4.99        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 9.03/4.99       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
% 9.03/4.99         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4174_norm__diff__triangle__less,axiom,
% 9.03/4.99      ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
% 9.03/4.99        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 9.03/4.99       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
% 9.03/4.99         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4175_norm__triangle__le__diff,axiom,
% 9.03/4.99      ! [X: real,Y: real,E2: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
% 9.03/4.99       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4176_norm__triangle__le__diff,axiom,
% 9.03/4.99      ! [X: complex,Y: complex,E2: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
% 9.03/4.99       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4177_norm__diff__triangle__le,axiom,
% 9.03/4.99      ! [X: real,Y: real,E1: real,Z: real,E22: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
% 9.03/4.99         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4178_norm__diff__triangle__le,axiom,
% 9.03/4.99      ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
% 9.03/4.99         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4179_norm__triangle__ineq4,axiom,
% 9.03/4.99      ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4180_norm__triangle__ineq4,axiom,
% 9.03/4.99      ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4181_norm__triangle__sub,axiom,
% 9.03/4.99      ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4182_norm__triangle__sub,axiom,
% 9.03/4.99      ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4183_power__eq__1__iff,axiom,
% 9.03/4.99      ! [W: real,N: nat] :
% 9.03/4.99        ( ( ( power_power_real @ W @ N )
% 9.03/4.99          = one_one_real )
% 9.03/4.99       => ( ( ( real_V7735802525324610683m_real @ W )
% 9.03/4.99            = one_one_real )
% 9.03/4.99          | ( N = zero_zero_nat ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4184_power__eq__1__iff,axiom,
% 9.03/4.99      ! [W: complex,N: nat] :
% 9.03/4.99        ( ( ( power_power_complex @ W @ N )
% 9.03/4.99          = one_one_complex )
% 9.03/4.99       => ( ( ( real_V1022390504157884413omplex @ W )
% 9.03/4.99            = one_one_real )
% 9.03/4.99          | ( N = zero_zero_nat ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4185_norm__diff__triangle__ineq,axiom,
% 9.03/4.99      ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4186_norm__diff__triangle__ineq,axiom,
% 9.03/4.99      ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4187_square__norm__one,axiom,
% 9.03/4.99      ! [X: real] :
% 9.03/4.99        ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99          = one_one_real )
% 9.03/4.99       => ( ( real_V7735802525324610683m_real @ X )
% 9.03/4.99          = one_one_real ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4188_square__norm__one,axiom,
% 9.03/4.99      ! [X: complex] :
% 9.03/4.99        ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/4.99          = one_one_complex )
% 9.03/4.99       => ( ( real_V1022390504157884413omplex @ X )
% 9.03/4.99          = one_one_real ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4189_norm__power__diff,axiom,
% 9.03/4.99      ! [Z: real,W: real,M: nat] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 9.03/4.99         => ( 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 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4190_norm__power__diff,axiom,
% 9.03/4.99      ! [Z: complex,W: complex,M: nat] :
% 9.03/4.99        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 9.03/4.99       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 9.03/4.99         => ( 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 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4191_both__member__options__ding,axiom,
% 9.03/4.99      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
% 9.03/4.99        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 9.03/4.99       => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.03/4.99         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.03/4.99           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4192_linear__plus__1__le__power,axiom,
% 9.03/4.99      ! [X: real,N: nat] :
% 9.03/4.99        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/4.99       => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4193_zdiff__int__split,axiom,
% 9.03/4.99      ! [P: int > $o,X: nat,Y: nat] :
% 9.03/4.99        ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
% 9.03/4.99        = ( ( ( ord_less_eq_nat @ Y @ X )
% 9.03/4.99           => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
% 9.03/4.99          & ( ( ord_less_nat @ X @ Y )
% 9.03/4.99           => ( P @ zero_zero_int ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4194_ln__2__less__1,axiom,
% 9.03/4.99      ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4195_decr__mult__lemma,axiom,
% 9.03/4.99      ! [D: int,P: int > $o,K: int] :
% 9.03/4.99        ( ( ord_less_int @ zero_zero_int @ D )
% 9.03/4.99       => ( ! [X3: int] :
% 9.03/4.99              ( ( P @ X3 )
% 9.03/4.99             => ( P @ ( minus_minus_int @ X3 @ D ) ) )
% 9.03/4.99         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.03/4.99           => ! [X2: int] :
% 9.03/4.99                ( ( P @ X2 )
% 9.03/4.99               => ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4196_incr__mult__lemma,axiom,
% 9.03/4.99      ! [D: int,P: int > $o,K: int] :
% 9.03/4.99        ( ( ord_less_int @ zero_zero_int @ D )
% 9.03/4.99       => ( ! [X3: int] :
% 9.03/4.99              ( ( P @ X3 )
% 9.03/4.99             => ( P @ ( plus_plus_int @ X3 @ D ) ) )
% 9.03/4.99         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.03/4.99           => ! [X2: int] :
% 9.03/4.99                ( ( P @ X2 )
% 9.03/4.99               => ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4197_inthall,axiom,
% 9.03/4.99      ! [Xs: list_complex,P: complex > $o,N: nat] :
% 9.03/4.99        ( ! [X3: complex] :
% 9.03/4.99            ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
% 9.03/4.99           => ( P @ X3 ) )
% 9.03/4.99       => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
% 9.03/4.99         => ( P @ ( nth_complex @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4198_inthall,axiom,
% 9.03/4.99      ! [Xs: list_real,P: real > $o,N: nat] :
% 9.03/4.99        ( ! [X3: real] :
% 9.03/4.99            ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.03/4.99           => ( P @ X3 ) )
% 9.03/4.99       => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 9.03/4.99         => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4199_inthall,axiom,
% 9.03/4.99      ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
% 9.03/4.99        ( ! [X3: set_nat] :
% 9.03/4.99            ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 9.03/4.99           => ( P @ X3 ) )
% 9.03/4.99       => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.03/4.99         => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4200_inthall,axiom,
% 9.03/4.99      ! [Xs: list_nat,P: nat > $o,N: nat] :
% 9.03/4.99        ( ! [X3: nat] :
% 9.03/4.99            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 9.03/4.99           => ( P @ X3 ) )
% 9.03/4.99       => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.03/4.99         => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4201_inthall,axiom,
% 9.03/4.99      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 9.03/4.99        ( ! [X3: vEBT_VEBT] :
% 9.03/4.99            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.03/4.99           => ( P @ X3 ) )
% 9.03/4.99       => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.03/4.99         => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4202_inthall,axiom,
% 9.03/4.99      ! [Xs: list_o,P: $o > $o,N: nat] :
% 9.03/4.99        ( ! [X3: $o] :
% 9.03/4.99            ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 9.03/4.99           => ( P @ X3 ) )
% 9.03/4.99       => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.03/4.99         => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4203_inthall,axiom,
% 9.03/4.99      ! [Xs: list_int,P: int > $o,N: nat] :
% 9.03/4.99        ( ! [X3: int] :
% 9.03/4.99            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.03/4.99           => ( P @ X3 ) )
% 9.03/4.99       => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.03/4.99         => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4204_nth__equalityI,axiom,
% 9.03/4.99      ! [Xs: list_nat,Ys2: list_nat] :
% 9.03/4.99        ( ( ( size_size_list_nat @ Xs )
% 9.03/4.99          = ( size_size_list_nat @ Ys2 ) )
% 9.03/4.99       => ( ! [I2: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 9.03/4.99             => ( ( nth_nat @ Xs @ I2 )
% 9.03/4.99                = ( nth_nat @ Ys2 @ I2 ) ) )
% 9.03/4.99         => ( Xs = Ys2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4205_nth__equalityI,axiom,
% 9.03/4.99      ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 9.03/4.99        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.03/4.99          = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 9.03/4.99       => ( ! [I2: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.03/4.99             => ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 9.03/4.99                = ( nth_VEBT_VEBT @ Ys2 @ I2 ) ) )
% 9.03/4.99         => ( Xs = Ys2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4206_nth__equalityI,axiom,
% 9.03/4.99      ! [Xs: list_o,Ys2: list_o] :
% 9.03/4.99        ( ( ( size_size_list_o @ Xs )
% 9.03/4.99          = ( size_size_list_o @ Ys2 ) )
% 9.03/4.99       => ( ! [I2: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 9.03/4.99             => ( ( nth_o @ Xs @ I2 )
% 9.03/4.99                = ( nth_o @ Ys2 @ I2 ) ) )
% 9.03/4.99         => ( Xs = Ys2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4207_nth__equalityI,axiom,
% 9.03/4.99      ! [Xs: list_int,Ys2: list_int] :
% 9.03/4.99        ( ( ( size_size_list_int @ Xs )
% 9.03/4.99          = ( size_size_list_int @ Ys2 ) )
% 9.03/4.99       => ( ! [I2: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 9.03/4.99             => ( ( nth_int @ Xs @ I2 )
% 9.03/4.99                = ( nth_int @ Ys2 @ I2 ) ) )
% 9.03/4.99         => ( Xs = Ys2 ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4208_Skolem__list__nth,axiom,
% 9.03/4.99      ! [K: nat,P: nat > nat > $o] :
% 9.03/4.99        ( ( ! [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ K )
% 9.03/4.99             => ? [X6: nat] : ( P @ I3 @ X6 ) ) )
% 9.03/4.99        = ( ? [Xs3: list_nat] :
% 9.03/4.99              ( ( ( size_size_list_nat @ Xs3 )
% 9.03/4.99                = K )
% 9.03/4.99              & ! [I3: nat] :
% 9.03/4.99                  ( ( ord_less_nat @ I3 @ K )
% 9.03/4.99                 => ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4209_Skolem__list__nth,axiom,
% 9.03/4.99      ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 9.03/4.99        ( ( ! [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ K )
% 9.03/4.99             => ? [X6: vEBT_VEBT] : ( P @ I3 @ X6 ) ) )
% 9.03/4.99        = ( ? [Xs3: list_VEBT_VEBT] :
% 9.03/4.99              ( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 9.03/4.99                = K )
% 9.03/4.99              & ! [I3: nat] :
% 9.03/4.99                  ( ( ord_less_nat @ I3 @ K )
% 9.03/4.99                 => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs3 @ I3 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4210_Skolem__list__nth,axiom,
% 9.03/4.99      ! [K: nat,P: nat > $o > $o] :
% 9.03/4.99        ( ( ! [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ K )
% 9.03/4.99             => ? [X6: $o] : ( P @ I3 @ X6 ) ) )
% 9.03/4.99        = ( ? [Xs3: list_o] :
% 9.03/4.99              ( ( ( size_size_list_o @ Xs3 )
% 9.03/4.99                = K )
% 9.03/4.99              & ! [I3: nat] :
% 9.03/4.99                  ( ( ord_less_nat @ I3 @ K )
% 9.03/4.99                 => ( P @ I3 @ ( nth_o @ Xs3 @ I3 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4211_Skolem__list__nth,axiom,
% 9.03/4.99      ! [K: nat,P: nat > int > $o] :
% 9.03/4.99        ( ( ! [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ K )
% 9.03/4.99             => ? [X6: int] : ( P @ I3 @ X6 ) ) )
% 9.03/4.99        = ( ? [Xs3: list_int] :
% 9.03/4.99              ( ( ( size_size_list_int @ Xs3 )
% 9.03/4.99                = K )
% 9.03/4.99              & ! [I3: nat] :
% 9.03/4.99                  ( ( ord_less_nat @ I3 @ K )
% 9.03/4.99                 => ( P @ I3 @ ( nth_int @ Xs3 @ I3 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4212_list__eq__iff__nth__eq,axiom,
% 9.03/4.99      ( ( ^ [Y3: list_nat,Z2: list_nat] : ( Y3 = Z2 ) )
% 9.03/4.99      = ( ^ [Xs3: list_nat,Ys3: list_nat] :
% 9.03/4.99            ( ( ( size_size_list_nat @ Xs3 )
% 9.03/4.99              = ( size_size_list_nat @ Ys3 ) )
% 9.03/4.99            & ! [I3: nat] :
% 9.03/4.99                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
% 9.03/4.99               => ( ( nth_nat @ Xs3 @ I3 )
% 9.03/4.99                  = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4213_list__eq__iff__nth__eq,axiom,
% 9.03/4.99      ( ( ^ [Y3: list_VEBT_VEBT,Z2: list_VEBT_VEBT] : ( Y3 = Z2 ) )
% 9.03/4.99      = ( ^ [Xs3: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 9.03/4.99            ( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 9.03/4.99              = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 9.03/4.99            & ! [I3: nat] :
% 9.03/4.99                ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 9.03/4.99               => ( ( nth_VEBT_VEBT @ Xs3 @ I3 )
% 9.03/4.99                  = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4214_list__eq__iff__nth__eq,axiom,
% 9.03/4.99      ( ( ^ [Y3: list_o,Z2: list_o] : ( Y3 = Z2 ) )
% 9.03/4.99      = ( ^ [Xs3: list_o,Ys3: list_o] :
% 9.03/4.99            ( ( ( size_size_list_o @ Xs3 )
% 9.03/4.99              = ( size_size_list_o @ Ys3 ) )
% 9.03/4.99            & ! [I3: nat] :
% 9.03/4.99                ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs3 ) )
% 9.03/4.99               => ( ( nth_o @ Xs3 @ I3 )
% 9.03/4.99                  = ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4215_list__eq__iff__nth__eq,axiom,
% 9.03/4.99      ( ( ^ [Y3: list_int,Z2: list_int] : ( Y3 = Z2 ) )
% 9.03/4.99      = ( ^ [Xs3: list_int,Ys3: list_int] :
% 9.03/4.99            ( ( ( size_size_list_int @ Xs3 )
% 9.03/4.99              = ( size_size_list_int @ Ys3 ) )
% 9.03/4.99            & ! [I3: nat] :
% 9.03/4.99                ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs3 ) )
% 9.03/4.99               => ( ( nth_int @ Xs3 @ I3 )
% 9.03/4.99                  = ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4216_all__set__conv__all__nth,axiom,
% 9.03/4.99      ! [Xs: list_nat,P: nat > $o] :
% 9.03/4.99        ( ( ! [X4: nat] :
% 9.03/4.99              ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 9.03/4.99             => ( P @ X4 ) ) )
% 9.03/4.99        = ( ! [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 9.03/4.99             => ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4217_all__set__conv__all__nth,axiom,
% 9.03/4.99      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.03/4.99        ( ( ! [X4: vEBT_VEBT] :
% 9.03/4.99              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.03/4.99             => ( P @ X4 ) ) )
% 9.03/4.99        = ( ! [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.03/4.99             => ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4218_all__set__conv__all__nth,axiom,
% 9.03/4.99      ! [Xs: list_o,P: $o > $o] :
% 9.03/4.99        ( ( ! [X4: $o] :
% 9.03/4.99              ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 9.03/4.99             => ( P @ X4 ) ) )
% 9.03/4.99        = ( ! [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 9.03/4.99             => ( P @ ( nth_o @ Xs @ I3 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4219_all__set__conv__all__nth,axiom,
% 9.03/4.99      ! [Xs: list_int,P: int > $o] :
% 9.03/4.99        ( ( ! [X4: int] :
% 9.03/4.99              ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 9.03/4.99             => ( P @ X4 ) ) )
% 9.03/4.99        = ( ! [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 9.03/4.99             => ( P @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4220_all__nth__imp__all__set,axiom,
% 9.03/4.99      ! [Xs: list_complex,P: complex > $o,X: complex] :
% 9.03/4.99        ( ! [I2: nat] :
% 9.03/4.99            ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) )
% 9.03/4.99           => ( P @ ( nth_complex @ Xs @ I2 ) ) )
% 9.03/4.99       => ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 9.03/4.99         => ( P @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4221_all__nth__imp__all__set,axiom,
% 9.03/4.99      ! [Xs: list_real,P: real > $o,X: real] :
% 9.03/4.99        ( ! [I2: nat] :
% 9.03/4.99            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
% 9.03/4.99           => ( P @ ( nth_real @ Xs @ I2 ) ) )
% 9.03/4.99       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.03/4.99         => ( P @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4222_all__nth__imp__all__set,axiom,
% 9.03/4.99      ! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
% 9.03/4.99        ( ! [I2: nat] :
% 9.03/4.99            ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.03/4.99           => ( P @ ( nth_set_nat @ Xs @ I2 ) ) )
% 9.03/4.99       => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.03/4.99         => ( P @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4223_all__nth__imp__all__set,axiom,
% 9.03/4.99      ! [Xs: list_nat,P: nat > $o,X: nat] :
% 9.03/4.99        ( ! [I2: nat] :
% 9.03/4.99            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 9.03/4.99           => ( P @ ( nth_nat @ Xs @ I2 ) ) )
% 9.03/4.99       => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.03/4.99         => ( P @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4224_all__nth__imp__all__set,axiom,
% 9.03/4.99      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 9.03/4.99        ( ! [I2: nat] :
% 9.03/4.99            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.03/4.99           => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
% 9.03/4.99       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.03/4.99         => ( P @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4225_all__nth__imp__all__set,axiom,
% 9.03/4.99      ! [Xs: list_o,P: $o > $o,X: $o] :
% 9.03/4.99        ( ! [I2: nat] :
% 9.03/4.99            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 9.03/4.99           => ( P @ ( nth_o @ Xs @ I2 ) ) )
% 9.03/4.99       => ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.03/4.99         => ( P @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4226_all__nth__imp__all__set,axiom,
% 9.03/4.99      ! [Xs: list_int,P: int > $o,X: int] :
% 9.03/4.99        ( ! [I2: nat] :
% 9.03/4.99            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 9.03/4.99           => ( P @ ( nth_int @ Xs @ I2 ) ) )
% 9.03/4.99       => ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.03/4.99         => ( P @ X ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4227_in__set__conv__nth,axiom,
% 9.03/4.99      ! [X: complex,Xs: list_complex] :
% 9.03/4.99        ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 9.03/4.99        = ( ? [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
% 9.03/4.99              & ( ( nth_complex @ Xs @ I3 )
% 9.03/4.99                = X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4228_in__set__conv__nth,axiom,
% 9.03/4.99      ! [X: real,Xs: list_real] :
% 9.03/4.99        ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.03/4.99        = ( ? [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
% 9.03/4.99              & ( ( nth_real @ Xs @ I3 )
% 9.03/4.99                = X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4229_in__set__conv__nth,axiom,
% 9.03/4.99      ! [X: set_nat,Xs: list_set_nat] :
% 9.03/4.99        ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.03/4.99        = ( ? [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.03/4.99              & ( ( nth_set_nat @ Xs @ I3 )
% 9.03/4.99                = X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4230_in__set__conv__nth,axiom,
% 9.03/4.99      ! [X: nat,Xs: list_nat] :
% 9.03/4.99        ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.03/4.99        = ( ? [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 9.03/4.99              & ( ( nth_nat @ Xs @ I3 )
% 9.03/4.99                = X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4231_in__set__conv__nth,axiom,
% 9.03/4.99      ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.03/4.99        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.03/4.99        = ( ? [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.03/4.99              & ( ( nth_VEBT_VEBT @ Xs @ I3 )
% 9.03/4.99                = X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4232_in__set__conv__nth,axiom,
% 9.03/4.99      ! [X: $o,Xs: list_o] :
% 9.03/4.99        ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.03/4.99        = ( ? [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 9.03/4.99              & ( ( nth_o @ Xs @ I3 )
% 9.03/4.99                = X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4233_in__set__conv__nth,axiom,
% 9.03/4.99      ! [X: int,Xs: list_int] :
% 9.03/4.99        ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.03/4.99        = ( ? [I3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 9.03/4.99              & ( ( nth_int @ Xs @ I3 )
% 9.03/4.99                = X ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4234_list__ball__nth,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_nat,P: nat > $o] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.03/4.99       => ( ! [X3: nat] :
% 9.03/4.99              ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 9.03/4.99             => ( P @ X3 ) )
% 9.03/4.99         => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4235_list__ball__nth,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.03/4.99       => ( ! [X3: vEBT_VEBT] :
% 9.03/4.99              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.03/4.99             => ( P @ X3 ) )
% 9.03/4.99         => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4236_list__ball__nth,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_o,P: $o > $o] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.03/4.99       => ( ! [X3: $o] :
% 9.03/4.99              ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 9.03/4.99             => ( P @ X3 ) )
% 9.03/4.99         => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4237_list__ball__nth,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_int,P: int > $o] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.03/4.99       => ( ! [X3: int] :
% 9.03/4.99              ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.03/4.99             => ( P @ X3 ) )
% 9.03/4.99         => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4238_nth__mem,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_complex] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
% 9.03/4.99       => ( member_complex @ ( nth_complex @ Xs @ N ) @ ( set_complex2 @ Xs ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4239_nth__mem,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_real] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 9.03/4.99       => ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4240_nth__mem,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_set_nat] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.03/4.99       => ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4241_nth__mem,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_nat] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.03/4.99       => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4242_nth__mem,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_VEBT_VEBT] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.03/4.99       => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4243_nth__mem,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_o] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.03/4.99       => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4244_nth__mem,axiom,
% 9.03/4.99      ! [N: nat,Xs: list_int] :
% 9.03/4.99        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.03/4.99       => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4245_minf_I7_J,axiom,
% 9.03/4.99      ! [T: real] :
% 9.03/4.99      ? [Z3: real] :
% 9.03/4.99      ! [X2: real] :
% 9.03/4.99        ( ( ord_less_real @ X2 @ Z3 )
% 9.03/4.99       => ~ ( ord_less_real @ T @ X2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4246_minf_I7_J,axiom,
% 9.03/4.99      ! [T: rat] :
% 9.03/4.99      ? [Z3: rat] :
% 9.03/4.99      ! [X2: rat] :
% 9.03/4.99        ( ( ord_less_rat @ X2 @ Z3 )
% 9.03/4.99       => ~ ( ord_less_rat @ T @ X2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4247_minf_I7_J,axiom,
% 9.03/4.99      ! [T: num] :
% 9.03/4.99      ? [Z3: num] :
% 9.03/4.99      ! [X2: num] :
% 9.03/4.99        ( ( ord_less_num @ X2 @ Z3 )
% 9.03/4.99       => ~ ( ord_less_num @ T @ X2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4248_minf_I7_J,axiom,
% 9.03/4.99      ! [T: nat] :
% 9.03/4.99      ? [Z3: nat] :
% 9.03/4.99      ! [X2: nat] :
% 9.03/4.99        ( ( ord_less_nat @ X2 @ Z3 )
% 9.03/4.99       => ~ ( ord_less_nat @ T @ X2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4249_minf_I7_J,axiom,
% 9.03/4.99      ! [T: int] :
% 9.03/4.99      ? [Z3: int] :
% 9.03/4.99      ! [X2: int] :
% 9.03/4.99        ( ( ord_less_int @ X2 @ Z3 )
% 9.03/4.99       => ~ ( ord_less_int @ T @ X2 ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4250_minf_I5_J,axiom,
% 9.03/4.99      ! [T: real] :
% 9.03/4.99      ? [Z3: real] :
% 9.03/4.99      ! [X2: real] :
% 9.03/4.99        ( ( ord_less_real @ X2 @ Z3 )
% 9.03/4.99       => ( ord_less_real @ X2 @ T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4251_minf_I5_J,axiom,
% 9.03/4.99      ! [T: rat] :
% 9.03/4.99      ? [Z3: rat] :
% 9.03/4.99      ! [X2: rat] :
% 9.03/4.99        ( ( ord_less_rat @ X2 @ Z3 )
% 9.03/4.99       => ( ord_less_rat @ X2 @ T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4252_minf_I5_J,axiom,
% 9.03/4.99      ! [T: num] :
% 9.03/4.99      ? [Z3: num] :
% 9.03/4.99      ! [X2: num] :
% 9.03/4.99        ( ( ord_less_num @ X2 @ Z3 )
% 9.03/4.99       => ( ord_less_num @ X2 @ T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4253_minf_I5_J,axiom,
% 9.03/4.99      ! [T: nat] :
% 9.03/4.99      ? [Z3: nat] :
% 9.03/4.99      ! [X2: nat] :
% 9.03/4.99        ( ( ord_less_nat @ X2 @ Z3 )
% 9.03/4.99       => ( ord_less_nat @ X2 @ T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4254_minf_I5_J,axiom,
% 9.03/4.99      ! [T: int] :
% 9.03/4.99      ? [Z3: int] :
% 9.03/4.99      ! [X2: int] :
% 9.03/4.99        ( ( ord_less_int @ X2 @ Z3 )
% 9.03/4.99       => ( ord_less_int @ X2 @ T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4255_minf_I4_J,axiom,
% 9.03/4.99      ! [T: real] :
% 9.03/4.99      ? [Z3: real] :
% 9.03/4.99      ! [X2: real] :
% 9.03/4.99        ( ( ord_less_real @ X2 @ Z3 )
% 9.03/4.99       => ( X2 != T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4256_minf_I4_J,axiom,
% 9.03/4.99      ! [T: rat] :
% 9.03/4.99      ? [Z3: rat] :
% 9.03/4.99      ! [X2: rat] :
% 9.03/4.99        ( ( ord_less_rat @ X2 @ Z3 )
% 9.03/4.99       => ( X2 != T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4257_minf_I4_J,axiom,
% 9.03/4.99      ! [T: num] :
% 9.03/4.99      ? [Z3: num] :
% 9.03/4.99      ! [X2: num] :
% 9.03/4.99        ( ( ord_less_num @ X2 @ Z3 )
% 9.03/4.99       => ( X2 != T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4258_minf_I4_J,axiom,
% 9.03/4.99      ! [T: nat] :
% 9.03/4.99      ? [Z3: nat] :
% 9.03/4.99      ! [X2: nat] :
% 9.03/4.99        ( ( ord_less_nat @ X2 @ Z3 )
% 9.03/4.99       => ( X2 != T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4259_minf_I4_J,axiom,
% 9.03/4.99      ! [T: int] :
% 9.03/4.99      ? [Z3: int] :
% 9.03/4.99      ! [X2: int] :
% 9.03/4.99        ( ( ord_less_int @ X2 @ Z3 )
% 9.03/4.99       => ( X2 != T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4260_minf_I3_J,axiom,
% 9.03/4.99      ! [T: real] :
% 9.03/4.99      ? [Z3: real] :
% 9.03/4.99      ! [X2: real] :
% 9.03/4.99        ( ( ord_less_real @ X2 @ Z3 )
% 9.03/4.99       => ( X2 != T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4261_minf_I3_J,axiom,
% 9.03/4.99      ! [T: rat] :
% 9.03/4.99      ? [Z3: rat] :
% 9.03/4.99      ! [X2: rat] :
% 9.03/4.99        ( ( ord_less_rat @ X2 @ Z3 )
% 9.03/4.99       => ( X2 != T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4262_minf_I3_J,axiom,
% 9.03/4.99      ! [T: num] :
% 9.03/4.99      ? [Z3: num] :
% 9.03/4.99      ! [X2: num] :
% 9.03/4.99        ( ( ord_less_num @ X2 @ Z3 )
% 9.03/4.99       => ( X2 != T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4263_minf_I3_J,axiom,
% 9.03/4.99      ! [T: nat] :
% 9.03/4.99      ? [Z3: nat] :
% 9.03/4.99      ! [X2: nat] :
% 9.03/4.99        ( ( ord_less_nat @ X2 @ Z3 )
% 9.03/4.99       => ( X2 != T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4264_minf_I3_J,axiom,
% 9.03/4.99      ! [T: int] :
% 9.03/4.99      ? [Z3: int] :
% 9.03/4.99      ! [X2: int] :
% 9.03/4.99        ( ( ord_less_int @ X2 @ Z3 )
% 9.03/4.99       => ( X2 != T ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4265_minf_I2_J,axiom,
% 9.03/4.99      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 9.03/4.99        ( ? [Z5: real] :
% 9.03/4.99          ! [X3: real] :
% 9.03/4.99            ( ( ord_less_real @ X3 @ Z5 )
% 9.03/4.99           => ( ( P @ X3 )
% 9.03/4.99              = ( P6 @ X3 ) ) )
% 9.03/4.99       => ( ? [Z5: real] :
% 9.03/4.99            ! [X3: real] :
% 9.03/4.99              ( ( ord_less_real @ X3 @ Z5 )
% 9.03/4.99             => ( ( Q @ X3 )
% 9.03/4.99                = ( Q6 @ X3 ) ) )
% 9.03/4.99         => ? [Z3: real] :
% 9.03/4.99            ! [X2: real] :
% 9.03/4.99              ( ( ord_less_real @ X2 @ Z3 )
% 9.03/4.99             => ( ( ( P @ X2 )
% 9.03/4.99                  | ( Q @ X2 ) )
% 9.03/4.99                = ( ( P6 @ X2 )
% 9.03/4.99                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4266_minf_I2_J,axiom,
% 9.03/4.99      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 9.03/4.99        ( ? [Z5: rat] :
% 9.03/4.99          ! [X3: rat] :
% 9.03/4.99            ( ( ord_less_rat @ X3 @ Z5 )
% 9.03/4.99           => ( ( P @ X3 )
% 9.03/4.99              = ( P6 @ X3 ) ) )
% 9.03/4.99       => ( ? [Z5: rat] :
% 9.03/4.99            ! [X3: rat] :
% 9.03/4.99              ( ( ord_less_rat @ X3 @ Z5 )
% 9.03/4.99             => ( ( Q @ X3 )
% 9.03/4.99                = ( Q6 @ X3 ) ) )
% 9.03/4.99         => ? [Z3: rat] :
% 9.03/4.99            ! [X2: rat] :
% 9.03/4.99              ( ( ord_less_rat @ X2 @ Z3 )
% 9.03/4.99             => ( ( ( P @ X2 )
% 9.03/4.99                  | ( Q @ X2 ) )
% 9.03/4.99                = ( ( P6 @ X2 )
% 9.03/4.99                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4267_minf_I2_J,axiom,
% 9.03/4.99      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 9.03/4.99        ( ? [Z5: num] :
% 9.03/4.99          ! [X3: num] :
% 9.03/4.99            ( ( ord_less_num @ X3 @ Z5 )
% 9.03/4.99           => ( ( P @ X3 )
% 9.03/4.99              = ( P6 @ X3 ) ) )
% 9.03/4.99       => ( ? [Z5: num] :
% 9.03/4.99            ! [X3: num] :
% 9.03/4.99              ( ( ord_less_num @ X3 @ Z5 )
% 9.03/4.99             => ( ( Q @ X3 )
% 9.03/4.99                = ( Q6 @ X3 ) ) )
% 9.03/4.99         => ? [Z3: num] :
% 9.03/4.99            ! [X2: num] :
% 9.03/4.99              ( ( ord_less_num @ X2 @ Z3 )
% 9.03/4.99             => ( ( ( P @ X2 )
% 9.03/4.99                  | ( Q @ X2 ) )
% 9.03/4.99                = ( ( P6 @ X2 )
% 9.03/4.99                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4268_minf_I2_J,axiom,
% 9.03/4.99      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 9.03/4.99        ( ? [Z5: nat] :
% 9.03/4.99          ! [X3: nat] :
% 9.03/4.99            ( ( ord_less_nat @ X3 @ Z5 )
% 9.03/4.99           => ( ( P @ X3 )
% 9.03/4.99              = ( P6 @ X3 ) ) )
% 9.03/4.99       => ( ? [Z5: nat] :
% 9.03/4.99            ! [X3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ X3 @ Z5 )
% 9.03/4.99             => ( ( Q @ X3 )
% 9.03/4.99                = ( Q6 @ X3 ) ) )
% 9.03/4.99         => ? [Z3: nat] :
% 9.03/4.99            ! [X2: nat] :
% 9.03/4.99              ( ( ord_less_nat @ X2 @ Z3 )
% 9.03/4.99             => ( ( ( P @ X2 )
% 9.03/4.99                  | ( Q @ X2 ) )
% 9.03/4.99                = ( ( P6 @ X2 )
% 9.03/4.99                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4269_minf_I2_J,axiom,
% 9.03/4.99      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 9.03/4.99        ( ? [Z5: int] :
% 9.03/4.99          ! [X3: int] :
% 9.03/4.99            ( ( ord_less_int @ X3 @ Z5 )
% 9.03/4.99           => ( ( P @ X3 )
% 9.03/4.99              = ( P6 @ X3 ) ) )
% 9.03/4.99       => ( ? [Z5: int] :
% 9.03/4.99            ! [X3: int] :
% 9.03/4.99              ( ( ord_less_int @ X3 @ Z5 )
% 9.03/4.99             => ( ( Q @ X3 )
% 9.03/4.99                = ( Q6 @ X3 ) ) )
% 9.03/4.99         => ? [Z3: int] :
% 9.03/4.99            ! [X2: int] :
% 9.03/4.99              ( ( ord_less_int @ X2 @ Z3 )
% 9.03/4.99             => ( ( ( P @ X2 )
% 9.03/4.99                  | ( Q @ X2 ) )
% 9.03/4.99                = ( ( P6 @ X2 )
% 9.03/4.99                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4270_minf_I1_J,axiom,
% 9.03/4.99      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 9.03/4.99        ( ? [Z5: real] :
% 9.03/4.99          ! [X3: real] :
% 9.03/4.99            ( ( ord_less_real @ X3 @ Z5 )
% 9.03/4.99           => ( ( P @ X3 )
% 9.03/4.99              = ( P6 @ X3 ) ) )
% 9.03/4.99       => ( ? [Z5: real] :
% 9.03/4.99            ! [X3: real] :
% 9.03/4.99              ( ( ord_less_real @ X3 @ Z5 )
% 9.03/4.99             => ( ( Q @ X3 )
% 9.03/4.99                = ( Q6 @ X3 ) ) )
% 9.03/4.99         => ? [Z3: real] :
% 9.03/4.99            ! [X2: real] :
% 9.03/4.99              ( ( ord_less_real @ X2 @ Z3 )
% 9.03/4.99             => ( ( ( P @ X2 )
% 9.03/4.99                  & ( Q @ X2 ) )
% 9.03/4.99                = ( ( P6 @ X2 )
% 9.03/4.99                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4271_minf_I1_J,axiom,
% 9.03/4.99      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 9.03/4.99        ( ? [Z5: rat] :
% 9.03/4.99          ! [X3: rat] :
% 9.03/4.99            ( ( ord_less_rat @ X3 @ Z5 )
% 9.03/4.99           => ( ( P @ X3 )
% 9.03/4.99              = ( P6 @ X3 ) ) )
% 9.03/4.99       => ( ? [Z5: rat] :
% 9.03/4.99            ! [X3: rat] :
% 9.03/4.99              ( ( ord_less_rat @ X3 @ Z5 )
% 9.03/4.99             => ( ( Q @ X3 )
% 9.03/4.99                = ( Q6 @ X3 ) ) )
% 9.03/4.99         => ? [Z3: rat] :
% 9.03/4.99            ! [X2: rat] :
% 9.03/4.99              ( ( ord_less_rat @ X2 @ Z3 )
% 9.03/4.99             => ( ( ( P @ X2 )
% 9.03/4.99                  & ( Q @ X2 ) )
% 9.03/4.99                = ( ( P6 @ X2 )
% 9.03/4.99                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4272_minf_I1_J,axiom,
% 9.03/4.99      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 9.03/4.99        ( ? [Z5: num] :
% 9.03/4.99          ! [X3: num] :
% 9.03/4.99            ( ( ord_less_num @ X3 @ Z5 )
% 9.03/4.99           => ( ( P @ X3 )
% 9.03/4.99              = ( P6 @ X3 ) ) )
% 9.03/4.99       => ( ? [Z5: num] :
% 9.03/4.99            ! [X3: num] :
% 9.03/4.99              ( ( ord_less_num @ X3 @ Z5 )
% 9.03/4.99             => ( ( Q @ X3 )
% 9.03/4.99                = ( Q6 @ X3 ) ) )
% 9.03/4.99         => ? [Z3: num] :
% 9.03/4.99            ! [X2: num] :
% 9.03/4.99              ( ( ord_less_num @ X2 @ Z3 )
% 9.03/4.99             => ( ( ( P @ X2 )
% 9.03/4.99                  & ( Q @ X2 ) )
% 9.03/4.99                = ( ( P6 @ X2 )
% 9.03/4.99                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4273_minf_I1_J,axiom,
% 9.03/4.99      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 9.03/4.99        ( ? [Z5: nat] :
% 9.03/4.99          ! [X3: nat] :
% 9.03/4.99            ( ( ord_less_nat @ X3 @ Z5 )
% 9.03/4.99           => ( ( P @ X3 )
% 9.03/4.99              = ( P6 @ X3 ) ) )
% 9.03/4.99       => ( ? [Z5: nat] :
% 9.03/4.99            ! [X3: nat] :
% 9.03/4.99              ( ( ord_less_nat @ X3 @ Z5 )
% 9.03/4.99             => ( ( Q @ X3 )
% 9.03/4.99                = ( Q6 @ X3 ) ) )
% 9.03/4.99         => ? [Z3: nat] :
% 9.03/4.99            ! [X2: nat] :
% 9.03/4.99              ( ( ord_less_nat @ X2 @ Z3 )
% 9.03/4.99             => ( ( ( P @ X2 )
% 9.03/4.99                  & ( Q @ X2 ) )
% 9.03/4.99                = ( ( P6 @ X2 )
% 9.03/4.99                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/4.99  
% 9.03/4.99  thf(fact_4274_minf_I1_J,axiom,
% 9.03/4.99      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 9.03/4.99        ( ? [Z5: int] :
% 9.03/4.99          ! [X3: int] :
% 9.03/4.99            ( ( ord_less_int @ X3 @ Z5 )
% 9.03/4.99           => ( ( P @ X3 )
% 9.03/4.99              = ( P6 @ X3 ) ) )
% 9.03/4.99       => ( ? [Z5: int] :
% 9.03/4.99            ! [X3: int] :
% 9.03/4.99              ( ( ord_less_int @ X3 @ Z5 )
% 9.03/5.00             => ( ( Q @ X3 )
% 9.03/5.00                = ( Q6 @ X3 ) ) )
% 9.03/5.00         => ? [Z3: int] :
% 9.03/5.00            ! [X2: int] :
% 9.03/5.00              ( ( ord_less_int @ X2 @ Z3 )
% 9.03/5.00             => ( ( ( P @ X2 )
% 9.03/5.00                  & ( Q @ X2 ) )
% 9.03/5.00                = ( ( P6 @ X2 )
% 9.03/5.00                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4275_pinf_I7_J,axiom,
% 9.03/5.00      ! [T: real] :
% 9.03/5.00      ? [Z3: real] :
% 9.03/5.00      ! [X2: real] :
% 9.03/5.00        ( ( ord_less_real @ Z3 @ X2 )
% 9.03/5.00       => ( ord_less_real @ T @ X2 ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4276_pinf_I7_J,axiom,
% 9.03/5.00      ! [T: rat] :
% 9.03/5.00      ? [Z3: rat] :
% 9.03/5.00      ! [X2: rat] :
% 9.03/5.00        ( ( ord_less_rat @ Z3 @ X2 )
% 9.03/5.00       => ( ord_less_rat @ T @ X2 ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4277_pinf_I7_J,axiom,
% 9.03/5.00      ! [T: num] :
% 9.03/5.00      ? [Z3: num] :
% 9.03/5.00      ! [X2: num] :
% 9.03/5.00        ( ( ord_less_num @ Z3 @ X2 )
% 9.03/5.00       => ( ord_less_num @ T @ X2 ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4278_pinf_I7_J,axiom,
% 9.03/5.00      ! [T: nat] :
% 9.03/5.00      ? [Z3: nat] :
% 9.03/5.00      ! [X2: nat] :
% 9.03/5.00        ( ( ord_less_nat @ Z3 @ X2 )
% 9.03/5.00       => ( ord_less_nat @ T @ X2 ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4279_pinf_I7_J,axiom,
% 9.03/5.00      ! [T: int] :
% 9.03/5.00      ? [Z3: int] :
% 9.03/5.00      ! [X2: int] :
% 9.03/5.00        ( ( ord_less_int @ Z3 @ X2 )
% 9.03/5.00       => ( ord_less_int @ T @ X2 ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4280_pinf_I5_J,axiom,
% 9.03/5.00      ! [T: real] :
% 9.03/5.00      ? [Z3: real] :
% 9.03/5.00      ! [X2: real] :
% 9.03/5.00        ( ( ord_less_real @ Z3 @ X2 )
% 9.03/5.00       => ~ ( ord_less_real @ X2 @ T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4281_pinf_I5_J,axiom,
% 9.03/5.00      ! [T: rat] :
% 9.03/5.00      ? [Z3: rat] :
% 9.03/5.00      ! [X2: rat] :
% 9.03/5.00        ( ( ord_less_rat @ Z3 @ X2 )
% 9.03/5.00       => ~ ( ord_less_rat @ X2 @ T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4282_pinf_I5_J,axiom,
% 9.03/5.00      ! [T: num] :
% 9.03/5.00      ? [Z3: num] :
% 9.03/5.00      ! [X2: num] :
% 9.03/5.00        ( ( ord_less_num @ Z3 @ X2 )
% 9.03/5.00       => ~ ( ord_less_num @ X2 @ T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4283_pinf_I5_J,axiom,
% 9.03/5.00      ! [T: nat] :
% 9.03/5.00      ? [Z3: nat] :
% 9.03/5.00      ! [X2: nat] :
% 9.03/5.00        ( ( ord_less_nat @ Z3 @ X2 )
% 9.03/5.00       => ~ ( ord_less_nat @ X2 @ T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4284_pinf_I5_J,axiom,
% 9.03/5.00      ! [T: int] :
% 9.03/5.00      ? [Z3: int] :
% 9.03/5.00      ! [X2: int] :
% 9.03/5.00        ( ( ord_less_int @ Z3 @ X2 )
% 9.03/5.00       => ~ ( ord_less_int @ X2 @ T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4285_pinf_I4_J,axiom,
% 9.03/5.00      ! [T: real] :
% 9.03/5.00      ? [Z3: real] :
% 9.03/5.00      ! [X2: real] :
% 9.03/5.00        ( ( ord_less_real @ Z3 @ X2 )
% 9.03/5.00       => ( X2 != T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4286_pinf_I4_J,axiom,
% 9.03/5.00      ! [T: rat] :
% 9.03/5.00      ? [Z3: rat] :
% 9.03/5.00      ! [X2: rat] :
% 9.03/5.00        ( ( ord_less_rat @ Z3 @ X2 )
% 9.03/5.00       => ( X2 != T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4287_pinf_I4_J,axiom,
% 9.03/5.00      ! [T: num] :
% 9.03/5.00      ? [Z3: num] :
% 9.03/5.00      ! [X2: num] :
% 9.03/5.00        ( ( ord_less_num @ Z3 @ X2 )
% 9.03/5.00       => ( X2 != T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4288_pinf_I4_J,axiom,
% 9.03/5.00      ! [T: nat] :
% 9.03/5.00      ? [Z3: nat] :
% 9.03/5.00      ! [X2: nat] :
% 9.03/5.00        ( ( ord_less_nat @ Z3 @ X2 )
% 9.03/5.00       => ( X2 != T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4289_pinf_I4_J,axiom,
% 9.03/5.00      ! [T: int] :
% 9.03/5.00      ? [Z3: int] :
% 9.03/5.00      ! [X2: int] :
% 9.03/5.00        ( ( ord_less_int @ Z3 @ X2 )
% 9.03/5.00       => ( X2 != T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4290_pinf_I3_J,axiom,
% 9.03/5.00      ! [T: real] :
% 9.03/5.00      ? [Z3: real] :
% 9.03/5.00      ! [X2: real] :
% 9.03/5.00        ( ( ord_less_real @ Z3 @ X2 )
% 9.03/5.00       => ( X2 != T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4291_pinf_I3_J,axiom,
% 9.03/5.00      ! [T: rat] :
% 9.03/5.00      ? [Z3: rat] :
% 9.03/5.00      ! [X2: rat] :
% 9.03/5.00        ( ( ord_less_rat @ Z3 @ X2 )
% 9.03/5.00       => ( X2 != T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4292_pinf_I3_J,axiom,
% 9.03/5.00      ! [T: num] :
% 9.03/5.00      ? [Z3: num] :
% 9.03/5.00      ! [X2: num] :
% 9.03/5.00        ( ( ord_less_num @ Z3 @ X2 )
% 9.03/5.00       => ( X2 != T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4293_pinf_I3_J,axiom,
% 9.03/5.00      ! [T: nat] :
% 9.03/5.00      ? [Z3: nat] :
% 9.03/5.00      ! [X2: nat] :
% 9.03/5.00        ( ( ord_less_nat @ Z3 @ X2 )
% 9.03/5.00       => ( X2 != T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4294_pinf_I3_J,axiom,
% 9.03/5.00      ! [T: int] :
% 9.03/5.00      ? [Z3: int] :
% 9.03/5.00      ! [X2: int] :
% 9.03/5.00        ( ( ord_less_int @ Z3 @ X2 )
% 9.03/5.00       => ( X2 != T ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4295_pinf_I2_J,axiom,
% 9.03/5.00      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 9.03/5.00        ( ? [Z5: real] :
% 9.03/5.00          ! [X3: real] :
% 9.03/5.00            ( ( ord_less_real @ Z5 @ X3 )
% 9.03/5.00           => ( ( P @ X3 )
% 9.03/5.00              = ( P6 @ X3 ) ) )
% 9.03/5.00       => ( ? [Z5: real] :
% 9.03/5.00            ! [X3: real] :
% 9.03/5.00              ( ( ord_less_real @ Z5 @ X3 )
% 9.03/5.00             => ( ( Q @ X3 )
% 9.03/5.00                = ( Q6 @ X3 ) ) )
% 9.03/5.00         => ? [Z3: real] :
% 9.03/5.00            ! [X2: real] :
% 9.03/5.00              ( ( ord_less_real @ Z3 @ X2 )
% 9.03/5.00             => ( ( ( P @ X2 )
% 9.03/5.00                  | ( Q @ X2 ) )
% 9.03/5.00                = ( ( P6 @ X2 )
% 9.03/5.00                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4296_pinf_I2_J,axiom,
% 9.03/5.00      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 9.03/5.00        ( ? [Z5: rat] :
% 9.03/5.00          ! [X3: rat] :
% 9.03/5.00            ( ( ord_less_rat @ Z5 @ X3 )
% 9.03/5.00           => ( ( P @ X3 )
% 9.03/5.00              = ( P6 @ X3 ) ) )
% 9.03/5.00       => ( ? [Z5: rat] :
% 9.03/5.00            ! [X3: rat] :
% 9.03/5.00              ( ( ord_less_rat @ Z5 @ X3 )
% 9.03/5.00             => ( ( Q @ X3 )
% 9.03/5.00                = ( Q6 @ X3 ) ) )
% 9.03/5.00         => ? [Z3: rat] :
% 9.03/5.00            ! [X2: rat] :
% 9.03/5.00              ( ( ord_less_rat @ Z3 @ X2 )
% 9.03/5.00             => ( ( ( P @ X2 )
% 9.03/5.00                  | ( Q @ X2 ) )
% 9.03/5.00                = ( ( P6 @ X2 )
% 9.03/5.00                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4297_pinf_I2_J,axiom,
% 9.03/5.00      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 9.03/5.00        ( ? [Z5: num] :
% 9.03/5.00          ! [X3: num] :
% 9.03/5.00            ( ( ord_less_num @ Z5 @ X3 )
% 9.03/5.00           => ( ( P @ X3 )
% 9.03/5.00              = ( P6 @ X3 ) ) )
% 9.03/5.00       => ( ? [Z5: num] :
% 9.03/5.00            ! [X3: num] :
% 9.03/5.00              ( ( ord_less_num @ Z5 @ X3 )
% 9.03/5.00             => ( ( Q @ X3 )
% 9.03/5.00                = ( Q6 @ X3 ) ) )
% 9.03/5.00         => ? [Z3: num] :
% 9.03/5.00            ! [X2: num] :
% 9.03/5.00              ( ( ord_less_num @ Z3 @ X2 )
% 9.03/5.00             => ( ( ( P @ X2 )
% 9.03/5.00                  | ( Q @ X2 ) )
% 9.03/5.00                = ( ( P6 @ X2 )
% 9.03/5.00                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4298_pinf_I2_J,axiom,
% 9.03/5.00      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 9.03/5.00        ( ? [Z5: nat] :
% 9.03/5.00          ! [X3: nat] :
% 9.03/5.00            ( ( ord_less_nat @ Z5 @ X3 )
% 9.03/5.00           => ( ( P @ X3 )
% 9.03/5.00              = ( P6 @ X3 ) ) )
% 9.03/5.00       => ( ? [Z5: nat] :
% 9.03/5.00            ! [X3: nat] :
% 9.03/5.00              ( ( ord_less_nat @ Z5 @ X3 )
% 9.03/5.00             => ( ( Q @ X3 )
% 9.03/5.00                = ( Q6 @ X3 ) ) )
% 9.03/5.00         => ? [Z3: nat] :
% 9.03/5.00            ! [X2: nat] :
% 9.03/5.00              ( ( ord_less_nat @ Z3 @ X2 )
% 9.03/5.00             => ( ( ( P @ X2 )
% 9.03/5.00                  | ( Q @ X2 ) )
% 9.03/5.00                = ( ( P6 @ X2 )
% 9.03/5.00                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4299_pinf_I2_J,axiom,
% 9.03/5.00      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 9.03/5.00        ( ? [Z5: int] :
% 9.03/5.00          ! [X3: int] :
% 9.03/5.00            ( ( ord_less_int @ Z5 @ X3 )
% 9.03/5.00           => ( ( P @ X3 )
% 9.03/5.00              = ( P6 @ X3 ) ) )
% 9.03/5.00       => ( ? [Z5: int] :
% 9.03/5.00            ! [X3: int] :
% 9.03/5.00              ( ( ord_less_int @ Z5 @ X3 )
% 9.03/5.00             => ( ( Q @ X3 )
% 9.03/5.00                = ( Q6 @ X3 ) ) )
% 9.03/5.00         => ? [Z3: int] :
% 9.03/5.00            ! [X2: int] :
% 9.03/5.00              ( ( ord_less_int @ Z3 @ X2 )
% 9.03/5.00             => ( ( ( P @ X2 )
% 9.03/5.00                  | ( Q @ X2 ) )
% 9.03/5.00                = ( ( P6 @ X2 )
% 9.03/5.00                  | ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4300_pinf_I1_J,axiom,
% 9.03/5.00      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 9.03/5.00        ( ? [Z5: real] :
% 9.03/5.00          ! [X3: real] :
% 9.03/5.00            ( ( ord_less_real @ Z5 @ X3 )
% 9.03/5.00           => ( ( P @ X3 )
% 9.03/5.00              = ( P6 @ X3 ) ) )
% 9.03/5.00       => ( ? [Z5: real] :
% 9.03/5.00            ! [X3: real] :
% 9.03/5.00              ( ( ord_less_real @ Z5 @ X3 )
% 9.03/5.00             => ( ( Q @ X3 )
% 9.03/5.00                = ( Q6 @ X3 ) ) )
% 9.03/5.00         => ? [Z3: real] :
% 9.03/5.00            ! [X2: real] :
% 9.03/5.00              ( ( ord_less_real @ Z3 @ X2 )
% 9.03/5.00             => ( ( ( P @ X2 )
% 9.03/5.00                  & ( Q @ X2 ) )
% 9.03/5.00                = ( ( P6 @ X2 )
% 9.03/5.00                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4301_pinf_I1_J,axiom,
% 9.03/5.00      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 9.03/5.00        ( ? [Z5: rat] :
% 9.03/5.00          ! [X3: rat] :
% 9.03/5.00            ( ( ord_less_rat @ Z5 @ X3 )
% 9.03/5.00           => ( ( P @ X3 )
% 9.03/5.00              = ( P6 @ X3 ) ) )
% 9.03/5.00       => ( ? [Z5: rat] :
% 9.03/5.00            ! [X3: rat] :
% 9.03/5.00              ( ( ord_less_rat @ Z5 @ X3 )
% 9.03/5.00             => ( ( Q @ X3 )
% 9.03/5.00                = ( Q6 @ X3 ) ) )
% 9.03/5.00         => ? [Z3: rat] :
% 9.03/5.00            ! [X2: rat] :
% 9.03/5.00              ( ( ord_less_rat @ Z3 @ X2 )
% 9.03/5.00             => ( ( ( P @ X2 )
% 9.03/5.00                  & ( Q @ X2 ) )
% 9.03/5.00                = ( ( P6 @ X2 )
% 9.03/5.00                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.00  
% 9.03/5.00  thf(fact_4302_pinf_I1_J,axiom,
% 9.03/5.00      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 9.03/5.00        ( ? [Z5: num] :
% 9.03/5.00          ! [X3: num] :
% 9.03/5.00            ( ( ord_less_num @ Z5 @ X3 )
% 9.03/5.00           => ( ( P @ X3 )
% 9.03/5.00              = ( P6 @ X3 ) ) )
% 9.03/5.00       => ( ? [Z5: num] :
% 9.03/5.00            ! [X3: num] :
% 9.03/5.00              ( ( ord_less_num @ Z5 @ X3 )
% 9.03/5.00             => ( ( Q @ X3 )
% 9.03/5.00                = ( Q6 @ X3 ) ) )
% 9.03/5.00         => ? [Z3: num] :
% 9.03/5.00            ! [X2: num] :
% 9.03/5.00              ( ( ord_less_num @ Z3 @ X2 )
% 9.03/5.00             => ( ( ( P @ X2 )
% 9.03/5.00                  & ( Q @ X2 ) )
% 9.03/5.00                = ( ( P6 @ X2 )
% 9.03/5.00                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4303_pinf_I1_J,axiom,
% 9.03/5.01      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 9.03/5.01        ( ? [Z5: nat] :
% 9.03/5.01          ! [X3: nat] :
% 9.03/5.01            ( ( ord_less_nat @ Z5 @ X3 )
% 9.03/5.01           => ( ( P @ X3 )
% 9.03/5.01              = ( P6 @ X3 ) ) )
% 9.03/5.01       => ( ? [Z5: nat] :
% 9.03/5.01            ! [X3: nat] :
% 9.03/5.01              ( ( ord_less_nat @ Z5 @ X3 )
% 9.03/5.01             => ( ( Q @ X3 )
% 9.03/5.01                = ( Q6 @ X3 ) ) )
% 9.03/5.01         => ? [Z3: nat] :
% 9.03/5.01            ! [X2: nat] :
% 9.03/5.01              ( ( ord_less_nat @ Z3 @ X2 )
% 9.03/5.01             => ( ( ( P @ X2 )
% 9.03/5.01                  & ( Q @ X2 ) )
% 9.03/5.01                = ( ( P6 @ X2 )
% 9.03/5.01                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4304_pinf_I1_J,axiom,
% 9.03/5.01      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 9.03/5.01        ( ? [Z5: int] :
% 9.03/5.01          ! [X3: int] :
% 9.03/5.01            ( ( ord_less_int @ Z5 @ X3 )
% 9.03/5.01           => ( ( P @ X3 )
% 9.03/5.01              = ( P6 @ X3 ) ) )
% 9.03/5.01       => ( ? [Z5: int] :
% 9.03/5.01            ! [X3: int] :
% 9.03/5.01              ( ( ord_less_int @ Z5 @ X3 )
% 9.03/5.01             => ( ( Q @ X3 )
% 9.03/5.01                = ( Q6 @ X3 ) ) )
% 9.03/5.01         => ? [Z3: int] :
% 9.03/5.01            ! [X2: int] :
% 9.03/5.01              ( ( ord_less_int @ Z3 @ X2 )
% 9.03/5.01             => ( ( ( P @ X2 )
% 9.03/5.01                  & ( Q @ X2 ) )
% 9.03/5.01                = ( ( P6 @ X2 )
% 9.03/5.01                  & ( Q6 @ X2 ) ) ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4305_minf_I8_J,axiom,
% 9.03/5.01      ! [T: real] :
% 9.03/5.01      ? [Z3: real] :
% 9.03/5.01      ! [X2: real] :
% 9.03/5.01        ( ( ord_less_real @ X2 @ Z3 )
% 9.03/5.01       => ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4306_minf_I8_J,axiom,
% 9.03/5.01      ! [T: rat] :
% 9.03/5.01      ? [Z3: rat] :
% 9.03/5.01      ! [X2: rat] :
% 9.03/5.01        ( ( ord_less_rat @ X2 @ Z3 )
% 9.03/5.01       => ~ ( ord_less_eq_rat @ T @ X2 ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4307_minf_I8_J,axiom,
% 9.03/5.01      ! [T: num] :
% 9.03/5.01      ? [Z3: num] :
% 9.03/5.01      ! [X2: num] :
% 9.03/5.01        ( ( ord_less_num @ X2 @ Z3 )
% 9.03/5.01       => ~ ( ord_less_eq_num @ T @ X2 ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4308_minf_I8_J,axiom,
% 9.03/5.01      ! [T: nat] :
% 9.03/5.01      ? [Z3: nat] :
% 9.03/5.01      ! [X2: nat] :
% 9.03/5.01        ( ( ord_less_nat @ X2 @ Z3 )
% 9.03/5.01       => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4309_minf_I8_J,axiom,
% 9.03/5.01      ! [T: int] :
% 9.03/5.01      ? [Z3: int] :
% 9.03/5.01      ! [X2: int] :
% 9.03/5.01        ( ( ord_less_int @ X2 @ Z3 )
% 9.03/5.01       => ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4310_minf_I6_J,axiom,
% 9.03/5.01      ! [T: real] :
% 9.03/5.01      ? [Z3: real] :
% 9.03/5.01      ! [X2: real] :
% 9.03/5.01        ( ( ord_less_real @ X2 @ Z3 )
% 9.03/5.01       => ( ord_less_eq_real @ X2 @ T ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4311_minf_I6_J,axiom,
% 9.03/5.01      ! [T: rat] :
% 9.03/5.01      ? [Z3: rat] :
% 9.03/5.01      ! [X2: rat] :
% 9.03/5.01        ( ( ord_less_rat @ X2 @ Z3 )
% 9.03/5.01       => ( ord_less_eq_rat @ X2 @ T ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4312_minf_I6_J,axiom,
% 9.03/5.01      ! [T: num] :
% 9.03/5.01      ? [Z3: num] :
% 9.03/5.01      ! [X2: num] :
% 9.03/5.01        ( ( ord_less_num @ X2 @ Z3 )
% 9.03/5.01       => ( ord_less_eq_num @ X2 @ T ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4313_minf_I6_J,axiom,
% 9.03/5.01      ! [T: nat] :
% 9.03/5.01      ? [Z3: nat] :
% 9.03/5.01      ! [X2: nat] :
% 9.03/5.01        ( ( ord_less_nat @ X2 @ Z3 )
% 9.03/5.01       => ( ord_less_eq_nat @ X2 @ T ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4314_minf_I6_J,axiom,
% 9.03/5.01      ! [T: int] :
% 9.03/5.01      ? [Z3: int] :
% 9.03/5.01      ! [X2: int] :
% 9.03/5.01        ( ( ord_less_int @ X2 @ Z3 )
% 9.03/5.01       => ( ord_less_eq_int @ X2 @ T ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4315_pinf_I8_J,axiom,
% 9.03/5.01      ! [T: real] :
% 9.03/5.01      ? [Z3: real] :
% 9.03/5.01      ! [X2: real] :
% 9.03/5.01        ( ( ord_less_real @ Z3 @ X2 )
% 9.03/5.01       => ( ord_less_eq_real @ T @ X2 ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4316_pinf_I8_J,axiom,
% 9.03/5.01      ! [T: rat] :
% 9.03/5.01      ? [Z3: rat] :
% 9.03/5.01      ! [X2: rat] :
% 9.03/5.01        ( ( ord_less_rat @ Z3 @ X2 )
% 9.03/5.01       => ( ord_less_eq_rat @ T @ X2 ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4317_pinf_I8_J,axiom,
% 9.03/5.01      ! [T: num] :
% 9.03/5.01      ? [Z3: num] :
% 9.03/5.01      ! [X2: num] :
% 9.03/5.01        ( ( ord_less_num @ Z3 @ X2 )
% 9.03/5.01       => ( ord_less_eq_num @ T @ X2 ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4318_pinf_I8_J,axiom,
% 9.03/5.01      ! [T: nat] :
% 9.03/5.01      ? [Z3: nat] :
% 9.03/5.01      ! [X2: nat] :
% 9.03/5.01        ( ( ord_less_nat @ Z3 @ X2 )
% 9.03/5.01       => ( ord_less_eq_nat @ T @ X2 ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4319_pinf_I8_J,axiom,
% 9.03/5.01      ! [T: int] :
% 9.03/5.01      ? [Z3: int] :
% 9.03/5.01      ! [X2: int] :
% 9.03/5.01        ( ( ord_less_int @ Z3 @ X2 )
% 9.03/5.01       => ( ord_less_eq_int @ T @ X2 ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4320_pinf_I6_J,axiom,
% 9.03/5.01      ! [T: real] :
% 9.03/5.01      ? [Z3: real] :
% 9.03/5.01      ! [X2: real] :
% 9.03/5.01        ( ( ord_less_real @ Z3 @ X2 )
% 9.03/5.01       => ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4321_pinf_I6_J,axiom,
% 9.03/5.01      ! [T: rat] :
% 9.03/5.01      ? [Z3: rat] :
% 9.03/5.01      ! [X2: rat] :
% 9.03/5.01        ( ( ord_less_rat @ Z3 @ X2 )
% 9.03/5.01       => ~ ( ord_less_eq_rat @ X2 @ T ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4322_pinf_I6_J,axiom,
% 9.03/5.01      ! [T: num] :
% 9.03/5.01      ? [Z3: num] :
% 9.03/5.01      ! [X2: num] :
% 9.03/5.01        ( ( ord_less_num @ Z3 @ X2 )
% 9.03/5.01       => ~ ( ord_less_eq_num @ X2 @ T ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4323_pinf_I6_J,axiom,
% 9.03/5.01      ! [T: nat] :
% 9.03/5.01      ? [Z3: nat] :
% 9.03/5.01      ! [X2: nat] :
% 9.03/5.01        ( ( ord_less_nat @ Z3 @ X2 )
% 9.03/5.01       => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4324_pinf_I6_J,axiom,
% 9.03/5.01      ! [T: int] :
% 9.03/5.01      ? [Z3: int] :
% 9.03/5.01      ! [X2: int] :
% 9.03/5.01        ( ( ord_less_int @ Z3 @ X2 )
% 9.03/5.01       => ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4325_imp__le__cong,axiom,
% 9.03/5.01      ! [X: int,X7: int,P: $o,P6: $o] :
% 9.03/5.01        ( ( X = X7 )
% 9.03/5.01       => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 9.03/5.01           => ( P = P6 ) )
% 9.03/5.01         => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.03/5.01             => P )
% 9.03/5.01            = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 9.03/5.01             => P6 ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4326_conj__le__cong,axiom,
% 9.03/5.01      ! [X: int,X7: int,P: $o,P6: $o] :
% 9.03/5.01        ( ( X = X7 )
% 9.03/5.01       => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 9.03/5.01           => ( P = P6 ) )
% 9.03/5.01         => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.03/5.01              & P )
% 9.03/5.01            = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 9.03/5.01              & P6 ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4327_Bolzano,axiom,
% 9.03/5.01      ! [A: real,B: real,P: real > real > $o] :
% 9.03/5.01        ( ( ord_less_eq_real @ A @ B )
% 9.03/5.01       => ( ! [A5: real,B6: real,C4: real] :
% 9.03/5.01              ( ( P @ A5 @ B6 )
% 9.03/5.01             => ( ( P @ B6 @ C4 )
% 9.03/5.01               => ( ( ord_less_eq_real @ A5 @ B6 )
% 9.03/5.01                 => ( ( ord_less_eq_real @ B6 @ C4 )
% 9.03/5.01                   => ( P @ A5 @ C4 ) ) ) ) )
% 9.03/5.01         => ( ! [X3: real] :
% 9.03/5.01                ( ( ord_less_eq_real @ A @ X3 )
% 9.03/5.01               => ( ( ord_less_eq_real @ X3 @ B )
% 9.03/5.01                 => ? [D4: real] :
% 9.03/5.01                      ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.03/5.01                      & ! [A5: real,B6: real] :
% 9.03/5.01                          ( ( ( ord_less_eq_real @ A5 @ X3 )
% 9.03/5.01                            & ( ord_less_eq_real @ X3 @ B6 )
% 9.03/5.01                            & ( ord_less_real @ ( minus_minus_real @ B6 @ A5 ) @ D4 ) )
% 9.03/5.01                         => ( P @ A5 @ B6 ) ) ) ) )
% 9.03/5.01           => ( P @ A @ B ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4328_minusinfinity,axiom,
% 9.03/5.01      ! [D: int,P1: int > $o,P: int > $o] :
% 9.03/5.01        ( ( ord_less_int @ zero_zero_int @ D )
% 9.03/5.01       => ( ! [X3: int,K3: int] :
% 9.03/5.01              ( ( P1 @ X3 )
% 9.03/5.01              = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 9.03/5.01         => ( ? [Z5: int] :
% 9.03/5.01              ! [X3: int] :
% 9.03/5.01                ( ( ord_less_int @ X3 @ Z5 )
% 9.03/5.01               => ( ( P @ X3 )
% 9.03/5.01                  = ( P1 @ X3 ) ) )
% 9.03/5.01           => ( ? [X_1: int] : ( P1 @ X_1 )
% 9.03/5.01             => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4329_plusinfinity,axiom,
% 9.03/5.01      ! [D: int,P6: int > $o,P: int > $o] :
% 9.03/5.01        ( ( ord_less_int @ zero_zero_int @ D )
% 9.03/5.01       => ( ! [X3: int,K3: int] :
% 9.03/5.01              ( ( P6 @ X3 )
% 9.03/5.01              = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 9.03/5.01         => ( ? [Z5: int] :
% 9.03/5.01              ! [X3: int] :
% 9.03/5.01                ( ( ord_less_int @ Z5 @ X3 )
% 9.03/5.01               => ( ( P @ X3 )
% 9.03/5.01                  = ( P6 @ X3 ) ) )
% 9.03/5.01           => ( ? [X_1: int] : ( P6 @ X_1 )
% 9.03/5.01             => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4330_in__children__def,axiom,
% 9.03/5.01      ( vEBT_V5917875025757280293ildren
% 9.03/5.01      = ( ^ [N3: nat,TreeList2: list_VEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ N3 ) ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4331_low__def,axiom,
% 9.03/5.01      ( vEBT_VEBT_low
% 9.03/5.01      = ( ^ [X4: nat,N3: nat] : ( modulo_modulo_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4332_tanh__ln__real,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/5.01       => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 9.03/5.01          = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4333_ln__one__minus__pos__lower__bound,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/5.01       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.03/5.01         => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4334_arcosh__def,axiom,
% 9.03/5.01      ( arcosh_real
% 9.03/5.01      = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4335_abs__ln__one__plus__x__minus__x__bound,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.03/5.01       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4336_add_Oinverse__inverse,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 9.03/5.01        = A ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4337_add_Oinverse__inverse,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 9.03/5.01        = A ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4338_add_Oinverse__inverse,axiom,
% 9.03/5.01      ! [A: complex] :
% 9.03/5.01        ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 9.03/5.01        = A ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4339_add_Oinverse__inverse,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = A ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4340_add_Oinverse__inverse,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = A ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4341_neg__equal__iff__equal,axiom,
% 9.03/5.01      ! [A: real,B: real] :
% 9.03/5.01        ( ( ( uminus_uminus_real @ A )
% 9.03/5.01          = ( uminus_uminus_real @ B ) )
% 9.03/5.01        = ( A = B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4342_neg__equal__iff__equal,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( ( uminus_uminus_int @ A )
% 9.03/5.01          = ( uminus_uminus_int @ B ) )
% 9.03/5.01        = ( A = B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4343_neg__equal__iff__equal,axiom,
% 9.03/5.01      ! [A: complex,B: complex] :
% 9.03/5.01        ( ( ( uminus1482373934393186551omplex @ A )
% 9.03/5.01          = ( uminus1482373934393186551omplex @ B ) )
% 9.03/5.01        = ( A = B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4344_neg__equal__iff__equal,axiom,
% 9.03/5.01      ! [A: rat,B: rat] :
% 9.03/5.01        ( ( ( uminus_uminus_rat @ A )
% 9.03/5.01          = ( uminus_uminus_rat @ B ) )
% 9.03/5.01        = ( A = B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4345_neg__equal__iff__equal,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( ( uminus1351360451143612070nteger @ A )
% 9.03/5.01          = ( uminus1351360451143612070nteger @ B ) )
% 9.03/5.01        = ( A = B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4346_verit__minus__simplify_I4_J,axiom,
% 9.03/5.01      ! [B: real] :
% 9.03/5.01        ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4347_verit__minus__simplify_I4_J,axiom,
% 9.03/5.01      ! [B: int] :
% 9.03/5.01        ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4348_verit__minus__simplify_I4_J,axiom,
% 9.03/5.01      ! [B: complex] :
% 9.03/5.01        ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4349_verit__minus__simplify_I4_J,axiom,
% 9.03/5.01      ! [B: rat] :
% 9.03/5.01        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4350_verit__minus__simplify_I4_J,axiom,
% 9.03/5.01      ! [B: code_integer] :
% 9.03/5.01        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4351_abs__idempotent,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 9.03/5.01        = ( abs_abs_real @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4352_abs__idempotent,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 9.03/5.01        = ( abs_abs_int @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4353_abs__idempotent,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 9.03/5.01        = ( abs_abs_rat @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4354_abs__idempotent,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 9.03/5.01        = ( abs_abs_Code_integer @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4355_arsinh__minus__real,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( arsinh_real @ ( uminus_uminus_real @ X ) )
% 9.03/5.01        = ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4356_VEBT_Oinject_I2_J,axiom,
% 9.03/5.01      ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
% 9.03/5.01        ( ( ( vEBT_Leaf @ X21 @ X222 )
% 9.03/5.01          = ( vEBT_Leaf @ Y21 @ Y22 ) )
% 9.03/5.01        = ( ( X21 = Y21 )
% 9.03/5.01          & ( X222 = Y22 ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4357_tanh__real__eq__iff,axiom,
% 9.03/5.01      ! [X: real,Y: real] :
% 9.03/5.01        ( ( ( tanh_real @ X )
% 9.03/5.01          = ( tanh_real @ Y ) )
% 9.03/5.01        = ( X = Y ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4358_neg__le__iff__le,axiom,
% 9.03/5.01      ! [B: real,A: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 9.03/5.01        = ( ord_less_eq_real @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4359_neg__le__iff__le,axiom,
% 9.03/5.01      ! [B: code_integer,A: code_integer] :
% 9.03/5.01        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4360_neg__le__iff__le,axiom,
% 9.03/5.01      ! [B: rat,A: rat] :
% 9.03/5.01        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = ( ord_less_eq_rat @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4361_neg__le__iff__le,axiom,
% 9.03/5.01      ! [B: int,A: int] :
% 9.03/5.01        ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 9.03/5.01        = ( ord_less_eq_int @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4362_add_Oinverse__neutral,axiom,
% 9.03/5.01      ( ( uminus_uminus_real @ zero_zero_real )
% 9.03/5.01      = zero_zero_real ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4363_add_Oinverse__neutral,axiom,
% 9.03/5.01      ( ( uminus_uminus_int @ zero_zero_int )
% 9.03/5.01      = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4364_add_Oinverse__neutral,axiom,
% 9.03/5.01      ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 9.03/5.01      = zero_zero_complex ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4365_add_Oinverse__neutral,axiom,
% 9.03/5.01      ( ( uminus_uminus_rat @ zero_zero_rat )
% 9.03/5.01      = zero_zero_rat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4366_add_Oinverse__neutral,axiom,
% 9.03/5.01      ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 9.03/5.01      = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4367_neg__0__equal__iff__equal,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( zero_zero_real
% 9.03/5.01          = ( uminus_uminus_real @ A ) )
% 9.03/5.01        = ( zero_zero_real = A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4368_neg__0__equal__iff__equal,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( zero_zero_int
% 9.03/5.01          = ( uminus_uminus_int @ A ) )
% 9.03/5.01        = ( zero_zero_int = A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4369_neg__0__equal__iff__equal,axiom,
% 9.03/5.01      ! [A: complex] :
% 9.03/5.01        ( ( zero_zero_complex
% 9.03/5.01          = ( uminus1482373934393186551omplex @ A ) )
% 9.03/5.01        = ( zero_zero_complex = A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4370_neg__0__equal__iff__equal,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( zero_zero_rat
% 9.03/5.01          = ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = ( zero_zero_rat = A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4371_neg__0__equal__iff__equal,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( zero_z3403309356797280102nteger
% 9.03/5.01          = ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = ( zero_z3403309356797280102nteger = A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4372_neg__equal__0__iff__equal,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ( uminus_uminus_real @ A )
% 9.03/5.01          = zero_zero_real )
% 9.03/5.01        = ( A = zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4373_neg__equal__0__iff__equal,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ( uminus_uminus_int @ A )
% 9.03/5.01          = zero_zero_int )
% 9.03/5.01        = ( A = zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4374_neg__equal__0__iff__equal,axiom,
% 9.03/5.01      ! [A: complex] :
% 9.03/5.01        ( ( ( uminus1482373934393186551omplex @ A )
% 9.03/5.01          = zero_zero_complex )
% 9.03/5.01        = ( A = zero_zero_complex ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4375_neg__equal__0__iff__equal,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ( uminus_uminus_rat @ A )
% 9.03/5.01          = zero_zero_rat )
% 9.03/5.01        = ( A = zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4376_neg__equal__0__iff__equal,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ( uminus1351360451143612070nteger @ A )
% 9.03/5.01          = zero_z3403309356797280102nteger )
% 9.03/5.01        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4377_equal__neg__zero,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( A
% 9.03/5.01          = ( uminus_uminus_real @ A ) )
% 9.03/5.01        = ( A = zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4378_equal__neg__zero,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( A
% 9.03/5.01          = ( uminus_uminus_int @ A ) )
% 9.03/5.01        = ( A = zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4379_equal__neg__zero,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( A
% 9.03/5.01          = ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = ( A = zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4380_equal__neg__zero,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( A
% 9.03/5.01          = ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4381_neg__equal__zero,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ( uminus_uminus_real @ A )
% 9.03/5.01          = A )
% 9.03/5.01        = ( A = zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4382_neg__equal__zero,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ( uminus_uminus_int @ A )
% 9.03/5.01          = A )
% 9.03/5.01        = ( A = zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4383_neg__equal__zero,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ( uminus_uminus_rat @ A )
% 9.03/5.01          = A )
% 9.03/5.01        = ( A = zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4384_neg__equal__zero,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ( uminus1351360451143612070nteger @ A )
% 9.03/5.01          = A )
% 9.03/5.01        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4385_neg__less__iff__less,axiom,
% 9.03/5.01      ! [B: real,A: real] :
% 9.03/5.01        ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 9.03/5.01        = ( ord_less_real @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4386_neg__less__iff__less,axiom,
% 9.03/5.01      ! [B: int,A: int] :
% 9.03/5.01        ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 9.03/5.01        = ( ord_less_int @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4387_neg__less__iff__less,axiom,
% 9.03/5.01      ! [B: rat,A: rat] :
% 9.03/5.01        ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = ( ord_less_rat @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4388_neg__less__iff__less,axiom,
% 9.03/5.01      ! [B: code_integer,A: code_integer] :
% 9.03/5.01        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4389_neg__numeral__eq__iff,axiom,
% 9.03/5.01      ! [M: num,N: num] :
% 9.03/5.01        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 9.03/5.01          = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.01        = ( M = N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4390_neg__numeral__eq__iff,axiom,
% 9.03/5.01      ! [M: num,N: num] :
% 9.03/5.01        ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 9.03/5.01          = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.03/5.01        = ( M = N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4391_neg__numeral__eq__iff,axiom,
% 9.03/5.01      ! [M: num,N: num] :
% 9.03/5.01        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 9.03/5.01          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.03/5.01        = ( M = N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4392_neg__numeral__eq__iff,axiom,
% 9.03/5.01      ! [M: num,N: num] :
% 9.03/5.01        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 9.03/5.01          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.01        = ( M = N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4393_neg__numeral__eq__iff,axiom,
% 9.03/5.01      ! [M: num,N: num] :
% 9.03/5.01        ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 9.03/5.01          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.03/5.01        = ( M = N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4394_add__minus__cancel,axiom,
% 9.03/5.01      ! [A: real,B: real] :
% 9.03/5.01        ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4395_add__minus__cancel,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4396_add__minus__cancel,axiom,
% 9.03/5.01      ! [A: complex,B: complex] :
% 9.03/5.01        ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4397_add__minus__cancel,axiom,
% 9.03/5.01      ! [A: rat,B: rat] :
% 9.03/5.01        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4398_add__minus__cancel,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4399_minus__add__cancel,axiom,
% 9.03/5.01      ! [A: real,B: real] :
% 9.03/5.01        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4400_minus__add__cancel,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4401_minus__add__cancel,axiom,
% 9.03/5.01      ! [A: complex,B: complex] :
% 9.03/5.01        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4402_minus__add__cancel,axiom,
% 9.03/5.01      ! [A: rat,B: rat] :
% 9.03/5.01        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4403_minus__add__cancel,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.03/5.01        = B ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4404_minus__add__distrib,axiom,
% 9.03/5.01      ! [A: real,B: real] :
% 9.03/5.01        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 9.03/5.01        = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4405_minus__add__distrib,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 9.03/5.01        = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4406_minus__add__distrib,axiom,
% 9.03/5.01      ! [A: complex,B: complex] :
% 9.03/5.01        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 9.03/5.01        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4407_minus__add__distrib,axiom,
% 9.03/5.01      ! [A: rat,B: rat] :
% 9.03/5.01        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 9.03/5.01        = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4408_minus__add__distrib,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.03/5.01        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4409_minus__diff__eq,axiom,
% 9.03/5.01      ! [A: real,B: real] :
% 9.03/5.01        ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 9.03/5.01        = ( minus_minus_real @ B @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4410_minus__diff__eq,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 9.03/5.01        = ( minus_minus_int @ B @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4411_minus__diff__eq,axiom,
% 9.03/5.01      ! [A: complex,B: complex] :
% 9.03/5.01        ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 9.03/5.01        = ( minus_minus_complex @ B @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4412_minus__diff__eq,axiom,
% 9.03/5.01      ! [A: rat,B: rat] :
% 9.03/5.01        ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 9.03/5.01        = ( minus_minus_rat @ B @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4413_minus__diff__eq,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 9.03/5.01        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4414_div__minus__minus,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 9.03/5.01        = ( divide_divide_int @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4415_div__minus__minus,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 9.03/5.01        = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4416_abs__0__eq,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( zero_z3403309356797280102nteger
% 9.03/5.01          = ( abs_abs_Code_integer @ A ) )
% 9.03/5.01        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4417_abs__0__eq,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( zero_zero_real
% 9.03/5.01          = ( abs_abs_real @ A ) )
% 9.03/5.01        = ( A = zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4418_abs__0__eq,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( zero_zero_rat
% 9.03/5.01          = ( abs_abs_rat @ A ) )
% 9.03/5.01        = ( A = zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4419_abs__0__eq,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( zero_zero_int
% 9.03/5.01          = ( abs_abs_int @ A ) )
% 9.03/5.01        = ( A = zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4420_abs__eq__0,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ( abs_abs_Code_integer @ A )
% 9.03/5.01          = zero_z3403309356797280102nteger )
% 9.03/5.01        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4421_abs__eq__0,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ( abs_abs_real @ A )
% 9.03/5.01          = zero_zero_real )
% 9.03/5.01        = ( A = zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4422_abs__eq__0,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ( abs_abs_rat @ A )
% 9.03/5.01          = zero_zero_rat )
% 9.03/5.01        = ( A = zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4423_abs__eq__0,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ( abs_abs_int @ A )
% 9.03/5.01          = zero_zero_int )
% 9.03/5.01        = ( A = zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4424_abs__zero,axiom,
% 9.03/5.01      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 9.03/5.01      = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4425_abs__zero,axiom,
% 9.03/5.01      ( ( abs_abs_real @ zero_zero_real )
% 9.03/5.01      = zero_zero_real ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4426_abs__zero,axiom,
% 9.03/5.01      ( ( abs_abs_rat @ zero_zero_rat )
% 9.03/5.01      = zero_zero_rat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4427_abs__zero,axiom,
% 9.03/5.01      ( ( abs_abs_int @ zero_zero_int )
% 9.03/5.01      = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4428_abs__0,axiom,
% 9.03/5.01      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 9.03/5.01      = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4429_abs__0,axiom,
% 9.03/5.01      ( ( abs_abs_complex @ zero_zero_complex )
% 9.03/5.01      = zero_zero_complex ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4430_abs__0,axiom,
% 9.03/5.01      ( ( abs_abs_real @ zero_zero_real )
% 9.03/5.01      = zero_zero_real ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4431_abs__0,axiom,
% 9.03/5.01      ( ( abs_abs_rat @ zero_zero_rat )
% 9.03/5.01      = zero_zero_rat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4432_abs__0,axiom,
% 9.03/5.01      ( ( abs_abs_int @ zero_zero_int )
% 9.03/5.01      = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4433_bits__mod__0,axiom,
% 9.03/5.01      ! [A: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 9.03/5.01        = zero_zero_nat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4434_bits__mod__0,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4435_bits__mod__0,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4436_mod__self,axiom,
% 9.03/5.01      ! [A: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ A @ A )
% 9.03/5.01        = zero_zero_nat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4437_mod__self,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ A @ A )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4438_mod__self,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ A @ A )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4439_mod__by__0,axiom,
% 9.03/5.01      ! [A: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 9.03/5.01        = A ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4440_mod__by__0,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ A @ zero_zero_int )
% 9.03/5.01        = A ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4441_mod__by__0,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 9.03/5.01        = A ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4442_mod__0,axiom,
% 9.03/5.01      ! [A: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 9.03/5.01        = zero_zero_nat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4443_mod__0,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4444_mod__0,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4445_abs__numeral,axiom,
% 9.03/5.01      ! [N: num] :
% 9.03/5.01        ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 9.03/5.01        = ( numera6620942414471956472nteger @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4446_abs__numeral,axiom,
% 9.03/5.01      ! [N: num] :
% 9.03/5.01        ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 9.03/5.01        = ( numeral_numeral_real @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4447_abs__numeral,axiom,
% 9.03/5.01      ! [N: num] :
% 9.03/5.01        ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 9.03/5.01        = ( numeral_numeral_rat @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4448_abs__numeral,axiom,
% 9.03/5.01      ! [N: num] :
% 9.03/5.01        ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 9.03/5.01        = ( numeral_numeral_int @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4449_abs__1,axiom,
% 9.03/5.01      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 9.03/5.01      = one_one_Code_integer ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4450_abs__1,axiom,
% 9.03/5.01      ( ( abs_abs_complex @ one_one_complex )
% 9.03/5.01      = one_one_complex ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4451_abs__1,axiom,
% 9.03/5.01      ( ( abs_abs_real @ one_one_real )
% 9.03/5.01      = one_one_real ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4452_abs__1,axiom,
% 9.03/5.01      ( ( abs_abs_rat @ one_one_rat )
% 9.03/5.01      = one_one_rat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4453_abs__1,axiom,
% 9.03/5.01      ( ( abs_abs_int @ one_one_int )
% 9.03/5.01      = one_one_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4454_abs__add__abs,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 9.03/5.01        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4455_abs__add__abs,axiom,
% 9.03/5.01      ! [A: real,B: real] :
% 9.03/5.01        ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 9.03/5.01        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4456_abs__add__abs,axiom,
% 9.03/5.01      ! [A: rat,B: rat] :
% 9.03/5.01        ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 9.03/5.01        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4457_abs__add__abs,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 9.03/5.01        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4458_mod__add__self1,axiom,
% 9.03/5.01      ! [B: nat,A: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.03/5.01        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4459_mod__add__self1,axiom,
% 9.03/5.01      ! [B: int,A: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.03/5.01        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4460_mod__add__self1,axiom,
% 9.03/5.01      ! [B: code_integer,A: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 9.03/5.01        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4461_mod__add__self2,axiom,
% 9.03/5.01      ! [A: nat,B: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.03/5.01        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4462_mod__add__self2,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.03/5.01        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4463_mod__add__self2,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 9.03/5.01        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4464_abs__divide,axiom,
% 9.03/5.01      ! [A: complex,B: complex] :
% 9.03/5.01        ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.03/5.01        = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4465_abs__divide,axiom,
% 9.03/5.01      ! [A: real,B: real] :
% 9.03/5.01        ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 9.03/5.01        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4466_abs__divide,axiom,
% 9.03/5.01      ! [A: rat,B: rat] :
% 9.03/5.01        ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 9.03/5.01        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4467_abs__minus__cancel,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 9.03/5.01        = ( abs_abs_real @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4468_abs__minus__cancel,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 9.03/5.01        = ( abs_abs_int @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4469_abs__minus__cancel,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = ( abs_abs_rat @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4470_abs__minus__cancel,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = ( abs_abs_Code_integer @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4471_of__int__minus,axiom,
% 9.03/5.01      ! [Z: int] :
% 9.03/5.01        ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
% 9.03/5.01        = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4472_of__int__minus,axiom,
% 9.03/5.01      ! [Z: int] :
% 9.03/5.01        ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
% 9.03/5.01        = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4473_of__int__minus,axiom,
% 9.03/5.01      ! [Z: int] :
% 9.03/5.01        ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z ) )
% 9.03/5.01        = ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4474_of__int__minus,axiom,
% 9.03/5.01      ! [Z: int] :
% 9.03/5.01        ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z ) )
% 9.03/5.01        = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4475_of__int__minus,axiom,
% 9.03/5.01      ! [Z: int] :
% 9.03/5.01        ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z ) )
% 9.03/5.01        = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4476_abs__of__nat,axiom,
% 9.03/5.01      ! [N: nat] :
% 9.03/5.01        ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 9.03/5.01        = ( semiri4939895301339042750nteger @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4477_abs__of__nat,axiom,
% 9.03/5.01      ! [N: nat] :
% 9.03/5.01        ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/5.01        = ( semiri5074537144036343181t_real @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4478_abs__of__nat,axiom,
% 9.03/5.01      ! [N: nat] :
% 9.03/5.01        ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.03/5.01        = ( semiri681578069525770553at_rat @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4479_abs__of__nat,axiom,
% 9.03/5.01      ! [N: nat] :
% 9.03/5.01        ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.03/5.01        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4480_of__int__abs,axiom,
% 9.03/5.01      ! [X: int] :
% 9.03/5.01        ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
% 9.03/5.01        = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4481_of__int__abs,axiom,
% 9.03/5.01      ! [X: int] :
% 9.03/5.01        ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
% 9.03/5.01        = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4482_of__int__abs,axiom,
% 9.03/5.01      ! [X: int] :
% 9.03/5.01        ( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
% 9.03/5.01        = ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4483_of__int__abs,axiom,
% 9.03/5.01      ! [X: int] :
% 9.03/5.01        ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
% 9.03/5.01        = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4484_mod__less,axiom,
% 9.03/5.01      ! [M: nat,N: nat] :
% 9.03/5.01        ( ( ord_less_nat @ M @ N )
% 9.03/5.01       => ( ( modulo_modulo_nat @ M @ N )
% 9.03/5.01          = M ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4485_tanh__0,axiom,
% 9.03/5.01      ( ( tanh_complex @ zero_zero_complex )
% 9.03/5.01      = zero_zero_complex ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4486_tanh__0,axiom,
% 9.03/5.01      ( ( tanh_real @ zero_zero_real )
% 9.03/5.01      = zero_zero_real ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4487_tanh__minus,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( tanh_real @ ( uminus_uminus_real @ X ) )
% 9.03/5.01        = ( uminus_uminus_real @ ( tanh_real @ X ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4488_tanh__minus,axiom,
% 9.03/5.01      ! [X: complex] :
% 9.03/5.01        ( ( tanh_complex @ ( uminus1482373934393186551omplex @ X ) )
% 9.03/5.01        = ( uminus1482373934393186551omplex @ ( tanh_complex @ X ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4489_tanh__real__zero__iff,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( ( tanh_real @ X )
% 9.03/5.01          = zero_zero_real )
% 9.03/5.01        = ( X = zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4490_tanh__real__less__iff,axiom,
% 9.03/5.01      ! [X: real,Y: real] :
% 9.03/5.01        ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
% 9.03/5.01        = ( ord_less_real @ X @ Y ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4491_tanh__real__le__iff,axiom,
% 9.03/5.01      ! [X: real,Y: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
% 9.03/5.01        = ( ord_less_eq_real @ X @ Y ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4492_tanh__real__abs,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( tanh_real @ ( abs_abs_real @ X ) )
% 9.03/5.01        = ( abs_abs_real @ ( tanh_real @ X ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4493_neg__less__eq__nonneg,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 9.03/5.01        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4494_neg__less__eq__nonneg,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.03/5.01        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4495_neg__less__eq__nonneg,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.03/5.01        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4496_neg__less__eq__nonneg,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 9.03/5.01        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4497_less__eq__neg__nonpos,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 9.03/5.01        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4498_less__eq__neg__nonpos,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4499_less__eq__neg__nonpos,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4500_less__eq__neg__nonpos,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 9.03/5.01        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4501_neg__le__0__iff__le,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 9.03/5.01        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4502_neg__le__0__iff__le,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 9.03/5.01        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4503_neg__le__0__iff__le,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 9.03/5.01        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4504_neg__le__0__iff__le,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 9.03/5.01        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4505_neg__0__le__iff__le,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 9.03/5.01        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4506_neg__0__le__iff__le,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4507_neg__0__le__iff__le,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4508_neg__0__le__iff__le,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 9.03/5.01        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4509_less__neg__neg,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 9.03/5.01        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4510_less__neg__neg,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 9.03/5.01        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4511_less__neg__neg,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4512_less__neg__neg,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4513_neg__less__pos,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 9.03/5.01        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4514_neg__less__pos,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 9.03/5.01        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4515_neg__less__pos,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.03/5.01        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4516_neg__less__pos,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.03/5.01        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4517_neg__0__less__iff__less,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 9.03/5.01        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4518_neg__0__less__iff__less,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 9.03/5.01        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4519_neg__0__less__iff__less,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4520_neg__0__less__iff__less,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4521_neg__less__0__iff__less,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 9.03/5.01        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4522_neg__less__0__iff__less,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 9.03/5.01        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4523_neg__less__0__iff__less,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 9.03/5.01        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4524_neg__less__0__iff__less,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 9.03/5.01        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4525_add_Oright__inverse,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 9.03/5.01        = zero_zero_real ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4526_add_Oright__inverse,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4527_add_Oright__inverse,axiom,
% 9.03/5.01      ! [A: complex] :
% 9.03/5.01        ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 9.03/5.01        = zero_zero_complex ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4528_add_Oright__inverse,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.03/5.01        = zero_zero_rat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4529_add_Oright__inverse,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4530_ab__left__minus,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 9.03/5.01        = zero_zero_real ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4531_ab__left__minus,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4532_ab__left__minus,axiom,
% 9.03/5.01      ! [A: complex] :
% 9.03/5.01        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 9.03/5.01        = zero_zero_complex ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4533_ab__left__minus,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.03/5.01        = zero_zero_rat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4534_ab__left__minus,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4535_verit__minus__simplify_I3_J,axiom,
% 9.03/5.01      ! [B: real] :
% 9.03/5.01        ( ( minus_minus_real @ zero_zero_real @ B )
% 9.03/5.01        = ( uminus_uminus_real @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4536_verit__minus__simplify_I3_J,axiom,
% 9.03/5.01      ! [B: int] :
% 9.03/5.01        ( ( minus_minus_int @ zero_zero_int @ B )
% 9.03/5.01        = ( uminus_uminus_int @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4537_verit__minus__simplify_I3_J,axiom,
% 9.03/5.01      ! [B: complex] :
% 9.03/5.01        ( ( minus_minus_complex @ zero_zero_complex @ B )
% 9.03/5.01        = ( uminus1482373934393186551omplex @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4538_verit__minus__simplify_I3_J,axiom,
% 9.03/5.01      ! [B: rat] :
% 9.03/5.01        ( ( minus_minus_rat @ zero_zero_rat @ B )
% 9.03/5.01        = ( uminus_uminus_rat @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4539_verit__minus__simplify_I3_J,axiom,
% 9.03/5.01      ! [B: code_integer] :
% 9.03/5.01        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 9.03/5.01        = ( uminus1351360451143612070nteger @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4540_diff__0,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( minus_minus_real @ zero_zero_real @ A )
% 9.03/5.01        = ( uminus_uminus_real @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4541_diff__0,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( minus_minus_int @ zero_zero_int @ A )
% 9.03/5.01        = ( uminus_uminus_int @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4542_diff__0,axiom,
% 9.03/5.01      ! [A: complex] :
% 9.03/5.01        ( ( minus_minus_complex @ zero_zero_complex @ A )
% 9.03/5.01        = ( uminus1482373934393186551omplex @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4543_diff__0,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( minus_minus_rat @ zero_zero_rat @ A )
% 9.03/5.01        = ( uminus_uminus_rat @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4544_diff__0,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 9.03/5.01        = ( uminus1351360451143612070nteger @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4545_add__neg__numeral__simps_I3_J,axiom,
% 9.03/5.01      ! [M: num,N: num] :
% 9.03/5.01        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.01        = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4546_add__neg__numeral__simps_I3_J,axiom,
% 9.03/5.01      ! [M: num,N: num] :
% 9.03/5.01        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.03/5.01        = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4547_add__neg__numeral__simps_I3_J,axiom,
% 9.03/5.01      ! [M: num,N: num] :
% 9.03/5.01        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.03/5.01        = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4548_add__neg__numeral__simps_I3_J,axiom,
% 9.03/5.01      ! [M: num,N: num] :
% 9.03/5.01        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.01        = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4549_add__neg__numeral__simps_I3_J,axiom,
% 9.03/5.01      ! [M: num,N: num] :
% 9.03/5.01        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.03/5.01        = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4550_mult__minus1,axiom,
% 9.03/5.01      ! [Z: real] :
% 9.03/5.01        ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 9.03/5.01        = ( uminus_uminus_real @ Z ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4551_mult__minus1,axiom,
% 9.03/5.01      ! [Z: int] :
% 9.03/5.01        ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 9.03/5.01        = ( uminus_uminus_int @ Z ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4552_mult__minus1,axiom,
% 9.03/5.01      ! [Z: complex] :
% 9.03/5.01        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 9.03/5.01        = ( uminus1482373934393186551omplex @ Z ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4553_mult__minus1,axiom,
% 9.03/5.01      ! [Z: rat] :
% 9.03/5.01        ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 9.03/5.01        = ( uminus_uminus_rat @ Z ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4554_mult__minus1,axiom,
% 9.03/5.01      ! [Z: code_integer] :
% 9.03/5.01        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 9.03/5.01        = ( uminus1351360451143612070nteger @ Z ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4555_mult__minus1__right,axiom,
% 9.03/5.01      ! [Z: real] :
% 9.03/5.01        ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.01        = ( uminus_uminus_real @ Z ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4556_mult__minus1__right,axiom,
% 9.03/5.01      ! [Z: int] :
% 9.03/5.01        ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.01        = ( uminus_uminus_int @ Z ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4557_mult__minus1__right,axiom,
% 9.03/5.01      ! [Z: complex] :
% 9.03/5.01        ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.03/5.01        = ( uminus1482373934393186551omplex @ Z ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4558_mult__minus1__right,axiom,
% 9.03/5.01      ! [Z: rat] :
% 9.03/5.01        ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.01        = ( uminus_uminus_rat @ Z ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4559_mult__minus1__right,axiom,
% 9.03/5.01      ! [Z: code_integer] :
% 9.03/5.01        ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.01        = ( uminus1351360451143612070nteger @ Z ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4560_abs__le__zero__iff,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 9.03/5.01        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4561_abs__le__zero__iff,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 9.03/5.01        = ( A = zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4562_abs__le__zero__iff,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 9.03/5.01        = ( A = zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4563_abs__le__zero__iff,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 9.03/5.01        = ( A = zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4564_abs__le__self__iff,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 9.03/5.01        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4565_abs__le__self__iff,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 9.03/5.01        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4566_abs__le__self__iff,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 9.03/5.01        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4567_abs__le__self__iff,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 9.03/5.01        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4568_abs__of__nonneg,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.03/5.01       => ( ( abs_abs_Code_integer @ A )
% 9.03/5.01          = A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4569_abs__of__nonneg,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/5.01       => ( ( abs_abs_real @ A )
% 9.03/5.01          = A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4570_abs__of__nonneg,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/5.01       => ( ( abs_abs_rat @ A )
% 9.03/5.01          = A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4571_abs__of__nonneg,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/5.01       => ( ( abs_abs_int @ A )
% 9.03/5.01          = A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4572_uminus__add__conv__diff,axiom,
% 9.03/5.01      ! [A: real,B: real] :
% 9.03/5.01        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 9.03/5.01        = ( minus_minus_real @ B @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4573_uminus__add__conv__diff,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 9.03/5.01        = ( minus_minus_int @ B @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4574_uminus__add__conv__diff,axiom,
% 9.03/5.01      ! [A: complex,B: complex] :
% 9.03/5.01        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 9.03/5.01        = ( minus_minus_complex @ B @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4575_uminus__add__conv__diff,axiom,
% 9.03/5.01      ! [A: rat,B: rat] :
% 9.03/5.01        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.03/5.01        = ( minus_minus_rat @ B @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4576_uminus__add__conv__diff,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.03/5.01        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4577_diff__minus__eq__add,axiom,
% 9.03/5.01      ! [A: real,B: real] :
% 9.03/5.01        ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 9.03/5.01        = ( plus_plus_real @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4578_diff__minus__eq__add,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 9.03/5.01        = ( plus_plus_int @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4579_diff__minus__eq__add,axiom,
% 9.03/5.01      ! [A: complex,B: complex] :
% 9.03/5.01        ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 9.03/5.01        = ( plus_plus_complex @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4580_diff__minus__eq__add,axiom,
% 9.03/5.01      ! [A: rat,B: rat] :
% 9.03/5.01        ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.03/5.01        = ( plus_plus_rat @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4581_diff__minus__eq__add,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.03/5.01        = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4582_zero__less__abs__iff,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 9.03/5.01        = ( A != zero_z3403309356797280102nteger ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4583_zero__less__abs__iff,axiom,
% 9.03/5.01      ! [A: real] :
% 9.03/5.01        ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 9.03/5.01        = ( A != zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4584_zero__less__abs__iff,axiom,
% 9.03/5.01      ! [A: rat] :
% 9.03/5.01        ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 9.03/5.01        = ( A != zero_zero_rat ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4585_zero__less__abs__iff,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 9.03/5.01        = ( A != zero_zero_int ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4586_divide__minus1,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.01        = ( uminus_uminus_real @ X ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4587_divide__minus1,axiom,
% 9.03/5.01      ! [X: complex] :
% 9.03/5.01        ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.03/5.01        = ( uminus1482373934393186551omplex @ X ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4588_divide__minus1,axiom,
% 9.03/5.01      ! [X: rat] :
% 9.03/5.01        ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.01        = ( uminus_uminus_rat @ X ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4589_div__minus1__right,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.01        = ( uminus_uminus_int @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4590_div__minus1__right,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.01        = ( uminus1351360451143612070nteger @ A ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4591_mod__mult__self1__is__0,axiom,
% 9.03/5.01      ! [B: nat,A: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 9.03/5.01        = zero_zero_nat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4592_mod__mult__self1__is__0,axiom,
% 9.03/5.01      ! [B: int,A: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4593_mod__mult__self1__is__0,axiom,
% 9.03/5.01      ! [B: code_integer,A: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4594_mod__mult__self2__is__0,axiom,
% 9.03/5.01      ! [A: nat,B: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 9.03/5.01        = zero_zero_nat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4595_mod__mult__self2__is__0,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4596_mod__mult__self2__is__0,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4597_bits__mod__by__1,axiom,
% 9.03/5.01      ! [A: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 9.03/5.01        = zero_zero_nat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4598_bits__mod__by__1,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ A @ one_one_int )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4599_bits__mod__by__1,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4600_mod__by__1,axiom,
% 9.03/5.01      ! [A: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 9.03/5.01        = zero_zero_nat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4601_mod__by__1,axiom,
% 9.03/5.01      ! [A: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ A @ one_one_int )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4602_mod__by__1,axiom,
% 9.03/5.01      ! [A: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4603_mod__div__trivial,axiom,
% 9.03/5.01      ! [A: nat,B: nat] :
% 9.03/5.01        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 9.03/5.01        = zero_zero_nat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4604_mod__div__trivial,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4605_mod__div__trivial,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4606_bits__mod__div__trivial,axiom,
% 9.03/5.01      ! [A: nat,B: nat] :
% 9.03/5.01        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 9.03/5.01        = zero_zero_nat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4607_bits__mod__div__trivial,axiom,
% 9.03/5.01      ! [A: int,B: int] :
% 9.03/5.01        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 9.03/5.01        = zero_zero_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4608_bits__mod__div__trivial,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer] :
% 9.03/5.01        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 9.03/5.01        = zero_z3403309356797280102nteger ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4609_mod__mult__self4,axiom,
% 9.03/5.01      ! [B: nat,C: nat,A: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 9.03/5.01        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4610_mod__mult__self4,axiom,
% 9.03/5.01      ! [B: int,C: int,A: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 9.03/5.01        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4611_mod__mult__self4,axiom,
% 9.03/5.01      ! [B: code_integer,C: code_integer,A: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 9.03/5.01        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4612_mod__mult__self3,axiom,
% 9.03/5.01      ! [C: nat,B: nat,A: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 9.03/5.01        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4613_mod__mult__self3,axiom,
% 9.03/5.01      ! [C: int,B: int,A: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 9.03/5.01        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4614_mod__mult__self3,axiom,
% 9.03/5.01      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 9.03/5.01        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4615_mod__mult__self2,axiom,
% 9.03/5.01      ! [A: nat,B: nat,C: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 9.03/5.01        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4616_mod__mult__self2,axiom,
% 9.03/5.01      ! [A: int,B: int,C: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 9.03/5.01        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4617_mod__mult__self2,axiom,
% 9.03/5.01      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 9.03/5.01        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4618_mod__mult__self1,axiom,
% 9.03/5.01      ! [A: nat,C: nat,B: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 9.03/5.01        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4619_mod__mult__self1,axiom,
% 9.03/5.01      ! [A: int,C: int,B: int] :
% 9.03/5.01        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 9.03/5.01        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4620_mod__mult__self1,axiom,
% 9.03/5.01      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.03/5.01        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 9.03/5.01        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4621_abs__neg__numeral,axiom,
% 9.03/5.01      ! [N: num] :
% 9.03/5.01        ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.01        = ( numeral_numeral_real @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4622_abs__neg__numeral,axiom,
% 9.03/5.01      ! [N: num] :
% 9.03/5.01        ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.03/5.01        = ( numeral_numeral_int @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4623_abs__neg__numeral,axiom,
% 9.03/5.01      ! [N: num] :
% 9.03/5.01        ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.01        = ( numeral_numeral_rat @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4624_abs__neg__numeral,axiom,
% 9.03/5.01      ! [N: num] :
% 9.03/5.01        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.03/5.01        = ( numera6620942414471956472nteger @ N ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4625_abs__neg__one,axiom,
% 9.03/5.01      ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.01      = one_one_real ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4626_abs__neg__one,axiom,
% 9.03/5.01      ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.01      = one_one_int ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4627_abs__neg__one,axiom,
% 9.03/5.01      ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.01      = one_one_rat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4628_abs__neg__one,axiom,
% 9.03/5.01      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.01      = one_one_Code_integer ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4629_abs__power__minus,axiom,
% 9.03/5.01      ! [A: real,N: nat] :
% 9.03/5.01        ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 9.03/5.01        = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4630_abs__power__minus,axiom,
% 9.03/5.01      ! [A: int,N: nat] :
% 9.03/5.01        ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 9.03/5.01        = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4631_abs__power__minus,axiom,
% 9.03/5.01      ! [A: rat,N: nat] :
% 9.03/5.01        ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 9.03/5.01        = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4632_abs__power__minus,axiom,
% 9.03/5.01      ! [A: code_integer,N: nat] :
% 9.03/5.01        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 9.03/5.01        = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4633_of__real__eq__0__iff,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( ( real_V1803761363581548252l_real @ X )
% 9.03/5.01          = zero_zero_real )
% 9.03/5.01        = ( X = zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4634_of__real__eq__0__iff,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( ( real_V4546457046886955230omplex @ X )
% 9.03/5.01          = zero_zero_complex )
% 9.03/5.01        = ( X = zero_zero_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4635_of__real__0,axiom,
% 9.03/5.01      ( ( real_V1803761363581548252l_real @ zero_zero_real )
% 9.03/5.01      = zero_zero_real ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4636_of__real__0,axiom,
% 9.03/5.01      ( ( real_V4546457046886955230omplex @ zero_zero_real )
% 9.03/5.01      = zero_zero_complex ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4637_of__real__1,axiom,
% 9.03/5.01      ( ( real_V1803761363581548252l_real @ one_one_real )
% 9.03/5.01      = one_one_real ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4638_of__real__1,axiom,
% 9.03/5.01      ( ( real_V4546457046886955230omplex @ one_one_real )
% 9.03/5.01      = one_one_complex ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4639_of__real__eq__1__iff,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( ( real_V1803761363581548252l_real @ X )
% 9.03/5.01          = one_one_real )
% 9.03/5.01        = ( X = one_one_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4640_of__real__eq__1__iff,axiom,
% 9.03/5.01      ! [X: real] :
% 9.03/5.01        ( ( ( real_V4546457046886955230omplex @ X )
% 9.03/5.01          = one_one_complex )
% 9.03/5.01        = ( X = one_one_real ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4641_mod__by__Suc__0,axiom,
% 9.03/5.01      ! [M: nat] :
% 9.03/5.01        ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.03/5.01        = zero_zero_nat ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4642_of__real__numeral,axiom,
% 9.03/5.01      ! [W: num] :
% 9.03/5.01        ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
% 9.03/5.01        = ( numeral_numeral_real @ W ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4643_of__real__numeral,axiom,
% 9.03/5.01      ! [W: num] :
% 9.03/5.01        ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
% 9.03/5.01        = ( numera6690914467698888265omplex @ W ) ) ).
% 9.03/5.01  
% 9.03/5.01  thf(fact_4644_of__real__divide,axiom,
% 9.03/5.01      ! [X: real,Y: real] :
% 9.03/5.01        ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
% 9.03/5.01        = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4645_of__real__divide,axiom,
% 9.03/5.02      ! [X: real,Y: real] :
% 9.03/5.02        ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
% 9.03/5.02        = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4646_real__add__minus__iff,axiom,
% 9.03/5.02      ! [X: real,A: real] :
% 9.03/5.02        ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
% 9.03/5.02          = zero_zero_real )
% 9.03/5.02        = ( X = A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4647_of__real__add,axiom,
% 9.03/5.02      ! [X: real,Y: real] :
% 9.03/5.02        ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
% 9.03/5.02        = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4648_of__real__add,axiom,
% 9.03/5.02      ! [X: real,Y: real] :
% 9.03/5.02        ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X @ Y ) )
% 9.03/5.02        = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4649_of__real__power,axiom,
% 9.03/5.02      ! [X: real,N: nat] :
% 9.03/5.02        ( ( real_V1803761363581548252l_real @ ( power_power_real @ X @ N ) )
% 9.03/5.02        = ( power_power_real @ ( real_V1803761363581548252l_real @ X ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4650_of__real__power,axiom,
% 9.03/5.02      ! [X: real,N: nat] :
% 9.03/5.02        ( ( real_V4546457046886955230omplex @ ( power_power_real @ X @ N ) )
% 9.03/5.02        = ( power_power_complex @ ( real_V4546457046886955230omplex @ X ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4651_floor__uminus__of__int,axiom,
% 9.03/5.02      ! [Z: int] :
% 9.03/5.02        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) )
% 9.03/5.02        = ( uminus_uminus_int @ Z ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4652_floor__uminus__of__int,axiom,
% 9.03/5.02      ! [Z: int] :
% 9.03/5.02        ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) )
% 9.03/5.02        = ( uminus_uminus_int @ Z ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4653_of__real__of__nat__eq,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( real_V4546457046886955230omplex @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/5.02        = ( semiri8010041392384452111omplex @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4654_of__real__of__nat__eq,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( real_V1803761363581548252l_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.03/5.02        = ( semiri5074537144036343181t_real @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4655_of__real__of__int__eq,axiom,
% 9.03/5.02      ! [Z: int] :
% 9.03/5.02        ( ( real_V1803761363581548252l_real @ ( ring_1_of_int_real @ Z ) )
% 9.03/5.02        = ( ring_1_of_int_real @ Z ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4656_of__real__of__int__eq,axiom,
% 9.03/5.02      ! [Z: int] :
% 9.03/5.02        ( ( real_V4546457046886955230omplex @ ( ring_1_of_int_real @ Z ) )
% 9.03/5.02        = ( ring_17405671764205052669omplex @ Z ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4657_tanh__real__neg__iff,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
% 9.03/5.02        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4658_tanh__real__pos__iff,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
% 9.03/5.02        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4659_tanh__real__nonneg__iff,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
% 9.03/5.02        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4660_tanh__real__nonpos__iff,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
% 9.03/5.02        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4661_dbl__simps_I1_J,axiom,
% 9.03/5.02      ! [K: num] :
% 9.03/5.02        ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 9.03/5.02        = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4662_dbl__simps_I1_J,axiom,
% 9.03/5.02      ! [K: num] :
% 9.03/5.02        ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4663_dbl__simps_I1_J,axiom,
% 9.03/5.02      ! [K: num] :
% 9.03/5.02        ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 9.03/5.02        = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4664_dbl__simps_I1_J,axiom,
% 9.03/5.02      ! [K: num] :
% 9.03/5.02        ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 9.03/5.02        = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4665_dbl__simps_I1_J,axiom,
% 9.03/5.02      ! [K: num] :
% 9.03/5.02        ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 9.03/5.02        = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4666_add__neg__numeral__special_I8_J,axiom,
% 9.03/5.02      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 9.03/5.02      = zero_zero_real ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4667_add__neg__numeral__special_I8_J,axiom,
% 9.03/5.02      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 9.03/5.02      = zero_zero_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4668_add__neg__numeral__special_I8_J,axiom,
% 9.03/5.02      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 9.03/5.02      = zero_zero_complex ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4669_add__neg__numeral__special_I8_J,axiom,
% 9.03/5.02      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 9.03/5.02      = zero_zero_rat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4670_add__neg__numeral__special_I8_J,axiom,
% 9.03/5.02      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 9.03/5.02      = zero_z3403309356797280102nteger ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4671_add__neg__numeral__special_I7_J,axiom,
% 9.03/5.02      ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.02      = zero_zero_real ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4672_add__neg__numeral__special_I7_J,axiom,
% 9.03/5.02      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.02      = zero_zero_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4673_add__neg__numeral__special_I7_J,axiom,
% 9.03/5.02      ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.03/5.02      = zero_zero_complex ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4674_add__neg__numeral__special_I7_J,axiom,
% 9.03/5.02      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.02      = zero_zero_rat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4675_add__neg__numeral__special_I7_J,axiom,
% 9.03/5.02      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.02      = zero_z3403309356797280102nteger ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4676_diff__numeral__special_I12_J,axiom,
% 9.03/5.02      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.02      = zero_zero_real ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4677_diff__numeral__special_I12_J,axiom,
% 9.03/5.02      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.02      = zero_zero_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4678_diff__numeral__special_I12_J,axiom,
% 9.03/5.02      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.03/5.02      = zero_zero_complex ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4679_diff__numeral__special_I12_J,axiom,
% 9.03/5.02      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.02      = zero_zero_rat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4680_diff__numeral__special_I12_J,axiom,
% 9.03/5.02      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.02      = zero_z3403309356797280102nteger ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4681_neg__one__eq__numeral__iff,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( ( uminus_uminus_real @ one_one_real )
% 9.03/5.02          = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.02        = ( N = one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4682_neg__one__eq__numeral__iff,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( ( uminus_uminus_int @ one_one_int )
% 9.03/5.02          = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.03/5.02        = ( N = one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4683_neg__one__eq__numeral__iff,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 9.03/5.02          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.03/5.02        = ( N = one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4684_neg__one__eq__numeral__iff,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( ( uminus_uminus_rat @ one_one_rat )
% 9.03/5.02          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.02        = ( N = one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4685_neg__one__eq__numeral__iff,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 9.03/5.02          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.03/5.02        = ( N = one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4686_numeral__eq__neg__one__iff,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 9.03/5.02          = ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.02        = ( N = one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4687_numeral__eq__neg__one__iff,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 9.03/5.02          = ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.02        = ( N = one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4688_numeral__eq__neg__one__iff,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 9.03/5.02          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.03/5.02        = ( N = one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4689_numeral__eq__neg__one__iff,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 9.03/5.02          = ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.02        = ( N = one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4690_numeral__eq__neg__one__iff,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 9.03/5.02          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.02        = ( N = one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4691_zero__le__divide__abs__iff,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 9.03/5.02        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/5.02          | ( B = zero_zero_real ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4692_zero__le__divide__abs__iff,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 9.03/5.02        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/5.02          | ( B = zero_zero_rat ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4693_divide__le__0__abs__iff,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 9.03/5.02        = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/5.02          | ( B = zero_zero_real ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4694_divide__le__0__abs__iff,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 9.03/5.02        = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/5.02          | ( B = zero_zero_rat ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4695_abs__of__nonpos,axiom,
% 9.03/5.02      ! [A: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.03/5.02       => ( ( abs_abs_real @ A )
% 9.03/5.02          = ( uminus_uminus_real @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4696_abs__of__nonpos,axiom,
% 9.03/5.02      ! [A: code_integer] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 9.03/5.02       => ( ( abs_abs_Code_integer @ A )
% 9.03/5.02          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4697_abs__of__nonpos,axiom,
% 9.03/5.02      ! [A: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.03/5.02       => ( ( abs_abs_rat @ A )
% 9.03/5.02          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4698_abs__of__nonpos,axiom,
% 9.03/5.02      ! [A: int] :
% 9.03/5.02        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.03/5.02       => ( ( abs_abs_int @ A )
% 9.03/5.02          = ( uminus_uminus_int @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4699_minus__one__mult__self,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 9.03/5.02        = one_one_real ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4700_minus__one__mult__self,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 9.03/5.02        = one_one_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4701_minus__one__mult__self,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 9.03/5.02        = one_one_complex ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4702_minus__one__mult__self,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 9.03/5.02        = one_one_rat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4703_minus__one__mult__self,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 9.03/5.02        = one_one_Code_integer ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4704_left__minus__one__mult__self,axiom,
% 9.03/5.02      ! [N: nat,A: real] :
% 9.03/5.02        ( ( 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 ) )
% 9.03/5.02        = A ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4705_left__minus__one__mult__self,axiom,
% 9.03/5.02      ! [N: nat,A: int] :
% 9.03/5.02        ( ( 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 ) )
% 9.03/5.02        = A ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4706_left__minus__one__mult__self,axiom,
% 9.03/5.02      ! [N: nat,A: complex] :
% 9.03/5.02        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 9.03/5.02        = A ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4707_left__minus__one__mult__self,axiom,
% 9.03/5.02      ! [N: nat,A: rat] :
% 9.03/5.02        ( ( 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 ) )
% 9.03/5.02        = A ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4708_left__minus__one__mult__self,axiom,
% 9.03/5.02      ! [N: nat,A: code_integer] :
% 9.03/5.02        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 9.03/5.02        = A ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4709_mod__minus1__right,axiom,
% 9.03/5.02      ! [A: int] :
% 9.03/5.02        ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.02        = zero_zero_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4710_mod__minus1__right,axiom,
% 9.03/5.02      ! [A: code_integer] :
% 9.03/5.02        ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.02        = zero_z3403309356797280102nteger ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4711_norm__neg__numeral,axiom,
% 9.03/5.02      ! [W: num] :
% 9.03/5.02        ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.03/5.02        = ( numeral_numeral_real @ W ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4712_norm__neg__numeral,axiom,
% 9.03/5.02      ! [W: num] :
% 9.03/5.02        ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.03/5.02        = ( numeral_numeral_real @ W ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4713_floor__neg__numeral,axiom,
% 9.03/5.02      ! [V: num] :
% 9.03/5.02        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4714_floor__neg__numeral,axiom,
% 9.03/5.02      ! [V: num] :
% 9.03/5.02        ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4715_semiring__norm_I168_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: real] :
% 9.03/5.02        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 9.03/5.02        = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4716_semiring__norm_I168_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: int] :
% 9.03/5.02        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 9.03/5.02        = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4717_semiring__norm_I168_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: complex] :
% 9.03/5.02        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 9.03/5.02        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4718_semiring__norm_I168_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: rat] :
% 9.03/5.02        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 9.03/5.02        = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4719_semiring__norm_I168_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: code_integer] :
% 9.03/5.02        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 9.03/5.02        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4720_ceiling__neg__numeral,axiom,
% 9.03/5.02      ! [V: num] :
% 9.03/5.02        ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4721_ceiling__neg__numeral,axiom,
% 9.03/5.02      ! [V: num] :
% 9.03/5.02        ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4722_diff__numeral__simps_I2_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.02        = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4723_diff__numeral__simps_I2_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.03/5.02        = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4724_diff__numeral__simps_I2_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.03/5.02        = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4725_diff__numeral__simps_I2_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.02        = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4726_diff__numeral__simps_I2_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.03/5.02        = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4727_diff__numeral__simps_I3_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 9.03/5.02        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4728_diff__numeral__simps_I3_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4729_diff__numeral__simps_I3_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 9.03/5.02        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4730_diff__numeral__simps_I3_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 9.03/5.02        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4731_diff__numeral__simps_I3_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 9.03/5.02        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4732_Suc__mod__mult__self1,axiom,
% 9.03/5.02      ! [M: nat,K: nat,N: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 9.03/5.02        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4733_Suc__mod__mult__self2,axiom,
% 9.03/5.02      ! [M: nat,N: nat,K: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 9.03/5.02        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4734_Suc__mod__mult__self3,axiom,
% 9.03/5.02      ! [K: nat,N: nat,M: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 9.03/5.02        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4735_Suc__mod__mult__self4,axiom,
% 9.03/5.02      ! [N: nat,K: nat,M: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 9.03/5.02        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4736_semiring__norm_I172_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: real] :
% 9.03/5.02        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 9.03/5.02        = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4737_semiring__norm_I172_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: int] :
% 9.03/5.02        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 9.03/5.02        = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4738_semiring__norm_I172_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: complex] :
% 9.03/5.02        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 9.03/5.02        = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4739_semiring__norm_I172_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: rat] :
% 9.03/5.02        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 9.03/5.02        = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4740_semiring__norm_I172_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: code_integer] :
% 9.03/5.02        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 9.03/5.02        = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4741_semiring__norm_I171_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: real] :
% 9.03/5.02        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 9.03/5.02        = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4742_semiring__norm_I171_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: int] :
% 9.03/5.02        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 9.03/5.02        = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4743_semiring__norm_I171_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: complex] :
% 9.03/5.02        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 9.03/5.02        = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4744_semiring__norm_I171_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: rat] :
% 9.03/5.02        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 9.03/5.02        = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4745_semiring__norm_I171_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: code_integer] :
% 9.03/5.02        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 9.03/5.02        = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4746_semiring__norm_I170_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: real] :
% 9.03/5.02        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
% 9.03/5.02        = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4747_semiring__norm_I170_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: int] :
% 9.03/5.02        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
% 9.03/5.02        = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4748_semiring__norm_I170_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: complex] :
% 9.03/5.02        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
% 9.03/5.02        = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4749_semiring__norm_I170_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: rat] :
% 9.03/5.02        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
% 9.03/5.02        = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4750_semiring__norm_I170_J,axiom,
% 9.03/5.02      ! [V: num,W: num,Y: code_integer] :
% 9.03/5.02        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
% 9.03/5.02        = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4751_mult__neg__numeral__simps_I3_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.02        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4752_mult__neg__numeral__simps_I3_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4753_mult__neg__numeral__simps_I3_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.03/5.02        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4754_mult__neg__numeral__simps_I3_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.02        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4755_mult__neg__numeral__simps_I3_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.03/5.02        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4756_mult__neg__numeral__simps_I2_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 9.03/5.02        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4757_mult__neg__numeral__simps_I2_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4758_mult__neg__numeral__simps_I2_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 9.03/5.02        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4759_mult__neg__numeral__simps_I2_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 9.03/5.02        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4760_mult__neg__numeral__simps_I2_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 9.03/5.02        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4761_mult__neg__numeral__simps_I1_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.02        = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4762_mult__neg__numeral__simps_I1_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.03/5.02        = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4763_mult__neg__numeral__simps_I1_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.03/5.02        = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4764_mult__neg__numeral__simps_I1_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.02        = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4765_mult__neg__numeral__simps_I1_J,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.03/5.02        = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4766_artanh__minus__real,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.03/5.02       => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
% 9.03/5.02          = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4767_neg__numeral__le__iff,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.02        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4768_neg__numeral__le__iff,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.03/5.02        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4769_neg__numeral__le__iff,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.02        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4770_neg__numeral__le__iff,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.03/5.02        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4771_neg__numeral__less__iff,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.02        = ( ord_less_num @ N @ M ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4772_neg__numeral__less__iff,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.03/5.02        = ( ord_less_num @ N @ M ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4773_neg__numeral__less__iff,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.02        = ( ord_less_num @ N @ M ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4774_neg__numeral__less__iff,axiom,
% 9.03/5.02      ! [M: num,N: num] :
% 9.03/5.02        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.03/5.02        = ( ord_less_num @ N @ M ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4775_round__neg__numeral,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4776_round__neg__numeral,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4777_norm__of__int,axiom,
% 9.03/5.02      ! [Z: int] :
% 9.03/5.02        ( ( real_V7735802525324610683m_real @ ( ring_1_of_int_real @ Z ) )
% 9.03/5.02        = ( abs_abs_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4778_norm__of__int,axiom,
% 9.03/5.02      ! [Z: int] :
% 9.03/5.02        ( ( real_V1022390504157884413omplex @ ( ring_17405671764205052669omplex @ Z ) )
% 9.03/5.02        = ( abs_abs_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4779_not__neg__one__le__neg__numeral__iff,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 9.03/5.02        = ( M != one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4780_not__neg__one__le__neg__numeral__iff,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 9.03/5.02        = ( M != one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4781_not__neg__one__le__neg__numeral__iff,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 9.03/5.02        = ( M != one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4782_not__neg__one__le__neg__numeral__iff,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 9.03/5.02        = ( M != one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4783_le__divide__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [A: real,B: real,W: num] :
% 9.03/5.02        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 9.03/5.02        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4784_le__divide__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [A: rat,B: rat,W: num] :
% 9.03/5.02        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 9.03/5.02        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4785_divide__le__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [B: real,W: num,A: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 9.03/5.02        = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4786_divide__le__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [B: rat,W: num,A: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 9.03/5.02        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4787_eq__divide__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [A: real,B: real,W: num] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 9.03/5.02        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.03/5.02             != zero_zero_real )
% 9.03/5.02           => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.03/5.02              = B ) )
% 9.03/5.02          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.03/5.02              = zero_zero_real )
% 9.03/5.02           => ( A = zero_zero_real ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4788_eq__divide__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [A: complex,B: complex,W: num] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 9.03/5.02        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.03/5.02             != zero_zero_complex )
% 9.03/5.02           => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.03/5.02              = B ) )
% 9.03/5.02          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.03/5.02              = zero_zero_complex )
% 9.03/5.02           => ( A = zero_zero_complex ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4789_eq__divide__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [A: rat,B: rat,W: num] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 9.03/5.02        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.03/5.02             != zero_zero_rat )
% 9.03/5.02           => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.03/5.02              = B ) )
% 9.03/5.02          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.03/5.02              = zero_zero_rat )
% 9.03/5.02           => ( A = zero_zero_rat ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4790_divide__eq__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [B: real,W: num,A: real] :
% 9.03/5.02        ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.03/5.02          = A )
% 9.03/5.02        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.03/5.02             != zero_zero_real )
% 9.03/5.02           => ( B
% 9.03/5.02              = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 9.03/5.02          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.03/5.02              = zero_zero_real )
% 9.03/5.02           => ( A = zero_zero_real ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4791_divide__eq__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [B: complex,W: num,A: complex] :
% 9.03/5.02        ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.03/5.02          = A )
% 9.03/5.02        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.03/5.02             != zero_zero_complex )
% 9.03/5.02           => ( B
% 9.03/5.02              = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 9.03/5.02          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.03/5.02              = zero_zero_complex )
% 9.03/5.02           => ( A = zero_zero_complex ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4792_divide__eq__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [B: rat,W: num,A: rat] :
% 9.03/5.02        ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.03/5.02          = A )
% 9.03/5.02        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.03/5.02             != zero_zero_rat )
% 9.03/5.02           => ( B
% 9.03/5.02              = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 9.03/5.02          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.03/5.02              = zero_zero_rat )
% 9.03/5.02           => ( A = zero_zero_rat ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4793_neg__numeral__less__neg__one__iff,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.02        = ( M != one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4794_neg__numeral__less__neg__one__iff,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.02        = ( M != one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4795_neg__numeral__less__neg__one__iff,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.02        = ( M != one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4796_neg__numeral__less__neg__one__iff,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.02        = ( M != one ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4797_less__divide__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [A: real,B: real,W: num] :
% 9.03/5.02        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 9.03/5.02        = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4798_less__divide__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [A: rat,B: rat,W: num] :
% 9.03/5.02        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 9.03/5.02        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4799_divide__less__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [B: real,W: num,A: real] :
% 9.03/5.02        ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 9.03/5.02        = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4800_divide__less__eq__numeral1_I2_J,axiom,
% 9.03/5.02      ! [B: rat,W: num,A: rat] :
% 9.03/5.02        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 9.03/5.02        = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4801_bits__one__mod__two__eq__one,axiom,
% 9.03/5.02      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02      = one_one_nat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4802_bits__one__mod__two__eq__one,axiom,
% 9.03/5.02      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/5.02      = one_one_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4803_bits__one__mod__two__eq__one,axiom,
% 9.03/5.02      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.03/5.02      = one_one_Code_integer ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4804_one__mod__two__eq__one,axiom,
% 9.03/5.02      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02      = one_one_nat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4805_one__mod__two__eq__one,axiom,
% 9.03/5.02      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/5.02      = one_one_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4806_one__mod__two__eq__one,axiom,
% 9.03/5.02      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.03/5.02      = one_one_Code_integer ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4807_zero__less__power__abs__iff,axiom,
% 9.03/5.02      ! [A: code_integer,N: nat] :
% 9.03/5.02        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 9.03/5.02        = ( ( A != zero_z3403309356797280102nteger )
% 9.03/5.02          | ( N = zero_zero_nat ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4808_zero__less__power__abs__iff,axiom,
% 9.03/5.02      ! [A: real,N: nat] :
% 9.03/5.02        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 9.03/5.02        = ( ( A != zero_zero_real )
% 9.03/5.02          | ( N = zero_zero_nat ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4809_zero__less__power__abs__iff,axiom,
% 9.03/5.02      ! [A: rat,N: nat] :
% 9.03/5.02        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 9.03/5.02        = ( ( A != zero_zero_rat )
% 9.03/5.02          | ( N = zero_zero_nat ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4810_zero__less__power__abs__iff,axiom,
% 9.03/5.02      ! [A: int,N: nat] :
% 9.03/5.02        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 9.03/5.02        = ( ( A != zero_zero_int )
% 9.03/5.02          | ( N = zero_zero_nat ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4811_power2__minus,axiom,
% 9.03/5.02      ! [A: real] :
% 9.03/5.02        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4812_power2__minus,axiom,
% 9.03/5.02      ! [A: int] :
% 9.03/5.02        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4813_power2__minus,axiom,
% 9.03/5.02      ! [A: complex] :
% 9.03/5.02        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4814_power2__minus,axiom,
% 9.03/5.02      ! [A: rat] :
% 9.03/5.02        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4815_power2__minus,axiom,
% 9.03/5.02      ! [A: code_integer] :
% 9.03/5.02        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4816_abs__power2,axiom,
% 9.03/5.02      ! [A: rat] :
% 9.03/5.02        ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/5.02        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4817_abs__power2,axiom,
% 9.03/5.02      ! [A: code_integer] :
% 9.03/5.02        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/5.02        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4818_abs__power2,axiom,
% 9.03/5.02      ! [A: real] :
% 9.03/5.02        ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/5.02        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4819_abs__power2,axiom,
% 9.03/5.02      ! [A: int] :
% 9.03/5.02        ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/5.02        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4820_power2__abs,axiom,
% 9.03/5.02      ! [A: rat] :
% 9.03/5.02        ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4821_power2__abs,axiom,
% 9.03/5.02      ! [A: code_integer] :
% 9.03/5.02        ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4822_power2__abs,axiom,
% 9.03/5.02      ! [A: real] :
% 9.03/5.02        ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4823_power2__abs,axiom,
% 9.03/5.02      ! [A: int] :
% 9.03/5.02        ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4824_mod2__Suc__Suc,axiom,
% 9.03/5.02      ! [M: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4825_of__real__neg__numeral,axiom,
% 9.03/5.02      ! [W: num] :
% 9.03/5.02        ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.03/5.02        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4826_of__real__neg__numeral,axiom,
% 9.03/5.02      ! [W: num] :
% 9.03/5.02        ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.03/5.02        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4827_Suc__times__numeral__mod__eq,axiom,
% 9.03/5.02      ! [K: num,N: nat] :
% 9.03/5.02        ( ( ( numeral_numeral_nat @ K )
% 9.03/5.02         != one_one_nat )
% 9.03/5.02       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 9.03/5.02          = one_one_nat ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4828_add__neg__numeral__special_I9_J,axiom,
% 9.03/5.02      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.02      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4829_add__neg__numeral__special_I9_J,axiom,
% 9.03/5.02      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.02      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4830_add__neg__numeral__special_I9_J,axiom,
% 9.03/5.02      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.03/5.02      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4831_add__neg__numeral__special_I9_J,axiom,
% 9.03/5.02      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.02      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4832_add__neg__numeral__special_I9_J,axiom,
% 9.03/5.02      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.02      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4833_diff__numeral__special_I11_J,axiom,
% 9.03/5.02      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.02      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4834_diff__numeral__special_I11_J,axiom,
% 9.03/5.02      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.02      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4835_diff__numeral__special_I11_J,axiom,
% 9.03/5.02      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.03/5.02      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4836_diff__numeral__special_I11_J,axiom,
% 9.03/5.02      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.02      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4837_diff__numeral__special_I11_J,axiom,
% 9.03/5.02      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.02      = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4838_diff__numeral__special_I10_J,axiom,
% 9.03/5.02      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 9.03/5.02      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4839_diff__numeral__special_I10_J,axiom,
% 9.03/5.02      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 9.03/5.02      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4840_diff__numeral__special_I10_J,axiom,
% 9.03/5.02      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 9.03/5.02      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4841_diff__numeral__special_I10_J,axiom,
% 9.03/5.02      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 9.03/5.02      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4842_diff__numeral__special_I10_J,axiom,
% 9.03/5.02      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 9.03/5.02      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4843_minus__1__div__2__eq,axiom,
% 9.03/5.02      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/5.02      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4844_minus__1__div__2__eq,axiom,
% 9.03/5.02      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.03/5.02      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4845_not__mod__2__eq__0__eq__1,axiom,
% 9.03/5.02      ! [A: nat] :
% 9.03/5.02        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02         != zero_zero_nat )
% 9.03/5.02        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02          = one_one_nat ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4846_not__mod__2__eq__0__eq__1,axiom,
% 9.03/5.02      ! [A: int] :
% 9.03/5.02        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/5.02         != zero_zero_int )
% 9.03/5.02        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/5.02          = one_one_int ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4847_not__mod__2__eq__0__eq__1,axiom,
% 9.03/5.02      ! [A: code_integer] :
% 9.03/5.02        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.03/5.02         != zero_z3403309356797280102nteger )
% 9.03/5.02        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.03/5.02          = one_one_Code_integer ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4848_not__mod__2__eq__1__eq__0,axiom,
% 9.03/5.02      ! [A: nat] :
% 9.03/5.02        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02         != one_one_nat )
% 9.03/5.02        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02          = zero_zero_nat ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4849_not__mod__2__eq__1__eq__0,axiom,
% 9.03/5.02      ! [A: int] :
% 9.03/5.02        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/5.02         != one_one_int )
% 9.03/5.02        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/5.02          = zero_zero_int ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4850_not__mod__2__eq__1__eq__0,axiom,
% 9.03/5.02      ! [A: code_integer] :
% 9.03/5.02        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.03/5.02         != one_one_Code_integer )
% 9.03/5.02        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.03/5.02          = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4851_bits__minus__1__mod__2__eq,axiom,
% 9.03/5.02      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/5.02      = one_one_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4852_bits__minus__1__mod__2__eq,axiom,
% 9.03/5.02      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.03/5.02      = one_one_Code_integer ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4853_minus__1__mod__2__eq,axiom,
% 9.03/5.02      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.03/5.02      = one_one_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4854_minus__1__mod__2__eq,axiom,
% 9.03/5.02      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.03/5.02      = one_one_Code_integer ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4855_Power_Oring__1__class_Opower__minus__even,axiom,
% 9.03/5.02      ! [A: real,N: nat] :
% 9.03/5.02        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/5.02        = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4856_Power_Oring__1__class_Opower__minus__even,axiom,
% 9.03/5.02      ! [A: int,N: nat] :
% 9.03/5.02        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/5.02        = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4857_Power_Oring__1__class_Opower__minus__even,axiom,
% 9.03/5.02      ! [A: complex,N: nat] :
% 9.03/5.02        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/5.02        = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4858_Power_Oring__1__class_Opower__minus__even,axiom,
% 9.03/5.02      ! [A: rat,N: nat] :
% 9.03/5.02        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/5.02        = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4859_Power_Oring__1__class_Opower__minus__even,axiom,
% 9.03/5.02      ! [A: code_integer,N: nat] :
% 9.03/5.02        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/5.02        = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4860_not__mod2__eq__Suc__0__eq__0,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02         != ( suc @ zero_zero_nat ) )
% 9.03/5.02        = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02          = zero_zero_nat ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4861_diff__numeral__special_I4_J,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 9.03/5.02        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4862_diff__numeral__special_I4_J,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 9.03/5.02        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4863_diff__numeral__special_I4_J,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 9.03/5.02        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4864_diff__numeral__special_I4_J,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 9.03/5.02        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4865_diff__numeral__special_I4_J,axiom,
% 9.03/5.02      ! [M: num] :
% 9.03/5.02        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 9.03/5.02        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4866_diff__numeral__special_I3_J,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.03/5.02        = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4867_diff__numeral__special_I3_J,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.03/5.02        = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4868_diff__numeral__special_I3_J,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.03/5.02        = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4869_diff__numeral__special_I3_J,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.03/5.02        = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4870_diff__numeral__special_I3_J,axiom,
% 9.03/5.02      ! [N: num] :
% 9.03/5.02        ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.03/5.02        = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4871_add__self__mod__2,axiom,
% 9.03/5.02      ! [M: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02        = zero_zero_nat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4872_neg__numeral__le__floor,axiom,
% 9.03/5.02      ! [V: num,X: real] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 9.03/5.02        = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4873_neg__numeral__le__floor,axiom,
% 9.03/5.02      ! [V: num,X: rat] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/5.02        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4874_floor__less__neg__numeral,axiom,
% 9.03/5.02      ! [X: real,V: num] :
% 9.03/5.02        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.03/5.02        = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4875_floor__less__neg__numeral,axiom,
% 9.03/5.02      ! [X: rat,V: num] :
% 9.03/5.02        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.03/5.02        = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4876_ceiling__le__neg__numeral,axiom,
% 9.03/5.02      ! [X: real,V: num] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.03/5.02        = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4877_ceiling__le__neg__numeral,axiom,
% 9.03/5.02      ! [X: rat,V: num] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.03/5.02        = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4878_ceiling__less__zero,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 9.03/5.02        = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4879_ceiling__less__zero,axiom,
% 9.03/5.02      ! [X: rat] :
% 9.03/5.02        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 9.03/5.02        = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4880_zero__le__ceiling,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 9.03/5.02        = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4881_zero__le__ceiling,axiom,
% 9.03/5.02      ! [X: rat] :
% 9.03/5.02        ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/5.02        = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4882_neg__numeral__less__ceiling,axiom,
% 9.03/5.02      ! [V: num,X: real] :
% 9.03/5.02        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 9.03/5.02        = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4883_neg__numeral__less__ceiling,axiom,
% 9.03/5.02      ! [V: num,X: rat] :
% 9.03/5.02        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/5.02        = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4884_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [Y: int,X: num,N: nat] :
% 9.03/5.02        ( ( ( ring_1_of_int_real @ Y )
% 9.03/5.02          = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 9.03/5.02        = ( Y
% 9.03/5.02          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4885_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [Y: int,X: num,N: nat] :
% 9.03/5.02        ( ( ( ring_1_of_int_int @ Y )
% 9.03/5.02          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 9.03/5.02        = ( Y
% 9.03/5.02          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4886_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [Y: int,X: num,N: nat] :
% 9.03/5.02        ( ( ( ring_17405671764205052669omplex @ Y )
% 9.03/5.02          = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
% 9.03/5.02        = ( Y
% 9.03/5.02          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4887_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [Y: int,X: num,N: nat] :
% 9.03/5.02        ( ( ( ring_1_of_int_rat @ Y )
% 9.03/5.02          = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 9.03/5.02        = ( Y
% 9.03/5.02          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4888_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [Y: int,X: num,N: nat] :
% 9.03/5.02        ( ( ( ring_18347121197199848620nteger @ Y )
% 9.03/5.02          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 9.03/5.02        = ( Y
% 9.03/5.02          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4889_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,Y: int] :
% 9.03/5.02        ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
% 9.03/5.02          = ( ring_1_of_int_real @ Y ) )
% 9.03/5.02        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 9.03/5.02          = Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4890_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,Y: int] :
% 9.03/5.02        ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 9.03/5.02          = ( ring_1_of_int_int @ Y ) )
% 9.03/5.02        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 9.03/5.02          = Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4891_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,Y: int] :
% 9.03/5.02        ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
% 9.03/5.02          = ( ring_17405671764205052669omplex @ Y ) )
% 9.03/5.02        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 9.03/5.02          = Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4892_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,Y: int] :
% 9.03/5.02        ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
% 9.03/5.02          = ( ring_1_of_int_rat @ Y ) )
% 9.03/5.02        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 9.03/5.02          = Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4893_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,Y: int] :
% 9.03/5.02        ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
% 9.03/5.02          = ( ring_18347121197199848620nteger @ Y ) )
% 9.03/5.02        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 9.03/5.02          = Y ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4894_dbl__simps_I4_J,axiom,
% 9.03/5.02      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.02      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4895_dbl__simps_I4_J,axiom,
% 9.03/5.02      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.03/5.02      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4896_dbl__simps_I4_J,axiom,
% 9.03/5.02      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.03/5.02      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4897_dbl__simps_I4_J,axiom,
% 9.03/5.02      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.02      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4898_dbl__simps_I4_J,axiom,
% 9.03/5.02      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.03/5.02      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4899_power__minus1__even,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/5.02        = one_one_real ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4900_power__minus1__even,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/5.02        = one_one_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4901_power__minus1__even,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/5.02        = one_one_complex ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4902_power__minus1__even,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/5.02        = one_one_rat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4903_power__minus1__even,axiom,
% 9.03/5.02      ! [N: nat] :
% 9.03/5.02        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.03/5.02        = one_one_Code_integer ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4904_mod2__gr__0,axiom,
% 9.03/5.02      ! [M: nat] :
% 9.03/5.02        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/5.02        = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.02          = one_one_nat ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4905_norm__of__real__add1,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ one_one_real ) )
% 9.03/5.02        = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4906_norm__of__real__add1,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ one_one_complex ) )
% 9.03/5.02        = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4907_norm__of__real__addn,axiom,
% 9.03/5.02      ! [X: real,B: num] :
% 9.03/5.02        ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B ) ) )
% 9.03/5.02        = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4908_norm__of__real__addn,axiom,
% 9.03/5.02      ! [X: real,B: num] :
% 9.03/5.02        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B ) ) )
% 9.03/5.02        = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4909_neg__numeral__less__floor,axiom,
% 9.03/5.02      ! [V: num,X: real] :
% 9.03/5.02        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 9.03/5.02        = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4910_neg__numeral__less__floor,axiom,
% 9.03/5.02      ! [V: num,X: rat] :
% 9.03/5.02        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 9.03/5.02        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4911_floor__le__neg__numeral,axiom,
% 9.03/5.02      ! [X: real,V: num] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.03/5.02        = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4912_floor__le__neg__numeral,axiom,
% 9.03/5.02      ! [X: rat,V: num] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.03/5.02        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4913_ceiling__less__neg__numeral,axiom,
% 9.03/5.02      ! [X: real,V: num] :
% 9.03/5.02        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.03/5.02        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4914_ceiling__less__neg__numeral,axiom,
% 9.03/5.02      ! [X: rat,V: num] :
% 9.03/5.02        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.03/5.02        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4915_neg__numeral__le__ceiling,axiom,
% 9.03/5.02      ! [V: num,X: real] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 9.03/5.02        = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4916_neg__numeral__le__ceiling,axiom,
% 9.03/5.02      ! [V: num,X: rat] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 9.03/5.02        = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4917_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [A: int,X: num,N: nat] :
% 9.03/5.02        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 9.03/5.02        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4918_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [A: int,X: num,N: nat] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 9.03/5.02        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4919_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [A: int,X: num,N: nat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 9.03/5.02        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4920_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [A: int,X: num,N: nat] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 9.03/5.02        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4921_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,A: int] :
% 9.03/5.02        ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 9.03/5.02        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4922_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,A: int] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 9.03/5.02        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4923_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,A: int] :
% 9.03/5.02        ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 9.03/5.02        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4924_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,A: int] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 9.03/5.02        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4925_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [A: int,X: num,N: nat] :
% 9.03/5.02        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 9.03/5.02        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4926_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [A: int,X: num,N: nat] :
% 9.03/5.02        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 9.03/5.02        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4927_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [A: int,X: num,N: nat] :
% 9.03/5.02        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 9.03/5.02        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4928_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.03/5.02      ! [A: int,X: num,N: nat] :
% 9.03/5.02        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 9.03/5.02        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4929_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,A: int] :
% 9.03/5.02        ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 9.03/5.02        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4930_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,A: int] :
% 9.03/5.02        ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 9.03/5.02        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4931_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,A: int] :
% 9.03/5.02        ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 9.03/5.02        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4932_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.03/5.02      ! [X: num,N: nat,A: int] :
% 9.03/5.02        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 9.03/5.02        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4933_square__powr__half,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.03/5.02        = ( abs_abs_real @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4934_abs__less__iff,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 9.03/5.02        = ( ( ord_less_real @ A @ B )
% 9.03/5.02          & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4935_abs__less__iff,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 9.03/5.02        = ( ( ord_less_int @ A @ B )
% 9.03/5.02          & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4936_abs__less__iff,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 9.03/5.02        = ( ( ord_less_rat @ A @ B )
% 9.03/5.02          & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4937_abs__less__iff,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.03/5.02        = ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.03/5.02          & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4938_equation__minus__iff,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( uminus_uminus_real @ B ) )
% 9.03/5.02        = ( B
% 9.03/5.02          = ( uminus_uminus_real @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4939_equation__minus__iff,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( uminus_uminus_int @ B ) )
% 9.03/5.02        = ( B
% 9.03/5.02          = ( uminus_uminus_int @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4940_equation__minus__iff,axiom,
% 9.03/5.02      ! [A: complex,B: complex] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( uminus1482373934393186551omplex @ B ) )
% 9.03/5.02        = ( B
% 9.03/5.02          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4941_equation__minus__iff,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( uminus_uminus_rat @ B ) )
% 9.03/5.02        = ( B
% 9.03/5.02          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4942_equation__minus__iff,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( uminus1351360451143612070nteger @ B ) )
% 9.03/5.02        = ( B
% 9.03/5.02          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4943_minus__equation__iff,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ( uminus_uminus_real @ A )
% 9.03/5.02          = B )
% 9.03/5.02        = ( ( uminus_uminus_real @ B )
% 9.03/5.02          = A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4944_minus__equation__iff,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( ( uminus_uminus_int @ A )
% 9.03/5.02          = B )
% 9.03/5.02        = ( ( uminus_uminus_int @ B )
% 9.03/5.02          = A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4945_minus__equation__iff,axiom,
% 9.03/5.02      ! [A: complex,B: complex] :
% 9.03/5.02        ( ( ( uminus1482373934393186551omplex @ A )
% 9.03/5.02          = B )
% 9.03/5.02        = ( ( uminus1482373934393186551omplex @ B )
% 9.03/5.02          = A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4946_minus__equation__iff,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ( uminus_uminus_rat @ A )
% 9.03/5.02          = B )
% 9.03/5.02        = ( ( uminus_uminus_rat @ B )
% 9.03/5.02          = A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4947_minus__equation__iff,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( ( uminus1351360451143612070nteger @ A )
% 9.03/5.02          = B )
% 9.03/5.02        = ( ( uminus1351360451143612070nteger @ B )
% 9.03/5.02          = A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4948_verit__negate__coefficient_I3_J,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( A = B )
% 9.03/5.02       => ( ( uminus_uminus_real @ A )
% 9.03/5.02          = ( uminus_uminus_real @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4949_verit__negate__coefficient_I3_J,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( A = B )
% 9.03/5.02       => ( ( uminus_uminus_int @ A )
% 9.03/5.02          = ( uminus_uminus_int @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4950_verit__negate__coefficient_I3_J,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( A = B )
% 9.03/5.02       => ( ( uminus_uminus_rat @ A )
% 9.03/5.02          = ( uminus_uminus_rat @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4951_verit__negate__coefficient_I3_J,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( A = B )
% 9.03/5.02       => ( ( uminus1351360451143612070nteger @ A )
% 9.03/5.02          = ( uminus1351360451143612070nteger @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4952_abs__leI,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ A @ B )
% 9.03/5.02       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 9.03/5.02         => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4953_abs__leI,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ A @ B )
% 9.03/5.02       => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.03/5.02         => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4954_abs__leI,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ A @ B )
% 9.03/5.02       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.03/5.02         => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4955_abs__leI,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( ord_less_eq_int @ A @ B )
% 9.03/5.02       => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 9.03/5.02         => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4956_abs__le__D2,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 9.03/5.02       => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4957_abs__le__D2,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.03/5.02       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4958_abs__le__D2,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 9.03/5.02       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4959_abs__le__D2,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 9.03/5.02       => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4960_abs__le__iff,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 9.03/5.02        = ( ( ord_less_eq_real @ A @ B )
% 9.03/5.02          & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4961_abs__le__iff,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.03/5.02        = ( ( ord_le3102999989581377725nteger @ A @ B )
% 9.03/5.02          & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4962_abs__le__iff,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 9.03/5.02        = ( ( ord_less_eq_rat @ A @ B )
% 9.03/5.02          & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4963_abs__le__iff,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 9.03/5.02        = ( ( ord_less_eq_int @ A @ B )
% 9.03/5.02          & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4964_abs__ge__minus__self,axiom,
% 9.03/5.02      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4965_abs__ge__minus__self,axiom,
% 9.03/5.02      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4966_abs__ge__minus__self,axiom,
% 9.03/5.02      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4967_abs__ge__minus__self,axiom,
% 9.03/5.02      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4968_of__nat__mod,axiom,
% 9.03/5.02      ! [M: nat,N: nat] :
% 9.03/5.02        ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 9.03/5.02        = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4969_of__nat__mod,axiom,
% 9.03/5.02      ! [M: nat,N: nat] :
% 9.03/5.02        ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 9.03/5.02        = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4970_of__nat__mod,axiom,
% 9.03/5.02      ! [M: nat,N: nat] :
% 9.03/5.02        ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 9.03/5.02        = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4971_artanh__tanh__real,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( artanh_real @ ( tanh_real @ X ) )
% 9.03/5.02        = X ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4972_abs__minus__le__zero,axiom,
% 9.03/5.02      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4973_abs__minus__le__zero,axiom,
% 9.03/5.02      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4974_abs__minus__le__zero,axiom,
% 9.03/5.02      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4975_abs__minus__le__zero,axiom,
% 9.03/5.02      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4976_eq__abs__iff_H,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( abs_abs_real @ B ) )
% 9.03/5.02        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/5.02          & ( ( B = A )
% 9.03/5.02            | ( B
% 9.03/5.02              = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4977_eq__abs__iff_H,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( abs_abs_Code_integer @ B ) )
% 9.03/5.02        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.03/5.02          & ( ( B = A )
% 9.03/5.02            | ( B
% 9.03/5.02              = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4978_eq__abs__iff_H,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( abs_abs_rat @ B ) )
% 9.03/5.02        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/5.02          & ( ( B = A )
% 9.03/5.02            | ( B
% 9.03/5.02              = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4979_eq__abs__iff_H,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( A
% 9.03/5.02          = ( abs_abs_int @ B ) )
% 9.03/5.02        = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/5.02          & ( ( B = A )
% 9.03/5.02            | ( B
% 9.03/5.02              = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4980_abs__eq__iff_H,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ( abs_abs_real @ A )
% 9.03/5.02          = B )
% 9.03/5.02        = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/5.02          & ( ( A = B )
% 9.03/5.02            | ( A
% 9.03/5.02              = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4981_abs__eq__iff_H,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( ( abs_abs_Code_integer @ A )
% 9.03/5.02          = B )
% 9.03/5.02        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 9.03/5.02          & ( ( A = B )
% 9.03/5.02            | ( A
% 9.03/5.02              = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4982_abs__eq__iff_H,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ( abs_abs_rat @ A )
% 9.03/5.02          = B )
% 9.03/5.02        = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/5.02          & ( ( A = B )
% 9.03/5.02            | ( A
% 9.03/5.02              = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4983_abs__eq__iff_H,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( ( abs_abs_int @ A )
% 9.03/5.02          = B )
% 9.03/5.02        = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/5.02          & ( ( A = B )
% 9.03/5.02            | ( A
% 9.03/5.02              = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4984_abs__of__neg,axiom,
% 9.03/5.02      ! [A: real] :
% 9.03/5.02        ( ( ord_less_real @ A @ zero_zero_real )
% 9.03/5.02       => ( ( abs_abs_real @ A )
% 9.03/5.02          = ( uminus_uminus_real @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4985_abs__of__neg,axiom,
% 9.03/5.02      ! [A: int] :
% 9.03/5.02        ( ( ord_less_int @ A @ zero_zero_int )
% 9.03/5.02       => ( ( abs_abs_int @ A )
% 9.03/5.02          = ( uminus_uminus_int @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4986_abs__of__neg,axiom,
% 9.03/5.02      ! [A: rat] :
% 9.03/5.02        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.03/5.02       => ( ( abs_abs_rat @ A )
% 9.03/5.02          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4987_abs__of__neg,axiom,
% 9.03/5.02      ! [A: code_integer] :
% 9.03/5.02        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 9.03/5.02       => ( ( abs_abs_Code_integer @ A )
% 9.03/5.02          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4988_abs__if__raw,axiom,
% 9.03/5.02      ( abs_abs_real
% 9.03/5.02      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4989_abs__if__raw,axiom,
% 9.03/5.02      ( abs_abs_int
% 9.03/5.02      = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4990_abs__if__raw,axiom,
% 9.03/5.02      ( abs_abs_rat
% 9.03/5.02      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4991_abs__if__raw,axiom,
% 9.03/5.02      ( abs_abs_Code_integer
% 9.03/5.02      = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4992_abs__if,axiom,
% 9.03/5.02      ( abs_abs_real
% 9.03/5.02      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4993_abs__if,axiom,
% 9.03/5.02      ( abs_abs_int
% 9.03/5.02      = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4994_abs__if,axiom,
% 9.03/5.02      ( abs_abs_rat
% 9.03/5.02      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4995_abs__if,axiom,
% 9.03/5.02      ( abs_abs_Code_integer
% 9.03/5.02      = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4996_abs__real__def,axiom,
% 9.03/5.02      ( abs_abs_real
% 9.03/5.02      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4997_abs__ge__self,axiom,
% 9.03/5.02      ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4998_abs__ge__self,axiom,
% 9.03/5.02      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_4999_abs__ge__self,axiom,
% 9.03/5.02      ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5000_abs__ge__self,axiom,
% 9.03/5.02      ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5001_abs__le__D1,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 9.03/5.02       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5002_abs__le__D1,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.03/5.02       => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5003_abs__le__D1,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 9.03/5.02       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5004_abs__le__D1,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 9.03/5.02       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5005_abs__eq__0__iff,axiom,
% 9.03/5.02      ! [A: code_integer] :
% 9.03/5.02        ( ( ( abs_abs_Code_integer @ A )
% 9.03/5.02          = zero_z3403309356797280102nteger )
% 9.03/5.02        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5006_abs__eq__0__iff,axiom,
% 9.03/5.02      ! [A: complex] :
% 9.03/5.02        ( ( ( abs_abs_complex @ A )
% 9.03/5.02          = zero_zero_complex )
% 9.03/5.02        = ( A = zero_zero_complex ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5007_abs__eq__0__iff,axiom,
% 9.03/5.02      ! [A: real] :
% 9.03/5.02        ( ( ( abs_abs_real @ A )
% 9.03/5.02          = zero_zero_real )
% 9.03/5.02        = ( A = zero_zero_real ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5008_abs__eq__0__iff,axiom,
% 9.03/5.02      ! [A: rat] :
% 9.03/5.02        ( ( ( abs_abs_rat @ A )
% 9.03/5.02          = zero_zero_rat )
% 9.03/5.02        = ( A = zero_zero_rat ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5009_abs__eq__0__iff,axiom,
% 9.03/5.02      ! [A: int] :
% 9.03/5.02        ( ( ( abs_abs_int @ A )
% 9.03/5.02          = zero_zero_int )
% 9.03/5.02        = ( A = zero_zero_int ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5010_abs__one,axiom,
% 9.03/5.02      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 9.03/5.02      = one_one_Code_integer ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5011_abs__one,axiom,
% 9.03/5.02      ( ( abs_abs_real @ one_one_real )
% 9.03/5.02      = one_one_real ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5012_abs__one,axiom,
% 9.03/5.02      ( ( abs_abs_rat @ one_one_rat )
% 9.03/5.02      = one_one_rat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5013_abs__one,axiom,
% 9.03/5.02      ( ( abs_abs_int @ one_one_int )
% 9.03/5.02      = one_one_int ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5014_abs__minus__commute,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 9.03/5.02        = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5015_abs__minus__commute,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 9.03/5.02        = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5016_abs__minus__commute,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 9.03/5.02        = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5017_abs__minus__commute,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 9.03/5.02        = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5018_tanh__real__gt__neg1,axiom,
% 9.03/5.02      ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5019_power__abs,axiom,
% 9.03/5.02      ! [A: rat,N: nat] :
% 9.03/5.02        ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
% 9.03/5.02        = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5020_power__abs,axiom,
% 9.03/5.02      ! [A: code_integer,N: nat] :
% 9.03/5.02        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 9.03/5.02        = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5021_power__abs,axiom,
% 9.03/5.02      ! [A: real,N: nat] :
% 9.03/5.02        ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 9.03/5.02        = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5022_power__abs,axiom,
% 9.03/5.02      ! [A: int,N: nat] :
% 9.03/5.02        ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 9.03/5.02        = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5023_mod__add__eq,axiom,
% 9.03/5.02      ! [A: nat,C: nat,B: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 9.03/5.02        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5024_mod__add__eq,axiom,
% 9.03/5.02      ! [A: int,C: int,B: int] :
% 9.03/5.02        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 9.03/5.02        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5025_mod__add__eq,axiom,
% 9.03/5.02      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.03/5.02        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 9.03/5.02        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5026_mod__add__cong,axiom,
% 9.03/5.02      ! [A: nat,C: nat,A6: nat,B: nat,B7: nat] :
% 9.03/5.02        ( ( ( modulo_modulo_nat @ A @ C )
% 9.03/5.02          = ( modulo_modulo_nat @ A6 @ C ) )
% 9.03/5.02       => ( ( ( modulo_modulo_nat @ B @ C )
% 9.03/5.02            = ( modulo_modulo_nat @ B7 @ C ) )
% 9.03/5.02         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.03/5.02            = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B7 ) @ C ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5027_mod__add__cong,axiom,
% 9.03/5.02      ! [A: int,C: int,A6: int,B: int,B7: int] :
% 9.03/5.02        ( ( ( modulo_modulo_int @ A @ C )
% 9.03/5.02          = ( modulo_modulo_int @ A6 @ C ) )
% 9.03/5.02       => ( ( ( modulo_modulo_int @ B @ C )
% 9.03/5.02            = ( modulo_modulo_int @ B7 @ C ) )
% 9.03/5.02         => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.03/5.02            = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B7 ) @ C ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5028_mod__add__cong,axiom,
% 9.03/5.02      ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B7: code_integer] :
% 9.03/5.02        ( ( ( modulo364778990260209775nteger @ A @ C )
% 9.03/5.02          = ( modulo364778990260209775nteger @ A6 @ C ) )
% 9.03/5.02       => ( ( ( modulo364778990260209775nteger @ B @ C )
% 9.03/5.02            = ( modulo364778990260209775nteger @ B7 @ C ) )
% 9.03/5.02         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 9.03/5.02            = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A6 @ B7 ) @ C ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5029_mod__add__left__eq,axiom,
% 9.03/5.02      ! [A: nat,C: nat,B: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 9.03/5.02        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5030_mod__add__left__eq,axiom,
% 9.03/5.02      ! [A: int,C: int,B: int] :
% 9.03/5.02        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 9.03/5.02        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5031_mod__add__left__eq,axiom,
% 9.03/5.02      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.03/5.02        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 9.03/5.02        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5032_mod__add__right__eq,axiom,
% 9.03/5.02      ! [A: nat,B: nat,C: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 9.03/5.02        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5033_mod__add__right__eq,axiom,
% 9.03/5.02      ! [A: int,B: int,C: int] :
% 9.03/5.02        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 9.03/5.02        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5034_mod__add__right__eq,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.03/5.02        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 9.03/5.02        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5035_ceiling__def,axiom,
% 9.03/5.02      ( archim7802044766580827645g_real
% 9.03/5.02      = ( ^ [X4: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5036_ceiling__def,axiom,
% 9.03/5.02      ( archim2889992004027027881ng_rat
% 9.03/5.02      = ( ^ [X4: rat] : ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X4 ) ) ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5037_floor__minus,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5038_floor__minus,axiom,
% 9.03/5.02      ! [X: rat] :
% 9.03/5.02        ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5039_ceiling__minus,axiom,
% 9.03/5.02      ! [X: real] :
% 9.03/5.02        ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ X ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( archim6058952711729229775r_real @ X ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5040_ceiling__minus,axiom,
% 9.03/5.02      ! [X: rat] :
% 9.03/5.02        ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ X ) )
% 9.03/5.02        = ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ X ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5041_power__mod,axiom,
% 9.03/5.02      ! [A: nat,B: nat,N: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
% 9.03/5.02        = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5042_power__mod,axiom,
% 9.03/5.02      ! [A: int,B: int,N: nat] :
% 9.03/5.02        ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
% 9.03/5.02        = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5043_power__mod,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer,N: nat] :
% 9.03/5.02        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
% 9.03/5.02        = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5044_VEBT_Osize_I4_J,axiom,
% 9.03/5.02      ! [X21: $o,X222: $o] :
% 9.03/5.02        ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 9.03/5.02        = zero_zero_nat ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5045_mod__Suc__Suc__eq,axiom,
% 9.03/5.02      ! [M: nat,N: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 9.03/5.02        = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5046_mod__Suc__eq,axiom,
% 9.03/5.02      ! [M: nat,N: nat] :
% 9.03/5.02        ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 9.03/5.02        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5047_le__imp__neg__le,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ A @ B )
% 9.03/5.02       => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5048_le__imp__neg__le,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ A @ B )
% 9.03/5.02       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5049_le__imp__neg__le,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ A @ B )
% 9.03/5.02       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5050_le__imp__neg__le,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( ord_less_eq_int @ A @ B )
% 9.03/5.02       => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5051_minus__le__iff,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 9.03/5.02        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5052_minus__le__iff,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.03/5.02        = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5053_minus__le__iff,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.03/5.02        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5054_minus__le__iff,axiom,
% 9.03/5.02      ! [A: int,B: int] :
% 9.03/5.02        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 9.03/5.02        = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5055_le__minus__iff,axiom,
% 9.03/5.02      ! [A: real,B: real] :
% 9.03/5.02        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 9.03/5.02        = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5056_le__minus__iff,axiom,
% 9.03/5.02      ! [A: code_integer,B: code_integer] :
% 9.03/5.02        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.03/5.02        = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.03/5.02  
% 9.03/5.02  thf(fact_5057_le__minus__iff,axiom,
% 9.03/5.02      ! [A: rat,B: rat] :
% 9.03/5.02        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.03/5.02        = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5058_le__minus__iff,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 9.03/5.03        = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5059_less__minus__iff,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 9.03/5.03        = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5060_less__minus__iff,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 9.03/5.03        = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5061_less__minus__iff,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.03/5.03        = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5062_less__minus__iff,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.03/5.03        = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5063_minus__less__iff,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 9.03/5.03        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5064_minus__less__iff,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 9.03/5.03        = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5065_minus__less__iff,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.03/5.03        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5066_minus__less__iff,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.03/5.03        = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5067_verit__negate__coefficient_I2_J,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( ord_less_real @ A @ B )
% 9.03/5.03       => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5068_verit__negate__coefficient_I2_J,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ord_less_int @ A @ B )
% 9.03/5.03       => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5069_verit__negate__coefficient_I2_J,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( ord_less_rat @ A @ B )
% 9.03/5.03       => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5070_verit__negate__coefficient_I2_J,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.03/5.03       => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5071_neg__numeral__neq__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 9.03/5.03       != ( numeral_numeral_real @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5072_neg__numeral__neq__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 9.03/5.03       != ( numeral_numeral_int @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5073_neg__numeral__neq__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 9.03/5.03       != ( numera6690914467698888265omplex @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5074_neg__numeral__neq__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 9.03/5.03       != ( numeral_numeral_rat @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5075_neg__numeral__neq__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 9.03/5.03       != ( numera6620942414471956472nteger @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5076_numeral__neq__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( numeral_numeral_real @ M )
% 9.03/5.03       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5077_numeral__neq__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( numeral_numeral_int @ M )
% 9.03/5.03       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5078_numeral__neq__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( numera6690914467698888265omplex @ M )
% 9.03/5.03       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5079_numeral__neq__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( numeral_numeral_rat @ M )
% 9.03/5.03       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5080_numeral__neq__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( numera6620942414471956472nteger @ M )
% 9.03/5.03       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5081_mod__less__eq__dividend,axiom,
% 9.03/5.03      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5082_one__neq__neg__one,axiom,
% 9.03/5.03      ( one_one_real
% 9.03/5.03     != ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5083_one__neq__neg__one,axiom,
% 9.03/5.03      ( one_one_int
% 9.03/5.03     != ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5084_one__neq__neg__one,axiom,
% 9.03/5.03      ( one_one_complex
% 9.03/5.03     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5085_one__neq__neg__one,axiom,
% 9.03/5.03      ( one_one_rat
% 9.03/5.03     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5086_one__neq__neg__one,axiom,
% 9.03/5.03      ( one_one_Code_integer
% 9.03/5.03     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5087_is__num__normalize_I8_J,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 9.03/5.03        = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5088_is__num__normalize_I8_J,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 9.03/5.03        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5089_is__num__normalize_I8_J,axiom,
% 9.03/5.03      ! [A: complex,B: complex] :
% 9.03/5.03        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 9.03/5.03        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5090_is__num__normalize_I8_J,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 9.03/5.03        = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5091_is__num__normalize_I8_J,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.03/5.03        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5092_group__cancel_Oneg1,axiom,
% 9.03/5.03      ! [A2: real,K: real,A: real] :
% 9.03/5.03        ( ( A2
% 9.03/5.03          = ( plus_plus_real @ K @ A ) )
% 9.03/5.03       => ( ( uminus_uminus_real @ A2 )
% 9.03/5.03          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5093_group__cancel_Oneg1,axiom,
% 9.03/5.03      ! [A2: int,K: int,A: int] :
% 9.03/5.03        ( ( A2
% 9.03/5.03          = ( plus_plus_int @ K @ A ) )
% 9.03/5.03       => ( ( uminus_uminus_int @ A2 )
% 9.03/5.03          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5094_group__cancel_Oneg1,axiom,
% 9.03/5.03      ! [A2: complex,K: complex,A: complex] :
% 9.03/5.03        ( ( A2
% 9.03/5.03          = ( plus_plus_complex @ K @ A ) )
% 9.03/5.03       => ( ( uminus1482373934393186551omplex @ A2 )
% 9.03/5.03          = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5095_group__cancel_Oneg1,axiom,
% 9.03/5.03      ! [A2: rat,K: rat,A: rat] :
% 9.03/5.03        ( ( A2
% 9.03/5.03          = ( plus_plus_rat @ K @ A ) )
% 9.03/5.03       => ( ( uminus_uminus_rat @ A2 )
% 9.03/5.03          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5096_group__cancel_Oneg1,axiom,
% 9.03/5.03      ! [A2: code_integer,K: code_integer,A: code_integer] :
% 9.03/5.03        ( ( A2
% 9.03/5.03          = ( plus_p5714425477246183910nteger @ K @ A ) )
% 9.03/5.03       => ( ( uminus1351360451143612070nteger @ A2 )
% 9.03/5.03          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5097_add_Oinverse__distrib__swap,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 9.03/5.03        = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5098_add_Oinverse__distrib__swap,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 9.03/5.03        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5099_add_Oinverse__distrib__swap,axiom,
% 9.03/5.03      ! [A: complex,B: complex] :
% 9.03/5.03        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 9.03/5.03        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5100_add_Oinverse__distrib__swap,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 9.03/5.03        = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5101_add_Oinverse__distrib__swap,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.03/5.03        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5102_minus__diff__commute,axiom,
% 9.03/5.03      ! [B: real,A: real] :
% 9.03/5.03        ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 9.03/5.03        = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5103_minus__diff__commute,axiom,
% 9.03/5.03      ! [B: int,A: int] :
% 9.03/5.03        ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 9.03/5.03        = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5104_minus__diff__commute,axiom,
% 9.03/5.03      ! [B: complex,A: complex] :
% 9.03/5.03        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 9.03/5.03        = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5105_minus__diff__commute,axiom,
% 9.03/5.03      ! [B: rat,A: rat] :
% 9.03/5.03        ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 9.03/5.03        = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5106_minus__diff__commute,axiom,
% 9.03/5.03      ! [B: code_integer,A: code_integer] :
% 9.03/5.03        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 9.03/5.03        = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5107_minus__diff__minus,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 9.03/5.03        = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5108_minus__diff__minus,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 9.03/5.03        = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5109_minus__diff__minus,axiom,
% 9.03/5.03      ! [A: complex,B: complex] :
% 9.03/5.03        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 9.03/5.03        = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5110_minus__diff__minus,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 9.03/5.03        = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5111_minus__diff__minus,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 9.03/5.03        = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5112_div__minus__right,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 9.03/5.03        = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5113_div__minus__right,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.03/5.03        = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5114_minus__divide__left,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 9.03/5.03        = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5115_minus__divide__left,axiom,
% 9.03/5.03      ! [A: complex,B: complex] :
% 9.03/5.03        ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.03/5.03        = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5116_minus__divide__left,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 9.03/5.03        = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5117_minus__divide__divide,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 9.03/5.03        = ( divide_divide_real @ A @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5118_minus__divide__divide,axiom,
% 9.03/5.03      ! [A: complex,B: complex] :
% 9.03/5.03        ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 9.03/5.03        = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5119_minus__divide__divide,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 9.03/5.03        = ( divide_divide_rat @ A @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5120_minus__divide__right,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 9.03/5.03        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5121_minus__divide__right,axiom,
% 9.03/5.03      ! [A: complex,B: complex] :
% 9.03/5.03        ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.03/5.03        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5122_minus__divide__right,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 9.03/5.03        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5123_VEBT__internal_Oheight_Ocases,axiom,
% 9.03/5.03      ! [X: vEBT_VEBT] :
% 9.03/5.03        ( ! [A5: $o,B6: $o] :
% 9.03/5.03            ( X
% 9.03/5.03           != ( vEBT_Leaf @ A5 @ B6 ) )
% 9.03/5.03       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.03/5.03              ( X
% 9.03/5.03             != ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5124_VEBT_Odistinct_I1_J,axiom,
% 9.03/5.03      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 9.03/5.03        ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 9.03/5.03       != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5125_VEBT_Oexhaust,axiom,
% 9.03/5.03      ! [Y: vEBT_VEBT] :
% 9.03/5.03        ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 9.03/5.03            ( Y
% 9.03/5.03           != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 9.03/5.03       => ~ ! [X212: $o,X223: $o] :
% 9.03/5.03              ( Y
% 9.03/5.03             != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5126_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 9.03/5.03      ! [Uu: $o,Uv: $o,Uw: nat] :
% 9.03/5.03        ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5127_of__int__neg__numeral,axiom,
% 9.03/5.03      ! [K: num] :
% 9.03/5.03        ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.03/5.03        = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5128_of__int__neg__numeral,axiom,
% 9.03/5.03      ! [K: num] :
% 9.03/5.03        ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.03/5.03        = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5129_of__int__neg__numeral,axiom,
% 9.03/5.03      ! [K: num] :
% 9.03/5.03        ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.03/5.03        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5130_of__int__neg__numeral,axiom,
% 9.03/5.03      ! [K: num] :
% 9.03/5.03        ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.03/5.03        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5131_of__int__neg__numeral,axiom,
% 9.03/5.03      ! [K: num] :
% 9.03/5.03        ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.03/5.03        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5132_norm__of__real__diff,axiom,
% 9.03/5.03      ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5133_norm__of__real__diff,axiom,
% 9.03/5.03      ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5134_abs__ge__zero,axiom,
% 9.03/5.03      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5135_abs__ge__zero,axiom,
% 9.03/5.03      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5136_abs__ge__zero,axiom,
% 9.03/5.03      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5137_abs__ge__zero,axiom,
% 9.03/5.03      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5138_abs__of__pos,axiom,
% 9.03/5.03      ! [A: code_integer] :
% 9.03/5.03        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 9.03/5.03       => ( ( abs_abs_Code_integer @ A )
% 9.03/5.03          = A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5139_abs__of__pos,axiom,
% 9.03/5.03      ! [A: real] :
% 9.03/5.03        ( ( ord_less_real @ zero_zero_real @ A )
% 9.03/5.03       => ( ( abs_abs_real @ A )
% 9.03/5.03          = A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5140_abs__of__pos,axiom,
% 9.03/5.03      ! [A: rat] :
% 9.03/5.03        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.03/5.03       => ( ( abs_abs_rat @ A )
% 9.03/5.03          = A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5141_abs__of__pos,axiom,
% 9.03/5.03      ! [A: int] :
% 9.03/5.03        ( ( ord_less_int @ zero_zero_int @ A )
% 9.03/5.03       => ( ( abs_abs_int @ A )
% 9.03/5.03          = A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5142_abs__not__less__zero,axiom,
% 9.03/5.03      ! [A: code_integer] :
% 9.03/5.03        ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5143_abs__not__less__zero,axiom,
% 9.03/5.03      ! [A: real] :
% 9.03/5.03        ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5144_abs__not__less__zero,axiom,
% 9.03/5.03      ! [A: rat] :
% 9.03/5.03        ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5145_abs__not__less__zero,axiom,
% 9.03/5.03      ! [A: int] :
% 9.03/5.03        ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5146_abs__triangle__ineq,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5147_abs__triangle__ineq,axiom,
% 9.03/5.03      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5148_abs__triangle__ineq,axiom,
% 9.03/5.03      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5149_abs__triangle__ineq,axiom,
% 9.03/5.03      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5150_abs__mult__less,axiom,
% 9.03/5.03      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 9.03/5.03        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 9.03/5.03       => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 9.03/5.03         => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5151_abs__mult__less,axiom,
% 9.03/5.03      ! [A: real,C: real,B: real,D: real] :
% 9.03/5.03        ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 9.03/5.03       => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 9.03/5.03         => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5152_abs__mult__less,axiom,
% 9.03/5.03      ! [A: rat,C: rat,B: rat,D: rat] :
% 9.03/5.03        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 9.03/5.03       => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 9.03/5.03         => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5153_abs__mult__less,axiom,
% 9.03/5.03      ! [A: int,C: int,B: int,D: int] :
% 9.03/5.03        ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 9.03/5.03       => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 9.03/5.03         => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5154_abs__triangle__ineq2,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5155_abs__triangle__ineq2,axiom,
% 9.03/5.03      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5156_abs__triangle__ineq2,axiom,
% 9.03/5.03      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5157_abs__triangle__ineq2,axiom,
% 9.03/5.03      ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5158_abs__triangle__ineq3,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5159_abs__triangle__ineq3,axiom,
% 9.03/5.03      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5160_abs__triangle__ineq3,axiom,
% 9.03/5.03      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5161_abs__triangle__ineq3,axiom,
% 9.03/5.03      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5162_abs__triangle__ineq2__sym,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5163_abs__triangle__ineq2__sym,axiom,
% 9.03/5.03      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5164_abs__triangle__ineq2__sym,axiom,
% 9.03/5.03      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5165_abs__triangle__ineq2__sym,axiom,
% 9.03/5.03      ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5166_nonzero__abs__divide,axiom,
% 9.03/5.03      ! [B: real,A: real] :
% 9.03/5.03        ( ( B != zero_zero_real )
% 9.03/5.03       => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 9.03/5.03          = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5167_nonzero__abs__divide,axiom,
% 9.03/5.03      ! [B: rat,A: rat] :
% 9.03/5.03        ( ( B != zero_zero_rat )
% 9.03/5.03       => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 9.03/5.03          = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5168_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.03/5.03       => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5169_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/5.03       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5170_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/5.03       => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5171_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.03/5.03      ! [B: nat,A: nat] :
% 9.03/5.03        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.03/5.03       => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5172_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.03/5.03      ! [B: int,A: int] :
% 9.03/5.03        ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/5.03       => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5173_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.03/5.03      ! [B: code_integer,A: code_integer] :
% 9.03/5.03        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.03/5.03       => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5174_mod__eq__self__iff__div__eq__0,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( ( ( modulo_modulo_nat @ A @ B )
% 9.03/5.03          = A )
% 9.03/5.03        = ( ( divide_divide_nat @ A @ B )
% 9.03/5.03          = zero_zero_nat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5175_mod__eq__self__iff__div__eq__0,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ( modulo_modulo_int @ A @ B )
% 9.03/5.03          = A )
% 9.03/5.03        = ( ( divide_divide_int @ A @ B )
% 9.03/5.03          = zero_zero_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5176_mod__eq__self__iff__div__eq__0,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ( modulo364778990260209775nteger @ A @ B )
% 9.03/5.03          = A )
% 9.03/5.03        = ( ( divide6298287555418463151nteger @ A @ B )
% 9.03/5.03          = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5177_cong__exp__iff__simps_I9_J,axiom,
% 9.03/5.03      ! [M: num,Q3: num,N: num] :
% 9.03/5.03        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.03/5.03          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 9.03/5.03        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 9.03/5.03          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5178_cong__exp__iff__simps_I9_J,axiom,
% 9.03/5.03      ! [M: num,Q3: num,N: num] :
% 9.03/5.03        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.03/5.03          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 9.03/5.03        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 9.03/5.03          = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5179_cong__exp__iff__simps_I9_J,axiom,
% 9.03/5.03      ! [M: num,Q3: num,N: num] :
% 9.03/5.03        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.03/5.03          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 9.03/5.03        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 9.03/5.03          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5180_cong__exp__iff__simps_I4_J,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 9.03/5.03        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5181_cong__exp__iff__simps_I4_J,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 9.03/5.03        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5182_cong__exp__iff__simps_I4_J,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 9.03/5.03        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5183_mod__eqE,axiom,
% 9.03/5.03      ! [A: int,C: int,B: int] :
% 9.03/5.03        ( ( ( modulo_modulo_int @ A @ C )
% 9.03/5.03          = ( modulo_modulo_int @ B @ C ) )
% 9.03/5.03       => ~ ! [D5: int] :
% 9.03/5.03              ( B
% 9.03/5.03             != ( plus_plus_int @ A @ ( times_times_int @ C @ D5 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5184_mod__eqE,axiom,
% 9.03/5.03      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ( modulo364778990260209775nteger @ A @ C )
% 9.03/5.03          = ( modulo364778990260209775nteger @ B @ C ) )
% 9.03/5.03       => ~ ! [D5: code_integer] :
% 9.03/5.03              ( B
% 9.03/5.03             != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D5 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5185_div__add1__eq,axiom,
% 9.03/5.03      ! [A: nat,B: nat,C: nat] :
% 9.03/5.03        ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.03/5.03        = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5186_div__add1__eq,axiom,
% 9.03/5.03      ! [A: int,B: int,C: int] :
% 9.03/5.03        ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.03/5.03        = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5187_div__add1__eq,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.03/5.03        ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 9.03/5.03        = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5188_mod__Suc,axiom,
% 9.03/5.03      ! [M: nat,N: nat] :
% 9.03/5.03        ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 9.03/5.03            = N )
% 9.03/5.03         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 9.03/5.03            = zero_zero_nat ) )
% 9.03/5.03        & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 9.03/5.03           != N )
% 9.03/5.03         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 9.03/5.03            = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5189_not__numeral__le__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5190_not__numeral__le__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5191_not__numeral__le__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5192_not__numeral__le__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5193_neg__numeral__le__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5194_neg__numeral__le__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5195_neg__numeral__le__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5196_neg__numeral__le__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5197_mod__induct,axiom,
% 9.03/5.03      ! [P: nat > $o,N: nat,P5: nat,M: nat] :
% 9.03/5.03        ( ( P @ N )
% 9.03/5.03       => ( ( ord_less_nat @ N @ P5 )
% 9.03/5.03         => ( ( ord_less_nat @ M @ P5 )
% 9.03/5.03           => ( ! [N2: nat] :
% 9.03/5.03                  ( ( ord_less_nat @ N2 @ P5 )
% 9.03/5.03                 => ( ( P @ N2 )
% 9.03/5.03                   => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P5 ) ) ) )
% 9.03/5.03             => ( P @ M ) ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5198_zero__neq__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( zero_zero_real
% 9.03/5.03       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5199_zero__neq__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( zero_zero_int
% 9.03/5.03       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5200_zero__neq__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( zero_zero_complex
% 9.03/5.03       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5201_zero__neq__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( zero_zero_rat
% 9.03/5.03       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5202_zero__neq__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( zero_z3403309356797280102nteger
% 9.03/5.03       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5203_mod__less__divisor,axiom,
% 9.03/5.03      ! [N: nat,M: nat] :
% 9.03/5.03        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/5.03       => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5204_not__numeral__less__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5205_not__numeral__less__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5206_not__numeral__less__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5207_not__numeral__less__neg__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] :
% 9.03/5.03        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5208_neg__numeral__less__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5209_neg__numeral__less__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5210_neg__numeral__less__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5211_neg__numeral__less__numeral,axiom,
% 9.03/5.03      ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5212_le__minus__one__simps_I2_J,axiom,
% 9.03/5.03      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5213_le__minus__one__simps_I2_J,axiom,
% 9.03/5.03      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5214_le__minus__one__simps_I2_J,axiom,
% 9.03/5.03      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5215_le__minus__one__simps_I2_J,axiom,
% 9.03/5.03      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5216_le__minus__one__simps_I4_J,axiom,
% 9.03/5.03      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5217_le__minus__one__simps_I4_J,axiom,
% 9.03/5.03      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5218_le__minus__one__simps_I4_J,axiom,
% 9.03/5.03      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5219_le__minus__one__simps_I4_J,axiom,
% 9.03/5.03      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5220_zero__neq__neg__one,axiom,
% 9.03/5.03      ( zero_zero_real
% 9.03/5.03     != ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5221_zero__neq__neg__one,axiom,
% 9.03/5.03      ( zero_zero_int
% 9.03/5.03     != ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5222_zero__neq__neg__one,axiom,
% 9.03/5.03      ( zero_zero_complex
% 9.03/5.03     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5223_zero__neq__neg__one,axiom,
% 9.03/5.03      ( zero_zero_rat
% 9.03/5.03     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5224_zero__neq__neg__one,axiom,
% 9.03/5.03      ( zero_z3403309356797280102nteger
% 9.03/5.03     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5225_add__eq__0__iff,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( ( plus_plus_real @ A @ B )
% 9.03/5.03          = zero_zero_real )
% 9.03/5.03        = ( B
% 9.03/5.03          = ( uminus_uminus_real @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5226_add__eq__0__iff,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ( plus_plus_int @ A @ B )
% 9.03/5.03          = zero_zero_int )
% 9.03/5.03        = ( B
% 9.03/5.03          = ( uminus_uminus_int @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5227_add__eq__0__iff,axiom,
% 9.03/5.03      ! [A: complex,B: complex] :
% 9.03/5.03        ( ( ( plus_plus_complex @ A @ B )
% 9.03/5.03          = zero_zero_complex )
% 9.03/5.03        = ( B
% 9.03/5.03          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5228_add__eq__0__iff,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( ( plus_plus_rat @ A @ B )
% 9.03/5.03          = zero_zero_rat )
% 9.03/5.03        = ( B
% 9.03/5.03          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5229_add__eq__0__iff,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.03/5.03          = zero_z3403309356797280102nteger )
% 9.03/5.03        = ( B
% 9.03/5.03          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5230_ab__group__add__class_Oab__left__minus,axiom,
% 9.03/5.03      ! [A: real] :
% 9.03/5.03        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 9.03/5.03        = zero_zero_real ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5231_ab__group__add__class_Oab__left__minus,axiom,
% 9.03/5.03      ! [A: int] :
% 9.03/5.03        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 9.03/5.03        = zero_zero_int ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5232_ab__group__add__class_Oab__left__minus,axiom,
% 9.03/5.03      ! [A: complex] :
% 9.03/5.03        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 9.03/5.03        = zero_zero_complex ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5233_ab__group__add__class_Oab__left__minus,axiom,
% 9.03/5.03      ! [A: rat] :
% 9.03/5.03        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.03/5.03        = zero_zero_rat ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5234_ab__group__add__class_Oab__left__minus,axiom,
% 9.03/5.03      ! [A: code_integer] :
% 9.03/5.03        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.03/5.03        = zero_z3403309356797280102nteger ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5235_add_Oinverse__unique,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( ( plus_plus_real @ A @ B )
% 9.03/5.03          = zero_zero_real )
% 9.03/5.03       => ( ( uminus_uminus_real @ A )
% 9.03/5.03          = B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5236_add_Oinverse__unique,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ( plus_plus_int @ A @ B )
% 9.03/5.03          = zero_zero_int )
% 9.03/5.03       => ( ( uminus_uminus_int @ A )
% 9.03/5.03          = B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5237_add_Oinverse__unique,axiom,
% 9.03/5.03      ! [A: complex,B: complex] :
% 9.03/5.03        ( ( ( plus_plus_complex @ A @ B )
% 9.03/5.03          = zero_zero_complex )
% 9.03/5.03       => ( ( uminus1482373934393186551omplex @ A )
% 9.03/5.03          = B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5238_add_Oinverse__unique,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( ( plus_plus_rat @ A @ B )
% 9.03/5.03          = zero_zero_rat )
% 9.03/5.03       => ( ( uminus_uminus_rat @ A )
% 9.03/5.03          = B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5239_add_Oinverse__unique,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.03/5.03          = zero_z3403309356797280102nteger )
% 9.03/5.03       => ( ( uminus1351360451143612070nteger @ A )
% 9.03/5.03          = B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5240_eq__neg__iff__add__eq__0,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( A
% 9.03/5.03          = ( uminus_uminus_real @ B ) )
% 9.03/5.03        = ( ( plus_plus_real @ A @ B )
% 9.03/5.03          = zero_zero_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5241_eq__neg__iff__add__eq__0,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( A
% 9.03/5.03          = ( uminus_uminus_int @ B ) )
% 9.03/5.03        = ( ( plus_plus_int @ A @ B )
% 9.03/5.03          = zero_zero_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5242_eq__neg__iff__add__eq__0,axiom,
% 9.03/5.03      ! [A: complex,B: complex] :
% 9.03/5.03        ( ( A
% 9.03/5.03          = ( uminus1482373934393186551omplex @ B ) )
% 9.03/5.03        = ( ( plus_plus_complex @ A @ B )
% 9.03/5.03          = zero_zero_complex ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5243_eq__neg__iff__add__eq__0,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( A
% 9.03/5.03          = ( uminus_uminus_rat @ B ) )
% 9.03/5.03        = ( ( plus_plus_rat @ A @ B )
% 9.03/5.03          = zero_zero_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5244_eq__neg__iff__add__eq__0,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( A
% 9.03/5.03          = ( uminus1351360451143612070nteger @ B ) )
% 9.03/5.03        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.03/5.03          = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5245_neg__eq__iff__add__eq__0,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( ( uminus_uminus_real @ A )
% 9.03/5.03          = B )
% 9.03/5.03        = ( ( plus_plus_real @ A @ B )
% 9.03/5.03          = zero_zero_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5246_neg__eq__iff__add__eq__0,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ( uminus_uminus_int @ A )
% 9.03/5.03          = B )
% 9.03/5.03        = ( ( plus_plus_int @ A @ B )
% 9.03/5.03          = zero_zero_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5247_neg__eq__iff__add__eq__0,axiom,
% 9.03/5.03      ! [A: complex,B: complex] :
% 9.03/5.03        ( ( ( uminus1482373934393186551omplex @ A )
% 9.03/5.03          = B )
% 9.03/5.03        = ( ( plus_plus_complex @ A @ B )
% 9.03/5.03          = zero_zero_complex ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5248_neg__eq__iff__add__eq__0,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( ( uminus_uminus_rat @ A )
% 9.03/5.03          = B )
% 9.03/5.03        = ( ( plus_plus_rat @ A @ B )
% 9.03/5.03          = zero_zero_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5249_neg__eq__iff__add__eq__0,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ( uminus1351360451143612070nteger @ A )
% 9.03/5.03          = B )
% 9.03/5.03        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.03/5.03          = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5250_mod__Suc__le__divisor,axiom,
% 9.03/5.03      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5251_less__minus__one__simps_I2_J,axiom,
% 9.03/5.03      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5252_less__minus__one__simps_I2_J,axiom,
% 9.03/5.03      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5253_less__minus__one__simps_I2_J,axiom,
% 9.03/5.03      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5254_less__minus__one__simps_I2_J,axiom,
% 9.03/5.03      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5255_less__minus__one__simps_I4_J,axiom,
% 9.03/5.03      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5256_less__minus__one__simps_I4_J,axiom,
% 9.03/5.03      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5257_less__minus__one__simps_I4_J,axiom,
% 9.03/5.03      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5258_less__minus__one__simps_I4_J,axiom,
% 9.03/5.03      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5259_numeral__times__minus__swap,axiom,
% 9.03/5.03      ! [W: num,X: real] :
% 9.03/5.03        ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
% 9.03/5.03        = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5260_numeral__times__minus__swap,axiom,
% 9.03/5.03      ! [W: num,X: int] :
% 9.03/5.03        ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
% 9.03/5.03        = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5261_numeral__times__minus__swap,axiom,
% 9.03/5.03      ! [W: num,X: complex] :
% 9.03/5.03        ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
% 9.03/5.03        = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5262_numeral__times__minus__swap,axiom,
% 9.03/5.03      ! [W: num,X: rat] :
% 9.03/5.03        ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
% 9.03/5.03        = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5263_numeral__times__minus__swap,axiom,
% 9.03/5.03      ! [W: num,X: code_integer] :
% 9.03/5.03        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
% 9.03/5.03        = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5264_nonzero__minus__divide__divide,axiom,
% 9.03/5.03      ! [B: real,A: real] :
% 9.03/5.03        ( ( B != zero_zero_real )
% 9.03/5.03       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 9.03/5.03          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5265_nonzero__minus__divide__divide,axiom,
% 9.03/5.03      ! [B: complex,A: complex] :
% 9.03/5.03        ( ( B != zero_zero_complex )
% 9.03/5.03       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 9.03/5.03          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5266_nonzero__minus__divide__divide,axiom,
% 9.03/5.03      ! [B: rat,A: rat] :
% 9.03/5.03        ( ( B != zero_zero_rat )
% 9.03/5.03       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 9.03/5.03          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5267_nonzero__minus__divide__right,axiom,
% 9.03/5.03      ! [B: real,A: real] :
% 9.03/5.03        ( ( B != zero_zero_real )
% 9.03/5.03       => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 9.03/5.03          = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5268_nonzero__minus__divide__right,axiom,
% 9.03/5.03      ! [B: complex,A: complex] :
% 9.03/5.03        ( ( B != zero_zero_complex )
% 9.03/5.03       => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.03/5.03          = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5269_nonzero__minus__divide__right,axiom,
% 9.03/5.03      ! [B: rat,A: rat] :
% 9.03/5.03        ( ( B != zero_zero_rat )
% 9.03/5.03       => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 9.03/5.03          = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5270_numeral__neq__neg__one,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( ( numeral_numeral_real @ N )
% 9.03/5.03       != ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5271_numeral__neq__neg__one,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( ( numeral_numeral_int @ N )
% 9.03/5.03       != ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5272_numeral__neq__neg__one,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( ( numera6690914467698888265omplex @ N )
% 9.03/5.03       != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5273_numeral__neq__neg__one,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( ( numeral_numeral_rat @ N )
% 9.03/5.03       != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5274_numeral__neq__neg__one,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( ( numera6620942414471956472nteger @ N )
% 9.03/5.03       != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5275_one__neq__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( one_one_real
% 9.03/5.03       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5276_one__neq__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( one_one_int
% 9.03/5.03       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5277_one__neq__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( one_one_complex
% 9.03/5.03       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5278_one__neq__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( one_one_rat
% 9.03/5.03       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5279_one__neq__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( one_one_Code_integer
% 9.03/5.03       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5280_square__eq__1__iff,axiom,
% 9.03/5.03      ! [X: real] :
% 9.03/5.03        ( ( ( times_times_real @ X @ X )
% 9.03/5.03          = one_one_real )
% 9.03/5.03        = ( ( X = one_one_real )
% 9.03/5.03          | ( X
% 9.03/5.03            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5281_square__eq__1__iff,axiom,
% 9.03/5.03      ! [X: int] :
% 9.03/5.03        ( ( ( times_times_int @ X @ X )
% 9.03/5.03          = one_one_int )
% 9.03/5.03        = ( ( X = one_one_int )
% 9.03/5.03          | ( X
% 9.03/5.03            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5282_square__eq__1__iff,axiom,
% 9.03/5.03      ! [X: complex] :
% 9.03/5.03        ( ( ( times_times_complex @ X @ X )
% 9.03/5.03          = one_one_complex )
% 9.03/5.03        = ( ( X = one_one_complex )
% 9.03/5.03          | ( X
% 9.03/5.03            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5283_square__eq__1__iff,axiom,
% 9.03/5.03      ! [X: rat] :
% 9.03/5.03        ( ( ( times_times_rat @ X @ X )
% 9.03/5.03          = one_one_rat )
% 9.03/5.03        = ( ( X = one_one_rat )
% 9.03/5.03          | ( X
% 9.03/5.03            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5284_square__eq__1__iff,axiom,
% 9.03/5.03      ! [X: code_integer] :
% 9.03/5.03        ( ( ( times_3573771949741848930nteger @ X @ X )
% 9.03/5.03          = one_one_Code_integer )
% 9.03/5.03        = ( ( X = one_one_Code_integer )
% 9.03/5.03          | ( X
% 9.03/5.03            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5285_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.03/5.03      ( minus_minus_real
% 9.03/5.03      = ( ^ [A3: real,B4: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5286_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.03/5.03      ( minus_minus_int
% 9.03/5.03      = ( ^ [A3: int,B4: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5287_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.03/5.03      ( minus_minus_complex
% 9.03/5.03      = ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5288_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.03/5.03      ( minus_minus_rat
% 9.03/5.03      = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5289_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.03/5.03      ( minus_8373710615458151222nteger
% 9.03/5.03      = ( ^ [A3: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5290_diff__conv__add__uminus,axiom,
% 9.03/5.03      ( minus_minus_real
% 9.03/5.03      = ( ^ [A3: real,B4: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5291_diff__conv__add__uminus,axiom,
% 9.03/5.03      ( minus_minus_int
% 9.03/5.03      = ( ^ [A3: int,B4: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5292_diff__conv__add__uminus,axiom,
% 9.03/5.03      ( minus_minus_complex
% 9.03/5.03      = ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5293_diff__conv__add__uminus,axiom,
% 9.03/5.03      ( minus_minus_rat
% 9.03/5.03      = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5294_diff__conv__add__uminus,axiom,
% 9.03/5.03      ( minus_8373710615458151222nteger
% 9.03/5.03      = ( ^ [A3: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5295_group__cancel_Osub2,axiom,
% 9.03/5.03      ! [B2: real,K: real,B: real,A: real] :
% 9.03/5.03        ( ( B2
% 9.03/5.03          = ( plus_plus_real @ K @ B ) )
% 9.03/5.03       => ( ( minus_minus_real @ A @ B2 )
% 9.03/5.03          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5296_group__cancel_Osub2,axiom,
% 9.03/5.03      ! [B2: int,K: int,B: int,A: int] :
% 9.03/5.03        ( ( B2
% 9.03/5.03          = ( plus_plus_int @ K @ B ) )
% 9.03/5.03       => ( ( minus_minus_int @ A @ B2 )
% 9.03/5.03          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5297_group__cancel_Osub2,axiom,
% 9.03/5.03      ! [B2: complex,K: complex,B: complex,A: complex] :
% 9.03/5.03        ( ( B2
% 9.03/5.03          = ( plus_plus_complex @ K @ B ) )
% 9.03/5.03       => ( ( minus_minus_complex @ A @ B2 )
% 9.03/5.03          = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5298_group__cancel_Osub2,axiom,
% 9.03/5.03      ! [B2: rat,K: rat,B: rat,A: rat] :
% 9.03/5.03        ( ( B2
% 9.03/5.03          = ( plus_plus_rat @ K @ B ) )
% 9.03/5.03       => ( ( minus_minus_rat @ A @ B2 )
% 9.03/5.03          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5299_group__cancel_Osub2,axiom,
% 9.03/5.03      ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 9.03/5.03        ( ( B2
% 9.03/5.03          = ( plus_p5714425477246183910nteger @ K @ B ) )
% 9.03/5.03       => ( ( minus_8373710615458151222nteger @ A @ B2 )
% 9.03/5.03          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5300_mod__eq__0D,axiom,
% 9.03/5.03      ! [M: nat,D: nat] :
% 9.03/5.03        ( ( ( modulo_modulo_nat @ M @ D )
% 9.03/5.03          = zero_zero_nat )
% 9.03/5.03       => ? [Q2: nat] :
% 9.03/5.03            ( M
% 9.03/5.03            = ( times_times_nat @ D @ Q2 ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5301_mod__geq,axiom,
% 9.03/5.03      ! [M: nat,N: nat] :
% 9.03/5.03        ( ~ ( ord_less_nat @ M @ N )
% 9.03/5.03       => ( ( modulo_modulo_nat @ M @ N )
% 9.03/5.03          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5302_mod__if,axiom,
% 9.03/5.03      ( modulo_modulo_nat
% 9.03/5.03      = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N3 ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5303_le__mod__geq,axiom,
% 9.03/5.03      ! [N: nat,M: nat] :
% 9.03/5.03        ( ( ord_less_eq_nat @ N @ M )
% 9.03/5.03       => ( ( modulo_modulo_nat @ M @ N )
% 9.03/5.03          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5304_nat__mod__eq__iff,axiom,
% 9.03/5.03      ! [X: nat,N: nat,Y: nat] :
% 9.03/5.03        ( ( ( modulo_modulo_nat @ X @ N )
% 9.03/5.03          = ( modulo_modulo_nat @ Y @ N ) )
% 9.03/5.03        = ( ? [Q1: nat,Q22: nat] :
% 9.03/5.03              ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
% 9.03/5.03              = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5305_real__minus__mult__self__le,axiom,
% 9.03/5.03      ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5306_vebt__buildup_Osimps_I1_J,axiom,
% 9.03/5.03      ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 9.03/5.03      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5307_minus__real__def,axiom,
% 9.03/5.03      ( minus_minus_real
% 9.03/5.03      = ( ^ [X4: real,Y6: real] : ( plus_plus_real @ X4 @ ( uminus_uminus_real @ Y6 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5308_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 9.03/5.03      ! [Uu: $o,Uv: $o,D: nat] :
% 9.03/5.03        ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 9.03/5.03        = ( D = one_one_nat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5309_VEBT__internal_Oheight_Osimps_I1_J,axiom,
% 9.03/5.03      ! [A: $o,B: $o] :
% 9.03/5.03        ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A @ B ) )
% 9.03/5.03        = zero_zero_nat ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5310_tanh__real__lt__1,axiom,
% 9.03/5.03      ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5311_dense__eq0__I,axiom,
% 9.03/5.03      ! [X: real] :
% 9.03/5.03        ( ! [E: real] :
% 9.03/5.03            ( ( ord_less_real @ zero_zero_real @ E )
% 9.03/5.03           => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
% 9.03/5.03       => ( X = zero_zero_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5312_dense__eq0__I,axiom,
% 9.03/5.03      ! [X: rat] :
% 9.03/5.03        ( ! [E: rat] :
% 9.03/5.03            ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.03/5.03           => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
% 9.03/5.03       => ( X = zero_zero_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5313_abs__eq__mult,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.03/5.03            | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 9.03/5.03          & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 9.03/5.03            | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 9.03/5.03       => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 9.03/5.03          = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5314_abs__eq__mult,axiom,
% 9.03/5.03      ! [A: real,B: real] :
% 9.03/5.03        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.03/5.03            | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.03/5.03          & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.03/5.03            | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 9.03/5.03       => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 9.03/5.03          = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5315_abs__eq__mult,axiom,
% 9.03/5.03      ! [A: rat,B: rat] :
% 9.03/5.03        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.03/5.03            | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.03/5.03          & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.03/5.03            | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 9.03/5.03       => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 9.03/5.03          = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5316_abs__eq__mult,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/5.03            | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.03/5.03          & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.03/5.03            | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 9.03/5.03       => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 9.03/5.03          = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5317_abs__mult__pos,axiom,
% 9.03/5.03      ! [X: code_integer,Y: code_integer] :
% 9.03/5.03        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 9.03/5.03       => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
% 9.03/5.03          = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5318_abs__mult__pos,axiom,
% 9.03/5.03      ! [X: real,Y: real] :
% 9.03/5.03        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.03/5.03       => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
% 9.03/5.03          = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5319_abs__mult__pos,axiom,
% 9.03/5.03      ! [X: rat,Y: rat] :
% 9.03/5.03        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.03/5.03       => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
% 9.03/5.03          = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5320_abs__mult__pos,axiom,
% 9.03/5.03      ! [X: int,Y: int] :
% 9.03/5.03        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.03/5.03       => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
% 9.03/5.03          = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5321_zero__le__power__abs,axiom,
% 9.03/5.03      ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5322_zero__le__power__abs,axiom,
% 9.03/5.03      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5323_zero__le__power__abs,axiom,
% 9.03/5.03      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5324_zero__le__power__abs,axiom,
% 9.03/5.03      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5325_abs__div__pos,axiom,
% 9.03/5.03      ! [Y: real,X: real] :
% 9.03/5.03        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.03/5.03       => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
% 9.03/5.03          = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5326_abs__div__pos,axiom,
% 9.03/5.03      ! [Y: rat,X: rat] :
% 9.03/5.03        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.03/5.03       => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
% 9.03/5.03          = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5327_abs__triangle__ineq4,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5328_abs__triangle__ineq4,axiom,
% 9.03/5.03      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5329_abs__triangle__ineq4,axiom,
% 9.03/5.03      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5330_abs__triangle__ineq4,axiom,
% 9.03/5.03      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5331_abs__diff__triangle__ineq,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5332_abs__diff__triangle__ineq,axiom,
% 9.03/5.03      ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5333_abs__diff__triangle__ineq,axiom,
% 9.03/5.03      ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5334_abs__diff__triangle__ineq,axiom,
% 9.03/5.03      ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5335_abs__diff__le__iff,axiom,
% 9.03/5.03      ! [X: code_integer,A: code_integer,R2: code_integer] :
% 9.03/5.03        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 9.03/5.03        = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 9.03/5.03          & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5336_abs__diff__le__iff,axiom,
% 9.03/5.03      ! [X: real,A: real,R2: real] :
% 9.03/5.03        ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 9.03/5.03        = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 9.03/5.03          & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5337_abs__diff__le__iff,axiom,
% 9.03/5.03      ! [X: rat,A: rat,R2: rat] :
% 9.03/5.03        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 9.03/5.03        = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 9.03/5.03          & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5338_abs__diff__le__iff,axiom,
% 9.03/5.03      ! [X: int,A: int,R2: int] :
% 9.03/5.03        ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 9.03/5.03        = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 9.03/5.03          & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5339_abs__diff__less__iff,axiom,
% 9.03/5.03      ! [X: code_integer,A: code_integer,R2: code_integer] :
% 9.03/5.03        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 9.03/5.03        = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 9.03/5.03          & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5340_abs__diff__less__iff,axiom,
% 9.03/5.03      ! [X: real,A: real,R2: real] :
% 9.03/5.03        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 9.03/5.03        = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 9.03/5.03          & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5341_abs__diff__less__iff,axiom,
% 9.03/5.03      ! [X: rat,A: rat,R2: rat] :
% 9.03/5.03        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 9.03/5.03        = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 9.03/5.03          & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5342_abs__diff__less__iff,axiom,
% 9.03/5.03      ! [X: int,A: int,R2: int] :
% 9.03/5.03        ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 9.03/5.03        = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 9.03/5.03          & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5343_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.03/5.03      ! [B: code_integer,A: code_integer] :
% 9.03/5.03        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.03/5.03       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5344_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.03/5.03      ! [B: nat,A: nat] :
% 9.03/5.03        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.03/5.03       => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5345_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.03/5.03      ! [B: int,A: int] :
% 9.03/5.03        ( ( ord_less_int @ zero_zero_int @ B )
% 9.03/5.03       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5346_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.03/5.03       => ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.03/5.03         => ( ( modulo364778990260209775nteger @ A @ B )
% 9.03/5.03            = A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5347_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.03/5.03       => ( ( ord_less_nat @ A @ B )
% 9.03/5.03         => ( ( modulo_modulo_nat @ A @ B )
% 9.03/5.03            = A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5348_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.03/5.03       => ( ( ord_less_int @ A @ B )
% 9.03/5.03         => ( ( modulo_modulo_int @ A @ B )
% 9.03/5.03            = A ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5349_nonzero__of__real__divide,axiom,
% 9.03/5.03      ! [Y: real,X: real] :
% 9.03/5.03        ( ( Y != zero_zero_real )
% 9.03/5.03       => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
% 9.03/5.03          = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5350_nonzero__of__real__divide,axiom,
% 9.03/5.03      ! [Y: real,X: real] :
% 9.03/5.03        ( ( Y != zero_zero_real )
% 9.03/5.03       => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
% 9.03/5.03          = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5351_cong__exp__iff__simps_I2_J,axiom,
% 9.03/5.03      ! [N: num,Q3: num] :
% 9.03/5.03        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.03/5.03          = zero_zero_nat )
% 9.03/5.03        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 9.03/5.03          = zero_zero_nat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5352_cong__exp__iff__simps_I2_J,axiom,
% 9.03/5.03      ! [N: num,Q3: num] :
% 9.03/5.03        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.03/5.03          = zero_zero_int )
% 9.03/5.03        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 9.03/5.03          = zero_zero_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5353_cong__exp__iff__simps_I2_J,axiom,
% 9.03/5.03      ! [N: num,Q3: num] :
% 9.03/5.03        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.03/5.03          = zero_z3403309356797280102nteger )
% 9.03/5.03        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 9.03/5.03          = zero_z3403309356797280102nteger ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5354_cong__exp__iff__simps_I1_J,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 9.03/5.03        = zero_zero_nat ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5355_cong__exp__iff__simps_I1_J,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 9.03/5.03        = zero_zero_int ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5356_cong__exp__iff__simps_I1_J,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 9.03/5.03        = zero_z3403309356797280102nteger ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5357_cong__exp__iff__simps_I8_J,axiom,
% 9.03/5.03      ! [M: num,Q3: num] :
% 9.03/5.03        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.03/5.03       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5358_cong__exp__iff__simps_I8_J,axiom,
% 9.03/5.03      ! [M: num,Q3: num] :
% 9.03/5.03        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.03/5.03       != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5359_cong__exp__iff__simps_I8_J,axiom,
% 9.03/5.03      ! [M: num,Q3: num] :
% 9.03/5.03        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.03/5.03       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5360_cong__exp__iff__simps_I6_J,axiom,
% 9.03/5.03      ! [Q3: num,N: num] :
% 9.03/5.03        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.03/5.03       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5361_cong__exp__iff__simps_I6_J,axiom,
% 9.03/5.03      ! [Q3: num,N: num] :
% 9.03/5.03        ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.03/5.03       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5362_cong__exp__iff__simps_I6_J,axiom,
% 9.03/5.03      ! [Q3: num,N: num] :
% 9.03/5.03        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.03/5.03       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5363_div__mult1__eq,axiom,
% 9.03/5.03      ! [A: nat,B: nat,C: nat] :
% 9.03/5.03        ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.03/5.03        = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5364_div__mult1__eq,axiom,
% 9.03/5.03      ! [A: int,B: int,C: int] :
% 9.03/5.03        ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 9.03/5.03        = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5365_div__mult1__eq,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.03/5.03        ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 9.03/5.03        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5366_cancel__div__mod__rules_I2_J,axiom,
% 9.03/5.03      ! [B: nat,A: nat,C: nat] :
% 9.03/5.03        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 9.03/5.03        = ( plus_plus_nat @ A @ C ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5367_cancel__div__mod__rules_I2_J,axiom,
% 9.03/5.03      ! [B: int,A: int,C: int] :
% 9.03/5.03        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 9.03/5.03        = ( plus_plus_int @ A @ C ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5368_cancel__div__mod__rules_I2_J,axiom,
% 9.03/5.03      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.03/5.03        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 9.03/5.03        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5369_cancel__div__mod__rules_I1_J,axiom,
% 9.03/5.03      ! [A: nat,B: nat,C: nat] :
% 9.03/5.03        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 9.03/5.03        = ( plus_plus_nat @ A @ C ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5370_cancel__div__mod__rules_I1_J,axiom,
% 9.03/5.03      ! [A: int,B: int,C: int] :
% 9.03/5.03        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 9.03/5.03        = ( plus_plus_int @ A @ C ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5371_cancel__div__mod__rules_I1_J,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.03/5.03        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 9.03/5.03        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5372_mod__div__decomp,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( A
% 9.03/5.03        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5373_mod__div__decomp,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( A
% 9.03/5.03        = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5374_mod__div__decomp,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( A
% 9.03/5.03        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5375_div__mult__mod__eq,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5376_div__mult__mod__eq,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5377_div__mult__mod__eq,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5378_mod__div__mult__eq,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5379_mod__div__mult__eq,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5380_mod__div__mult__eq,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5381_mod__mult__div__eq,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5382_mod__mult__div__eq,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5383_mod__mult__div__eq,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5384_mult__div__mod__eq,axiom,
% 9.03/5.03      ! [B: nat,A: nat] :
% 9.03/5.03        ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5385_mult__div__mod__eq,axiom,
% 9.03/5.03      ! [B: int,A: int] :
% 9.03/5.03        ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5386_mult__div__mod__eq,axiom,
% 9.03/5.03      ! [B: code_integer,A: code_integer] :
% 9.03/5.03        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 9.03/5.03        = A ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5387_minus__div__mult__eq__mod,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 9.03/5.03        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5388_minus__div__mult__eq__mod,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 9.03/5.03        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5389_minus__div__mult__eq__mod,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 9.03/5.03        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5390_minus__mod__eq__div__mult,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 9.03/5.03        = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5391_minus__mod__eq__div__mult,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 9.03/5.03        = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5392_minus__mod__eq__div__mult,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 9.03/5.03        = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5393_minus__mod__eq__mult__div,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 9.03/5.03        = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5394_minus__mod__eq__mult__div,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 9.03/5.03        = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5395_minus__mod__eq__mult__div,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 9.03/5.03        = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5396_minus__mult__div__eq__mod,axiom,
% 9.03/5.03      ! [A: nat,B: nat] :
% 9.03/5.03        ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 9.03/5.03        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5397_minus__mult__div__eq__mod,axiom,
% 9.03/5.03      ! [A: int,B: int] :
% 9.03/5.03        ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 9.03/5.03        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5398_minus__mult__div__eq__mod,axiom,
% 9.03/5.03      ! [A: code_integer,B: code_integer] :
% 9.03/5.03        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 9.03/5.03        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5399_not__zero__le__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5400_not__zero__le__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5401_not__zero__le__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5402_not__zero__le__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5403_neg__numeral__le__zero,axiom,
% 9.03/5.03      ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5404_neg__numeral__le__zero,axiom,
% 9.03/5.03      ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5405_neg__numeral__le__zero,axiom,
% 9.03/5.03      ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5406_neg__numeral__le__zero,axiom,
% 9.03/5.03      ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5407_not__zero__less__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5408_not__zero__less__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5409_not__zero__less__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5410_not__zero__less__neg__numeral,axiom,
% 9.03/5.03      ! [N: num] :
% 9.03/5.03        ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5411_neg__numeral__less__zero,axiom,
% 9.03/5.03      ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5412_neg__numeral__less__zero,axiom,
% 9.03/5.03      ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5413_neg__numeral__less__zero,axiom,
% 9.03/5.03      ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5414_neg__numeral__less__zero,axiom,
% 9.03/5.03      ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5415_le__minus__one__simps_I3_J,axiom,
% 9.03/5.03      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5416_le__minus__one__simps_I3_J,axiom,
% 9.03/5.03      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5417_le__minus__one__simps_I3_J,axiom,
% 9.03/5.03      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5418_le__minus__one__simps_I3_J,axiom,
% 9.03/5.03      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5419_le__minus__one__simps_I1_J,axiom,
% 9.03/5.03      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5420_le__minus__one__simps_I1_J,axiom,
% 9.03/5.03      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5421_le__minus__one__simps_I1_J,axiom,
% 9.03/5.03      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5422_le__minus__one__simps_I1_J,axiom,
% 9.03/5.03      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5423_less__minus__one__simps_I3_J,axiom,
% 9.03/5.03      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5424_less__minus__one__simps_I3_J,axiom,
% 9.03/5.03      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5425_less__minus__one__simps_I3_J,axiom,
% 9.03/5.03      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5426_less__minus__one__simps_I3_J,axiom,
% 9.03/5.03      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5427_less__minus__one__simps_I1_J,axiom,
% 9.03/5.03      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5428_less__minus__one__simps_I1_J,axiom,
% 9.03/5.03      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5429_less__minus__one__simps_I1_J,axiom,
% 9.03/5.03      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5430_less__minus__one__simps_I1_J,axiom,
% 9.03/5.03      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5431_neg__numeral__le__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5432_neg__numeral__le__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5433_neg__numeral__le__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5434_neg__numeral__le__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5435_neg__one__le__numeral,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5436_neg__one__le__numeral,axiom,
% 9.03/5.03      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5437_neg__one__le__numeral,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5438_neg__one__le__numeral,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5439_neg__numeral__le__neg__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5440_neg__numeral__le__neg__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5441_neg__numeral__le__neg__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5442_neg__numeral__le__neg__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5443_not__numeral__le__neg__one,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5444_not__numeral__le__neg__one,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5445_not__numeral__le__neg__one,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5446_not__numeral__le__neg__one,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5447_not__one__le__neg__numeral,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5448_not__one__le__neg__numeral,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5449_not__one__le__neg__numeral,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5450_not__one__le__neg__numeral,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5451_neg__numeral__less__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5452_neg__numeral__less__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5453_neg__numeral__less__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5454_neg__numeral__less__one,axiom,
% 9.03/5.03      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5455_neg__one__less__numeral,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5456_neg__one__less__numeral,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5457_neg__one__less__numeral,axiom,
% 9.03/5.03      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5458_neg__one__less__numeral,axiom,
% 9.03/5.03      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5459_not__numeral__less__neg__one,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5460_not__numeral__less__neg__one,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5461_not__numeral__less__neg__one,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5462_not__numeral__less__neg__one,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5463_not__one__less__neg__numeral,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5464_not__one__less__neg__numeral,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5465_not__one__less__neg__numeral,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5466_not__one__less__neg__numeral,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5467_not__neg__one__less__neg__numeral,axiom,
% 9.03/5.03      ! [M: num] :
% 9.03/5.03        ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.03/5.03  
% 9.03/5.03  thf(fact_5468_not__neg__one__less__neg__numeral,axiom,
% 9.03/5.04      ! [M: num] :
% 9.03/5.04        ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5469_not__neg__one__less__neg__numeral,axiom,
% 9.03/5.04      ! [M: num] :
% 9.03/5.04        ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5470_not__neg__one__less__neg__numeral,axiom,
% 9.03/5.04      ! [M: num] :
% 9.03/5.04        ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5471_mod__le__divisor,axiom,
% 9.03/5.04      ! [N: nat,M: nat] :
% 9.03/5.04        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/5.04       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5472_nonzero__neg__divide__eq__eq2,axiom,
% 9.03/5.04      ! [B: real,C: real,A: real] :
% 9.03/5.04        ( ( B != zero_zero_real )
% 9.03/5.04       => ( ( C
% 9.03/5.04            = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 9.03/5.04          = ( ( times_times_real @ C @ B )
% 9.03/5.04            = ( uminus_uminus_real @ A ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5473_nonzero__neg__divide__eq__eq2,axiom,
% 9.03/5.04      ! [B: complex,C: complex,A: complex] :
% 9.03/5.04        ( ( B != zero_zero_complex )
% 9.03/5.04       => ( ( C
% 9.03/5.04            = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 9.03/5.04          = ( ( times_times_complex @ C @ B )
% 9.03/5.04            = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5474_nonzero__neg__divide__eq__eq2,axiom,
% 9.03/5.04      ! [B: rat,C: rat,A: rat] :
% 9.03/5.04        ( ( B != zero_zero_rat )
% 9.03/5.04       => ( ( C
% 9.03/5.04            = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 9.03/5.04          = ( ( times_times_rat @ C @ B )
% 9.03/5.04            = ( uminus_uminus_rat @ A ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5475_nonzero__neg__divide__eq__eq,axiom,
% 9.03/5.04      ! [B: real,A: real,C: real] :
% 9.03/5.04        ( ( B != zero_zero_real )
% 9.03/5.04       => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 9.03/5.04            = C )
% 9.03/5.04          = ( ( uminus_uminus_real @ A )
% 9.03/5.04            = ( times_times_real @ C @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5476_nonzero__neg__divide__eq__eq,axiom,
% 9.03/5.04      ! [B: complex,A: complex,C: complex] :
% 9.03/5.04        ( ( B != zero_zero_complex )
% 9.03/5.04       => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.03/5.04            = C )
% 9.03/5.04          = ( ( uminus1482373934393186551omplex @ A )
% 9.03/5.04            = ( times_times_complex @ C @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5477_nonzero__neg__divide__eq__eq,axiom,
% 9.03/5.04      ! [B: rat,A: rat,C: rat] :
% 9.03/5.04        ( ( B != zero_zero_rat )
% 9.03/5.04       => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 9.03/5.04            = C )
% 9.03/5.04          = ( ( uminus_uminus_rat @ A )
% 9.03/5.04            = ( times_times_rat @ C @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5478_minus__divide__eq__eq,axiom,
% 9.03/5.04      ! [B: real,C: real,A: real] :
% 9.03/5.04        ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 9.03/5.04          = A )
% 9.03/5.04        = ( ( ( C != zero_zero_real )
% 9.03/5.04           => ( ( uminus_uminus_real @ B )
% 9.03/5.04              = ( times_times_real @ A @ C ) ) )
% 9.03/5.04          & ( ( C = zero_zero_real )
% 9.03/5.04           => ( A = zero_zero_real ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5479_minus__divide__eq__eq,axiom,
% 9.03/5.04      ! [B: complex,C: complex,A: complex] :
% 9.03/5.04        ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 9.03/5.04          = A )
% 9.03/5.04        = ( ( ( C != zero_zero_complex )
% 9.03/5.04           => ( ( uminus1482373934393186551omplex @ B )
% 9.03/5.04              = ( times_times_complex @ A @ C ) ) )
% 9.03/5.04          & ( ( C = zero_zero_complex )
% 9.03/5.04           => ( A = zero_zero_complex ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5480_minus__divide__eq__eq,axiom,
% 9.03/5.04      ! [B: rat,C: rat,A: rat] :
% 9.03/5.04        ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 9.03/5.04          = A )
% 9.03/5.04        = ( ( ( C != zero_zero_rat )
% 9.03/5.04           => ( ( uminus_uminus_rat @ B )
% 9.03/5.04              = ( times_times_rat @ A @ C ) ) )
% 9.03/5.04          & ( ( C = zero_zero_rat )
% 9.03/5.04           => ( A = zero_zero_rat ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5481_eq__minus__divide__eq,axiom,
% 9.03/5.04      ! [A: real,B: real,C: real] :
% 9.03/5.04        ( ( A
% 9.03/5.04          = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.03/5.04        = ( ( ( C != zero_zero_real )
% 9.03/5.04           => ( ( times_times_real @ A @ C )
% 9.03/5.04              = ( uminus_uminus_real @ B ) ) )
% 9.03/5.04          & ( ( C = zero_zero_real )
% 9.03/5.04           => ( A = zero_zero_real ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5482_eq__minus__divide__eq,axiom,
% 9.03/5.04      ! [A: complex,B: complex,C: complex] :
% 9.03/5.04        ( ( A
% 9.03/5.04          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 9.03/5.04        = ( ( ( C != zero_zero_complex )
% 9.03/5.04           => ( ( times_times_complex @ A @ C )
% 9.03/5.04              = ( uminus1482373934393186551omplex @ B ) ) )
% 9.03/5.04          & ( ( C = zero_zero_complex )
% 9.03/5.04           => ( A = zero_zero_complex ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5483_eq__minus__divide__eq,axiom,
% 9.03/5.04      ! [A: rat,B: rat,C: rat] :
% 9.03/5.04        ( ( A
% 9.03/5.04          = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.03/5.04        = ( ( ( C != zero_zero_rat )
% 9.03/5.04           => ( ( times_times_rat @ A @ C )
% 9.03/5.04              = ( uminus_uminus_rat @ B ) ) )
% 9.03/5.04          & ( ( C = zero_zero_rat )
% 9.03/5.04           => ( A = zero_zero_rat ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5484_divide__eq__minus__1__iff,axiom,
% 9.03/5.04      ! [A: real,B: real] :
% 9.03/5.04        ( ( ( divide_divide_real @ A @ B )
% 9.03/5.04          = ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.04        = ( ( B != zero_zero_real )
% 9.03/5.04          & ( A
% 9.03/5.04            = ( uminus_uminus_real @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5485_divide__eq__minus__1__iff,axiom,
% 9.03/5.04      ! [A: complex,B: complex] :
% 9.03/5.04        ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.03/5.04          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.03/5.04        = ( ( B != zero_zero_complex )
% 9.03/5.04          & ( A
% 9.03/5.04            = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5486_divide__eq__minus__1__iff,axiom,
% 9.03/5.04      ! [A: rat,B: rat] :
% 9.03/5.04        ( ( ( divide_divide_rat @ A @ B )
% 9.03/5.04          = ( uminus_uminus_rat @ one_one_rat ) )
% 9.03/5.04        = ( ( B != zero_zero_rat )
% 9.03/5.04          & ( A
% 9.03/5.04            = ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5487_mult__1s__ring__1_I2_J,axiom,
% 9.03/5.04      ! [B: real] :
% 9.03/5.04        ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 9.03/5.04        = ( uminus_uminus_real @ B ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5488_mult__1s__ring__1_I2_J,axiom,
% 9.03/5.04      ! [B: int] :
% 9.03/5.04        ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 9.03/5.04        = ( uminus_uminus_int @ B ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5489_mult__1s__ring__1_I2_J,axiom,
% 9.03/5.04      ! [B: complex] :
% 9.03/5.04        ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 9.03/5.04        = ( uminus1482373934393186551omplex @ B ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5490_mult__1s__ring__1_I2_J,axiom,
% 9.03/5.04      ! [B: rat] :
% 9.03/5.04        ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 9.03/5.04        = ( uminus_uminus_rat @ B ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5491_mult__1s__ring__1_I2_J,axiom,
% 9.03/5.04      ! [B: code_integer] :
% 9.03/5.04        ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 9.03/5.04        = ( uminus1351360451143612070nteger @ B ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5492_mult__1s__ring__1_I1_J,axiom,
% 9.03/5.04      ! [B: real] :
% 9.03/5.04        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 9.03/5.04        = ( uminus_uminus_real @ B ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5493_mult__1s__ring__1_I1_J,axiom,
% 9.03/5.04      ! [B: int] :
% 9.03/5.04        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 9.03/5.04        = ( uminus_uminus_int @ B ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5494_mult__1s__ring__1_I1_J,axiom,
% 9.03/5.04      ! [B: complex] :
% 9.03/5.04        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 9.03/5.04        = ( uminus1482373934393186551omplex @ B ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5495_mult__1s__ring__1_I1_J,axiom,
% 9.03/5.04      ! [B: rat] :
% 9.03/5.04        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 9.03/5.04        = ( uminus_uminus_rat @ B ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5496_mult__1s__ring__1_I1_J,axiom,
% 9.03/5.04      ! [B: code_integer] :
% 9.03/5.04        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 9.03/5.04        = ( uminus1351360451143612070nteger @ B ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5497_uminus__numeral__One,axiom,
% 9.03/5.04      ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 9.03/5.04      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5498_uminus__numeral__One,axiom,
% 9.03/5.04      ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 9.03/5.04      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5499_uminus__numeral__One,axiom,
% 9.03/5.04      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 9.03/5.04      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5500_uminus__numeral__One,axiom,
% 9.03/5.04      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 9.03/5.04      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5501_uminus__numeral__One,axiom,
% 9.03/5.04      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 9.03/5.04      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5502_power__minus,axiom,
% 9.03/5.04      ! [A: real,N: nat] :
% 9.03/5.04        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 9.03/5.04        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5503_power__minus,axiom,
% 9.03/5.04      ! [A: int,N: nat] :
% 9.03/5.04        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 9.03/5.04        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5504_power__minus,axiom,
% 9.03/5.04      ! [A: complex,N: nat] :
% 9.03/5.04        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 9.03/5.04        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5505_power__minus,axiom,
% 9.03/5.04      ! [A: rat,N: nat] :
% 9.03/5.04        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 9.03/5.04        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5506_power__minus,axiom,
% 9.03/5.04      ! [A: code_integer,N: nat] :
% 9.03/5.04        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 9.03/5.04        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5507_div__less__mono,axiom,
% 9.03/5.04      ! [A2: nat,B2: nat,N: nat] :
% 9.03/5.04        ( ( ord_less_nat @ A2 @ B2 )
% 9.03/5.04       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.03/5.04         => ( ( ( modulo_modulo_nat @ A2 @ N )
% 9.03/5.04              = zero_zero_nat )
% 9.03/5.04           => ( ( ( modulo_modulo_nat @ B2 @ N )
% 9.03/5.04                = zero_zero_nat )
% 9.03/5.04             => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5508_mod__eq__nat1E,axiom,
% 9.03/5.04      ! [M: nat,Q3: nat,N: nat] :
% 9.03/5.04        ( ( ( modulo_modulo_nat @ M @ Q3 )
% 9.03/5.04          = ( modulo_modulo_nat @ N @ Q3 ) )
% 9.03/5.04       => ( ( ord_less_eq_nat @ N @ M )
% 9.03/5.04         => ~ ! [S3: nat] :
% 9.03/5.04                ( M
% 9.03/5.04               != ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5509_mod__eq__nat2E,axiom,
% 9.03/5.04      ! [M: nat,Q3: nat,N: nat] :
% 9.03/5.04        ( ( ( modulo_modulo_nat @ M @ Q3 )
% 9.03/5.04          = ( modulo_modulo_nat @ N @ Q3 ) )
% 9.03/5.04       => ( ( ord_less_eq_nat @ M @ N )
% 9.03/5.04         => ~ ! [S3: nat] :
% 9.03/5.04                ( N
% 9.03/5.04               != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5510_nat__mod__eq__lemma,axiom,
% 9.03/5.04      ! [X: nat,N: nat,Y: nat] :
% 9.03/5.04        ( ( ( modulo_modulo_nat @ X @ N )
% 9.03/5.04          = ( modulo_modulo_nat @ Y @ N ) )
% 9.03/5.04       => ( ( ord_less_eq_nat @ Y @ X )
% 9.03/5.04         => ? [Q2: nat] :
% 9.03/5.04              ( X
% 9.03/5.04              = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q2 ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5511_lemma__interval__lt,axiom,
% 9.03/5.04      ! [A: real,X: real,B: real] :
% 9.03/5.04        ( ( ord_less_real @ A @ X )
% 9.03/5.04       => ( ( ord_less_real @ X @ B )
% 9.03/5.04         => ? [D5: real] :
% 9.03/5.04              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.03/5.04              & ! [Y5: real] :
% 9.03/5.04                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D5 )
% 9.03/5.04                 => ( ( ord_less_real @ A @ Y5 )
% 9.03/5.04                    & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5512_power__minus__Bit0,axiom,
% 9.03/5.04      ! [X: real,K: num] :
% 9.03/5.04        ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.03/5.04        = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5513_power__minus__Bit0,axiom,
% 9.03/5.04      ! [X: int,K: num] :
% 9.03/5.04        ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.03/5.04        = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5514_power__minus__Bit0,axiom,
% 9.03/5.04      ! [X: complex,K: num] :
% 9.03/5.04        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.03/5.04        = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5515_power__minus__Bit0,axiom,
% 9.03/5.04      ! [X: rat,K: num] :
% 9.03/5.04        ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.03/5.04        = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5516_power__minus__Bit0,axiom,
% 9.03/5.04      ! [X: code_integer,K: num] :
% 9.03/5.04        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.03/5.04        = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5517_norm__uminus__minus,axiom,
% 9.03/5.04      ! [X: real,Y: real] :
% 9.03/5.04        ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
% 9.03/5.04        = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5518_norm__uminus__minus,axiom,
% 9.03/5.04      ! [X: complex,Y: complex] :
% 9.03/5.04        ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
% 9.03/5.04        = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5519_invar__vebt_Ointros_I1_J,axiom,
% 9.03/5.04      ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5520_mod__mult2__eq,axiom,
% 9.03/5.04      ! [M: nat,N: nat,Q3: nat] :
% 9.03/5.04        ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q3 ) )
% 9.03/5.04        = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5521_div__mod__decomp,axiom,
% 9.03/5.04      ! [A2: nat,N: nat] :
% 9.03/5.04        ( A2
% 9.03/5.04        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5522_modulo__nat__def,axiom,
% 9.03/5.04      ( modulo_modulo_nat
% 9.03/5.04      = ( ^ [M6: nat,N3: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N3 ) @ N3 ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5523_sin__bound__lemma,axiom,
% 9.03/5.04      ! [X: real,Y: real,U: real,V: real] :
% 9.03/5.04        ( ( X = Y )
% 9.03/5.04       => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 9.03/5.04         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5524_powr__minus__divide,axiom,
% 9.03/5.04      ! [X: real,A: real] :
% 9.03/5.04        ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
% 9.03/5.04        = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5525_real__0__less__add__iff,axiom,
% 9.03/5.04      ! [X: real,Y: real] :
% 9.03/5.04        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 9.03/5.04        = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5526_real__add__less__0__iff,axiom,
% 9.03/5.04      ! [X: real,Y: real] :
% 9.03/5.04        ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 9.03/5.04        = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5527_real__add__le__0__iff,axiom,
% 9.03/5.04      ! [X: real,Y: real] :
% 9.03/5.04        ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 9.03/5.04        = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5528_real__0__le__add__iff,axiom,
% 9.03/5.04      ! [X: real,Y: real] :
% 9.03/5.04        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 9.03/5.04        = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5529_vebt__buildup_Osimps_I2_J,axiom,
% 9.03/5.04      ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 9.03/5.04      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5530_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 9.03/5.04      ! [A: $o,B: $o,X: nat] :
% 9.03/5.04        ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 9.03/5.04        = ( ( ( X = zero_zero_nat )
% 9.03/5.04           => A )
% 9.03/5.04          & ( ( X != zero_zero_nat )
% 9.03/5.04           => ( ( ( X = one_one_nat )
% 9.03/5.04               => B )
% 9.03/5.04              & ( X = one_one_nat ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5531_divide__powr__uminus,axiom,
% 9.03/5.04      ! [A: real,B: real,C: real] :
% 9.03/5.04        ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 9.03/5.04        = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5532_abs__add__one__gt__zero,axiom,
% 9.03/5.04      ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5533_abs__add__one__gt__zero,axiom,
% 9.03/5.04      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5534_abs__add__one__gt__zero,axiom,
% 9.03/5.04      ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5535_abs__add__one__gt__zero,axiom,
% 9.03/5.04      ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5536_norm__less__p1,axiom,
% 9.03/5.04      ! [X: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X ) ) @ one_one_real ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5537_norm__less__p1,axiom,
% 9.03/5.04      ! [X: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X ) ) @ one_one_complex ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5538_of__int__leD,axiom,
% 9.03/5.04      ! [N: int,X: real] :
% 9.03/5.04        ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 9.03/5.04       => ( ( N = zero_zero_int )
% 9.03/5.04          | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5539_of__int__leD,axiom,
% 9.03/5.04      ! [N: int,X: code_integer] :
% 9.03/5.04        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 9.03/5.04       => ( ( N = zero_zero_int )
% 9.03/5.04          | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5540_of__int__leD,axiom,
% 9.03/5.04      ! [N: int,X: rat] :
% 9.03/5.04        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 9.03/5.04       => ( ( N = zero_zero_int )
% 9.03/5.04          | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5541_of__int__leD,axiom,
% 9.03/5.04      ! [N: int,X: int] :
% 9.03/5.04        ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 9.03/5.04       => ( ( N = zero_zero_int )
% 9.03/5.04          | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5542_of__int__lessD,axiom,
% 9.03/5.04      ! [N: int,X: code_integer] :
% 9.03/5.04        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 9.03/5.04       => ( ( N = zero_zero_int )
% 9.03/5.04          | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5543_of__int__lessD,axiom,
% 9.03/5.04      ! [N: int,X: real] :
% 9.03/5.04        ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 9.03/5.04       => ( ( N = zero_zero_int )
% 9.03/5.04          | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5544_of__int__lessD,axiom,
% 9.03/5.04      ! [N: int,X: rat] :
% 9.03/5.04        ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 9.03/5.04       => ( ( N = zero_zero_int )
% 9.03/5.04          | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5545_of__int__lessD,axiom,
% 9.03/5.04      ! [N: int,X: int] :
% 9.03/5.04        ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 9.03/5.04       => ( ( N = zero_zero_int )
% 9.03/5.04          | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5546_mod__mult2__eq_H,axiom,
% 9.03/5.04      ! [A: code_integer,M: nat,N: nat] :
% 9.03/5.04        ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 9.03/5.04        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5547_mod__mult2__eq_H,axiom,
% 9.03/5.04      ! [A: nat,M: nat,N: nat] :
% 9.03/5.04        ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 9.03/5.04        = ( 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 ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5548_mod__mult2__eq_H,axiom,
% 9.03/5.04      ! [A: int,M: nat,N: nat] :
% 9.03/5.04        ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.03/5.04        = ( 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 ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5549_pos__minus__divide__less__eq,axiom,
% 9.03/5.04      ! [C: real,B: real,A: real] :
% 9.03/5.04        ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.03/5.04          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5550_pos__minus__divide__less__eq,axiom,
% 9.03/5.04      ! [C: rat,B: rat,A: rat] :
% 9.03/5.04        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.03/5.04          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5551_pos__less__minus__divide__eq,axiom,
% 9.03/5.04      ! [C: real,A: real,B: real] :
% 9.03/5.04        ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.03/5.04          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5552_pos__less__minus__divide__eq,axiom,
% 9.03/5.04      ! [C: rat,A: rat,B: rat] :
% 9.03/5.04        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.03/5.04          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5553_neg__minus__divide__less__eq,axiom,
% 9.03/5.04      ! [C: real,B: real,A: real] :
% 9.03/5.04        ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.03/5.04          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5554_neg__minus__divide__less__eq,axiom,
% 9.03/5.04      ! [C: rat,B: rat,A: rat] :
% 9.03/5.04        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.03/5.04          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5555_neg__less__minus__divide__eq,axiom,
% 9.03/5.04      ! [C: real,A: real,B: real] :
% 9.03/5.04        ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.03/5.04          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5556_neg__less__minus__divide__eq,axiom,
% 9.03/5.04      ! [C: rat,A: rat,B: rat] :
% 9.03/5.04        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.03/5.04          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5557_minus__divide__less__eq,axiom,
% 9.03/5.04      ! [B: real,C: real,A: real] :
% 9.03/5.04        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.03/5.04        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.03/5.04          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.03/5.04              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5558_minus__divide__less__eq,axiom,
% 9.03/5.04      ! [B: rat,C: rat,A: rat] :
% 9.03/5.04        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.03/5.04        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.03/5.04          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.03/5.04              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5559_less__minus__divide__eq,axiom,
% 9.03/5.04      ! [A: real,B: real,C: real] :
% 9.03/5.04        ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.03/5.04        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.03/5.04          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.03/5.04              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5560_less__minus__divide__eq,axiom,
% 9.03/5.04      ! [A: rat,B: rat,C: rat] :
% 9.03/5.04        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.03/5.04        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.03/5.04          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.03/5.04              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5561_eq__divide__eq__numeral_I2_J,axiom,
% 9.03/5.04      ! [W: num,B: real,C: real] :
% 9.03/5.04        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.03/5.04          = ( divide_divide_real @ B @ C ) )
% 9.03/5.04        = ( ( ( C != zero_zero_real )
% 9.03/5.04           => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 9.03/5.04              = B ) )
% 9.03/5.04          & ( ( C = zero_zero_real )
% 9.03/5.04           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.03/5.04              = zero_zero_real ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5562_eq__divide__eq__numeral_I2_J,axiom,
% 9.03/5.04      ! [W: num,B: complex,C: complex] :
% 9.03/5.04        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.03/5.04          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.03/5.04        = ( ( ( C != zero_zero_complex )
% 9.03/5.04           => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 9.03/5.04              = B ) )
% 9.03/5.04          & ( ( C = zero_zero_complex )
% 9.03/5.04           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.03/5.04              = zero_zero_complex ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5563_eq__divide__eq__numeral_I2_J,axiom,
% 9.03/5.04      ! [W: num,B: rat,C: rat] :
% 9.03/5.04        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.03/5.04          = ( divide_divide_rat @ B @ C ) )
% 9.03/5.04        = ( ( ( C != zero_zero_rat )
% 9.03/5.04           => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 9.03/5.04              = B ) )
% 9.03/5.04          & ( ( C = zero_zero_rat )
% 9.03/5.04           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.03/5.04              = zero_zero_rat ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5564_divide__eq__eq__numeral_I2_J,axiom,
% 9.03/5.04      ! [B: real,C: real,W: num] :
% 9.03/5.04        ( ( ( divide_divide_real @ B @ C )
% 9.03/5.04          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.03/5.04        = ( ( ( C != zero_zero_real )
% 9.03/5.04           => ( B
% 9.03/5.04              = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 9.03/5.04          & ( ( C = zero_zero_real )
% 9.03/5.04           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.03/5.04              = zero_zero_real ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5565_divide__eq__eq__numeral_I2_J,axiom,
% 9.03/5.04      ! [B: complex,C: complex,W: num] :
% 9.03/5.04        ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.03/5.04          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.03/5.04        = ( ( ( C != zero_zero_complex )
% 9.03/5.04           => ( B
% 9.03/5.04              = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 9.03/5.04          & ( ( C = zero_zero_complex )
% 9.03/5.04           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.03/5.04              = zero_zero_complex ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5566_divide__eq__eq__numeral_I2_J,axiom,
% 9.03/5.04      ! [B: rat,C: rat,W: num] :
% 9.03/5.04        ( ( ( divide_divide_rat @ B @ C )
% 9.03/5.04          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.03/5.04        = ( ( ( C != zero_zero_rat )
% 9.03/5.04           => ( B
% 9.03/5.04              = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 9.03/5.04          & ( ( C = zero_zero_rat )
% 9.03/5.04           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.03/5.04              = zero_zero_rat ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5567_add__divide__eq__if__simps_I3_J,axiom,
% 9.03/5.04      ! [Z: real,A: real,B: real] :
% 9.03/5.04        ( ( ( Z = zero_zero_real )
% 9.03/5.04         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 9.03/5.04            = B ) )
% 9.03/5.04        & ( ( Z != zero_zero_real )
% 9.03/5.04         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 9.03/5.04            = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5568_add__divide__eq__if__simps_I3_J,axiom,
% 9.03/5.04      ! [Z: complex,A: complex,B: complex] :
% 9.03/5.04        ( ( ( Z = zero_zero_complex )
% 9.03/5.04         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 9.03/5.04            = B ) )
% 9.03/5.04        & ( ( Z != zero_zero_complex )
% 9.03/5.04         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 9.03/5.04            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5569_add__divide__eq__if__simps_I3_J,axiom,
% 9.03/5.04      ! [Z: rat,A: rat,B: rat] :
% 9.03/5.04        ( ( ( Z = zero_zero_rat )
% 9.03/5.04         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 9.03/5.04            = B ) )
% 9.03/5.04        & ( ( Z != zero_zero_rat )
% 9.03/5.04         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 9.03/5.04            = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5570_minus__divide__add__eq__iff,axiom,
% 9.03/5.04      ! [Z: real,X: real,Y: real] :
% 9.03/5.04        ( ( Z != zero_zero_real )
% 9.03/5.04       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
% 9.03/5.04          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5571_minus__divide__add__eq__iff,axiom,
% 9.03/5.04      ! [Z: complex,X: complex,Y: complex] :
% 9.03/5.04        ( ( Z != zero_zero_complex )
% 9.03/5.04       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
% 9.03/5.04          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5572_minus__divide__add__eq__iff,axiom,
% 9.03/5.04      ! [Z: rat,X: rat,Y: rat] :
% 9.03/5.04        ( ( Z != zero_zero_rat )
% 9.03/5.04       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
% 9.03/5.04          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5573_add__divide__eq__if__simps_I6_J,axiom,
% 9.03/5.04      ! [Z: real,A: real,B: real] :
% 9.03/5.04        ( ( ( Z = zero_zero_real )
% 9.03/5.04         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 9.03/5.04            = ( uminus_uminus_real @ B ) ) )
% 9.03/5.04        & ( ( Z != zero_zero_real )
% 9.03/5.04         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 9.03/5.04            = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5574_add__divide__eq__if__simps_I6_J,axiom,
% 9.03/5.04      ! [Z: complex,A: complex,B: complex] :
% 9.03/5.04        ( ( ( Z = zero_zero_complex )
% 9.03/5.04         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 9.03/5.04            = ( uminus1482373934393186551omplex @ B ) ) )
% 9.03/5.04        & ( ( Z != zero_zero_complex )
% 9.03/5.04         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 9.03/5.04            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5575_add__divide__eq__if__simps_I6_J,axiom,
% 9.03/5.04      ! [Z: rat,A: rat,B: rat] :
% 9.03/5.04        ( ( ( Z = zero_zero_rat )
% 9.03/5.04         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 9.03/5.04            = ( uminus_uminus_rat @ B ) ) )
% 9.03/5.04        & ( ( Z != zero_zero_rat )
% 9.03/5.04         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 9.03/5.04            = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5576_add__divide__eq__if__simps_I5_J,axiom,
% 9.03/5.04      ! [Z: real,A: real,B: real] :
% 9.03/5.04        ( ( ( Z = zero_zero_real )
% 9.03/5.04         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 9.03/5.04            = ( uminus_uminus_real @ B ) ) )
% 9.03/5.04        & ( ( Z != zero_zero_real )
% 9.03/5.04         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 9.03/5.04            = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5577_add__divide__eq__if__simps_I5_J,axiom,
% 9.03/5.04      ! [Z: complex,A: complex,B: complex] :
% 9.03/5.04        ( ( ( Z = zero_zero_complex )
% 9.03/5.04         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 9.03/5.04            = ( uminus1482373934393186551omplex @ B ) ) )
% 9.03/5.04        & ( ( Z != zero_zero_complex )
% 9.03/5.04         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 9.03/5.04            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5578_add__divide__eq__if__simps_I5_J,axiom,
% 9.03/5.04      ! [Z: rat,A: rat,B: rat] :
% 9.03/5.04        ( ( ( Z = zero_zero_rat )
% 9.03/5.04         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 9.03/5.04            = ( uminus_uminus_rat @ B ) ) )
% 9.03/5.04        & ( ( Z != zero_zero_rat )
% 9.03/5.04         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 9.03/5.04            = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5579_minus__divide__diff__eq__iff,axiom,
% 9.03/5.04      ! [Z: real,X: real,Y: real] :
% 9.03/5.04        ( ( Z != zero_zero_real )
% 9.03/5.04       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
% 9.03/5.04          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5580_minus__divide__diff__eq__iff,axiom,
% 9.03/5.04      ! [Z: complex,X: complex,Y: complex] :
% 9.03/5.04        ( ( Z != zero_zero_complex )
% 9.03/5.04       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
% 9.03/5.04          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5581_minus__divide__diff__eq__iff,axiom,
% 9.03/5.04      ! [Z: rat,X: rat,Y: rat] :
% 9.03/5.04        ( ( Z != zero_zero_rat )
% 9.03/5.04       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
% 9.03/5.04          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5582_field__char__0__class_Oof__nat__div,axiom,
% 9.03/5.04      ! [M: nat,N: nat] :
% 9.03/5.04        ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 9.03/5.04        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5583_field__char__0__class_Oof__nat__div,axiom,
% 9.03/5.04      ! [M: nat,N: nat] :
% 9.03/5.04        ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 9.03/5.04        = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5584_field__char__0__class_Oof__nat__div,axiom,
% 9.03/5.04      ! [M: nat,N: nat] :
% 9.03/5.04        ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 9.03/5.04        = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5585_split__mod,axiom,
% 9.03/5.04      ! [P: nat > $o,M: nat,N: nat] :
% 9.03/5.04        ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 9.03/5.04        = ( ( ( N = zero_zero_nat )
% 9.03/5.04           => ( P @ M ) )
% 9.03/5.04          & ( ( N != zero_zero_nat )
% 9.03/5.04           => ! [I3: nat,J2: nat] :
% 9.03/5.04                ( ( ord_less_nat @ J2 @ N )
% 9.03/5.04               => ( ( M
% 9.03/5.04                    = ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J2 ) )
% 9.03/5.04                 => ( P @ J2 ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5586_lemma__interval,axiom,
% 9.03/5.04      ! [A: real,X: real,B: real] :
% 9.03/5.04        ( ( ord_less_real @ A @ X )
% 9.03/5.04       => ( ( ord_less_real @ X @ B )
% 9.03/5.04         => ? [D5: real] :
% 9.03/5.04              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.03/5.04              & ! [Y5: real] :
% 9.03/5.04                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D5 )
% 9.03/5.04                 => ( ( ord_less_eq_real @ A @ Y5 )
% 9.03/5.04                    & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5587_power2__eq__iff,axiom,
% 9.03/5.04      ! [X: real,Y: real] :
% 9.03/5.04        ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/5.04        = ( ( X = Y )
% 9.03/5.04          | ( X
% 9.03/5.04            = ( uminus_uminus_real @ Y ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5588_power2__eq__iff,axiom,
% 9.03/5.04      ! [X: int,Y: int] :
% 9.03/5.04        ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/5.04        = ( ( X = Y )
% 9.03/5.04          | ( X
% 9.03/5.04            = ( uminus_uminus_int @ Y ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5589_power2__eq__iff,axiom,
% 9.03/5.04      ! [X: complex,Y: complex] :
% 9.03/5.04        ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/5.04        = ( ( X = Y )
% 9.03/5.04          | ( X
% 9.03/5.04            = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5590_power2__eq__iff,axiom,
% 9.03/5.04      ! [X: rat,Y: rat] :
% 9.03/5.04        ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/5.04        = ( ( X = Y )
% 9.03/5.04          | ( X
% 9.03/5.04            = ( uminus_uminus_rat @ Y ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5591_power2__eq__iff,axiom,
% 9.03/5.04      ! [X: code_integer,Y: code_integer] :
% 9.03/5.04        ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.03/5.04        = ( ( X = Y )
% 9.03/5.04          | ( X
% 9.03/5.04            = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5592_norm__triangle__ineq3,axiom,
% 9.03/5.04      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5593_norm__triangle__ineq3,axiom,
% 9.03/5.04      ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5594_round__diff__minimal,axiom,
% 9.03/5.04      ! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5595_round__diff__minimal,axiom,
% 9.03/5.04      ! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5596_real__of__nat__div__aux,axiom,
% 9.03/5.04      ! [X: nat,D: nat] :
% 9.03/5.04        ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
% 9.03/5.04        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5597_abs__le__square__iff,axiom,
% 9.03/5.04      ! [X: code_integer,Y: code_integer] :
% 9.03/5.04        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
% 9.03/5.04        = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5598_abs__le__square__iff,axiom,
% 9.03/5.04      ! [X: real,Y: real] :
% 9.03/5.04        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
% 9.03/5.04        = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5599_abs__le__square__iff,axiom,
% 9.03/5.04      ! [X: rat,Y: rat] :
% 9.03/5.04        ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
% 9.03/5.04        = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5600_abs__le__square__iff,axiom,
% 9.03/5.04      ! [X: int,Y: int] :
% 9.03/5.04        ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
% 9.03/5.04        = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5601_abs__square__eq__1,axiom,
% 9.03/5.04      ! [X: code_integer] :
% 9.03/5.04        ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = one_one_Code_integer )
% 9.03/5.04        = ( ( abs_abs_Code_integer @ X )
% 9.03/5.04          = one_one_Code_integer ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5602_abs__square__eq__1,axiom,
% 9.03/5.04      ! [X: rat] :
% 9.03/5.04        ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = one_one_rat )
% 9.03/5.04        = ( ( abs_abs_rat @ X )
% 9.03/5.04          = one_one_rat ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5603_abs__square__eq__1,axiom,
% 9.03/5.04      ! [X: real] :
% 9.03/5.04        ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = one_one_real )
% 9.03/5.04        = ( ( abs_abs_real @ X )
% 9.03/5.04          = one_one_real ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5604_abs__square__eq__1,axiom,
% 9.03/5.04      ! [X: int] :
% 9.03/5.04        ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = one_one_int )
% 9.03/5.04        = ( ( abs_abs_int @ X )
% 9.03/5.04          = one_one_int ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5605_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.03/5.04      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.03/5.04        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 9.03/5.04       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.03/5.04          = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5606_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.03/5.04      ! [C: nat,A: nat,B: nat] :
% 9.03/5.04        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.03/5.04       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.03/5.04          = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5607_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.03/5.04      ! [C: int,A: int,B: int] :
% 9.03/5.04        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.03/5.04       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 9.03/5.04          = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5608_pos__minus__divide__le__eq,axiom,
% 9.03/5.04      ! [C: real,B: real,A: real] :
% 9.03/5.04        ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.03/5.04          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5609_pos__minus__divide__le__eq,axiom,
% 9.03/5.04      ! [C: rat,B: rat,A: rat] :
% 9.03/5.04        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.03/5.04          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5610_pos__le__minus__divide__eq,axiom,
% 9.03/5.04      ! [C: real,A: real,B: real] :
% 9.03/5.04        ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.03/5.04          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5611_pos__le__minus__divide__eq,axiom,
% 9.03/5.04      ! [C: rat,A: rat,B: rat] :
% 9.03/5.04        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.03/5.04          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5612_neg__minus__divide__le__eq,axiom,
% 9.03/5.04      ! [C: real,B: real,A: real] :
% 9.03/5.04        ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.03/5.04          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5613_neg__minus__divide__le__eq,axiom,
% 9.03/5.04      ! [C: rat,B: rat,A: rat] :
% 9.03/5.04        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.03/5.04          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5614_neg__le__minus__divide__eq,axiom,
% 9.03/5.04      ! [C: real,A: real,B: real] :
% 9.03/5.04        ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.03/5.04          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5615_neg__le__minus__divide__eq,axiom,
% 9.03/5.04      ! [C: rat,A: rat,B: rat] :
% 9.03/5.04        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.03/5.04          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5616_minus__divide__le__eq,axiom,
% 9.03/5.04      ! [B: real,C: real,A: real] :
% 9.03/5.04        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.03/5.04        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.03/5.04          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.03/5.04              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5617_minus__divide__le__eq,axiom,
% 9.03/5.04      ! [B: rat,C: rat,A: rat] :
% 9.03/5.04        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.03/5.04        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.03/5.04          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.03/5.04              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5618_le__minus__divide__eq,axiom,
% 9.03/5.04      ! [A: real,B: real,C: real] :
% 9.03/5.04        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.03/5.04        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.03/5.04          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.03/5.04              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5619_le__minus__divide__eq,axiom,
% 9.03/5.04      ! [A: rat,B: rat,C: rat] :
% 9.03/5.04        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.03/5.04        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.03/5.04          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.03/5.04              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5620_less__divide__eq__numeral_I2_J,axiom,
% 9.03/5.04      ! [W: num,B: real,C: real] :
% 9.03/5.04        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 9.03/5.04        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 9.03/5.04          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 9.03/5.04              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5621_less__divide__eq__numeral_I2_J,axiom,
% 9.03/5.04      ! [W: num,B: rat,C: rat] :
% 9.03/5.04        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 9.03/5.04        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 9.03/5.04          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 9.03/5.04              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5622_divide__less__eq__numeral_I2_J,axiom,
% 9.03/5.04      ! [B: real,C: real,W: num] :
% 9.03/5.04        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.03/5.04        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 9.03/5.04          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.03/5.04           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 9.03/5.04              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.03/5.04               => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5623_divide__less__eq__numeral_I2_J,axiom,
% 9.03/5.04      ! [B: rat,C: rat,W: num] :
% 9.03/5.04        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.03/5.04        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 9.03/5.04          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.03/5.04           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 9.03/5.04              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.03/5.04               => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5624_Suc__times__mod__eq,axiom,
% 9.03/5.04      ! [M: nat,N: nat] :
% 9.03/5.04        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.03/5.04       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 9.03/5.04          = one_one_nat ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5625_power2__eq__1__iff,axiom,
% 9.03/5.04      ! [A: real] :
% 9.03/5.04        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = one_one_real )
% 9.03/5.04        = ( ( A = one_one_real )
% 9.03/5.04          | ( A
% 9.03/5.04            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5626_power2__eq__1__iff,axiom,
% 9.03/5.04      ! [A: int] :
% 9.03/5.04        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = one_one_int )
% 9.03/5.04        = ( ( A = one_one_int )
% 9.03/5.04          | ( A
% 9.03/5.04            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5627_power2__eq__1__iff,axiom,
% 9.03/5.04      ! [A: complex] :
% 9.03/5.04        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = one_one_complex )
% 9.03/5.04        = ( ( A = one_one_complex )
% 9.03/5.04          | ( A
% 9.03/5.04            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5628_power2__eq__1__iff,axiom,
% 9.03/5.04      ! [A: rat] :
% 9.03/5.04        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = one_one_rat )
% 9.03/5.04        = ( ( A = one_one_rat )
% 9.03/5.04          | ( A
% 9.03/5.04            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5629_power2__eq__1__iff,axiom,
% 9.03/5.04      ! [A: code_integer] :
% 9.03/5.04        ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.03/5.04          = one_one_Code_integer )
% 9.03/5.04        = ( ( A = one_one_Code_integer )
% 9.03/5.04          | ( A
% 9.03/5.04            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5630_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.03/5.04      ! [K: nat,N: nat] :
% 9.03/5.04        ( ( ord_less_eq_nat @ K @ N )
% 9.03/5.04       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 9.03/5.04          = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5631_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.03/5.04      ! [K: nat,N: nat] :
% 9.03/5.04        ( ( ord_less_eq_nat @ K @ N )
% 9.03/5.04       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 9.03/5.04          = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5632_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.03/5.04      ! [K: nat,N: nat] :
% 9.03/5.04        ( ( ord_less_eq_nat @ K @ N )
% 9.03/5.04       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 9.03/5.04          = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5633_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.03/5.04      ! [K: nat,N: nat] :
% 9.03/5.04        ( ( ord_less_eq_nat @ K @ N )
% 9.03/5.04       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 9.03/5.04          = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5634_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.03/5.04      ! [K: nat,N: nat] :
% 9.03/5.04        ( ( ord_less_eq_nat @ K @ N )
% 9.03/5.04       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 9.03/5.04          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5635_realpow__square__minus__le,axiom,
% 9.03/5.04      ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5636_powr__neg__one,axiom,
% 9.03/5.04      ! [X: real] :
% 9.03/5.04        ( ( ord_less_real @ zero_zero_real @ X )
% 9.03/5.04       => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 9.03/5.04          = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5637_ln__add__one__self__le__self2,axiom,
% 9.03/5.04      ! [X: real] :
% 9.03/5.04        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.03/5.04       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 9.03/5.04  
% 9.03/5.04  thf(fact_5638_power2__le__iff__abs__le,axiom,
% 9.03/5.04      ! [Y: code_integer,X: code_integer] :
% 9.03/5.04        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 9.04/5.04       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.04          = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5639_power2__le__iff__abs__le,axiom,
% 9.04/5.04      ! [Y: real,X: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.04       => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.04          = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5640_power2__le__iff__abs__le,axiom,
% 9.04/5.04      ! [Y: rat,X: rat] :
% 9.04/5.04        ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.04/5.04       => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.04          = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5641_power2__le__iff__abs__le,axiom,
% 9.04/5.04      ! [Y: int,X: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.04       => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.04          = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5642_abs__square__le__1,axiom,
% 9.04/5.04      ! [X: code_integer] :
% 9.04/5.04        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 9.04/5.04        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5643_abs__square__le__1,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 9.04/5.04        = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5644_abs__square__le__1,axiom,
% 9.04/5.04      ! [X: rat] :
% 9.04/5.04        ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 9.04/5.04        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5645_abs__square__le__1,axiom,
% 9.04/5.04      ! [X: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 9.04/5.04        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5646_abs__square__less__1,axiom,
% 9.04/5.04      ! [X: code_integer] :
% 9.04/5.04        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 9.04/5.04        = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5647_abs__square__less__1,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 9.04/5.04        = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5648_abs__square__less__1,axiom,
% 9.04/5.04      ! [X: rat] :
% 9.04/5.04        ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 9.04/5.04        = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5649_abs__square__less__1,axiom,
% 9.04/5.04      ! [X: int] :
% 9.04/5.04        ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 9.04/5.04        = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5650_divmod__digit__0_I2_J,axiom,
% 9.04/5.04      ! [B: nat,A: nat] :
% 9.04/5.04        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.04/5.04       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.04/5.04         => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 9.04/5.04            = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5651_divmod__digit__0_I2_J,axiom,
% 9.04/5.04      ! [B: int,A: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.04       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.04/5.04         => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 9.04/5.04            = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5652_divmod__digit__0_I2_J,axiom,
% 9.04/5.04      ! [B: code_integer,A: code_integer] :
% 9.04/5.04        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.04/5.04       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.04/5.04         => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 9.04/5.04            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5653_bits__stable__imp__add__self,axiom,
% 9.04/5.04      ! [A: nat] :
% 9.04/5.04        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.04          = A )
% 9.04/5.04       => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.04          = zero_zero_nat ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5654_bits__stable__imp__add__self,axiom,
% 9.04/5.04      ! [A: int] :
% 9.04/5.04        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.04          = A )
% 9.04/5.04       => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.04/5.04          = zero_zero_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5655_bits__stable__imp__add__self,axiom,
% 9.04/5.04      ! [A: code_integer] :
% 9.04/5.04        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.04          = A )
% 9.04/5.04       => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 9.04/5.04          = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5656_div__exp__mod__exp__eq,axiom,
% 9.04/5.04      ! [A: nat,N: nat,M: nat] :
% 9.04/5.04        ( ( 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 ) )
% 9.04/5.04        = ( 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 ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5657_div__exp__mod__exp__eq,axiom,
% 9.04/5.04      ! [A: int,N: nat,M: nat] :
% 9.04/5.04        ( ( 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 ) )
% 9.04/5.04        = ( 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 ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5658_div__exp__mod__exp__eq,axiom,
% 9.04/5.04      ! [A: code_integer,N: nat,M: nat] :
% 9.04/5.04        ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.04        = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5659_le__divide__eq__numeral_I2_J,axiom,
% 9.04/5.04      ! [W: num,B: real,C: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 9.04/5.04        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.04/5.04           => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 9.04/5.04          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.04/5.04           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.04/5.04               => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 9.04/5.04              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.04/5.04               => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5660_le__divide__eq__numeral_I2_J,axiom,
% 9.04/5.04      ! [W: num,B: rat,C: rat] :
% 9.04/5.04        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 9.04/5.04        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.04/5.04           => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 9.04/5.04          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.04/5.04           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.04/5.04               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 9.04/5.04              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.04/5.04               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5661_divide__le__eq__numeral_I2_J,axiom,
% 9.04/5.04      ! [B: real,C: real,W: num] :
% 9.04/5.04        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.04/5.04        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.04/5.04           => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 9.04/5.04          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.04/5.04           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.04/5.04               => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 9.04/5.04              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.04/5.04               => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5662_divide__le__eq__numeral_I2_J,axiom,
% 9.04/5.04      ! [B: rat,C: rat,W: num] :
% 9.04/5.04        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.04/5.04        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.04/5.04           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 9.04/5.04          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.04/5.04           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.04/5.04               => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 9.04/5.04              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.04/5.04               => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5663_square__le__1,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.04       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.04         => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5664_square__le__1,axiom,
% 9.04/5.04      ! [X: code_integer] :
% 9.04/5.04        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 9.04/5.04       => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 9.04/5.04         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5665_square__le__1,axiom,
% 9.04/5.04      ! [X: rat] :
% 9.04/5.04        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 9.04/5.04       => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 9.04/5.04         => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5666_square__le__1,axiom,
% 9.04/5.04      ! [X: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 9.04/5.04       => ( ( ord_less_eq_int @ X @ one_one_int )
% 9.04/5.04         => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5667_verit__le__mono__div,axiom,
% 9.04/5.04      ! [A2: nat,B2: nat,N: nat] :
% 9.04/5.04        ( ( ord_less_nat @ A2 @ B2 )
% 9.04/5.04       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.04         => ( ord_less_eq_nat
% 9.04/5.04            @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
% 9.04/5.04              @ ( if_nat
% 9.04/5.04                @ ( ( modulo_modulo_nat @ B2 @ N )
% 9.04/5.04                  = zero_zero_nat )
% 9.04/5.04                @ one_one_nat
% 9.04/5.04                @ zero_zero_nat ) )
% 9.04/5.04            @ ( divide_divide_nat @ B2 @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5668_minus__power__mult__self,axiom,
% 9.04/5.04      ! [A: real,N: nat] :
% 9.04/5.04        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 9.04/5.04        = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5669_minus__power__mult__self,axiom,
% 9.04/5.04      ! [A: int,N: nat] :
% 9.04/5.04        ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 9.04/5.04        = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5670_minus__power__mult__self,axiom,
% 9.04/5.04      ! [A: complex,N: nat] :
% 9.04/5.04        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 9.04/5.04        = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5671_minus__power__mult__self,axiom,
% 9.04/5.04      ! [A: rat,N: nat] :
% 9.04/5.04        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 9.04/5.04        = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5672_minus__power__mult__self,axiom,
% 9.04/5.04      ! [A: code_integer,N: nat] :
% 9.04/5.04        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 9.04/5.04        = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5673_ln__one__minus__pos__upper__bound,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.04       => ( ( ord_less_real @ X @ one_one_real )
% 9.04/5.04         => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5674_Bernoulli__inequality,axiom,
% 9.04/5.04      ! [X: real,N: nat] :
% 9.04/5.04        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.04       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5675_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.04       => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.04/5.04         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5676_divmod__digit__0_I1_J,axiom,
% 9.04/5.04      ! [B: nat,A: nat] :
% 9.04/5.04        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.04/5.04       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.04/5.04         => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 9.04/5.04            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5677_divmod__digit__0_I1_J,axiom,
% 9.04/5.04      ! [B: int,A: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.04       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.04/5.04         => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 9.04/5.04            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5678_divmod__digit__0_I1_J,axiom,
% 9.04/5.04      ! [B: code_integer,A: code_integer] :
% 9.04/5.04        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.04/5.04       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.04/5.04         => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 9.04/5.04            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5679_mult__exp__mod__exp__eq,axiom,
% 9.04/5.04      ! [M: nat,N: nat,A: nat] :
% 9.04/5.04        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.04       => ( ( 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 ) )
% 9.04/5.04          = ( 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 ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5680_mult__exp__mod__exp__eq,axiom,
% 9.04/5.04      ! [M: nat,N: nat,A: int] :
% 9.04/5.04        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.04       => ( ( 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 ) )
% 9.04/5.04          = ( 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 ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5681_mult__exp__mod__exp__eq,axiom,
% 9.04/5.04      ! [M: nat,N: nat,A: code_integer] :
% 9.04/5.04        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.04       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.04          = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5682_power__minus1__odd,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.04        = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5683_power__minus1__odd,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.04        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5684_power__minus1__odd,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.04        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5685_power__minus1__odd,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.04        = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5686_power__minus1__odd,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.04        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5687_mod__double__modulus,axiom,
% 9.04/5.04      ! [M: code_integer,X: code_integer] :
% 9.04/5.04        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 9.04/5.04       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 9.04/5.04         => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.04              = ( modulo364778990260209775nteger @ X @ M ) )
% 9.04/5.04            | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.04              = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5688_mod__double__modulus,axiom,
% 9.04/5.04      ! [M: nat,X: nat] :
% 9.04/5.04        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.04       => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 9.04/5.04         => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.04              = ( modulo_modulo_nat @ X @ M ) )
% 9.04/5.04            | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.04              = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5689_mod__double__modulus,axiom,
% 9.04/5.04      ! [M: int,X: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ M )
% 9.04/5.04       => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.04         => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.04              = ( modulo_modulo_int @ X @ M ) )
% 9.04/5.04            | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.04              = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5690_divmod__digit__1_I2_J,axiom,
% 9.04/5.04      ! [A: code_integer,B: code_integer] :
% 9.04/5.04        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.04/5.04       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.04/5.04         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 9.04/5.04           => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.04/5.04              = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5691_divmod__digit__1_I2_J,axiom,
% 9.04/5.04      ! [A: nat,B: nat] :
% 9.04/5.04        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.04/5.04       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.04/5.04         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 9.04/5.04           => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.04/5.04              = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5692_divmod__digit__1_I2_J,axiom,
% 9.04/5.04      ! [A: int,B: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.04/5.04       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.04         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 9.04/5.04           => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.04/5.04              = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5693_log__minus__eq__powr,axiom,
% 9.04/5.04      ! [B: real,X: real,Y: real] :
% 9.04/5.04        ( ( ord_less_real @ zero_zero_real @ B )
% 9.04/5.04       => ( ( B != one_one_real )
% 9.04/5.04         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.04           => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
% 9.04/5.04              = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5694_unset__bit__Suc,axiom,
% 9.04/5.04      ! [N: nat,A: code_integer] :
% 9.04/5.04        ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 9.04/5.04        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5695_unset__bit__Suc,axiom,
% 9.04/5.04      ! [N: nat,A: int] :
% 9.04/5.04        ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 9.04/5.04        = ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5696_unset__bit__Suc,axiom,
% 9.04/5.04      ! [N: nat,A: nat] :
% 9.04/5.04        ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 9.04/5.04        = ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5697_set__bit__Suc,axiom,
% 9.04/5.04      ! [N: nat,A: code_integer] :
% 9.04/5.04        ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 9.04/5.04        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5698_set__bit__Suc,axiom,
% 9.04/5.04      ! [N: nat,A: int] :
% 9.04/5.04        ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 9.04/5.04        = ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5699_set__bit__Suc,axiom,
% 9.04/5.04      ! [N: nat,A: nat] :
% 9.04/5.04        ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 9.04/5.04        = ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5700_powr__neg__numeral,axiom,
% 9.04/5.04      ! [X: real,N: num] :
% 9.04/5.04        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.04       => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.04/5.04          = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5701_of__int__round__abs__le,axiom,
% 9.04/5.04      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5702_of__int__round__abs__le,axiom,
% 9.04/5.04      ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5703_round__unique_H,axiom,
% 9.04/5.04      ! [X: real,N: int] :
% 9.04/5.04        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.04       => ( ( archim8280529875227126926d_real @ X )
% 9.04/5.04          = N ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5704_round__unique_H,axiom,
% 9.04/5.04      ! [X: rat,N: int] :
% 9.04/5.04        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.04/5.04       => ( ( archim7778729529865785530nd_rat @ X )
% 9.04/5.04          = N ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5705_divmod__digit__1_I1_J,axiom,
% 9.04/5.04      ! [A: code_integer,B: code_integer] :
% 9.04/5.04        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.04/5.04       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.04/5.04         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 9.04/5.04           => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 9.04/5.04              = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5706_divmod__digit__1_I1_J,axiom,
% 9.04/5.04      ! [A: nat,B: nat] :
% 9.04/5.04        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.04/5.04       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.04/5.04         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 9.04/5.04           => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
% 9.04/5.04              = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5707_divmod__digit__1_I1_J,axiom,
% 9.04/5.04      ! [A: int,B: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.04/5.04       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.04         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 9.04/5.04           => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
% 9.04/5.04              = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5708_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.04       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.04         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5709_VEBT__internal_Oheight_Oelims,axiom,
% 9.04/5.04      ! [X: vEBT_VEBT,Y: nat] :
% 9.04/5.04        ( ( ( vEBT_VEBT_height @ X )
% 9.04/5.04          = Y )
% 9.04/5.04       => ( ( ? [A5: $o,B6: $o] :
% 9.04/5.04                ( X
% 9.04/5.04                = ( vEBT_Leaf @ A5 @ B6 ) )
% 9.04/5.04           => ( Y != zero_zero_nat ) )
% 9.04/5.04         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.04/5.04                ( ( X
% 9.04/5.04                  = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.04               => ( Y
% 9.04/5.04                 != ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5710_arsinh__def,axiom,
% 9.04/5.04      ( arsinh_real
% 9.04/5.04      = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5711_abs__sqrt__wlog,axiom,
% 9.04/5.04      ! [P: code_integer > code_integer > $o,X: code_integer] :
% 9.04/5.04        ( ! [X3: code_integer] :
% 9.04/5.04            ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 9.04/5.04           => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.04       => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5712_abs__sqrt__wlog,axiom,
% 9.04/5.04      ! [P: real > real > $o,X: real] :
% 9.04/5.04        ( ! [X3: real] :
% 9.04/5.04            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.04/5.04           => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.04       => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5713_abs__sqrt__wlog,axiom,
% 9.04/5.04      ! [P: rat > rat > $o,X: rat] :
% 9.04/5.04        ( ! [X3: rat] :
% 9.04/5.04            ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.04/5.04           => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.04       => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5714_abs__sqrt__wlog,axiom,
% 9.04/5.04      ! [P: int > int > $o,X: int] :
% 9.04/5.04        ( ! [X3: int] :
% 9.04/5.04            ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.04/5.04           => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.04       => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5715_compl__le__compl__iff,axiom,
% 9.04/5.04      ! [X: set_int,Y: set_int] :
% 9.04/5.04        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
% 9.04/5.04        = ( ord_less_eq_set_int @ Y @ X ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5716_flip__bit__Suc,axiom,
% 9.04/5.04      ! [N: nat,A: code_integer] :
% 9.04/5.04        ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 9.04/5.04        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5717_flip__bit__Suc,axiom,
% 9.04/5.04      ! [N: nat,A: int] :
% 9.04/5.04        ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 9.04/5.04        = ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5718_flip__bit__Suc,axiom,
% 9.04/5.04      ! [N: nat,A: nat] :
% 9.04/5.04        ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 9.04/5.04        = ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5719_signed__take__bit__rec,axiom,
% 9.04/5.04      ( bit_ri6519982836138164636nteger
% 9.04/5.04      = ( ^ [N3: nat,A3: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5720_signed__take__bit__rec,axiom,
% 9.04/5.04      ( bit_ri631733984087533419it_int
% 9.04/5.04      = ( ^ [N3: nat,A3: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5721_round__altdef,axiom,
% 9.04/5.04      ( archim8280529875227126926d_real
% 9.04/5.04      = ( ^ [X4: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X4 ) ) @ ( archim7802044766580827645g_real @ X4 ) @ ( archim6058952711729229775r_real @ X4 ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5722_round__altdef,axiom,
% 9.04/5.04      ( archim7778729529865785530nd_rat
% 9.04/5.04      = ( ^ [X4: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X4 ) ) @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim3151403230148437115or_rat @ X4 ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5723_arctan__double,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.04       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 9.04/5.04          = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5724_pochhammer__double,axiom,
% 9.04/5.04      ! [Z: complex,N: nat] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.04        = ( 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 ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5725_pochhammer__double,axiom,
% 9.04/5.04      ! [Z: real,N: nat] :
% 9.04/5.04        ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.04        = ( 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 ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5726_pochhammer__double,axiom,
% 9.04/5.04      ! [Z: rat,N: nat] :
% 9.04/5.04        ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.04        = ( 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 ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5727_Compl__subset__Compl__iff,axiom,
% 9.04/5.04      ! [A2: set_int,B2: set_int] :
% 9.04/5.04        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B2 ) )
% 9.04/5.04        = ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5728_Compl__anti__mono,axiom,
% 9.04/5.04      ! [A2: set_int,B2: set_int] :
% 9.04/5.04        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.04/5.04       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5729_flip__bit__nonnegative__int__iff,axiom,
% 9.04/5.04      ! [N: nat,K: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 9.04/5.04        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5730_flip__bit__negative__int__iff,axiom,
% 9.04/5.04      ! [N: nat,K: int] :
% 9.04/5.04        ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 9.04/5.04        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5731_signed__take__bit__of__0,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 9.04/5.04        = zero_zero_int ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5732_arctan__eq__zero__iff,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ( arctan @ X )
% 9.04/5.04          = zero_zero_real )
% 9.04/5.04        = ( X = zero_zero_real ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5733_arctan__zero__zero,axiom,
% 9.04/5.04      ( ( arctan @ zero_zero_real )
% 9.04/5.04      = zero_zero_real ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5734_mod__pos__pos__trivial,axiom,
% 9.04/5.04      ! [K: int,L: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.04       => ( ( ord_less_int @ K @ L )
% 9.04/5.04         => ( ( modulo_modulo_int @ K @ L )
% 9.04/5.04            = K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5735_mod__neg__neg__trivial,axiom,
% 9.04/5.04      ! [K: int,L: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.04/5.04       => ( ( ord_less_int @ L @ K )
% 9.04/5.04         => ( ( modulo_modulo_int @ K @ L )
% 9.04/5.04            = K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5736_negative__eq__positive,axiom,
% 9.04/5.04      ! [N: nat,M: nat] :
% 9.04/5.04        ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.04/5.04          = ( semiri1314217659103216013at_int @ M ) )
% 9.04/5.04        = ( ( N = zero_zero_nat )
% 9.04/5.04          & ( M = zero_zero_nat ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5737_signed__take__bit__of__minus__1,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.04        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5738_signed__take__bit__of__minus__1,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.04        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5739_signed__take__bit__Suc__1,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 9.04/5.04        = one_one_int ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5740_signed__take__bit__numeral__of__1,axiom,
% 9.04/5.04      ! [K: num] :
% 9.04/5.04        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 9.04/5.04        = one_one_int ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5741_pochhammer__0,axiom,
% 9.04/5.04      ! [A: complex] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 9.04/5.04        = one_one_complex ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5742_pochhammer__0,axiom,
% 9.04/5.04      ! [A: real] :
% 9.04/5.04        ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 9.04/5.04        = one_one_real ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5743_pochhammer__0,axiom,
% 9.04/5.04      ! [A: rat] :
% 9.04/5.04        ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 9.04/5.04        = one_one_rat ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5744_pochhammer__0,axiom,
% 9.04/5.04      ! [A: nat] :
% 9.04/5.04        ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 9.04/5.04        = one_one_nat ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5745_pochhammer__0,axiom,
% 9.04/5.04      ! [A: int] :
% 9.04/5.04        ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 9.04/5.04        = one_one_int ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5746_negative__zle,axiom,
% 9.04/5.04      ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5747_zero__less__arctan__iff,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
% 9.04/5.04        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5748_arctan__less__zero__iff,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
% 9.04/5.04        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5749_zero__le__arctan__iff,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
% 9.04/5.04        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5750_arctan__le__zero__iff,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
% 9.04/5.04        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5751_frac__of__int,axiom,
% 9.04/5.04      ! [Z: int] :
% 9.04/5.04        ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
% 9.04/5.04        = zero_zero_real ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5752_frac__of__int,axiom,
% 9.04/5.04      ! [Z: int] :
% 9.04/5.04        ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.04/5.04        = zero_zero_rat ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5753_zmod__numeral__Bit0,axiom,
% 9.04/5.04      ! [V: num,W: num] :
% 9.04/5.04        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 9.04/5.04        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5754_negative__zless,axiom,
% 9.04/5.04      ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5755_zabs__less__one__iff,axiom,
% 9.04/5.04      ! [Z: int] :
% 9.04/5.04        ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 9.04/5.04        = ( Z = zero_zero_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5756_signed__take__bit__Suc__bit0,axiom,
% 9.04/5.04      ! [N: nat,K: num] :
% 9.04/5.04        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 9.04/5.04        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5757_signed__take__bit__Suc__minus__bit0,axiom,
% 9.04/5.04      ! [N: nat,K: num] :
% 9.04/5.04        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.04/5.04        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5758_ceiling__divide__eq__div__numeral,axiom,
% 9.04/5.04      ! [A: num,B: num] :
% 9.04/5.04        ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 9.04/5.04        = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5759_signed__take__bit__0,axiom,
% 9.04/5.04      ! [A: code_integer] :
% 9.04/5.04        ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 9.04/5.04        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5760_signed__take__bit__0,axiom,
% 9.04/5.04      ! [A: int] :
% 9.04/5.04        ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 9.04/5.04        = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5761_floor__minus__divide__eq__div__numeral,axiom,
% 9.04/5.04      ! [A: num,B: num] :
% 9.04/5.04        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 9.04/5.04        = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5762_ceiling__minus__divide__eq__div__numeral,axiom,
% 9.04/5.04      ! [A: num,B: num] :
% 9.04/5.04        ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 9.04/5.04        = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5763_floor__minus__one__divide__eq__div__numeral,axiom,
% 9.04/5.04      ! [B: num] :
% 9.04/5.04        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 9.04/5.04        = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5764_zmod__zminus1__not__zero,axiom,
% 9.04/5.04      ! [K: int,L: int] :
% 9.04/5.04        ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 9.04/5.04         != zero_zero_int )
% 9.04/5.04       => ( ( modulo_modulo_int @ K @ L )
% 9.04/5.04         != zero_zero_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5765_zmod__zminus2__not__zero,axiom,
% 9.04/5.04      ! [K: int,L: int] :
% 9.04/5.04        ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
% 9.04/5.04         != zero_zero_int )
% 9.04/5.04       => ( ( modulo_modulo_int @ K @ L )
% 9.04/5.04         != zero_zero_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5766_pochhammer__of__real,axiom,
% 9.04/5.04      ! [X: real,N: nat] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ ( real_V4546457046886955230omplex @ X ) @ N )
% 9.04/5.04        = ( real_V4546457046886955230omplex @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5767_arctan__eq__iff,axiom,
% 9.04/5.04      ! [X: real,Y: real] :
% 9.04/5.04        ( ( ( arctan @ X )
% 9.04/5.04          = ( arctan @ Y ) )
% 9.04/5.04        = ( X = Y ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5768_pochhammer__of__nat,axiom,
% 9.04/5.04      ! [X: nat,N: nat] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ ( semiri8010041392384452111omplex @ X ) @ N )
% 9.04/5.04        = ( semiri8010041392384452111omplex @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5769_pochhammer__of__nat,axiom,
% 9.04/5.04      ! [X: nat,N: nat] :
% 9.04/5.04        ( ( comm_s7457072308508201937r_real @ ( semiri5074537144036343181t_real @ X ) @ N )
% 9.04/5.04        = ( semiri5074537144036343181t_real @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5770_pochhammer__of__nat,axiom,
% 9.04/5.04      ! [X: nat,N: nat] :
% 9.04/5.04        ( ( comm_s4028243227959126397er_rat @ ( semiri681578069525770553at_rat @ X ) @ N )
% 9.04/5.04        = ( semiri681578069525770553at_rat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5771_pochhammer__of__nat,axiom,
% 9.04/5.04      ! [X: nat,N: nat] :
% 9.04/5.04        ( ( comm_s4663373288045622133er_nat @ ( semiri1316708129612266289at_nat @ X ) @ N )
% 9.04/5.04        = ( semiri1316708129612266289at_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5772_pochhammer__of__nat,axiom,
% 9.04/5.04      ! [X: nat,N: nat] :
% 9.04/5.04        ( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X ) @ N )
% 9.04/5.04        = ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5773_pochhammer__of__int,axiom,
% 9.04/5.04      ! [X: int,N: nat] :
% 9.04/5.04        ( ( comm_s7457072308508201937r_real @ ( ring_1_of_int_real @ X ) @ N )
% 9.04/5.04        = ( ring_1_of_int_real @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5774_pochhammer__of__int,axiom,
% 9.04/5.04      ! [X: int,N: nat] :
% 9.04/5.04        ( ( comm_s4028243227959126397er_rat @ ( ring_1_of_int_rat @ X ) @ N )
% 9.04/5.04        = ( ring_1_of_int_rat @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5775_pochhammer__of__int,axiom,
% 9.04/5.04      ! [X: int,N: nat] :
% 9.04/5.04        ( ( comm_s8582702949713902594nteger @ ( ring_18347121197199848620nteger @ X ) @ N )
% 9.04/5.04        = ( ring_18347121197199848620nteger @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5776_pochhammer__of__int,axiom,
% 9.04/5.04      ! [X: int,N: nat] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ ( ring_17405671764205052669omplex @ X ) @ N )
% 9.04/5.04        = ( ring_17405671764205052669omplex @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5777_abs__mod__less,axiom,
% 9.04/5.04      ! [L: int,K: int] :
% 9.04/5.04        ( ( L != zero_zero_int )
% 9.04/5.04       => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5778_zabs__def,axiom,
% 9.04/5.04      ( abs_abs_int
% 9.04/5.04      = ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5779_zmod__zminus1__eq__if,axiom,
% 9.04/5.04      ! [A: int,B: int] :
% 9.04/5.04        ( ( ( ( modulo_modulo_int @ A @ B )
% 9.04/5.04            = zero_zero_int )
% 9.04/5.04         => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 9.04/5.04            = zero_zero_int ) )
% 9.04/5.04        & ( ( ( modulo_modulo_int @ A @ B )
% 9.04/5.04           != zero_zero_int )
% 9.04/5.04         => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 9.04/5.04            = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5780_zmod__zminus2__eq__if,axiom,
% 9.04/5.04      ! [A: int,B: int] :
% 9.04/5.04        ( ( ( ( modulo_modulo_int @ A @ B )
% 9.04/5.04            = zero_zero_int )
% 9.04/5.04         => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 9.04/5.04            = zero_zero_int ) )
% 9.04/5.04        & ( ( ( modulo_modulo_int @ A @ B )
% 9.04/5.04           != zero_zero_int )
% 9.04/5.04         => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 9.04/5.04            = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5781_signed__take__bit__add,axiom,
% 9.04/5.04      ! [N: nat,K: int,L: int] :
% 9.04/5.04        ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 9.04/5.04        = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5782_uminus__int__code_I1_J,axiom,
% 9.04/5.04      ( ( uminus_uminus_int @ zero_zero_int )
% 9.04/5.04      = zero_zero_int ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5783_arctan__monotone,axiom,
% 9.04/5.04      ! [X: real,Y: real] :
% 9.04/5.04        ( ( ord_less_real @ X @ Y )
% 9.04/5.04       => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5784_arctan__less__iff,axiom,
% 9.04/5.04      ! [X: real,Y: real] :
% 9.04/5.04        ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 9.04/5.04        = ( ord_less_real @ X @ Y ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5785_arctan__le__iff,axiom,
% 9.04/5.04      ! [X: real,Y: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 9.04/5.04        = ( ord_less_eq_real @ X @ Y ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5786_arctan__monotone_H,axiom,
% 9.04/5.04      ! [X: real,Y: real] :
% 9.04/5.04        ( ( ord_less_eq_real @ X @ Y )
% 9.04/5.04       => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5787_int__cases2,axiom,
% 9.04/5.04      ! [Z: int] :
% 9.04/5.04        ( ! [N2: nat] :
% 9.04/5.04            ( Z
% 9.04/5.04           != ( semiri1314217659103216013at_int @ N2 ) )
% 9.04/5.04       => ~ ! [N2: nat] :
% 9.04/5.04              ( Z
% 9.04/5.04             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5788_arctan__minus,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( arctan @ ( uminus_uminus_real @ X ) )
% 9.04/5.04        = ( uminus_uminus_real @ ( arctan @ X ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5789_minus__mod__int__eq,axiom,
% 9.04/5.04      ! [L: int,K: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ zero_zero_int @ L )
% 9.04/5.04       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 9.04/5.04          = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5790_zmod__minus1,axiom,
% 9.04/5.04      ! [B: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.04       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 9.04/5.04          = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5791_zdiv__zminus2__eq__if,axiom,
% 9.04/5.04      ! [B: int,A: int] :
% 9.04/5.04        ( ( B != zero_zero_int )
% 9.04/5.04       => ( ( ( ( modulo_modulo_int @ A @ B )
% 9.04/5.04              = zero_zero_int )
% 9.04/5.04           => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 9.04/5.04              = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 9.04/5.04          & ( ( ( modulo_modulo_int @ A @ B )
% 9.04/5.04             != zero_zero_int )
% 9.04/5.04           => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 9.04/5.04              = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5792_zdiv__zminus1__eq__if,axiom,
% 9.04/5.04      ! [B: int,A: int] :
% 9.04/5.04        ( ( B != zero_zero_int )
% 9.04/5.04       => ( ( ( ( modulo_modulo_int @ A @ B )
% 9.04/5.04              = zero_zero_int )
% 9.04/5.04           => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 9.04/5.04              = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 9.04/5.04          & ( ( ( modulo_modulo_int @ A @ B )
% 9.04/5.04             != zero_zero_int )
% 9.04/5.04           => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 9.04/5.04              = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5793_zmod__le__nonneg__dividend,axiom,
% 9.04/5.04      ! [M: int,K: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ zero_zero_int @ M )
% 9.04/5.04       => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5794_Euclidean__Division_Opos__mod__bound,axiom,
% 9.04/5.04      ! [L: int,K: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ L )
% 9.04/5.04       => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5795_neg__mod__bound,axiom,
% 9.04/5.04      ! [L: int,K: int] :
% 9.04/5.04        ( ( ord_less_int @ L @ zero_zero_int )
% 9.04/5.04       => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5796_zmod__eq__0D,axiom,
% 9.04/5.04      ! [M: int,D: int] :
% 9.04/5.04        ( ( ( modulo_modulo_int @ M @ D )
% 9.04/5.04          = zero_zero_int )
% 9.04/5.04       => ? [Q2: int] :
% 9.04/5.04            ( M
% 9.04/5.04            = ( times_times_int @ D @ Q2 ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5797_zmod__eq__0__iff,axiom,
% 9.04/5.04      ! [M: int,D: int] :
% 9.04/5.04        ( ( ( modulo_modulo_int @ M @ D )
% 9.04/5.04          = zero_zero_int )
% 9.04/5.04        = ( ? [Q4: int] :
% 9.04/5.04              ( M
% 9.04/5.04              = ( times_times_int @ D @ Q4 ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5798_pochhammer__pos,axiom,
% 9.04/5.04      ! [X: real,N: nat] :
% 9.04/5.04        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.04       => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5799_pochhammer__pos,axiom,
% 9.04/5.04      ! [X: rat,N: nat] :
% 9.04/5.04        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.04/5.04       => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5800_pochhammer__pos,axiom,
% 9.04/5.04      ! [X: nat,N: nat] :
% 9.04/5.04        ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.04/5.04       => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5801_pochhammer__pos,axiom,
% 9.04/5.04      ! [X: int,N: nat] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ X )
% 9.04/5.04       => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5802_abs__zmult__eq__1,axiom,
% 9.04/5.04      ! [M: int,N: int] :
% 9.04/5.04        ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 9.04/5.04          = one_one_int )
% 9.04/5.04       => ( ( abs_abs_int @ M )
% 9.04/5.04          = one_one_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5803_pochhammer__eq__0__mono,axiom,
% 9.04/5.04      ! [A: complex,N: nat,M: nat] :
% 9.04/5.04        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 9.04/5.04          = zero_zero_complex )
% 9.04/5.04       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.04         => ( ( comm_s2602460028002588243omplex @ A @ M )
% 9.04/5.04            = zero_zero_complex ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5804_pochhammer__eq__0__mono,axiom,
% 9.04/5.04      ! [A: real,N: nat,M: nat] :
% 9.04/5.04        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 9.04/5.04          = zero_zero_real )
% 9.04/5.04       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.04         => ( ( comm_s7457072308508201937r_real @ A @ M )
% 9.04/5.04            = zero_zero_real ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5805_pochhammer__eq__0__mono,axiom,
% 9.04/5.04      ! [A: rat,N: nat,M: nat] :
% 9.04/5.04        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 9.04/5.04          = zero_zero_rat )
% 9.04/5.04       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.04         => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 9.04/5.04            = zero_zero_rat ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5806_pochhammer__neq__0__mono,axiom,
% 9.04/5.04      ! [A: complex,M: nat,N: nat] :
% 9.04/5.04        ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 9.04/5.04         != zero_zero_complex )
% 9.04/5.04       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.04         => ( ( comm_s2602460028002588243omplex @ A @ N )
% 9.04/5.04           != zero_zero_complex ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5807_pochhammer__neq__0__mono,axiom,
% 9.04/5.04      ! [A: real,M: nat,N: nat] :
% 9.04/5.04        ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 9.04/5.04         != zero_zero_real )
% 9.04/5.04       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.04         => ( ( comm_s7457072308508201937r_real @ A @ N )
% 9.04/5.04           != zero_zero_real ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5808_pochhammer__neq__0__mono,axiom,
% 9.04/5.04      ! [A: rat,M: nat,N: nat] :
% 9.04/5.04        ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 9.04/5.04         != zero_zero_rat )
% 9.04/5.04       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.04         => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 9.04/5.04           != zero_zero_rat ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5809_zmod__int,axiom,
% 9.04/5.04      ! [A: nat,B: nat] :
% 9.04/5.04        ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
% 9.04/5.04        = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5810_int__of__nat__induct,axiom,
% 9.04/5.04      ! [P: int > $o,Z: int] :
% 9.04/5.04        ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.04/5.04       => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
% 9.04/5.04         => ( P @ Z ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5811_int__cases,axiom,
% 9.04/5.04      ! [Z: int] :
% 9.04/5.04        ( ! [N2: nat] :
% 9.04/5.04            ( Z
% 9.04/5.04           != ( semiri1314217659103216013at_int @ N2 ) )
% 9.04/5.04       => ~ ! [N2: nat] :
% 9.04/5.04              ( Z
% 9.04/5.04             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5812_pos__zmult__eq__1__iff__lemma,axiom,
% 9.04/5.04      ! [M: int,N: int] :
% 9.04/5.04        ( ( ( times_times_int @ M @ N )
% 9.04/5.04          = one_one_int )
% 9.04/5.04       => ( ( M = one_one_int )
% 9.04/5.04          | ( M
% 9.04/5.04            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5813_zmult__eq__1__iff,axiom,
% 9.04/5.04      ! [M: int,N: int] :
% 9.04/5.04        ( ( ( times_times_int @ M @ N )
% 9.04/5.04          = one_one_int )
% 9.04/5.04        = ( ( ( M = one_one_int )
% 9.04/5.04            & ( N = one_one_int ) )
% 9.04/5.04          | ( ( M
% 9.04/5.04              = ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.04            & ( N
% 9.04/5.04              = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5814_minus__int__code_I2_J,axiom,
% 9.04/5.04      ! [L: int] :
% 9.04/5.04        ( ( minus_minus_int @ zero_zero_int @ L )
% 9.04/5.04        = ( uminus_uminus_int @ L ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5815_not__int__zless__negative,axiom,
% 9.04/5.04      ! [N: nat,M: nat] :
% 9.04/5.04        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5816_signed__take__bit__int__less__eq__self__iff,axiom,
% 9.04/5.04      ! [N: nat,K: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 9.04/5.04        = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5817_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 9.04/5.04      ! [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 ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5818_signed__take__bit__int__greater__self__iff,axiom,
% 9.04/5.04      ! [K: int,N: nat] :
% 9.04/5.04        ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.04/5.04        = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5819_frac__ge__0,axiom,
% 9.04/5.04      ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5820_frac__ge__0,axiom,
% 9.04/5.04      ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5821_frac__lt__1,axiom,
% 9.04/5.04      ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5822_frac__lt__1,axiom,
% 9.04/5.04      ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5823_frac__1__eq,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
% 9.04/5.04        = ( archim2898591450579166408c_real @ X ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5824_frac__1__eq,axiom,
% 9.04/5.04      ! [X: rat] :
% 9.04/5.04        ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
% 9.04/5.04        = ( archimedean_frac_rat @ X ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5825_signed__take__bit__int__eq__self,axiom,
% 9.04/5.04      ! [N: nat,K: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 9.04/5.04       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.04         => ( ( bit_ri631733984087533419it_int @ N @ K )
% 9.04/5.04            = K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5826_signed__take__bit__int__eq__self__iff,axiom,
% 9.04/5.04      ! [N: nat,K: int] :
% 9.04/5.04        ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 9.04/5.04          = K )
% 9.04/5.04        = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 9.04/5.04          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5827_zmod__trivial__iff,axiom,
% 9.04/5.04      ! [I: int,K: int] :
% 9.04/5.04        ( ( ( modulo_modulo_int @ I @ K )
% 9.04/5.04          = I )
% 9.04/5.04        = ( ( K = zero_zero_int )
% 9.04/5.04          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.04/5.04            & ( ord_less_int @ I @ K ) )
% 9.04/5.04          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 9.04/5.04            & ( ord_less_int @ K @ I ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5828_pos__mod__conj,axiom,
% 9.04/5.04      ! [B: int,A: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.04       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 9.04/5.04          & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5829_neg__mod__conj,axiom,
% 9.04/5.04      ! [B: int,A: int] :
% 9.04/5.04        ( ( ord_less_int @ B @ zero_zero_int )
% 9.04/5.04       => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 9.04/5.04          & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5830_Euclidean__Division_Opos__mod__sign,axiom,
% 9.04/5.04      ! [L: int,K: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ L )
% 9.04/5.04       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5831_neg__mod__sign,axiom,
% 9.04/5.04      ! [L: int,K: int] :
% 9.04/5.04        ( ( ord_less_int @ L @ zero_zero_int )
% 9.04/5.04       => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5832_pochhammer__nonneg,axiom,
% 9.04/5.04      ! [X: real,N: nat] :
% 9.04/5.04        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.04       => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5833_pochhammer__nonneg,axiom,
% 9.04/5.04      ! [X: rat,N: nat] :
% 9.04/5.04        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.04/5.04       => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5834_pochhammer__nonneg,axiom,
% 9.04/5.04      ! [X: nat,N: nat] :
% 9.04/5.04        ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.04/5.04       => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5835_pochhammer__nonneg,axiom,
% 9.04/5.04      ! [X: int,N: nat] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ X )
% 9.04/5.04       => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5836_zdiv__mono__strict,axiom,
% 9.04/5.04      ! [A2: int,B2: int,N: int] :
% 9.04/5.04        ( ( ord_less_int @ A2 @ B2 )
% 9.04/5.04       => ( ( ord_less_int @ zero_zero_int @ N )
% 9.04/5.04         => ( ( ( modulo_modulo_int @ A2 @ N )
% 9.04/5.04              = zero_zero_int )
% 9.04/5.04           => ( ( ( modulo_modulo_int @ B2 @ N )
% 9.04/5.04                = zero_zero_int )
% 9.04/5.04             => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B2 @ N ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5837_pochhammer__0__left,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( ( N = zero_zero_nat )
% 9.04/5.04         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 9.04/5.04            = one_one_complex ) )
% 9.04/5.04        & ( ( N != zero_zero_nat )
% 9.04/5.04         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 9.04/5.04            = zero_zero_complex ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5838_pochhammer__0__left,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( ( N = zero_zero_nat )
% 9.04/5.04         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 9.04/5.04            = one_one_real ) )
% 9.04/5.04        & ( ( N != zero_zero_nat )
% 9.04/5.04         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 9.04/5.04            = zero_zero_real ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5839_pochhammer__0__left,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( ( N = zero_zero_nat )
% 9.04/5.04         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 9.04/5.04            = one_one_rat ) )
% 9.04/5.04        & ( ( N != zero_zero_nat )
% 9.04/5.04         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 9.04/5.04            = zero_zero_rat ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5840_pochhammer__0__left,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( ( N = zero_zero_nat )
% 9.04/5.04         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 9.04/5.04            = one_one_nat ) )
% 9.04/5.04        & ( ( N != zero_zero_nat )
% 9.04/5.04         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 9.04/5.04            = zero_zero_nat ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5841_pochhammer__0__left,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( ( N = zero_zero_nat )
% 9.04/5.04         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 9.04/5.04            = one_one_int ) )
% 9.04/5.04        & ( ( N != zero_zero_nat )
% 9.04/5.04         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 9.04/5.04            = zero_zero_int ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5842_div__mod__decomp__int,axiom,
% 9.04/5.04      ! [A2: int,N: int] :
% 9.04/5.04        ( A2
% 9.04/5.04        = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5843_int__cases4,axiom,
% 9.04/5.04      ! [M: int] :
% 9.04/5.04        ( ! [N2: nat] :
% 9.04/5.04            ( M
% 9.04/5.04           != ( semiri1314217659103216013at_int @ N2 ) )
% 9.04/5.04       => ~ ! [N2: nat] :
% 9.04/5.04              ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.04/5.04             => ( M
% 9.04/5.04               != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5844_int__zle__neg,axiom,
% 9.04/5.04      ! [N: nat,M: nat] :
% 9.04/5.04        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 9.04/5.04        = ( ( N = zero_zero_nat )
% 9.04/5.04          & ( M = zero_zero_nat ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5845_negative__zle__0,axiom,
% 9.04/5.04      ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5846_nonpos__int__cases,axiom,
% 9.04/5.04      ! [K: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.04/5.04       => ~ ! [N2: nat] :
% 9.04/5.04              ( K
% 9.04/5.04             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5847_signed__take__bit__int__greater__eq,axiom,
% 9.04/5.04      ! [K: int,N: nat] :
% 9.04/5.04        ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.04       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5848_mod__pos__neg__trivial,axiom,
% 9.04/5.04      ! [K: int,L: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ K )
% 9.04/5.04       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 9.04/5.04         => ( ( modulo_modulo_int @ K @ L )
% 9.04/5.04            = ( plus_plus_int @ K @ L ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5849_mod__pos__geq,axiom,
% 9.04/5.04      ! [L: int,K: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ L )
% 9.04/5.04       => ( ( ord_less_eq_int @ L @ K )
% 9.04/5.04         => ( ( modulo_modulo_int @ K @ L )
% 9.04/5.04            = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5850_frac__def,axiom,
% 9.04/5.04      ( archim2898591450579166408c_real
% 9.04/5.04      = ( ^ [X4: real] : ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5851_frac__def,axiom,
% 9.04/5.04      ( archimedean_frac_rat
% 9.04/5.04      = ( ^ [X4: rat] : ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5852_pochhammer__rec,axiom,
% 9.04/5.04      ! [A: complex,N: nat] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5853_pochhammer__rec,axiom,
% 9.04/5.04      ! [A: real,N: nat] :
% 9.04/5.04        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5854_pochhammer__rec,axiom,
% 9.04/5.04      ! [A: rat,N: nat] :
% 9.04/5.04        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5855_pochhammer__rec,axiom,
% 9.04/5.04      ! [A: nat,N: nat] :
% 9.04/5.04        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5856_pochhammer__rec,axiom,
% 9.04/5.04      ! [A: int,N: nat] :
% 9.04/5.04        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5857_pochhammer__rec_H,axiom,
% 9.04/5.04      ! [Z: complex,N: nat] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5858_pochhammer__rec_H,axiom,
% 9.04/5.04      ! [Z: real,N: nat] :
% 9.04/5.04        ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5859_pochhammer__rec_H,axiom,
% 9.04/5.04      ! [Z: rat,N: nat] :
% 9.04/5.04        ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5860_pochhammer__rec_H,axiom,
% 9.04/5.04      ! [Z: nat,N: nat] :
% 9.04/5.04        ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5861_pochhammer__rec_H,axiom,
% 9.04/5.04      ! [Z: int,N: nat] :
% 9.04/5.04        ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5862_pochhammer__Suc,axiom,
% 9.04/5.04      ! [A: complex,N: nat] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5863_pochhammer__Suc,axiom,
% 9.04/5.04      ! [A: real,N: nat] :
% 9.04/5.04        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5864_pochhammer__Suc,axiom,
% 9.04/5.04      ! [A: rat,N: nat] :
% 9.04/5.04        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5865_pochhammer__Suc,axiom,
% 9.04/5.04      ! [A: nat,N: nat] :
% 9.04/5.04        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5866_pochhammer__Suc,axiom,
% 9.04/5.04      ! [A: int,N: nat] :
% 9.04/5.04        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 9.04/5.04        = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5867_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.04/5.04      ! [N: nat,K: nat] :
% 9.04/5.04        ( ( ord_less_nat @ N @ K )
% 9.04/5.04       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 9.04/5.04          = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5868_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.04/5.04      ! [N: nat,K: nat] :
% 9.04/5.04        ( ( ord_less_nat @ N @ K )
% 9.04/5.04       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 9.04/5.04          = zero_zero_complex ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5869_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.04/5.04      ! [N: nat,K: nat] :
% 9.04/5.04        ( ( ord_less_nat @ N @ K )
% 9.04/5.04       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 9.04/5.04          = zero_zero_real ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5870_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.04/5.04      ! [N: nat,K: nat] :
% 9.04/5.04        ( ( ord_less_nat @ N @ K )
% 9.04/5.04       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 9.04/5.04          = zero_zero_rat ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5871_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.04/5.04      ! [N: nat,K: nat] :
% 9.04/5.04        ( ( ord_less_nat @ N @ K )
% 9.04/5.04       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 9.04/5.04          = zero_zero_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5872_pochhammer__of__nat__eq__0__iff,axiom,
% 9.04/5.04      ! [N: nat,K: nat] :
% 9.04/5.04        ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 9.04/5.04          = zero_z3403309356797280102nteger )
% 9.04/5.04        = ( ord_less_nat @ N @ K ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5873_pochhammer__of__nat__eq__0__iff,axiom,
% 9.04/5.04      ! [N: nat,K: nat] :
% 9.04/5.04        ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 9.04/5.04          = zero_zero_complex )
% 9.04/5.04        = ( ord_less_nat @ N @ K ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5874_pochhammer__of__nat__eq__0__iff,axiom,
% 9.04/5.04      ! [N: nat,K: nat] :
% 9.04/5.04        ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 9.04/5.04          = zero_zero_real )
% 9.04/5.04        = ( ord_less_nat @ N @ K ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5875_pochhammer__of__nat__eq__0__iff,axiom,
% 9.04/5.04      ! [N: nat,K: nat] :
% 9.04/5.04        ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 9.04/5.04          = zero_zero_rat )
% 9.04/5.04        = ( ord_less_nat @ N @ K ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5876_pochhammer__of__nat__eq__0__iff,axiom,
% 9.04/5.04      ! [N: nat,K: nat] :
% 9.04/5.04        ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 9.04/5.04          = zero_zero_int )
% 9.04/5.04        = ( ord_less_nat @ N @ K ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5877_pochhammer__eq__0__iff,axiom,
% 9.04/5.04      ! [A: complex,N: nat] :
% 9.04/5.04        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 9.04/5.04          = zero_zero_complex )
% 9.04/5.04        = ( ? [K2: nat] :
% 9.04/5.04              ( ( ord_less_nat @ K2 @ N )
% 9.04/5.04              & ( A
% 9.04/5.04                = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5878_pochhammer__eq__0__iff,axiom,
% 9.04/5.04      ! [A: real,N: nat] :
% 9.04/5.04        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 9.04/5.04          = zero_zero_real )
% 9.04/5.04        = ( ? [K2: nat] :
% 9.04/5.04              ( ( ord_less_nat @ K2 @ N )
% 9.04/5.04              & ( A
% 9.04/5.04                = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5879_pochhammer__eq__0__iff,axiom,
% 9.04/5.04      ! [A: rat,N: nat] :
% 9.04/5.04        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 9.04/5.04          = zero_zero_rat )
% 9.04/5.04        = ( ? [K2: nat] :
% 9.04/5.04              ( ( ord_less_nat @ K2 @ N )
% 9.04/5.04              & ( A
% 9.04/5.04                = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5880_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.04/5.04      ! [K: nat,N: nat] :
% 9.04/5.04        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.04       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 9.04/5.04         != zero_z3403309356797280102nteger ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5881_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.04/5.04      ! [K: nat,N: nat] :
% 9.04/5.04        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.04       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 9.04/5.04         != zero_zero_complex ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5882_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.04/5.04      ! [K: nat,N: nat] :
% 9.04/5.04        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.04       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 9.04/5.04         != zero_zero_real ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5883_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.04/5.04      ! [K: nat,N: nat] :
% 9.04/5.04        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.04       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 9.04/5.04         != zero_zero_rat ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5884_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.04/5.04      ! [K: nat,N: nat] :
% 9.04/5.04        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.04       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 9.04/5.04         != zero_zero_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5885_pochhammer__product_H,axiom,
% 9.04/5.04      ! [Z: complex,N: nat,M: nat] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
% 9.04/5.04        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5886_pochhammer__product_H,axiom,
% 9.04/5.04      ! [Z: real,N: nat,M: nat] :
% 9.04/5.04        ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
% 9.04/5.04        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5887_pochhammer__product_H,axiom,
% 9.04/5.04      ! [Z: rat,N: nat,M: nat] :
% 9.04/5.04        ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
% 9.04/5.04        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5888_pochhammer__product_H,axiom,
% 9.04/5.04      ! [Z: nat,N: nat,M: nat] :
% 9.04/5.04        ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
% 9.04/5.04        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5889_pochhammer__product_H,axiom,
% 9.04/5.04      ! [Z: int,N: nat,M: nat] :
% 9.04/5.04        ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
% 9.04/5.04        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5890_int__cases3,axiom,
% 9.04/5.04      ! [K: int] :
% 9.04/5.04        ( ( K != zero_zero_int )
% 9.04/5.04       => ( ! [N2: nat] :
% 9.04/5.04              ( ( K
% 9.04/5.04                = ( semiri1314217659103216013at_int @ N2 ) )
% 9.04/5.04             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 9.04/5.04         => ~ ! [N2: nat] :
% 9.04/5.04                ( ( K
% 9.04/5.04                  = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 9.04/5.04               => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5891_not__zle__0__negative,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5892_negative__zless__0,axiom,
% 9.04/5.04      ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5893_negD,axiom,
% 9.04/5.04      ! [X: int] :
% 9.04/5.04        ( ( ord_less_int @ X @ zero_zero_int )
% 9.04/5.04       => ? [N2: nat] :
% 9.04/5.04            ( X
% 9.04/5.04            = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5894_real__of__int__div__aux,axiom,
% 9.04/5.04      ! [X: int,D: int] :
% 9.04/5.04        ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
% 9.04/5.04        = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5895_verit__less__mono__div__int2,axiom,
% 9.04/5.04      ! [A2: int,B2: int,N: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ A2 @ B2 )
% 9.04/5.04       => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 9.04/5.04         => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5896_div__eq__minus1,axiom,
% 9.04/5.04      ! [B: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.04       => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 9.04/5.04          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5897_ceiling__divide__eq__div,axiom,
% 9.04/5.04      ! [A: int,B: int] :
% 9.04/5.04        ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
% 9.04/5.04        = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5898_ceiling__divide__eq__div,axiom,
% 9.04/5.04      ! [A: int,B: int] :
% 9.04/5.04        ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
% 9.04/5.04        = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5899_frac__eq,axiom,
% 9.04/5.04      ! [X: real] :
% 9.04/5.04        ( ( ( archim2898591450579166408c_real @ X )
% 9.04/5.04          = X )
% 9.04/5.04        = ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.04          & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5900_frac__eq,axiom,
% 9.04/5.04      ! [X: rat] :
% 9.04/5.04        ( ( ( archimedean_frac_rat @ X )
% 9.04/5.04          = X )
% 9.04/5.04        = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.04/5.04          & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5901_split__zmod,axiom,
% 9.04/5.04      ! [P: int > $o,N: int,K: int] :
% 9.04/5.04        ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 9.04/5.04        = ( ( ( K = zero_zero_int )
% 9.04/5.04           => ( P @ N ) )
% 9.04/5.04          & ( ( ord_less_int @ zero_zero_int @ K )
% 9.04/5.04           => ! [I3: int,J2: int] :
% 9.04/5.04                ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
% 9.04/5.04                  & ( ord_less_int @ J2 @ K )
% 9.04/5.04                  & ( N
% 9.04/5.04                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 9.04/5.04               => ( P @ J2 ) ) )
% 9.04/5.04          & ( ( ord_less_int @ K @ zero_zero_int )
% 9.04/5.04           => ! [I3: int,J2: int] :
% 9.04/5.04                ( ( ( ord_less_int @ K @ J2 )
% 9.04/5.04                  & ( ord_less_eq_int @ J2 @ zero_zero_int )
% 9.04/5.04                  & ( N
% 9.04/5.04                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 9.04/5.04               => ( P @ J2 ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5902_int__mod__neg__eq,axiom,
% 9.04/5.04      ! [A: int,B: int,Q3: int,R2: int] :
% 9.04/5.04        ( ( A
% 9.04/5.04          = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 9.04/5.04       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.04/5.04         => ( ( ord_less_int @ B @ R2 )
% 9.04/5.04           => ( ( modulo_modulo_int @ A @ B )
% 9.04/5.04              = R2 ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5903_int__mod__pos__eq,axiom,
% 9.04/5.04      ! [A: int,B: int,Q3: int,R2: int] :
% 9.04/5.04        ( ( A
% 9.04/5.04          = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 9.04/5.04       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.04/5.04         => ( ( ord_less_int @ R2 @ B )
% 9.04/5.04           => ( ( modulo_modulo_int @ A @ B )
% 9.04/5.04              = R2 ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5904_frac__add,axiom,
% 9.04/5.04      ! [X: real,Y: real] :
% 9.04/5.04        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 9.04/5.04         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.04            = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
% 9.04/5.04        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 9.04/5.04         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.04            = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5905_frac__add,axiom,
% 9.04/5.04      ! [X: rat,Y: rat] :
% 9.04/5.04        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 9.04/5.04         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 9.04/5.04            = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
% 9.04/5.04        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 9.04/5.04         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 9.04/5.04            = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5906_signed__take__bit__int__less__exp,axiom,
% 9.04/5.04      ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5907_zmod__zmult2__eq,axiom,
% 9.04/5.04      ! [C: int,A: int,B: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.04/5.04       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 9.04/5.04          = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5908_pochhammer__product,axiom,
% 9.04/5.04      ! [M: nat,N: nat,Z: complex] :
% 9.04/5.04        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.04       => ( ( comm_s2602460028002588243omplex @ Z @ N )
% 9.04/5.04          = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5909_pochhammer__product,axiom,
% 9.04/5.04      ! [M: nat,N: nat,Z: real] :
% 9.04/5.04        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.04       => ( ( comm_s7457072308508201937r_real @ Z @ N )
% 9.04/5.04          = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5910_pochhammer__product,axiom,
% 9.04/5.04      ! [M: nat,N: nat,Z: rat] :
% 9.04/5.04        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.04       => ( ( comm_s4028243227959126397er_rat @ Z @ N )
% 9.04/5.04          = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5911_pochhammer__product,axiom,
% 9.04/5.04      ! [M: nat,N: nat,Z: nat] :
% 9.04/5.04        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.04       => ( ( comm_s4663373288045622133er_nat @ Z @ N )
% 9.04/5.04          = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5912_pochhammer__product,axiom,
% 9.04/5.04      ! [M: nat,N: nat,Z: int] :
% 9.04/5.04        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.04       => ( ( comm_s4660882817536571857er_int @ Z @ N )
% 9.04/5.04          = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5913_neg__int__cases,axiom,
% 9.04/5.04      ! [K: int] :
% 9.04/5.04        ( ( ord_less_int @ K @ zero_zero_int )
% 9.04/5.04       => ~ ! [N2: nat] :
% 9.04/5.04              ( ( K
% 9.04/5.04                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 9.04/5.04             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5914_nat__intermed__int__val,axiom,
% 9.04/5.04      ! [M: nat,N: nat,F: nat > int,K: int] :
% 9.04/5.04        ( ! [I2: nat] :
% 9.04/5.04            ( ( ( ord_less_eq_nat @ M @ I2 )
% 9.04/5.04              & ( ord_less_nat @ I2 @ N ) )
% 9.04/5.04           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 9.04/5.04       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.04         => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 9.04/5.04           => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.04/5.04             => ? [I2: nat] :
% 9.04/5.04                  ( ( ord_less_eq_nat @ M @ I2 )
% 9.04/5.04                  & ( ord_less_eq_nat @ I2 @ N )
% 9.04/5.04                  & ( ( F @ I2 )
% 9.04/5.04                    = K ) ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5915_decr__lemma,axiom,
% 9.04/5.04      ! [D: int,X: int,Z: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ D )
% 9.04/5.04       => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5916_incr__lemma,axiom,
% 9.04/5.04      ! [D: int,Z: int,X: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ D )
% 9.04/5.04       => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5917_verit__le__mono__div__int,axiom,
% 9.04/5.04      ! [A2: int,B2: int,N: int] :
% 9.04/5.04        ( ( ord_less_int @ A2 @ B2 )
% 9.04/5.04       => ( ( ord_less_int @ zero_zero_int @ N )
% 9.04/5.04         => ( ord_less_eq_int
% 9.04/5.04            @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
% 9.04/5.04              @ ( if_int
% 9.04/5.04                @ ( ( modulo_modulo_int @ B2 @ N )
% 9.04/5.04                  = zero_zero_int )
% 9.04/5.04                @ one_one_int
% 9.04/5.04                @ zero_zero_int ) )
% 9.04/5.04            @ ( divide_divide_int @ B2 @ N ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5918_split__neg__lemma,axiom,
% 9.04/5.04      ! [K: int,P: int > int > $o,N: int] :
% 9.04/5.04        ( ( ord_less_int @ K @ zero_zero_int )
% 9.04/5.04       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 9.04/5.04          = ( ! [I3: int,J2: int] :
% 9.04/5.04                ( ( ( ord_less_int @ K @ J2 )
% 9.04/5.04                  & ( ord_less_eq_int @ J2 @ zero_zero_int )
% 9.04/5.04                  & ( N
% 9.04/5.04                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 9.04/5.04               => ( P @ I3 @ J2 ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5919_split__pos__lemma,axiom,
% 9.04/5.04      ! [K: int,P: int > int > $o,N: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ K )
% 9.04/5.04       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 9.04/5.04          = ( ! [I3: int,J2: int] :
% 9.04/5.04                ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
% 9.04/5.04                  & ( ord_less_int @ J2 @ K )
% 9.04/5.04                  & ( N
% 9.04/5.04                    = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
% 9.04/5.04               => ( P @ I3 @ J2 ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5920_signed__take__bit__int__less__self__iff,axiom,
% 9.04/5.04      ! [N: nat,K: int] :
% 9.04/5.04        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 9.04/5.04        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5921_signed__take__bit__int__greater__eq__self__iff,axiom,
% 9.04/5.04      ! [K: int,N: nat] :
% 9.04/5.04        ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.04/5.04        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5922_pochhammer__absorb__comp,axiom,
% 9.04/5.04      ! [R2: code_integer,K: nat] :
% 9.04/5.04        ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
% 9.04/5.04        = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5923_pochhammer__absorb__comp,axiom,
% 9.04/5.04      ! [R2: complex,K: nat] :
% 9.04/5.04        ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
% 9.04/5.04        = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5924_pochhammer__absorb__comp,axiom,
% 9.04/5.04      ! [R2: real,K: nat] :
% 9.04/5.04        ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
% 9.04/5.04        = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5925_pochhammer__absorb__comp,axiom,
% 9.04/5.04      ! [R2: rat,K: nat] :
% 9.04/5.04        ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
% 9.04/5.04        = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5926_pochhammer__absorb__comp,axiom,
% 9.04/5.04      ! [R2: int,K: nat] :
% 9.04/5.04        ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
% 9.04/5.04        = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5927_minus__1__div__exp__eq__int,axiom,
% 9.04/5.04      ! [N: nat] :
% 9.04/5.04        ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.04        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5928_div__pos__neg__trivial,axiom,
% 9.04/5.04      ! [K: int,L: int] :
% 9.04/5.04        ( ( ord_less_int @ zero_zero_int @ K )
% 9.04/5.04       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 9.04/5.04         => ( ( divide_divide_int @ K @ L )
% 9.04/5.04            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5929_compl__le__swap2,axiom,
% 9.04/5.04      ! [Y: set_int,X: set_int] :
% 9.04/5.04        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
% 9.04/5.04       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5930_compl__le__swap1,axiom,
% 9.04/5.04      ! [Y: set_int,X: set_int] :
% 9.04/5.04        ( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
% 9.04/5.04       => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5931_compl__mono,axiom,
% 9.04/5.04      ! [X: set_int,Y: set_int] :
% 9.04/5.04        ( ( ord_less_eq_set_int @ X @ Y )
% 9.04/5.04       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5932_nat__ivt__aux,axiom,
% 9.04/5.04      ! [N: nat,F: nat > int,K: int] :
% 9.04/5.04        ( ! [I2: nat] :
% 9.04/5.04            ( ( ord_less_nat @ I2 @ N )
% 9.04/5.04           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 9.04/5.04       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 9.04/5.04         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.04/5.04           => ? [I2: nat] :
% 9.04/5.04                ( ( ord_less_eq_nat @ I2 @ N )
% 9.04/5.04                & ( ( F @ I2 )
% 9.04/5.04                  = K ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5933_complex__mod__minus__le__complex__mod,axiom,
% 9.04/5.04      ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5934_signed__take__bit__int__less__eq,axiom,
% 9.04/5.04      ! [N: nat,K: int] :
% 9.04/5.04        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 9.04/5.04       => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5935_pochhammer__minus_H,axiom,
% 9.04/5.04      ! [B: code_integer,K: nat] :
% 9.04/5.04        ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 9.04/5.04        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5936_pochhammer__minus_H,axiom,
% 9.04/5.04      ! [B: complex,K: nat] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 9.04/5.04        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5937_pochhammer__minus_H,axiom,
% 9.04/5.04      ! [B: real,K: nat] :
% 9.04/5.04        ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 9.04/5.04        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5938_pochhammer__minus_H,axiom,
% 9.04/5.04      ! [B: rat,K: nat] :
% 9.04/5.04        ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 9.04/5.04        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5939_pochhammer__minus_H,axiom,
% 9.04/5.04      ! [B: int,K: nat] :
% 9.04/5.04        ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 9.04/5.04        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5940_pochhammer__minus,axiom,
% 9.04/5.04      ! [B: code_integer,K: nat] :
% 9.04/5.04        ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 9.04/5.04        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5941_pochhammer__minus,axiom,
% 9.04/5.04      ! [B: complex,K: nat] :
% 9.04/5.04        ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 9.04/5.04        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5942_pochhammer__minus,axiom,
% 9.04/5.04      ! [B: real,K: nat] :
% 9.04/5.04        ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 9.04/5.04        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5943_pochhammer__minus,axiom,
% 9.04/5.04      ! [B: rat,K: nat] :
% 9.04/5.04        ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 9.04/5.04        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5944_pochhammer__minus,axiom,
% 9.04/5.04      ! [B: int,K: nat] :
% 9.04/5.04        ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 9.04/5.04        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5945_complex__mod__triangle__ineq2,axiom,
% 9.04/5.04      ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 9.04/5.04  
% 9.04/5.04  thf(fact_5946_int__bit__induct,axiom,
% 9.04/5.05      ! [P: int > $o,K: int] :
% 9.04/5.05        ( ( P @ zero_zero_int )
% 9.04/5.05       => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.05         => ( ! [K3: int] :
% 9.04/5.05                ( ( P @ K3 )
% 9.04/5.05               => ( ( K3 != zero_zero_int )
% 9.04/5.05                 => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.05           => ( ! [K3: int] :
% 9.04/5.05                  ( ( P @ K3 )
% 9.04/5.05                 => ( ( K3
% 9.04/5.05                     != ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.05                   => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 9.04/5.05             => ( P @ K ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5947_floor__add,axiom,
% 9.04/5.05      ! [X: real,Y: real] :
% 9.04/5.05        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 9.04/5.05         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.05            = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
% 9.04/5.05        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 9.04/5.05         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.05            = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5948_floor__add,axiom,
% 9.04/5.05      ! [X: rat,Y: rat] :
% 9.04/5.05        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 9.04/5.05         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 9.04/5.05            = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
% 9.04/5.05        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 9.04/5.05         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 9.04/5.05            = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5949_pos__zmod__mult__2,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.04/5.05       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.05          = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5950_nat0__intermed__int__val,axiom,
% 9.04/5.05      ! [N: nat,F: nat > int,K: int] :
% 9.04/5.05        ( ! [I2: nat] :
% 9.04/5.05            ( ( ord_less_nat @ I2 @ N )
% 9.04/5.05           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 9.04/5.05       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 9.04/5.05         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.04/5.05           => ? [I2: nat] :
% 9.04/5.05                ( ( ord_less_eq_nat @ I2 @ N )
% 9.04/5.05                & ( ( F @ I2 )
% 9.04/5.05                  = K ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5951_arctan__add,axiom,
% 9.04/5.05      ! [X: real,Y: real] :
% 9.04/5.05        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.05       => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.04/5.05         => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 9.04/5.05            = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5952_neg__zmod__mult__2,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.04/5.05       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.05          = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5953_signed__take__bit__Suc,axiom,
% 9.04/5.05      ! [N: nat,A: code_integer] :
% 9.04/5.05        ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 9.04/5.05        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5954_signed__take__bit__Suc,axiom,
% 9.04/5.05      ! [N: nat,A: int] :
% 9.04/5.05        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 9.04/5.05        = ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5955_central__binomial__lower__bound,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05       => ( 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 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5956_signed__take__bit__Suc__minus__bit1,axiom,
% 9.04/5.05      ! [N: nat,K: num] :
% 9.04/5.05        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.04/5.05        = ( 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 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5957_gcd__nat__induct,axiom,
% 9.04/5.05      ! [P: nat > nat > $o,M: nat,N: nat] :
% 9.04/5.05        ( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
% 9.04/5.05       => ( ! [M5: nat,N2: nat] :
% 9.04/5.05              ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.04/5.05             => ( ( P @ N2 @ ( modulo_modulo_nat @ M5 @ N2 ) )
% 9.04/5.05               => ( P @ M5 @ N2 ) ) )
% 9.04/5.05         => ( P @ M @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5958_option_Osize_I3_J,axiom,
% 9.04/5.05      ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 9.04/5.05      = ( suc @ zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5959_option_Osize_I3_J,axiom,
% 9.04/5.05      ( ( size_size_option_num @ none_num )
% 9.04/5.05      = ( suc @ zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5960_verit__eq__simplify_I9_J,axiom,
% 9.04/5.05      ! [X32: num,Y32: num] :
% 9.04/5.05        ( ( ( bit1 @ X32 )
% 9.04/5.05          = ( bit1 @ Y32 ) )
% 9.04/5.05        = ( X32 = Y32 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5961_semiring__norm_I90_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( ( bit1 @ M )
% 9.04/5.05          = ( bit1 @ N ) )
% 9.04/5.05        = ( M = N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5962_ComplI,axiom,
% 9.04/5.05      ! [C: complex,A2: set_complex] :
% 9.04/5.05        ( ~ ( member_complex @ C @ A2 )
% 9.04/5.05       => ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5963_ComplI,axiom,
% 9.04/5.05      ! [C: real,A2: set_real] :
% 9.04/5.05        ( ~ ( member_real @ C @ A2 )
% 9.04/5.05       => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5964_ComplI,axiom,
% 9.04/5.05      ! [C: set_nat,A2: set_set_nat] :
% 9.04/5.05        ( ~ ( member_set_nat @ C @ A2 )
% 9.04/5.05       => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5965_ComplI,axiom,
% 9.04/5.05      ! [C: nat,A2: set_nat] :
% 9.04/5.05        ( ~ ( member_nat @ C @ A2 )
% 9.04/5.05       => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5966_ComplI,axiom,
% 9.04/5.05      ! [C: int,A2: set_int] :
% 9.04/5.05        ( ~ ( member_int @ C @ A2 )
% 9.04/5.05       => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5967_Compl__iff,axiom,
% 9.04/5.05      ! [C: complex,A2: set_complex] :
% 9.04/5.05        ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
% 9.04/5.05        = ( ~ ( member_complex @ C @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5968_Compl__iff,axiom,
% 9.04/5.05      ! [C: real,A2: set_real] :
% 9.04/5.05        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 9.04/5.05        = ( ~ ( member_real @ C @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5969_Compl__iff,axiom,
% 9.04/5.05      ! [C: set_nat,A2: set_set_nat] :
% 9.04/5.05        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 9.04/5.05        = ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5970_Compl__iff,axiom,
% 9.04/5.05      ! [C: nat,A2: set_nat] :
% 9.04/5.05        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 9.04/5.05        = ( ~ ( member_nat @ C @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5971_Compl__iff,axiom,
% 9.04/5.05      ! [C: int,A2: set_int] :
% 9.04/5.05        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 9.04/5.05        = ( ~ ( member_int @ C @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5972_semiring__norm_I89_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( bit1 @ M )
% 9.04/5.05       != ( bit0 @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5973_semiring__norm_I88_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( bit0 @ M )
% 9.04/5.05       != ( bit1 @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5974_semiring__norm_I86_J,axiom,
% 9.04/5.05      ! [M: num] :
% 9.04/5.05        ( ( bit1 @ M )
% 9.04/5.05       != one ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5975_semiring__norm_I84_J,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( one
% 9.04/5.05       != ( bit1 @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5976_semiring__norm_I80_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.05        = ( ord_less_num @ M @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5977_semiring__norm_I73_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.05        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5978_semiring__norm_I7_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.05        = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5979_semiring__norm_I9_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.04/5.05        = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5980_semiring__norm_I14_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.05        = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5981_semiring__norm_I15_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.04/5.05        = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5982_semiring__norm_I81_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.04/5.05        = ( ord_less_num @ M @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5983_semiring__norm_I72_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.05        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5984_semiring__norm_I77_J,axiom,
% 9.04/5.05      ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5985_semiring__norm_I70_J,axiom,
% 9.04/5.05      ! [M: num] :
% 9.04/5.05        ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5986_zdiv__numeral__Bit1,axiom,
% 9.04/5.05      ! [V: num,W: num] :
% 9.04/5.05        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 9.04/5.05        = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5987_semiring__norm_I10_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.05        = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5988_semiring__norm_I8_J,axiom,
% 9.04/5.05      ! [M: num] :
% 9.04/5.05        ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 9.04/5.05        = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5989_semiring__norm_I5_J,axiom,
% 9.04/5.05      ! [M: num] :
% 9.04/5.05        ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 9.04/5.05        = ( bit1 @ M ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5990_semiring__norm_I4_J,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 9.04/5.05        = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5991_semiring__norm_I3_J,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 9.04/5.05        = ( bit1 @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5992_semiring__norm_I16_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.05        = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5993_semiring__norm_I74_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.04/5.05        = ( ord_less_num @ M @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5994_semiring__norm_I79_J,axiom,
% 9.04/5.05      ! [M: num,N: num] :
% 9.04/5.05        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.05        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5995_Suc__div__eq__add3__div__numeral,axiom,
% 9.04/5.05      ! [M: nat,V: num] :
% 9.04/5.05        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 9.04/5.05        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5996_div__Suc__eq__div__add3,axiom,
% 9.04/5.05      ! [M: nat,N: nat] :
% 9.04/5.05        ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 9.04/5.05        = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5997_Suc__mod__eq__add3__mod__numeral,axiom,
% 9.04/5.05      ! [M: nat,V: num] :
% 9.04/5.05        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 9.04/5.05        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5998_mod__Suc__eq__mod__add3,axiom,
% 9.04/5.05      ! [M: nat,N: nat] :
% 9.04/5.05        ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 9.04/5.05        = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_5999_zmod__numeral__Bit1,axiom,
% 9.04/5.05      ! [V: num,W: num] :
% 9.04/5.05        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 9.04/5.05        = ( 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 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6000_signed__take__bit__Suc__bit1,axiom,
% 9.04/5.05      ! [N: nat,K: num] :
% 9.04/5.05        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 9.04/5.05        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6001_ComplD,axiom,
% 9.04/5.05      ! [C: complex,A2: set_complex] :
% 9.04/5.05        ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
% 9.04/5.05       => ~ ( member_complex @ C @ A2 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6002_ComplD,axiom,
% 9.04/5.05      ! [C: real,A2: set_real] :
% 9.04/5.05        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 9.04/5.05       => ~ ( member_real @ C @ A2 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6003_ComplD,axiom,
% 9.04/5.05      ! [C: set_nat,A2: set_set_nat] :
% 9.04/5.05        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 9.04/5.05       => ~ ( member_set_nat @ C @ A2 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6004_ComplD,axiom,
% 9.04/5.05      ! [C: nat,A2: set_nat] :
% 9.04/5.05        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 9.04/5.05       => ~ ( member_nat @ C @ A2 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6005_ComplD,axiom,
% 9.04/5.05      ! [C: int,A2: set_int] :
% 9.04/5.05        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 9.04/5.05       => ~ ( member_int @ C @ A2 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6006_verit__eq__simplify_I14_J,axiom,
% 9.04/5.05      ! [X22: num,X32: num] :
% 9.04/5.05        ( ( bit0 @ X22 )
% 9.04/5.05       != ( bit1 @ X32 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6007_verit__eq__simplify_I12_J,axiom,
% 9.04/5.05      ! [X32: num] :
% 9.04/5.05        ( one
% 9.04/5.05       != ( bit1 @ X32 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6008_num_Oexhaust,axiom,
% 9.04/5.05      ! [Y: num] :
% 9.04/5.05        ( ( Y != one )
% 9.04/5.05       => ( ! [X23: num] :
% 9.04/5.05              ( Y
% 9.04/5.05             != ( bit0 @ X23 ) )
% 9.04/5.05         => ~ ! [X33: num] :
% 9.04/5.05                ( Y
% 9.04/5.05               != ( bit1 @ X33 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6009_numeral__Bit1,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 9.04/5.05        = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6010_numeral__Bit1,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 9.04/5.05        = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6011_numeral__Bit1,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 9.04/5.05        = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6012_numeral__Bit1,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 9.04/5.05        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6013_numeral__Bit1,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 9.04/5.05        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6014_eval__nat__numeral_I3_J,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 9.04/5.05        = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6015_cong__exp__iff__simps_I13_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num,N: num] :
% 9.04/5.05        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.04/5.05          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 9.04/5.05        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 9.04/5.05          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6016_cong__exp__iff__simps_I13_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num,N: num] :
% 9.04/5.05        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.04/5.05          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 9.04/5.05        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 9.04/5.05          = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6017_cong__exp__iff__simps_I13_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num,N: num] :
% 9.04/5.05        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.04/5.05          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 9.04/5.05        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 9.04/5.05          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6018_cong__exp__iff__simps_I12_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num,N: num] :
% 9.04/5.05        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.04/5.05       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6019_cong__exp__iff__simps_I12_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num,N: num] :
% 9.04/5.05        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.04/5.05       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6020_cong__exp__iff__simps_I12_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num,N: num] :
% 9.04/5.05        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.04/5.05       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6021_cong__exp__iff__simps_I10_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num,N: num] :
% 9.04/5.05        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.04/5.05       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6022_cong__exp__iff__simps_I10_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num,N: num] :
% 9.04/5.05        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.04/5.05       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6023_cong__exp__iff__simps_I10_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num,N: num] :
% 9.04/5.05        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.04/5.05       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6024_power__minus__Bit1,axiom,
% 9.04/5.05      ! [X: real,K: num] :
% 9.04/5.05        ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.04/5.05        = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6025_power__minus__Bit1,axiom,
% 9.04/5.05      ! [X: int,K: num] :
% 9.04/5.05        ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.04/5.05        = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6026_power__minus__Bit1,axiom,
% 9.04/5.05      ! [X: complex,K: num] :
% 9.04/5.05        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.04/5.05        = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6027_power__minus__Bit1,axiom,
% 9.04/5.05      ! [X: rat,K: num] :
% 9.04/5.05        ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.04/5.05        = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6028_power__minus__Bit1,axiom,
% 9.04/5.05      ! [X: code_integer,K: num] :
% 9.04/5.05        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.04/5.05        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6029_bounded__Max__nat,axiom,
% 9.04/5.05      ! [P: nat > $o,X: nat,M2: nat] :
% 9.04/5.05        ( ( P @ X )
% 9.04/5.05       => ( ! [X3: nat] :
% 9.04/5.05              ( ( P @ X3 )
% 9.04/5.05             => ( ord_less_eq_nat @ X3 @ M2 ) )
% 9.04/5.05         => ~ ! [M5: nat] :
% 9.04/5.05                ( ( P @ M5 )
% 9.04/5.05               => ~ ! [X2: nat] :
% 9.04/5.05                      ( ( P @ X2 )
% 9.04/5.05                     => ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6030_numeral__Bit1__div__2,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.05        = ( numeral_numeral_nat @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6031_numeral__Bit1__div__2,axiom,
% 9.04/5.05      ! [N: num] :
% 9.04/5.05        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.05        = ( numeral_numeral_int @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6032_cong__exp__iff__simps_I3_J,axiom,
% 9.04/5.05      ! [N: num,Q3: num] :
% 9.04/5.05        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.04/5.05       != zero_zero_nat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6033_cong__exp__iff__simps_I3_J,axiom,
% 9.04/5.05      ! [N: num,Q3: num] :
% 9.04/5.05        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.04/5.05       != zero_zero_int ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6034_cong__exp__iff__simps_I3_J,axiom,
% 9.04/5.05      ! [N: num,Q3: num] :
% 9.04/5.05        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.04/5.05       != zero_z3403309356797280102nteger ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6035_power3__eq__cube,axiom,
% 9.04/5.05      ! [A: complex] :
% 9.04/5.05        ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.04/5.05        = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6036_power3__eq__cube,axiom,
% 9.04/5.05      ! [A: real] :
% 9.04/5.05        ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.04/5.05        = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6037_power3__eq__cube,axiom,
% 9.04/5.05      ! [A: rat] :
% 9.04/5.05        ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.04/5.05        = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6038_power3__eq__cube,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.04/5.05        = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6039_power3__eq__cube,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.04/5.05        = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6040_numeral__3__eq__3,axiom,
% 9.04/5.05      ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 9.04/5.05      = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6041_Suc3__eq__add__3,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( suc @ ( suc @ ( suc @ N ) ) )
% 9.04/5.05        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6042_binomial__maximum_H,axiom,
% 9.04/5.05      ! [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 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6043_binomial__mono,axiom,
% 9.04/5.05      ! [K: nat,K5: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_eq_nat @ K @ K5 )
% 9.04/5.05       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 9.04/5.05         => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6044_binomial__antimono,axiom,
% 9.04/5.05      ! [K: nat,K5: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_eq_nat @ K @ K5 )
% 9.04/5.05       => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 9.04/5.05         => ( ( ord_less_eq_nat @ K5 @ N )
% 9.04/5.05           => ( ord_less_eq_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6045_binomial__maximum,axiom,
% 9.04/5.05      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6046_num_Osize_I6_J,axiom,
% 9.04/5.05      ! [X32: num] :
% 9.04/5.05        ( ( size_size_num @ ( bit1 @ X32 ) )
% 9.04/5.05        = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6047_cong__exp__iff__simps_I11_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num] :
% 9.04/5.05        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.04/5.05          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 9.04/5.05        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 9.04/5.05          = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6048_cong__exp__iff__simps_I11_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num] :
% 9.04/5.05        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.04/5.05          = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 9.04/5.05        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 9.04/5.05          = zero_zero_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6049_cong__exp__iff__simps_I11_J,axiom,
% 9.04/5.05      ! [M: num,Q3: num] :
% 9.04/5.05        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.04/5.05          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 9.04/5.05        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 9.04/5.05          = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6050_cong__exp__iff__simps_I7_J,axiom,
% 9.04/5.05      ! [Q3: num,N: num] :
% 9.04/5.05        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.04/5.05          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 9.04/5.05        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 9.04/5.05          = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6051_cong__exp__iff__simps_I7_J,axiom,
% 9.04/5.05      ! [Q3: num,N: num] :
% 9.04/5.05        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.04/5.05          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 9.04/5.05        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 9.04/5.05          = zero_zero_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6052_cong__exp__iff__simps_I7_J,axiom,
% 9.04/5.05      ! [Q3: num,N: num] :
% 9.04/5.05        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.04/5.05          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 9.04/5.05        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 9.04/5.05          = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6053_Suc__div__eq__add3__div,axiom,
% 9.04/5.05      ! [M: nat,N: nat] :
% 9.04/5.05        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 9.04/5.05        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6054_Suc__mod__eq__add3__mod,axiom,
% 9.04/5.05      ! [M: nat,N: nat] :
% 9.04/5.05        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 9.04/5.05        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6055_binomial__strict__mono,axiom,
% 9.04/5.05      ! [K: nat,K5: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_nat @ K @ K5 )
% 9.04/5.05       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 9.04/5.05         => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6056_binomial__strict__antimono,axiom,
% 9.04/5.05      ! [K: nat,K5: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_nat @ K @ K5 )
% 9.04/5.05       => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 9.04/5.05         => ( ( ord_less_eq_nat @ K5 @ N )
% 9.04/5.05           => ( ord_less_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6057_binomial__less__binomial__Suc,axiom,
% 9.04/5.05      ! [K: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.05       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6058_mod__exhaust__less__4,axiom,
% 9.04/5.05      ! [M: nat] :
% 9.04/5.05        ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.04/5.05          = zero_zero_nat )
% 9.04/5.05        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.04/5.05          = one_one_nat )
% 9.04/5.05        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.04/5.05          = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.05        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.04/5.05          = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6059_zero__notin__Suc__image,axiom,
% 9.04/5.05      ! [A2: set_nat] :
% 9.04/5.05        ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6060_Euclid__induct,axiom,
% 9.04/5.05      ! [P: nat > nat > $o,A: nat,B: nat] :
% 9.04/5.05        ( ! [A5: nat,B6: nat] :
% 9.04/5.05            ( ( P @ A5 @ B6 )
% 9.04/5.05            = ( P @ B6 @ A5 ) )
% 9.04/5.05       => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
% 9.04/5.05         => ( ! [A5: nat,B6: nat] :
% 9.04/5.05                ( ( P @ A5 @ B6 )
% 9.04/5.05               => ( P @ A5 @ ( plus_plus_nat @ A5 @ B6 ) ) )
% 9.04/5.05           => ( P @ A @ B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6061_zero__less__binomial__iff,axiom,
% 9.04/5.05      ! [N: nat,K: nat] :
% 9.04/5.05        ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 9.04/5.05        = ( ord_less_eq_nat @ K @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6062_choose__two,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.05        = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6063_binomial__n__0,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( binomial @ N @ zero_zero_nat )
% 9.04/5.05        = one_one_nat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6064_binomial__Suc__Suc,axiom,
% 9.04/5.05      ! [N: nat,K: nat] :
% 9.04/5.05        ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 9.04/5.05        = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6065_binomial__eq__0__iff,axiom,
% 9.04/5.05      ! [N: nat,K: nat] :
% 9.04/5.05        ( ( ( binomial @ N @ K )
% 9.04/5.05          = zero_zero_nat )
% 9.04/5.05        = ( ord_less_nat @ N @ K ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6066_binomial__1,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 9.04/5.05        = N ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6067_binomial__Suc__n,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( binomial @ ( suc @ N ) @ N )
% 9.04/5.05        = ( suc @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6068_binomial__n__n,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( binomial @ N @ N )
% 9.04/5.05        = one_one_nat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6069_binomial__0__Suc,axiom,
% 9.04/5.05      ! [K: nat] :
% 9.04/5.05        ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 9.04/5.05        = zero_zero_nat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6070_choose__one,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( binomial @ N @ one_one_nat )
% 9.04/5.05        = N ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6071_binomial__eq__0,axiom,
% 9.04/5.05      ! [N: nat,K: nat] :
% 9.04/5.05        ( ( ord_less_nat @ N @ K )
% 9.04/5.05       => ( ( binomial @ N @ K )
% 9.04/5.05          = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6072_Suc__times__binomial__eq,axiom,
% 9.04/5.05      ! [N: nat,K: nat] :
% 9.04/5.05        ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 9.04/5.05        = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6073_Suc__times__binomial,axiom,
% 9.04/5.05      ! [K: nat,N: nat] :
% 9.04/5.05        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 9.04/5.05        = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6074_binomial__symmetric,axiom,
% 9.04/5.05      ! [K: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.05       => ( ( binomial @ N @ K )
% 9.04/5.05          = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6075_choose__mult__lemma,axiom,
% 9.04/5.05      ! [M: nat,R2: nat,K: nat] :
% 9.04/5.05        ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 9.04/5.05        = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6076_binomial__le__pow,axiom,
% 9.04/5.05      ! [R2: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_eq_nat @ R2 @ N )
% 9.04/5.05       => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6077_zero__less__binomial,axiom,
% 9.04/5.05      ! [K: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.05       => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6078_Suc__times__binomial__add,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 9.04/5.05        = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6079_choose__mult,axiom,
% 9.04/5.05      ! [K: nat,M: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_eq_nat @ K @ M )
% 9.04/5.05       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.05         => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 9.04/5.05            = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6080_binomial__Suc__Suc__eq__times,axiom,
% 9.04/5.05      ! [N: nat,K: nat] :
% 9.04/5.05        ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 9.04/5.05        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6081_binomial__absorb__comp,axiom,
% 9.04/5.05      ! [N: nat,K: nat] :
% 9.04/5.05        ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 9.04/5.05        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6082_binomial__absorption,axiom,
% 9.04/5.05      ! [K: nat,N: nat] :
% 9.04/5.05        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 9.04/5.05        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6083_binomial__ge__n__over__k__pow__k,axiom,
% 9.04/5.05      ! [K: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.05       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6084_binomial__ge__n__over__k__pow__k,axiom,
% 9.04/5.05      ! [K: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.05       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6085_binomial__le__pow2,axiom,
% 9.04/5.05      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6086_choose__reduce__nat,axiom,
% 9.04/5.05      ! [N: nat,K: nat] :
% 9.04/5.05        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.04/5.05         => ( ( binomial @ N @ K )
% 9.04/5.05            = ( 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 ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6087_times__binomial__minus1__eq,axiom,
% 9.04/5.05      ! [K: nat,N: nat] :
% 9.04/5.05        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.04/5.05       => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 9.04/5.05          = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6088_binomial__addition__formula,axiom,
% 9.04/5.05      ! [N: nat,K: nat] :
% 9.04/5.05        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05       => ( ( binomial @ N @ ( suc @ K ) )
% 9.04/5.05          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6089_signed__take__bit__numeral__minus__bit1,axiom,
% 9.04/5.05      ! [L: num,K: num] :
% 9.04/5.05        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.04/5.05        = ( 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 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6090_dbl__dec__simps_I4_J,axiom,
% 9.04/5.05      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.04/5.05      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6091_dbl__dec__simps_I4_J,axiom,
% 9.04/5.05      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.05      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6092_dbl__dec__simps_I4_J,axiom,
% 9.04/5.05      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.04/5.05      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6093_dbl__dec__simps_I4_J,axiom,
% 9.04/5.05      ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.04/5.05      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6094_dbl__dec__simps_I4_J,axiom,
% 9.04/5.05      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.05      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6095_signed__take__bit__numeral__bit1,axiom,
% 9.04/5.05      ! [L: num,K: num] :
% 9.04/5.05        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 9.04/5.05        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6096_dbl__inc__simps_I3_J,axiom,
% 9.04/5.05      ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 9.04/5.05      = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6097_dbl__inc__simps_I3_J,axiom,
% 9.04/5.05      ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 9.04/5.05      = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6098_dbl__inc__simps_I3_J,axiom,
% 9.04/5.05      ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 9.04/5.05      = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6099_dbl__inc__simps_I3_J,axiom,
% 9.04/5.05      ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 9.04/5.05      = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6100_translation__Compl,axiom,
% 9.04/5.05      ! [A: real,T: set_real] :
% 9.04/5.05        ( ( image_real_real @ ( plus_plus_real @ A ) @ ( uminus612125837232591019t_real @ T ) )
% 9.04/5.05        = ( uminus612125837232591019t_real @ ( image_real_real @ ( plus_plus_real @ A ) @ T ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6101_translation__Compl,axiom,
% 9.04/5.05      ! [A: rat,T: set_rat] :
% 9.04/5.05        ( ( image_rat_rat @ ( plus_plus_rat @ A ) @ ( uminus2201863774496077783et_rat @ T ) )
% 9.04/5.05        = ( uminus2201863774496077783et_rat @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ T ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6102_translation__Compl,axiom,
% 9.04/5.05      ! [A: int,T: set_int] :
% 9.04/5.05        ( ( image_int_int @ ( plus_plus_int @ A ) @ ( uminus1532241313380277803et_int @ T ) )
% 9.04/5.05        = ( uminus1532241313380277803et_int @ ( image_int_int @ ( plus_plus_int @ A ) @ T ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6103_translation__Compl,axiom,
% 9.04/5.05      ! [A: complex,T: set_complex] :
% 9.04/5.05        ( ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ ( uminus8566677241136511917omplex @ T ) )
% 9.04/5.05        = ( uminus8566677241136511917omplex @ ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ T ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6104_concat__bit__Suc,axiom,
% 9.04/5.05      ! [N: nat,K: int,L: int] :
% 9.04/5.05        ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 9.04/5.05        = ( 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 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6105_concat__bit__0,axiom,
% 9.04/5.05      ! [K: int,L: int] :
% 9.04/5.05        ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 9.04/5.05        = L ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6106_dbl__dec__simps_I3_J,axiom,
% 9.04/5.05      ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 9.04/5.05      = one_one_complex ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6107_dbl__dec__simps_I3_J,axiom,
% 9.04/5.05      ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 9.04/5.05      = one_one_real ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6108_dbl__dec__simps_I3_J,axiom,
% 9.04/5.05      ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 9.04/5.05      = one_one_rat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6109_dbl__dec__simps_I3_J,axiom,
% 9.04/5.05      ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 9.04/5.05      = one_one_int ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6110_pred__numeral__simps_I1_J,axiom,
% 9.04/5.05      ( ( pred_numeral @ one )
% 9.04/5.05      = zero_zero_nat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6111_Suc__eq__numeral,axiom,
% 9.04/5.05      ! [N: nat,K: num] :
% 9.04/5.05        ( ( ( suc @ N )
% 9.04/5.05          = ( numeral_numeral_nat @ K ) )
% 9.04/5.05        = ( N
% 9.04/5.05          = ( pred_numeral @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6112_eq__numeral__Suc,axiom,
% 9.04/5.05      ! [K: num,N: nat] :
% 9.04/5.05        ( ( ( numeral_numeral_nat @ K )
% 9.04/5.05          = ( suc @ N ) )
% 9.04/5.05        = ( ( pred_numeral @ K )
% 9.04/5.05          = N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6113_concat__bit__nonnegative__iff,axiom,
% 9.04/5.05      ! [N: nat,K: int,L: int] :
% 9.04/5.05        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 9.04/5.05        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6114_concat__bit__negative__iff,axiom,
% 9.04/5.05      ! [N: nat,K: int,L: int] :
% 9.04/5.05        ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 9.04/5.05        = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6115_dbl__inc__simps_I2_J,axiom,
% 9.04/5.05      ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 9.04/5.05      = one_one_complex ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6116_dbl__inc__simps_I2_J,axiom,
% 9.04/5.05      ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 9.04/5.05      = one_one_real ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6117_dbl__inc__simps_I2_J,axiom,
% 9.04/5.05      ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 9.04/5.05      = one_one_rat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6118_dbl__inc__simps_I2_J,axiom,
% 9.04/5.05      ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 9.04/5.05      = one_one_int ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6119_dbl__inc__simps_I4_J,axiom,
% 9.04/5.05      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.04/5.05      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6120_dbl__inc__simps_I4_J,axiom,
% 9.04/5.05      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.05      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6121_dbl__inc__simps_I4_J,axiom,
% 9.04/5.05      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.04/5.05      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6122_dbl__inc__simps_I4_J,axiom,
% 9.04/5.05      ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.04/5.05      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6123_dbl__inc__simps_I4_J,axiom,
% 9.04/5.05      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.05      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6124_dbl__inc__simps_I5_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 9.04/5.05        = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6125_dbl__inc__simps_I5_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 9.04/5.05        = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6126_dbl__inc__simps_I5_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 9.04/5.05        = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6127_dbl__inc__simps_I5_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 9.04/5.05        = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6128_pred__numeral__simps_I3_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( pred_numeral @ ( bit1 @ K ) )
% 9.04/5.05        = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6129_less__Suc__numeral,axiom,
% 9.04/5.05      ! [N: nat,K: num] :
% 9.04/5.05        ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.04/5.05        = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6130_less__numeral__Suc,axiom,
% 9.04/5.05      ! [K: num,N: nat] :
% 9.04/5.05        ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.04/5.05        = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6131_le__Suc__numeral,axiom,
% 9.04/5.05      ! [N: nat,K: num] :
% 9.04/5.05        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.04/5.05        = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6132_le__numeral__Suc,axiom,
% 9.04/5.05      ! [K: num,N: nat] :
% 9.04/5.05        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.04/5.05        = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6133_diff__Suc__numeral,axiom,
% 9.04/5.05      ! [N: nat,K: num] :
% 9.04/5.05        ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.04/5.05        = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6134_diff__numeral__Suc,axiom,
% 9.04/5.05      ! [K: num,N: nat] :
% 9.04/5.05        ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.04/5.05        = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6135_dbl__dec__simps_I2_J,axiom,
% 9.04/5.05      ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 9.04/5.05      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6136_dbl__dec__simps_I2_J,axiom,
% 9.04/5.05      ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 9.04/5.05      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6137_dbl__dec__simps_I2_J,axiom,
% 9.04/5.05      ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 9.04/5.05      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6138_dbl__dec__simps_I2_J,axiom,
% 9.04/5.05      ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 9.04/5.05      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6139_dbl__dec__simps_I2_J,axiom,
% 9.04/5.05      ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 9.04/5.05      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6140_dbl__dec__simps_I1_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 9.04/5.05        = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6141_dbl__dec__simps_I1_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.04/5.05        = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6142_dbl__dec__simps_I1_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 9.04/5.05        = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6143_dbl__dec__simps_I1_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 9.04/5.05        = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6144_dbl__dec__simps_I1_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 9.04/5.05        = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6145_dbl__inc__simps_I1_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 9.04/5.05        = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6146_dbl__inc__simps_I1_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.04/5.05        = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6147_dbl__inc__simps_I1_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 9.04/5.05        = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6148_dbl__inc__simps_I1_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 9.04/5.05        = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6149_dbl__inc__simps_I1_J,axiom,
% 9.04/5.05      ! [K: num] :
% 9.04/5.05        ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 9.04/5.05        = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6150_signed__take__bit__numeral__bit0,axiom,
% 9.04/5.05      ! [L: num,K: num] :
% 9.04/5.05        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 9.04/5.05        = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6151_signed__take__bit__numeral__minus__bit0,axiom,
% 9.04/5.05      ! [L: num,K: num] :
% 9.04/5.05        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.04/5.05        = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6152_concat__bit__assoc,axiom,
% 9.04/5.05      ! [N: nat,K: int,M: nat,L: int,R2: int] :
% 9.04/5.05        ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
% 9.04/5.05        = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6153_numeral__eq__Suc,axiom,
% 9.04/5.05      ( numeral_numeral_nat
% 9.04/5.05      = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6154_pred__numeral__def,axiom,
% 9.04/5.05      ( pred_numeral
% 9.04/5.05      = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6155_dbl__inc__def,axiom,
% 9.04/5.05      ( neg_nu8295874005876285629c_real
% 9.04/5.05      = ( ^ [X4: real] : ( plus_plus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6156_dbl__inc__def,axiom,
% 9.04/5.05      ( neg_nu5219082963157363817nc_rat
% 9.04/5.05      = ( ^ [X4: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6157_dbl__inc__def,axiom,
% 9.04/5.05      ( neg_nu5851722552734809277nc_int
% 9.04/5.05      = ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6158_dbl__inc__def,axiom,
% 9.04/5.05      ( neg_nu8557863876264182079omplex
% 9.04/5.05      = ( ^ [X4: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6159_dbl__dec__def,axiom,
% 9.04/5.05      ( neg_nu6511756317524482435omplex
% 9.04/5.05      = ( ^ [X4: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6160_dbl__dec__def,axiom,
% 9.04/5.05      ( neg_nu6075765906172075777c_real
% 9.04/5.05      = ( ^ [X4: real] : ( minus_minus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6161_dbl__dec__def,axiom,
% 9.04/5.05      ( neg_nu3179335615603231917ec_rat
% 9.04/5.05      = ( ^ [X4: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6162_dbl__dec__def,axiom,
% 9.04/5.05      ( neg_nu3811975205180677377ec_int
% 9.04/5.05      = ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6163_translation__diff,axiom,
% 9.04/5.05      ! [A: real,S: set_real,T: set_real] :
% 9.04/5.05        ( ( image_real_real @ ( plus_plus_real @ A ) @ ( minus_minus_set_real @ S @ T ) )
% 9.04/5.05        = ( minus_minus_set_real @ ( image_real_real @ ( plus_plus_real @ A ) @ S ) @ ( image_real_real @ ( plus_plus_real @ A ) @ T ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6164_translation__diff,axiom,
% 9.04/5.05      ! [A: rat,S: set_rat,T: set_rat] :
% 9.04/5.05        ( ( image_rat_rat @ ( plus_plus_rat @ A ) @ ( minus_minus_set_rat @ S @ T ) )
% 9.04/5.05        = ( minus_minus_set_rat @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ S ) @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ T ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6165_translation__diff,axiom,
% 9.04/5.05      ! [A: int,S: set_int,T: set_int] :
% 9.04/5.05        ( ( image_int_int @ ( plus_plus_int @ A ) @ ( minus_minus_set_int @ S @ T ) )
% 9.04/5.05        = ( minus_minus_set_int @ ( image_int_int @ ( plus_plus_int @ A ) @ S ) @ ( image_int_int @ ( plus_plus_int @ A ) @ T ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6166_translation__diff,axiom,
% 9.04/5.05      ! [A: complex,S: set_complex,T: set_complex] :
% 9.04/5.05        ( ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ ( minus_811609699411566653omplex @ S @ T ) )
% 9.04/5.05        = ( minus_811609699411566653omplex @ ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ S ) @ ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ T ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6167_even__succ__mod__exp,axiom,
% 9.04/5.05      ! [A: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.05            = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6168_even__succ__mod__exp,axiom,
% 9.04/5.05      ! [A: int,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.05            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6169_even__succ__mod__exp,axiom,
% 9.04/5.05      ! [A: code_integer,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.05            = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6170_even__succ__div__exp,axiom,
% 9.04/5.05      ! [A: code_integer,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.05            = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6171_even__succ__div__exp,axiom,
% 9.04/5.05      ! [A: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05         => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.05            = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6172_even__succ__div__exp,axiom,
% 9.04/5.05      ! [A: int,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05         => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.05            = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6173_log__base__10__eq1,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.05       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.05          = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6174_add__scale__eq__noteq,axiom,
% 9.04/5.05      ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
% 9.04/5.05        ( ( R2 != zero_zero_complex )
% 9.04/5.05       => ( ( ( A = B )
% 9.04/5.05            & ( C != D ) )
% 9.04/5.05         => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
% 9.04/5.05           != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6175_add__scale__eq__noteq,axiom,
% 9.04/5.05      ! [R2: real,A: real,B: real,C: real,D: real] :
% 9.04/5.05        ( ( R2 != zero_zero_real )
% 9.04/5.05       => ( ( ( A = B )
% 9.04/5.05            & ( C != D ) )
% 9.04/5.05         => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 9.04/5.05           != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6176_add__scale__eq__noteq,axiom,
% 9.04/5.05      ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
% 9.04/5.05        ( ( R2 != zero_zero_rat )
% 9.04/5.05       => ( ( ( A = B )
% 9.04/5.05            & ( C != D ) )
% 9.04/5.05         => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 9.04/5.05           != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6177_add__scale__eq__noteq,axiom,
% 9.04/5.05      ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
% 9.04/5.05        ( ( R2 != zero_zero_nat )
% 9.04/5.05       => ( ( ( A = B )
% 9.04/5.05            & ( C != D ) )
% 9.04/5.05         => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 9.04/5.05           != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6178_add__scale__eq__noteq,axiom,
% 9.04/5.05      ! [R2: int,A: int,B: int,C: int,D: int] :
% 9.04/5.05        ( ( R2 != zero_zero_int )
% 9.04/5.05       => ( ( ( A = B )
% 9.04/5.05            & ( C != D ) )
% 9.04/5.05         => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 9.04/5.05           != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6179_nat__dvd__1__iff__1,axiom,
% 9.04/5.05      ! [M: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ M @ one_one_nat )
% 9.04/5.05        = ( M = one_one_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6180_int__dvd__int__iff,axiom,
% 9.04/5.05      ! [M: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.04/5.05        = ( dvd_dvd_nat @ M @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6181_exp__inj__iff,axiom,
% 9.04/5.05      ! [X: real,Y: real] :
% 9.04/5.05        ( ( ( exp_real @ X )
% 9.04/5.05          = ( exp_real @ Y ) )
% 9.04/5.05        = ( X = Y ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6182_dvd__0__right,axiom,
% 9.04/5.05      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6183_dvd__0__right,axiom,
% 9.04/5.05      ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6184_dvd__0__right,axiom,
% 9.04/5.05      ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6185_dvd__0__right,axiom,
% 9.04/5.05      ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6186_dvd__0__right,axiom,
% 9.04/5.05      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6187_dvd__0__right,axiom,
% 9.04/5.05      ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6188_dvd__0__left__iff,axiom,
% 9.04/5.05      ! [A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 9.04/5.05        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6189_dvd__0__left__iff,axiom,
% 9.04/5.05      ! [A: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 9.04/5.05        = ( A = zero_zero_complex ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6190_dvd__0__left__iff,axiom,
% 9.04/5.05      ! [A: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 9.04/5.05        = ( A = zero_zero_real ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6191_dvd__0__left__iff,axiom,
% 9.04/5.05      ! [A: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 9.04/5.05        = ( A = zero_zero_rat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6192_dvd__0__left__iff,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.04/5.05        = ( A = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6193_dvd__0__left__iff,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 9.04/5.05        = ( A = zero_zero_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6194_dvd__add__triv__right__iff,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 9.04/5.05        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6195_dvd__add__triv__right__iff,axiom,
% 9.04/5.05      ! [A: real,B: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 9.04/5.05        = ( dvd_dvd_real @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6196_dvd__add__triv__right__iff,axiom,
% 9.04/5.05      ! [A: rat,B: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 9.04/5.05        = ( dvd_dvd_rat @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6197_dvd__add__triv__right__iff,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 9.04/5.05        = ( dvd_dvd_nat @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6198_dvd__add__triv__right__iff,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 9.04/5.05        = ( dvd_dvd_int @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6199_dvd__add__triv__right__iff,axiom,
% 9.04/5.05      ! [A: complex,B: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ A ) )
% 9.04/5.05        = ( dvd_dvd_complex @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6200_dvd__add__triv__left__iff,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.04/5.05        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6201_dvd__add__triv__left__iff,axiom,
% 9.04/5.05      ! [A: real,B: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 9.04/5.05        = ( dvd_dvd_real @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6202_dvd__add__triv__left__iff,axiom,
% 9.04/5.05      ! [A: rat,B: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 9.04/5.05        = ( dvd_dvd_rat @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6203_dvd__add__triv__left__iff,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.04/5.05        = ( dvd_dvd_nat @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6204_dvd__add__triv__left__iff,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 9.04/5.05        = ( dvd_dvd_int @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6205_dvd__add__triv__left__iff,axiom,
% 9.04/5.05      ! [A: complex,B: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 9.04/5.05        = ( dvd_dvd_complex @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6206_dvd__1__iff__1,axiom,
% 9.04/5.05      ! [M: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.04/5.05        = ( M
% 9.04/5.05          = ( suc @ zero_zero_nat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6207_dvd__1__left,axiom,
% 9.04/5.05      ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6208_div__dvd__div,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ A @ C )
% 9.04/5.05         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 9.04/5.05            = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6209_div__dvd__div,axiom,
% 9.04/5.05      ! [A: nat,B: nat,C: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_nat @ A @ C )
% 9.04/5.05         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 9.04/5.05            = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6210_div__dvd__div,axiom,
% 9.04/5.05      ! [A: int,B: int,C: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_int @ A @ C )
% 9.04/5.05         => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 9.04/5.05            = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6211_nat__mult__dvd__cancel__disj,axiom,
% 9.04/5.05      ! [K: nat,M: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.04/5.05        = ( ( K = zero_zero_nat )
% 9.04/5.05          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6212_zdvd1__eq,axiom,
% 9.04/5.05      ! [X: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ X @ one_one_int )
% 9.04/5.05        = ( ( abs_abs_int @ X )
% 9.04/5.05          = one_one_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6213_exp__less__cancel__iff,axiom,
% 9.04/5.05      ! [X: real,Y: real] :
% 9.04/5.05        ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 9.04/5.05        = ( ord_less_real @ X @ Y ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6214_exp__less__mono,axiom,
% 9.04/5.05      ! [X: real,Y: real] :
% 9.04/5.05        ( ( ord_less_real @ X @ Y )
% 9.04/5.05       => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6215_exp__le__cancel__iff,axiom,
% 9.04/5.05      ! [X: real,Y: real] :
% 9.04/5.05        ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 9.04/5.05        = ( ord_less_eq_real @ X @ Y ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6216_abs__exp__cancel,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( abs_abs_real @ ( exp_real @ X ) )
% 9.04/5.05        = ( exp_real @ X ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6217_ln__exp,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( ln_ln_real @ ( exp_real @ X ) )
% 9.04/5.05        = X ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6218_dvd__times__right__cancel__iff,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.05        ( ( A != zero_z3403309356797280102nteger )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 9.04/5.05          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6219_dvd__times__right__cancel__iff,axiom,
% 9.04/5.05      ! [A: nat,B: nat,C: nat] :
% 9.04/5.05        ( ( A != zero_zero_nat )
% 9.04/5.05       => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 9.04/5.05          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6220_dvd__times__right__cancel__iff,axiom,
% 9.04/5.05      ! [A: int,B: int,C: int] :
% 9.04/5.05        ( ( A != zero_zero_int )
% 9.04/5.05       => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 9.04/5.05          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6221_dvd__times__left__cancel__iff,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.05        ( ( A != zero_z3403309356797280102nteger )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 9.04/5.05          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6222_dvd__times__left__cancel__iff,axiom,
% 9.04/5.05      ! [A: nat,B: nat,C: nat] :
% 9.04/5.05        ( ( A != zero_zero_nat )
% 9.04/5.05       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 9.04/5.05          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6223_dvd__times__left__cancel__iff,axiom,
% 9.04/5.05      ! [A: int,B: int,C: int] :
% 9.04/5.05        ( ( A != zero_zero_int )
% 9.04/5.05       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 9.04/5.05          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6224_dvd__mult__cancel__right,axiom,
% 9.04/5.05      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.04/5.05        = ( ( C = zero_z3403309356797280102nteger )
% 9.04/5.05          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6225_dvd__mult__cancel__right,axiom,
% 9.04/5.05      ! [A: complex,C: complex,B: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 9.04/5.05        = ( ( C = zero_zero_complex )
% 9.04/5.05          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6226_dvd__mult__cancel__right,axiom,
% 9.04/5.05      ! [A: real,C: real,B: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.04/5.05        = ( ( C = zero_zero_real )
% 9.04/5.05          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6227_dvd__mult__cancel__right,axiom,
% 9.04/5.05      ! [A: rat,C: rat,B: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.04/5.05        = ( ( C = zero_zero_rat )
% 9.04/5.05          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6228_dvd__mult__cancel__right,axiom,
% 9.04/5.05      ! [A: int,C: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.04/5.05        = ( ( C = zero_zero_int )
% 9.04/5.05          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6229_dvd__mult__cancel__left,axiom,
% 9.04/5.05      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.04/5.05        = ( ( C = zero_z3403309356797280102nteger )
% 9.04/5.05          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6230_dvd__mult__cancel__left,axiom,
% 9.04/5.05      ! [C: complex,A: complex,B: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.04/5.05        = ( ( C = zero_zero_complex )
% 9.04/5.05          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6231_dvd__mult__cancel__left,axiom,
% 9.04/5.05      ! [C: real,A: real,B: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.04/5.05        = ( ( C = zero_zero_real )
% 9.04/5.05          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6232_dvd__mult__cancel__left,axiom,
% 9.04/5.05      ! [C: rat,A: rat,B: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.04/5.05        = ( ( C = zero_zero_rat )
% 9.04/5.05          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6233_dvd__mult__cancel__left,axiom,
% 9.04/5.05      ! [C: int,A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.04/5.05        = ( ( C = zero_zero_int )
% 9.04/5.05          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6234_unit__prod,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.05         => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6235_unit__prod,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.05       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.05         => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6236_unit__prod,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.05       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.05         => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6237_dvd__add__times__triv__right__iff,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 9.04/5.05        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6238_dvd__add__times__triv__right__iff,axiom,
% 9.04/5.05      ! [A: complex,B: complex,C: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ ( times_times_complex @ C @ A ) ) )
% 9.04/5.05        = ( dvd_dvd_complex @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6239_dvd__add__times__triv__right__iff,axiom,
% 9.04/5.05      ! [A: real,B: real,C: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 9.04/5.05        = ( dvd_dvd_real @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6240_dvd__add__times__triv__right__iff,axiom,
% 9.04/5.05      ! [A: rat,B: rat,C: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 9.04/5.05        = ( dvd_dvd_rat @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6241_dvd__add__times__triv__right__iff,axiom,
% 9.04/5.05      ! [A: nat,B: nat,C: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 9.04/5.05        = ( dvd_dvd_nat @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6242_dvd__add__times__triv__right__iff,axiom,
% 9.04/5.05      ! [A: int,B: int,C: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 9.04/5.05        = ( dvd_dvd_int @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6243_dvd__add__times__triv__left__iff,axiom,
% 9.04/5.05      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 9.04/5.05        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6244_dvd__add__times__triv__left__iff,axiom,
% 9.04/5.05      ! [A: complex,C: complex,B: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ ( times_times_complex @ C @ A ) @ B ) )
% 9.04/5.05        = ( dvd_dvd_complex @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6245_dvd__add__times__triv__left__iff,axiom,
% 9.04/5.05      ! [A: real,C: real,B: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 9.04/5.05        = ( dvd_dvd_real @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6246_dvd__add__times__triv__left__iff,axiom,
% 9.04/5.05      ! [A: rat,C: rat,B: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 9.04/5.05        = ( dvd_dvd_rat @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6247_dvd__add__times__triv__left__iff,axiom,
% 9.04/5.05      ! [A: nat,C: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 9.04/5.05        = ( dvd_dvd_nat @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6248_dvd__add__times__triv__left__iff,axiom,
% 9.04/5.05      ! [A: int,C: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 9.04/5.05        = ( dvd_dvd_int @ A @ B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6249_dvd__mult__div__cancel,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.04/5.05       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 9.04/5.05          = B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6250_dvd__mult__div__cancel,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ B )
% 9.04/5.05       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 9.04/5.05          = B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6251_dvd__mult__div__cancel,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ B )
% 9.04/5.05       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 9.04/5.05          = B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6252_dvd__div__mult__self,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.04/5.05       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 9.04/5.05          = B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6253_dvd__div__mult__self,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ B )
% 9.04/5.05       => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 9.04/5.05          = B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6254_dvd__div__mult__self,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ B )
% 9.04/5.05       => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 9.04/5.05          = B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6255_unit__div__1__div__1,axiom,
% 9.04/5.05      ! [A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.05       => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 9.04/5.05          = A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6256_unit__div__1__div__1,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.05       => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 9.04/5.05          = A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6257_unit__div__1__div__1,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.05       => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 9.04/5.05          = A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6258_unit__div__1__unit,axiom,
% 9.04/5.05      ! [A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.05       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6259_unit__div__1__unit,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.05       => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6260_unit__div__1__unit,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.05       => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6261_unit__div,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.05         => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6262_unit__div,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.05       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.05         => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6263_unit__div,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.05       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.05         => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6264_div__add,axiom,
% 9.04/5.05      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.04/5.05         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 9.04/5.05            = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6265_div__add,axiom,
% 9.04/5.05      ! [C: nat,A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_nat @ C @ B )
% 9.04/5.05         => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.04/5.05            = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6266_div__add,axiom,
% 9.04/5.05      ! [C: int,A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_int @ C @ B )
% 9.04/5.05         => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.04/5.05            = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6267_div__diff,axiom,
% 9.04/5.05      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.04/5.05         => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 9.04/5.05            = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6268_div__diff,axiom,
% 9.04/5.05      ! [C: int,A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_int @ C @ B )
% 9.04/5.05         => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.04/5.05            = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6269_dvd__imp__mod__0,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ B )
% 9.04/5.05       => ( ( modulo_modulo_nat @ B @ A )
% 9.04/5.05          = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6270_dvd__imp__mod__0,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ B )
% 9.04/5.05       => ( ( modulo_modulo_int @ B @ A )
% 9.04/5.05          = zero_zero_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6271_dvd__imp__mod__0,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.04/5.05       => ( ( modulo364778990260209775nteger @ B @ A )
% 9.04/5.05          = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6272_exp__zero,axiom,
% 9.04/5.05      ( ( exp_complex @ zero_zero_complex )
% 9.04/5.05      = one_one_complex ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6273_exp__zero,axiom,
% 9.04/5.05      ( ( exp_real @ zero_zero_real )
% 9.04/5.05      = one_one_real ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6274_exp__eq__one__iff,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( ( exp_real @ X )
% 9.04/5.05          = one_one_real )
% 9.04/5.05        = ( X = zero_zero_real ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6275_unit__div__mult__self,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.05       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 9.04/5.05          = B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6276_unit__div__mult__self,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.05       => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 9.04/5.05          = B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6277_unit__div__mult__self,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.05       => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 9.04/5.05          = B ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6278_unit__mult__div__div,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.05       => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 9.04/5.05          = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6279_unit__mult__div__div,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.05       => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 9.04/5.05          = ( divide_divide_nat @ B @ A ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6280_unit__mult__div__div,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.05       => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 9.04/5.05          = ( divide_divide_int @ B @ A ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6281_even__Suc__Suc__iff,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 9.04/5.05        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6282_even__Suc,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 9.04/5.05        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6283_pow__divides__pow__iff,axiom,
% 9.04/5.05      ! [N: nat,A: nat,B: nat] :
% 9.04/5.05        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05       => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 9.04/5.05          = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6284_pow__divides__pow__iff,axiom,
% 9.04/5.05      ! [N: nat,A: int,B: int] :
% 9.04/5.05        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.05       => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 9.04/5.05          = ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6285_one__less__exp__iff,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 9.04/5.05        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6286_exp__less__one__iff,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 9.04/5.05        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6287_exp__le__one__iff,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
% 9.04/5.05        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6288_one__le__exp__iff,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
% 9.04/5.05        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6289_exp__ln__iff,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( ( exp_real @ ( ln_ln_real @ X ) )
% 9.04/5.05          = X )
% 9.04/5.05        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6290_exp__ln,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.05       => ( ( exp_real @ ( ln_ln_real @ X ) )
% 9.04/5.05          = X ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6291_even__mult__iff,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 9.04/5.05        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.05          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6292_even__mult__iff,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 9.04/5.05        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.05          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6293_even__mult__iff,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 9.04/5.05        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.05          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6294_even__add,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.04/5.05        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.05          = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6295_even__add,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 9.04/5.05        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.05          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6296_even__add,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 9.04/5.05        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.05          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6297_odd__add,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 9.04/5.05        = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.05         != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6298_odd__add,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 9.04/5.05        = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.05         != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6299_odd__add,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 9.04/5.05        = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.05         != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6300_power__minus__odd,axiom,
% 9.04/5.05      ! [N: nat,A: real] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 9.04/5.05          = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6301_power__minus__odd,axiom,
% 9.04/5.05      ! [N: nat,A: int] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 9.04/5.05          = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6302_power__minus__odd,axiom,
% 9.04/5.05      ! [N: nat,A: complex] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 9.04/5.05          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6303_power__minus__odd,axiom,
% 9.04/5.05      ! [N: nat,A: rat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 9.04/5.05          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6304_power__minus__odd,axiom,
% 9.04/5.05      ! [N: nat,A: code_integer] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 9.04/5.05          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6305_Parity_Oring__1__class_Opower__minus__even,axiom,
% 9.04/5.05      ! [N: nat,A: real] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 9.04/5.05          = ( power_power_real @ A @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6306_Parity_Oring__1__class_Opower__minus__even,axiom,
% 9.04/5.05      ! [N: nat,A: int] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 9.04/5.05          = ( power_power_int @ A @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6307_Parity_Oring__1__class_Opower__minus__even,axiom,
% 9.04/5.05      ! [N: nat,A: complex] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 9.04/5.05          = ( power_power_complex @ A @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6308_Parity_Oring__1__class_Opower__minus__even,axiom,
% 9.04/5.05      ! [N: nat,A: rat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 9.04/5.05          = ( power_power_rat @ A @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6309_Parity_Oring__1__class_Opower__minus__even,axiom,
% 9.04/5.05      ! [N: nat,A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 9.04/5.05          = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6310_even__mod__2__iff,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.05        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6311_even__mod__2__iff,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.04/5.05        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6312_even__mod__2__iff,axiom,
% 9.04/5.05      ! [A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 9.04/5.05        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6313_power__even__abs__numeral,axiom,
% 9.04/5.05      ! [W: num,A: rat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05       => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6314_power__even__abs__numeral,axiom,
% 9.04/5.05      ! [W: num,A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05       => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6315_power__even__abs__numeral,axiom,
% 9.04/5.05      ! [W: num,A: real] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05       => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6316_power__even__abs__numeral,axiom,
% 9.04/5.05      ! [W: num,A: int] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05       => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6317_odd__Suc__div__two,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6318_even__Suc__div__two,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6319_zero__le__power__eq__numeral,axiom,
% 9.04/5.05      ! [A: real,W: num] :
% 9.04/5.05        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.04/5.05        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6320_zero__le__power__eq__numeral,axiom,
% 9.04/5.05      ! [A: rat,W: num] :
% 9.04/5.05        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.04/5.05        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6321_zero__le__power__eq__numeral,axiom,
% 9.04/5.05      ! [A: int,W: num] :
% 9.04/5.05        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.04/5.05        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6322_power__less__zero__eq__numeral,axiom,
% 9.04/5.05      ! [A: real,W: num] :
% 9.04/5.05        ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 9.04/5.05        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6323_power__less__zero__eq__numeral,axiom,
% 9.04/5.05      ! [A: rat,W: num] :
% 9.04/5.05        ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 9.04/5.05        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6324_power__less__zero__eq__numeral,axiom,
% 9.04/5.05      ! [A: int,W: num] :
% 9.04/5.05        ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 9.04/5.05        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6325_power__less__zero__eq,axiom,
% 9.04/5.05      ! [A: real,N: nat] :
% 9.04/5.05        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 9.04/5.05        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6326_power__less__zero__eq,axiom,
% 9.04/5.05      ! [A: rat,N: nat] :
% 9.04/5.05        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 9.04/5.05        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6327_power__less__zero__eq,axiom,
% 9.04/5.05      ! [A: int,N: nat] :
% 9.04/5.05        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 9.04/5.05        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6328_even__plus__one__iff,axiom,
% 9.04/5.05      ! [A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 9.04/5.05        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6329_even__plus__one__iff,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 9.04/5.05        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6330_even__plus__one__iff,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 9.04/5.05        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6331_even__diff,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 9.04/5.05        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6332_even__diff,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 9.04/5.05        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6333_neg__one__even__power,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 9.04/5.05          = one_one_real ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6334_neg__one__even__power,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 9.04/5.05          = one_one_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6335_neg__one__even__power,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 9.04/5.05          = one_one_complex ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6336_neg__one__even__power,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 9.04/5.05          = one_one_rat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6337_neg__one__even__power,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 9.04/5.05          = one_one_Code_integer ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6338_neg__one__odd__power,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 9.04/5.05          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6339_neg__one__odd__power,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 9.04/5.05          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6340_neg__one__odd__power,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 9.04/5.05          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6341_neg__one__odd__power,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 9.04/5.05          = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6342_neg__one__odd__power,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 9.04/5.05          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6343_even__of__nat,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 9.04/5.05        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6344_even__of__nat,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.04/5.05        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6345_even__of__nat,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.04/5.05        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6346_odd__Suc__minus__one,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 9.04/5.05          = N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6347_even__diff__nat,axiom,
% 9.04/5.05      ! [M: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 9.04/5.05        = ( ( ord_less_nat @ M @ N )
% 9.04/5.05          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6348_zero__less__power__eq__numeral,axiom,
% 9.04/5.05      ! [A: real,W: num] :
% 9.04/5.05        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.04/5.05        = ( ( ( numeral_numeral_nat @ W )
% 9.04/5.05            = zero_zero_nat )
% 9.04/5.05          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05            & ( A != zero_zero_real ) )
% 9.04/5.05          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6349_zero__less__power__eq__numeral,axiom,
% 9.04/5.05      ! [A: rat,W: num] :
% 9.04/5.05        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.04/5.05        = ( ( ( numeral_numeral_nat @ W )
% 9.04/5.05            = zero_zero_nat )
% 9.04/5.05          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05            & ( A != zero_zero_rat ) )
% 9.04/5.05          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6350_zero__less__power__eq__numeral,axiom,
% 9.04/5.05      ! [A: int,W: num] :
% 9.04/5.05        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.04/5.05        = ( ( ( numeral_numeral_nat @ W )
% 9.04/5.05            = zero_zero_nat )
% 9.04/5.05          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05            & ( A != zero_zero_int ) )
% 9.04/5.05          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6351_even__succ__div__two,axiom,
% 9.04/5.05      ! [A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6352_even__succ__div__two,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6353_even__succ__div__two,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6354_odd__succ__div__two,axiom,
% 9.04/5.05      ! [A: code_integer] :
% 9.04/5.05        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6355_odd__succ__div__two,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6356_odd__succ__div__two,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6357_even__succ__div__2,axiom,
% 9.04/5.05      ! [A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6358_even__succ__div__2,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6359_even__succ__div__2,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.05          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6360_even__power,axiom,
% 9.04/5.05      ! [A: code_integer,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 9.04/5.05        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.05          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6361_even__power,axiom,
% 9.04/5.05      ! [A: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 9.04/5.05        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.05          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6362_even__power,axiom,
% 9.04/5.05      ! [A: int,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 9.04/5.05        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.05          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6363_odd__two__times__div__two__nat,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.05       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.05          = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6364_odd__two__times__div__two__succ,axiom,
% 9.04/5.05      ! [A: code_integer] :
% 9.04/5.05        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 9.04/5.05          = A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6365_odd__two__times__div__two__succ,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 9.04/5.05          = A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6366_odd__two__times__div__two__succ,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.05       => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 9.04/5.05          = A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6367_power__le__zero__eq__numeral,axiom,
% 9.04/5.05      ! [A: real,W: num] :
% 9.04/5.05        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 9.04/5.05        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.04/5.05            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05              & ( A = zero_zero_real ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6368_power__le__zero__eq__numeral,axiom,
% 9.04/5.05      ! [A: rat,W: num] :
% 9.04/5.05        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 9.04/5.05        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.04/5.05            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05              & ( A = zero_zero_rat ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6369_power__le__zero__eq__numeral,axiom,
% 9.04/5.05      ! [A: int,W: num] :
% 9.04/5.05        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 9.04/5.05        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.04/5.05            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.04/5.05              & ( A = zero_zero_int ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6370_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 9.04/5.05        = ( N = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6371_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 9.04/5.05        = ( N = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6372_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.04/5.05      ! [N: nat] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 9.04/5.05        = ( N = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6373_of__real__exp,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( real_V1803761363581548252l_real @ ( exp_real @ X ) )
% 9.04/5.05        = ( exp_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6374_of__real__exp,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( real_V4546457046886955230omplex @ ( exp_real @ X ) )
% 9.04/5.05        = ( exp_complex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6375_of__nat__dvd__iff,axiom,
% 9.04/5.05      ! [M: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 9.04/5.05        = ( dvd_dvd_nat @ M @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6376_of__nat__dvd__iff,axiom,
% 9.04/5.05      ! [M: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.04/5.05        = ( dvd_dvd_nat @ M @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6377_of__nat__dvd__iff,axiom,
% 9.04/5.05      ! [M: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.04/5.05        = ( dvd_dvd_nat @ M @ N ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6378_norm__exp,axiom,
% 9.04/5.05      ! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6379_norm__exp,axiom,
% 9.04/5.05      ! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6380_exp__less__cancel,axiom,
% 9.04/5.05      ! [X: real,Y: real] :
% 9.04/5.05        ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 9.04/5.05       => ( ord_less_real @ X @ Y ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6381_dvd__0__left,axiom,
% 9.04/5.05      ! [A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 9.04/5.05       => ( A = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6382_dvd__0__left,axiom,
% 9.04/5.05      ! [A: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 9.04/5.05       => ( A = zero_zero_complex ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6383_dvd__0__left,axiom,
% 9.04/5.05      ! [A: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 9.04/5.05       => ( A = zero_zero_real ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6384_dvd__0__left,axiom,
% 9.04/5.05      ! [A: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 9.04/5.05       => ( A = zero_zero_rat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6385_dvd__0__left,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.04/5.05       => ( A = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6386_dvd__0__left,axiom,
% 9.04/5.05      ! [A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 9.04/5.05       => ( A = zero_zero_int ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6387_dvd__field__iff,axiom,
% 9.04/5.05      ( dvd_dvd_complex
% 9.04/5.05      = ( ^ [A3: complex,B4: complex] :
% 9.04/5.05            ( ( A3 = zero_zero_complex )
% 9.04/5.05           => ( B4 = zero_zero_complex ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6388_dvd__field__iff,axiom,
% 9.04/5.05      ( dvd_dvd_real
% 9.04/5.05      = ( ^ [A3: real,B4: real] :
% 9.04/5.05            ( ( A3 = zero_zero_real )
% 9.04/5.05           => ( B4 = zero_zero_real ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6389_dvd__field__iff,axiom,
% 9.04/5.05      ( dvd_dvd_rat
% 9.04/5.05      = ( ^ [A3: rat,B4: rat] :
% 9.04/5.05            ( ( A3 = zero_zero_rat )
% 9.04/5.05           => ( B4 = zero_zero_rat ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6390_exp__not__eq__zero,axiom,
% 9.04/5.05      ! [X: complex] :
% 9.04/5.05        ( ( exp_complex @ X )
% 9.04/5.05       != zero_zero_complex ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6391_exp__not__eq__zero,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ( ( exp_real @ X )
% 9.04/5.05       != zero_zero_real ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6392_exp__times__arg__commute,axiom,
% 9.04/5.05      ! [A2: complex] :
% 9.04/5.05        ( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
% 9.04/5.05        = ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6393_exp__times__arg__commute,axiom,
% 9.04/5.05      ! [A2: real] :
% 9.04/5.05        ( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
% 9.04/5.05        = ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6394_dvd__unit__imp__unit,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.05         => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6395_dvd__unit__imp__unit,axiom,
% 9.04/5.05      ! [A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.05         => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6396_dvd__unit__imp__unit,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.05         => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6397_unit__imp__dvd,axiom,
% 9.04/5.05      ! [B: code_integer,A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.05       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6398_unit__imp__dvd,axiom,
% 9.04/5.05      ! [B: nat,A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.05       => ( dvd_dvd_nat @ B @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6399_unit__imp__dvd,axiom,
% 9.04/5.05      ! [B: int,A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.05       => ( dvd_dvd_int @ B @ A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6400_one__dvd,axiom,
% 9.04/5.05      ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6401_one__dvd,axiom,
% 9.04/5.05      ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6402_one__dvd,axiom,
% 9.04/5.05      ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6403_one__dvd,axiom,
% 9.04/5.05      ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6404_one__dvd,axiom,
% 9.04/5.05      ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6405_one__dvd,axiom,
% 9.04/5.05      ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6406_dvd__add__right__iff,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6407_dvd__add__right__iff,axiom,
% 9.04/5.05      ! [A: real,B: real,C: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_real @ A @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6408_dvd__add__right__iff,axiom,
% 9.04/5.05      ! [A: rat,B: rat,C: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6409_dvd__add__right__iff,axiom,
% 9.04/5.05      ! [A: nat,B: nat,C: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6410_dvd__add__right__iff,axiom,
% 9.04/5.05      ! [A: int,B: int,C: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6411_dvd__add__right__iff,axiom,
% 9.04/5.05      ! [A: complex,B: complex,C: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_complex @ A @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6412_dvd__add__left__iff,axiom,
% 9.04/5.05      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ C )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6413_dvd__add__left__iff,axiom,
% 9.04/5.05      ! [A: real,C: real,B: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ A @ C )
% 9.04/5.05       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_real @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6414_dvd__add__left__iff,axiom,
% 9.04/5.05      ! [A: rat,C: rat,B: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ A @ C )
% 9.04/5.05       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6415_dvd__add__left__iff,axiom,
% 9.04/5.05      ! [A: nat,C: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ C )
% 9.04/5.05       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6416_dvd__add__left__iff,axiom,
% 9.04/5.05      ! [A: int,C: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ C )
% 9.04/5.05       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6417_dvd__add__left__iff,axiom,
% 9.04/5.05      ! [A: complex,C: complex,B: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ A @ C )
% 9.04/5.05       => ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 9.04/5.05          = ( dvd_dvd_complex @ A @ B ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6418_dvd__add,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ A @ C )
% 9.04/5.05         => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6419_dvd__add,axiom,
% 9.04/5.05      ! [A: real,B: real,C: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_real @ A @ C )
% 9.04/5.05         => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6420_dvd__add,axiom,
% 9.04/5.05      ! [A: rat,B: rat,C: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_rat @ A @ C )
% 9.04/5.05         => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6421_dvd__add,axiom,
% 9.04/5.05      ! [A: nat,B: nat,C: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_nat @ A @ C )
% 9.04/5.05         => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6422_dvd__add,axiom,
% 9.04/5.05      ! [A: int,B: int,C: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_int @ A @ C )
% 9.04/5.05         => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6423_dvd__add,axiom,
% 9.04/5.05      ! [A: complex,B: complex,C: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_complex @ A @ C )
% 9.04/5.05         => ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6424_dvd__div__eq__iff,axiom,
% 9.04/5.05      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.04/5.05         => ( ( ( divide6298287555418463151nteger @ A @ C )
% 9.04/5.05              = ( divide6298287555418463151nteger @ B @ C ) )
% 9.04/5.05            = ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6425_dvd__div__eq__iff,axiom,
% 9.04/5.05      ! [C: complex,A: complex,B: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_complex @ C @ B )
% 9.04/5.05         => ( ( ( divide1717551699836669952omplex @ A @ C )
% 9.04/5.05              = ( divide1717551699836669952omplex @ B @ C ) )
% 9.04/5.05            = ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6426_dvd__div__eq__iff,axiom,
% 9.04/5.05      ! [C: real,A: real,B: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_real @ C @ B )
% 9.04/5.05         => ( ( ( divide_divide_real @ A @ C )
% 9.04/5.05              = ( divide_divide_real @ B @ C ) )
% 9.04/5.05            = ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6427_dvd__div__eq__iff,axiom,
% 9.04/5.05      ! [C: rat,A: rat,B: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_rat @ C @ B )
% 9.04/5.05         => ( ( ( divide_divide_rat @ A @ C )
% 9.04/5.05              = ( divide_divide_rat @ B @ C ) )
% 9.04/5.05            = ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6428_dvd__div__eq__iff,axiom,
% 9.04/5.05      ! [C: nat,A: nat,B: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_nat @ C @ B )
% 9.04/5.05         => ( ( ( divide_divide_nat @ A @ C )
% 9.04/5.05              = ( divide_divide_nat @ B @ C ) )
% 9.04/5.05            = ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6429_dvd__div__eq__iff,axiom,
% 9.04/5.05      ! [C: int,A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ C @ A )
% 9.04/5.05       => ( ( dvd_dvd_int @ C @ B )
% 9.04/5.05         => ( ( ( divide_divide_int @ A @ C )
% 9.04/5.05              = ( divide_divide_int @ B @ C ) )
% 9.04/5.05            = ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6430_dvd__div__eq__cancel,axiom,
% 9.04/5.05      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.04/5.05        ( ( ( divide6298287555418463151nteger @ A @ C )
% 9.04/5.05          = ( divide6298287555418463151nteger @ B @ C ) )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ C @ A )
% 9.04/5.05         => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.04/5.05           => ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6431_dvd__div__eq__cancel,axiom,
% 9.04/5.05      ! [A: complex,C: complex,B: complex] :
% 9.04/5.05        ( ( ( divide1717551699836669952omplex @ A @ C )
% 9.04/5.05          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.04/5.05       => ( ( dvd_dvd_complex @ C @ A )
% 9.04/5.05         => ( ( dvd_dvd_complex @ C @ B )
% 9.04/5.05           => ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6432_dvd__div__eq__cancel,axiom,
% 9.04/5.05      ! [A: real,C: real,B: real] :
% 9.04/5.05        ( ( ( divide_divide_real @ A @ C )
% 9.04/5.05          = ( divide_divide_real @ B @ C ) )
% 9.04/5.05       => ( ( dvd_dvd_real @ C @ A )
% 9.04/5.05         => ( ( dvd_dvd_real @ C @ B )
% 9.04/5.05           => ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6433_dvd__div__eq__cancel,axiom,
% 9.04/5.05      ! [A: rat,C: rat,B: rat] :
% 9.04/5.05        ( ( ( divide_divide_rat @ A @ C )
% 9.04/5.05          = ( divide_divide_rat @ B @ C ) )
% 9.04/5.05       => ( ( dvd_dvd_rat @ C @ A )
% 9.04/5.05         => ( ( dvd_dvd_rat @ C @ B )
% 9.04/5.05           => ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6434_dvd__div__eq__cancel,axiom,
% 9.04/5.05      ! [A: nat,C: nat,B: nat] :
% 9.04/5.05        ( ( ( divide_divide_nat @ A @ C )
% 9.04/5.05          = ( divide_divide_nat @ B @ C ) )
% 9.04/5.05       => ( ( dvd_dvd_nat @ C @ A )
% 9.04/5.05         => ( ( dvd_dvd_nat @ C @ B )
% 9.04/5.05           => ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6435_dvd__div__eq__cancel,axiom,
% 9.04/5.05      ! [A: int,C: int,B: int] :
% 9.04/5.05        ( ( ( divide_divide_int @ A @ C )
% 9.04/5.05          = ( divide_divide_int @ B @ C ) )
% 9.04/5.05       => ( ( dvd_dvd_int @ C @ A )
% 9.04/5.05         => ( ( dvd_dvd_int @ C @ B )
% 9.04/5.05           => ( A = B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6436_div__div__div__same,axiom,
% 9.04/5.05      ! [D: code_integer,B: code_integer,A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ D @ B )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ B @ A )
% 9.04/5.05         => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 9.04/5.05            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6437_div__div__div__same,axiom,
% 9.04/5.05      ! [D: nat,B: nat,A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ D @ B )
% 9.04/5.05       => ( ( dvd_dvd_nat @ B @ A )
% 9.04/5.05         => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 9.04/5.05            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6438_div__div__div__same,axiom,
% 9.04/5.05      ! [D: int,B: int,A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ D @ B )
% 9.04/5.05       => ( ( dvd_dvd_int @ B @ A )
% 9.04/5.05         => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 9.04/5.05            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6439_dvd__power__same,axiom,
% 9.04/5.05      ! [X: code_integer,Y: code_integer,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ X @ Y )
% 9.04/5.05       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6440_dvd__power__same,axiom,
% 9.04/5.05      ! [X: nat,Y: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ X @ Y )
% 9.04/5.05       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6441_dvd__power__same,axiom,
% 9.04/5.05      ! [X: real,Y: real,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_real @ X @ Y )
% 9.04/5.05       => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6442_dvd__power__same,axiom,
% 9.04/5.05      ! [X: int,Y: int,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_int @ X @ Y )
% 9.04/5.05       => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6443_dvd__power__same,axiom,
% 9.04/5.05      ! [X: complex,Y: complex,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_complex @ X @ Y )
% 9.04/5.05       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6444_gcd__nat_Oextremum__uniqueI,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.04/5.05       => ( A = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6445_gcd__nat_Onot__eq__extremum,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( A != zero_zero_nat )
% 9.04/5.05        = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
% 9.04/5.05          & ( A != zero_zero_nat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6446_gcd__nat_Oextremum__unique,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.04/5.05        = ( A = zero_zero_nat ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6447_gcd__nat_Oextremum__strict,axiom,
% 9.04/5.05      ! [A: nat] :
% 9.04/5.05        ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.04/5.05          & ( zero_zero_nat != A ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6448_gcd__nat_Oextremum,axiom,
% 9.04/5.05      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6449_ln__unique,axiom,
% 9.04/5.05      ! [Y: real,X: real] :
% 9.04/5.05        ( ( ( exp_real @ Y )
% 9.04/5.05          = X )
% 9.04/5.05       => ( ( ln_ln_real @ X )
% 9.04/5.05          = Y ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6450_dvd__diff__nat,axiom,
% 9.04/5.05      ! [K: nat,M: nat,N: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ K @ M )
% 9.04/5.05       => ( ( dvd_dvd_nat @ K @ N )
% 9.04/5.05         => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6451_zdvd__zdiffD,axiom,
% 9.04/5.05      ! [K: int,M: int,N: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 9.04/5.05       => ( ( dvd_dvd_int @ K @ N )
% 9.04/5.05         => ( dvd_dvd_int @ K @ M ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6452_zdvd__antisym__abs,axiom,
% 9.04/5.05      ! [A: int,B: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ B )
% 9.04/5.05       => ( ( dvd_dvd_int @ B @ A )
% 9.04/5.05         => ( ( abs_abs_int @ A )
% 9.04/5.05            = ( abs_abs_int @ B ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6453_not__exp__less__zero,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6454_exp__gt__zero,axiom,
% 9.04/5.05      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6455_exp__total,axiom,
% 9.04/5.05      ! [Y: real] :
% 9.04/5.05        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.04/5.05       => ? [X3: real] :
% 9.04/5.05            ( ( exp_real @ X3 )
% 9.04/5.05            = Y ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6456_not__exp__le__zero,axiom,
% 9.04/5.05      ! [X: real] :
% 9.04/5.05        ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6457_exp__ge__zero,axiom,
% 9.04/5.05      ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6458_not__is__unit__0,axiom,
% 9.04/5.05      ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6459_not__is__unit__0,axiom,
% 9.04/5.05      ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6460_not__is__unit__0,axiom,
% 9.04/5.05      ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6461_pinf_I9_J,axiom,
% 9.04/5.05      ! [D: code_integer,S: code_integer] :
% 9.04/5.05      ? [Z3: code_integer] :
% 9.04/5.05      ! [X2: code_integer] :
% 9.04/5.05        ( ( ord_le6747313008572928689nteger @ Z3 @ X2 )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 9.04/5.05          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6462_pinf_I9_J,axiom,
% 9.04/5.05      ! [D: real,S: real] :
% 9.04/5.05      ? [Z3: real] :
% 9.04/5.05      ! [X2: real] :
% 9.04/5.05        ( ( ord_less_real @ Z3 @ X2 )
% 9.04/5.05       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 9.04/5.05          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6463_pinf_I9_J,axiom,
% 9.04/5.05      ! [D: rat,S: rat] :
% 9.04/5.05      ? [Z3: rat] :
% 9.04/5.05      ! [X2: rat] :
% 9.04/5.05        ( ( ord_less_rat @ Z3 @ X2 )
% 9.04/5.05       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 9.04/5.05          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6464_pinf_I9_J,axiom,
% 9.04/5.05      ! [D: nat,S: nat] :
% 9.04/5.05      ? [Z3: nat] :
% 9.04/5.05      ! [X2: nat] :
% 9.04/5.05        ( ( ord_less_nat @ Z3 @ X2 )
% 9.04/5.05       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 9.04/5.05          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6465_pinf_I9_J,axiom,
% 9.04/5.05      ! [D: int,S: int] :
% 9.04/5.05      ? [Z3: int] :
% 9.04/5.05      ! [X2: int] :
% 9.04/5.05        ( ( ord_less_int @ Z3 @ X2 )
% 9.04/5.05       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 9.04/5.05          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6466_pinf_I10_J,axiom,
% 9.04/5.05      ! [D: code_integer,S: code_integer] :
% 9.04/5.05      ? [Z3: code_integer] :
% 9.04/5.05      ! [X2: code_integer] :
% 9.04/5.05        ( ( ord_le6747313008572928689nteger @ Z3 @ X2 )
% 9.04/5.05       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 9.04/5.05          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6467_pinf_I10_J,axiom,
% 9.04/5.05      ! [D: real,S: real] :
% 9.04/5.05      ? [Z3: real] :
% 9.04/5.05      ! [X2: real] :
% 9.04/5.05        ( ( ord_less_real @ Z3 @ X2 )
% 9.04/5.05       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 9.04/5.05          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6468_pinf_I10_J,axiom,
% 9.04/5.05      ! [D: rat,S: rat] :
% 9.04/5.05      ? [Z3: rat] :
% 9.04/5.05      ! [X2: rat] :
% 9.04/5.05        ( ( ord_less_rat @ Z3 @ X2 )
% 9.04/5.05       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 9.04/5.05          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6469_pinf_I10_J,axiom,
% 9.04/5.05      ! [D: nat,S: nat] :
% 9.04/5.05      ? [Z3: nat] :
% 9.04/5.05      ! [X2: nat] :
% 9.04/5.05        ( ( ord_less_nat @ Z3 @ X2 )
% 9.04/5.05       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 9.04/5.05          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6470_pinf_I10_J,axiom,
% 9.04/5.05      ! [D: int,S: int] :
% 9.04/5.05      ? [Z3: int] :
% 9.04/5.05      ! [X2: int] :
% 9.04/5.05        ( ( ord_less_int @ Z3 @ X2 )
% 9.04/5.05       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 9.04/5.05          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6471_minf_I9_J,axiom,
% 9.04/5.05      ! [D: code_integer,S: code_integer] :
% 9.04/5.05      ? [Z3: code_integer] :
% 9.04/5.05      ! [X2: code_integer] :
% 9.04/5.05        ( ( ord_le6747313008572928689nteger @ X2 @ Z3 )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 9.04/5.05          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6472_minf_I9_J,axiom,
% 9.04/5.05      ! [D: real,S: real] :
% 9.04/5.05      ? [Z3: real] :
% 9.04/5.05      ! [X2: real] :
% 9.04/5.05        ( ( ord_less_real @ X2 @ Z3 )
% 9.04/5.05       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 9.04/5.05          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6473_minf_I9_J,axiom,
% 9.04/5.05      ! [D: rat,S: rat] :
% 9.04/5.05      ? [Z3: rat] :
% 9.04/5.05      ! [X2: rat] :
% 9.04/5.05        ( ( ord_less_rat @ X2 @ Z3 )
% 9.04/5.05       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 9.04/5.05          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6474_minf_I9_J,axiom,
% 9.04/5.05      ! [D: nat,S: nat] :
% 9.04/5.05      ? [Z3: nat] :
% 9.04/5.05      ! [X2: nat] :
% 9.04/5.05        ( ( ord_less_nat @ X2 @ Z3 )
% 9.04/5.05       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 9.04/5.05          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6475_minf_I9_J,axiom,
% 9.04/5.05      ! [D: int,S: int] :
% 9.04/5.05      ? [Z3: int] :
% 9.04/5.05      ! [X2: int] :
% 9.04/5.05        ( ( ord_less_int @ X2 @ Z3 )
% 9.04/5.05       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 9.04/5.05          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6476_minf_I10_J,axiom,
% 9.04/5.05      ! [D: code_integer,S: code_integer] :
% 9.04/5.05      ? [Z3: code_integer] :
% 9.04/5.05      ! [X2: code_integer] :
% 9.04/5.05        ( ( ord_le6747313008572928689nteger @ X2 @ Z3 )
% 9.04/5.05       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 9.04/5.05          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6477_minf_I10_J,axiom,
% 9.04/5.05      ! [D: real,S: real] :
% 9.04/5.05      ? [Z3: real] :
% 9.04/5.05      ! [X2: real] :
% 9.04/5.05        ( ( ord_less_real @ X2 @ Z3 )
% 9.04/5.05       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 9.04/5.05          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6478_minf_I10_J,axiom,
% 9.04/5.05      ! [D: rat,S: rat] :
% 9.04/5.05      ? [Z3: rat] :
% 9.04/5.05      ! [X2: rat] :
% 9.04/5.05        ( ( ord_less_rat @ X2 @ Z3 )
% 9.04/5.05       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 9.04/5.05          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6479_minf_I10_J,axiom,
% 9.04/5.05      ! [D: nat,S: nat] :
% 9.04/5.05      ? [Z3: nat] :
% 9.04/5.05      ! [X2: nat] :
% 9.04/5.05        ( ( ord_less_nat @ X2 @ Z3 )
% 9.04/5.05       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 9.04/5.05          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6480_minf_I10_J,axiom,
% 9.04/5.05      ! [D: int,S: int] :
% 9.04/5.05      ? [Z3: int] :
% 9.04/5.05      ! [X2: int] :
% 9.04/5.05        ( ( ord_less_int @ X2 @ Z3 )
% 9.04/5.05       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 9.04/5.05          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6481_dvd__div__eq__0__iff,axiom,
% 9.04/5.05      ! [B: code_integer,A: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.04/5.05       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 9.04/5.05            = zero_z3403309356797280102nteger )
% 9.04/5.05          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6482_dvd__div__eq__0__iff,axiom,
% 9.04/5.05      ! [B: complex,A: complex] :
% 9.04/5.05        ( ( dvd_dvd_complex @ B @ A )
% 9.04/5.05       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.04/5.05            = zero_zero_complex )
% 9.04/5.05          = ( A = zero_zero_complex ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6483_dvd__div__eq__0__iff,axiom,
% 9.04/5.05      ! [B: real,A: real] :
% 9.04/5.05        ( ( dvd_dvd_real @ B @ A )
% 9.04/5.05       => ( ( ( divide_divide_real @ A @ B )
% 9.04/5.05            = zero_zero_real )
% 9.04/5.05          = ( A = zero_zero_real ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6484_dvd__div__eq__0__iff,axiom,
% 9.04/5.05      ! [B: rat,A: rat] :
% 9.04/5.05        ( ( dvd_dvd_rat @ B @ A )
% 9.04/5.05       => ( ( ( divide_divide_rat @ A @ B )
% 9.04/5.05            = zero_zero_rat )
% 9.04/5.05          = ( A = zero_zero_rat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6485_dvd__div__eq__0__iff,axiom,
% 9.04/5.05      ! [B: nat,A: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ B @ A )
% 9.04/5.05       => ( ( ( divide_divide_nat @ A @ B )
% 9.04/5.05            = zero_zero_nat )
% 9.04/5.05          = ( A = zero_zero_nat ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6486_dvd__div__eq__0__iff,axiom,
% 9.04/5.05      ! [B: int,A: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ B @ A )
% 9.04/5.05       => ( ( ( divide_divide_int @ A @ B )
% 9.04/5.05            = zero_zero_int )
% 9.04/5.05          = ( A = zero_zero_int ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6487_unit__mult__right__cancel,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.05       => ( ( ( times_3573771949741848930nteger @ B @ A )
% 9.04/5.05            = ( times_3573771949741848930nteger @ C @ A ) )
% 9.04/5.05          = ( B = C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6488_unit__mult__right__cancel,axiom,
% 9.04/5.05      ! [A: nat,B: nat,C: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.05       => ( ( ( times_times_nat @ B @ A )
% 9.04/5.05            = ( times_times_nat @ C @ A ) )
% 9.04/5.05          = ( B = C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6489_unit__mult__right__cancel,axiom,
% 9.04/5.05      ! [A: int,B: int,C: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.05       => ( ( ( times_times_int @ B @ A )
% 9.04/5.05            = ( times_times_int @ C @ A ) )
% 9.04/5.05          = ( B = C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6490_unit__mult__left__cancel,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.05       => ( ( ( times_3573771949741848930nteger @ A @ B )
% 9.04/5.05            = ( times_3573771949741848930nteger @ A @ C ) )
% 9.04/5.05          = ( B = C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6491_unit__mult__left__cancel,axiom,
% 9.04/5.05      ! [A: nat,B: nat,C: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.05       => ( ( ( times_times_nat @ A @ B )
% 9.04/5.05            = ( times_times_nat @ A @ C ) )
% 9.04/5.05          = ( B = C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6492_unit__mult__left__cancel,axiom,
% 9.04/5.05      ! [A: int,B: int,C: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.05       => ( ( ( times_times_int @ A @ B )
% 9.04/5.05            = ( times_times_int @ A @ C ) )
% 9.04/5.05          = ( B = C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6493_mult__unit__dvd__iff_H,axiom,
% 9.04/5.05      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.05        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.05       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 9.04/5.05          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6494_mult__unit__dvd__iff_H,axiom,
% 9.04/5.05      ! [A: nat,B: nat,C: nat] :
% 9.04/5.05        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.05       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.04/5.05          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 9.04/5.05  
% 9.04/5.05  thf(fact_6495_mult__unit__dvd__iff_H,axiom,
% 9.04/5.05      ! [A: int,B: int,C: int] :
% 9.04/5.05        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 9.04/5.06          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6496_dvd__mult__unit__iff_H,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.04/5.06          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6497_dvd__mult__unit__iff_H,axiom,
% 9.04/5.06      ! [B: nat,A: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.04/5.06          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6498_dvd__mult__unit__iff_H,axiom,
% 9.04/5.06      ! [B: int,A: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 9.04/5.06          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6499_mult__unit__dvd__iff,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 9.04/5.06          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6500_mult__unit__dvd__iff,axiom,
% 9.04/5.06      ! [B: nat,A: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.04/5.06          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6501_mult__unit__dvd__iff,axiom,
% 9.04/5.06      ! [B: int,A: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 9.04/5.06          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6502_dvd__mult__unit__iff,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.04/5.06          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6503_dvd__mult__unit__iff,axiom,
% 9.04/5.06      ! [B: nat,A: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 9.04/5.06          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6504_dvd__mult__unit__iff,axiom,
% 9.04/5.06      ! [B: int,A: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 9.04/5.06          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6505_is__unit__mult__iff,axiom,
% 9.04/5.06      ! [A: code_integer,B: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 9.04/5.06        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.06          & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6506_is__unit__mult__iff,axiom,
% 9.04/5.06      ! [A: nat,B: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 9.04/5.06        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.06          & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6507_is__unit__mult__iff,axiom,
% 9.04/5.06      ! [A: int,B: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 9.04/5.06        = ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.06          & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6508_mult__exp__exp,axiom,
% 9.04/5.06      ! [X: complex,Y: complex] :
% 9.04/5.06        ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
% 9.04/5.06        = ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6509_mult__exp__exp,axiom,
% 9.04/5.06      ! [X: real,Y: real] :
% 9.04/5.06        ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 9.04/5.06        = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6510_exp__add__commuting,axiom,
% 9.04/5.06      ! [X: complex,Y: complex] :
% 9.04/5.06        ( ( ( times_times_complex @ X @ Y )
% 9.04/5.06          = ( times_times_complex @ Y @ X ) )
% 9.04/5.06       => ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.04/5.06          = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6511_exp__add__commuting,axiom,
% 9.04/5.06      ! [X: real,Y: real] :
% 9.04/5.06        ( ( ( times_times_real @ X @ Y )
% 9.04/5.06          = ( times_times_real @ Y @ X ) )
% 9.04/5.06       => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.06          = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6512_div__mult__div__if__dvd,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ D @ C )
% 9.04/5.06         => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 9.04/5.06            = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6513_div__mult__div__if__dvd,axiom,
% 9.04/5.06      ! [B: nat,A: nat,D: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ A )
% 9.04/5.06       => ( ( dvd_dvd_nat @ D @ C )
% 9.04/5.06         => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 9.04/5.06            = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6514_div__mult__div__if__dvd,axiom,
% 9.04/5.06      ! [B: int,A: int,D: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ A )
% 9.04/5.06       => ( ( dvd_dvd_int @ D @ C )
% 9.04/5.06         => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 9.04/5.06            = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6515_dvd__mult__imp__div,axiom,
% 9.04/5.06      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 9.04/5.06       => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6516_dvd__mult__imp__div,axiom,
% 9.04/5.06      ! [A: nat,C: nat,B: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 9.04/5.06       => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6517_dvd__mult__imp__div,axiom,
% 9.04/5.06      ! [A: int,C: int,B: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 9.04/5.06       => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6518_dvd__div__mult2__eq,axiom,
% 9.04/5.06      ! [B: code_integer,C: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 9.04/5.06       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.04/5.06          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6519_dvd__div__mult2__eq,axiom,
% 9.04/5.06      ! [B: nat,C: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 9.04/5.06       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.04/5.06          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6520_dvd__div__mult2__eq,axiom,
% 9.04/5.06      ! [B: int,C: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 9.04/5.06       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 9.04/5.06          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6521_div__div__eq__right,axiom,
% 9.04/5.06      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ C @ B )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ B @ A )
% 9.04/5.06         => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 9.04/5.06            = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6522_div__div__eq__right,axiom,
% 9.04/5.06      ! [C: nat,B: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ C @ B )
% 9.04/5.06       => ( ( dvd_dvd_nat @ B @ A )
% 9.04/5.06         => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 9.04/5.06            = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6523_div__div__eq__right,axiom,
% 9.04/5.06      ! [C: int,B: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ C @ B )
% 9.04/5.06       => ( ( dvd_dvd_int @ B @ A )
% 9.04/5.06         => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 9.04/5.06            = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6524_div__mult__swap,axiom,
% 9.04/5.06      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ C @ B )
% 9.04/5.06       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 9.04/5.06          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6525_div__mult__swap,axiom,
% 9.04/5.06      ! [C: nat,B: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ C @ B )
% 9.04/5.06       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 9.04/5.06          = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6526_div__mult__swap,axiom,
% 9.04/5.06      ! [C: int,B: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ C @ B )
% 9.04/5.06       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 9.04/5.06          = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6527_dvd__div__mult,axiom,
% 9.04/5.06      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ C @ B )
% 9.04/5.06       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 9.04/5.06          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6528_dvd__div__mult,axiom,
% 9.04/5.06      ! [C: nat,B: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ C @ B )
% 9.04/5.06       => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 9.04/5.06          = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6529_dvd__div__mult,axiom,
% 9.04/5.06      ! [C: int,B: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ C @ B )
% 9.04/5.06       => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 9.04/5.06          = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6530_dvd__div__unit__iff,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 9.04/5.06          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6531_dvd__div__unit__iff,axiom,
% 9.04/5.06      ! [B: nat,A: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 9.04/5.06          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6532_dvd__div__unit__iff,axiom,
% 9.04/5.06      ! [B: int,A: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 9.04/5.06          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6533_div__unit__dvd__iff,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 9.04/5.06          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6534_div__unit__dvd__iff,axiom,
% 9.04/5.06      ! [B: nat,A: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 9.04/5.06          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6535_div__unit__dvd__iff,axiom,
% 9.04/5.06      ! [B: int,A: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 9.04/5.06          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6536_unit__div__cancel,axiom,
% 9.04/5.06      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.06       => ( ( ( divide6298287555418463151nteger @ B @ A )
% 9.04/5.06            = ( divide6298287555418463151nteger @ C @ A ) )
% 9.04/5.06          = ( B = C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6537_unit__div__cancel,axiom,
% 9.04/5.06      ! [A: nat,B: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.06       => ( ( ( divide_divide_nat @ B @ A )
% 9.04/5.06            = ( divide_divide_nat @ C @ A ) )
% 9.04/5.06          = ( B = C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6538_unit__div__cancel,axiom,
% 9.04/5.06      ! [A: int,B: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.06       => ( ( ( divide_divide_int @ B @ A )
% 9.04/5.06            = ( divide_divide_int @ C @ A ) )
% 9.04/5.06          = ( B = C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6539_div__plus__div__distrib__dvd__right,axiom,
% 9.04/5.06      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ C @ B )
% 9.04/5.06       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 9.04/5.06          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6540_div__plus__div__distrib__dvd__right,axiom,
% 9.04/5.06      ! [C: nat,B: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ C @ B )
% 9.04/5.06       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.04/5.06          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6541_div__plus__div__distrib__dvd__right,axiom,
% 9.04/5.06      ! [C: int,B: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ C @ B )
% 9.04/5.06       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.04/5.06          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6542_div__plus__div__distrib__dvd__left,axiom,
% 9.04/5.06      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.04/5.06       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 9.04/5.06          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6543_div__plus__div__distrib__dvd__left,axiom,
% 9.04/5.06      ! [C: nat,A: nat,B: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ C @ A )
% 9.04/5.06       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.04/5.06          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6544_div__plus__div__distrib__dvd__left,axiom,
% 9.04/5.06      ! [C: int,A: int,B: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ C @ A )
% 9.04/5.06       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.04/5.06          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6545_exp__diff,axiom,
% 9.04/5.06      ! [X: complex,Y: complex] :
% 9.04/5.06        ( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
% 9.04/5.06        = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6546_exp__diff,axiom,
% 9.04/5.06      ! [X: real,Y: real] :
% 9.04/5.06        ( ( exp_real @ ( minus_minus_real @ X @ Y ) )
% 9.04/5.06        = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6547_dvd__neg__div,axiom,
% 9.04/5.06      ! [B: real,A: real] :
% 9.04/5.06        ( ( dvd_dvd_real @ B @ A )
% 9.04/5.06       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 9.04/5.06          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6548_dvd__neg__div,axiom,
% 9.04/5.06      ! [B: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ A )
% 9.04/5.06       => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 9.04/5.06          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6549_dvd__neg__div,axiom,
% 9.04/5.06      ! [B: complex,A: complex] :
% 9.04/5.06        ( ( dvd_dvd_complex @ B @ A )
% 9.04/5.06       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 9.04/5.06          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6550_dvd__neg__div,axiom,
% 9.04/5.06      ! [B: rat,A: rat] :
% 9.04/5.06        ( ( dvd_dvd_rat @ B @ A )
% 9.04/5.06       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.04/5.06          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6551_dvd__neg__div,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.04/5.06       => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.04/5.06          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6552_dvd__div__neg,axiom,
% 9.04/5.06      ! [B: real,A: real] :
% 9.04/5.06        ( ( dvd_dvd_real @ B @ A )
% 9.04/5.06       => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 9.04/5.06          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6553_dvd__div__neg,axiom,
% 9.04/5.06      ! [B: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ A )
% 9.04/5.06       => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 9.04/5.06          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6554_dvd__div__neg,axiom,
% 9.04/5.06      ! [B: complex,A: complex] :
% 9.04/5.06        ( ( dvd_dvd_complex @ B @ A )
% 9.04/5.06       => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 9.04/5.06          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6555_dvd__div__neg,axiom,
% 9.04/5.06      ! [B: rat,A: rat] :
% 9.04/5.06        ( ( dvd_dvd_rat @ B @ A )
% 9.04/5.06       => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.04/5.06          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6556_dvd__div__neg,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.04/5.06       => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.04/5.06          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6557_div__power,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.04/5.06       => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
% 9.04/5.06          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6558_div__power,axiom,
% 9.04/5.06      ! [B: nat,A: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ A )
% 9.04/5.06       => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
% 9.04/5.06          = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6559_div__power,axiom,
% 9.04/5.06      ! [B: int,A: int,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ A )
% 9.04/5.06       => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
% 9.04/5.06          = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6560_mod__eq__0__iff__dvd,axiom,
% 9.04/5.06      ! [A: nat,B: nat] :
% 9.04/5.06        ( ( ( modulo_modulo_nat @ A @ B )
% 9.04/5.06          = zero_zero_nat )
% 9.04/5.06        = ( dvd_dvd_nat @ B @ A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6561_mod__eq__0__iff__dvd,axiom,
% 9.04/5.06      ! [A: int,B: int] :
% 9.04/5.06        ( ( ( modulo_modulo_int @ A @ B )
% 9.04/5.06          = zero_zero_int )
% 9.04/5.06        = ( dvd_dvd_int @ B @ A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6562_mod__eq__0__iff__dvd,axiom,
% 9.04/5.06      ! [A: code_integer,B: code_integer] :
% 9.04/5.06        ( ( ( modulo364778990260209775nteger @ A @ B )
% 9.04/5.06          = zero_z3403309356797280102nteger )
% 9.04/5.06        = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6563_dvd__eq__mod__eq__0,axiom,
% 9.04/5.06      ( dvd_dvd_nat
% 9.04/5.06      = ( ^ [A3: nat,B4: nat] :
% 9.04/5.06            ( ( modulo_modulo_nat @ B4 @ A3 )
% 9.04/5.06            = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6564_dvd__eq__mod__eq__0,axiom,
% 9.04/5.06      ( dvd_dvd_int
% 9.04/5.06      = ( ^ [A3: int,B4: int] :
% 9.04/5.06            ( ( modulo_modulo_int @ B4 @ A3 )
% 9.04/5.06            = zero_zero_int ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6565_dvd__eq__mod__eq__0,axiom,
% 9.04/5.06      ( dvd_dvd_Code_integer
% 9.04/5.06      = ( ^ [A3: code_integer,B4: code_integer] :
% 9.04/5.06            ( ( modulo364778990260209775nteger @ B4 @ A3 )
% 9.04/5.06            = zero_z3403309356797280102nteger ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6566_mod__0__imp__dvd,axiom,
% 9.04/5.06      ! [A: nat,B: nat] :
% 9.04/5.06        ( ( ( modulo_modulo_nat @ A @ B )
% 9.04/5.06          = zero_zero_nat )
% 9.04/5.06       => ( dvd_dvd_nat @ B @ A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6567_mod__0__imp__dvd,axiom,
% 9.04/5.06      ! [A: int,B: int] :
% 9.04/5.06        ( ( ( modulo_modulo_int @ A @ B )
% 9.04/5.06          = zero_zero_int )
% 9.04/5.06       => ( dvd_dvd_int @ B @ A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6568_mod__0__imp__dvd,axiom,
% 9.04/5.06      ! [A: code_integer,B: code_integer] :
% 9.04/5.06        ( ( ( modulo364778990260209775nteger @ A @ B )
% 9.04/5.06          = zero_z3403309356797280102nteger )
% 9.04/5.06       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6569_dvd__power__le,axiom,
% 9.04/5.06      ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ X @ Y )
% 9.04/5.06       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.06         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6570_dvd__power__le,axiom,
% 9.04/5.06      ! [X: nat,Y: nat,N: nat,M: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ X @ Y )
% 9.04/5.06       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.06         => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6571_dvd__power__le,axiom,
% 9.04/5.06      ! [X: real,Y: real,N: nat,M: nat] :
% 9.04/5.06        ( ( dvd_dvd_real @ X @ Y )
% 9.04/5.06       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.06         => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6572_dvd__power__le,axiom,
% 9.04/5.06      ! [X: int,Y: int,N: nat,M: nat] :
% 9.04/5.06        ( ( dvd_dvd_int @ X @ Y )
% 9.04/5.06       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.06         => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6573_dvd__power__le,axiom,
% 9.04/5.06      ! [X: complex,Y: complex,N: nat,M: nat] :
% 9.04/5.06        ( ( dvd_dvd_complex @ X @ Y )
% 9.04/5.06       => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.06         => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6574_power__le__dvd,axiom,
% 9.04/5.06      ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 9.04/5.06       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6575_power__le__dvd,axiom,
% 9.04/5.06      ! [A: nat,N: nat,B: nat,M: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 9.04/5.06       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06         => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6576_power__le__dvd,axiom,
% 9.04/5.06      ! [A: real,N: nat,B: real,M: nat] :
% 9.04/5.06        ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 9.04/5.06       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06         => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6577_power__le__dvd,axiom,
% 9.04/5.06      ! [A: int,N: nat,B: int,M: nat] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 9.04/5.06       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06         => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6578_power__le__dvd,axiom,
% 9.04/5.06      ! [A: complex,N: nat,B: complex,M: nat] :
% 9.04/5.06        ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 9.04/5.06       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06         => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6579_le__imp__power__dvd,axiom,
% 9.04/5.06      ! [M: nat,N: nat,A: code_integer] :
% 9.04/5.06        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6580_le__imp__power__dvd,axiom,
% 9.04/5.06      ! [M: nat,N: nat,A: nat] :
% 9.04/5.06        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6581_le__imp__power__dvd,axiom,
% 9.04/5.06      ! [M: nat,N: nat,A: real] :
% 9.04/5.06        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6582_le__imp__power__dvd,axiom,
% 9.04/5.06      ! [M: nat,N: nat,A: int] :
% 9.04/5.06        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6583_le__imp__power__dvd,axiom,
% 9.04/5.06      ! [M: nat,N: nat,A: complex] :
% 9.04/5.06        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6584_nat__dvd__not__less,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.06       => ( ( ord_less_nat @ M @ N )
% 9.04/5.06         => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6585_dvd__pos__nat,axiom,
% 9.04/5.06      ! [N: nat,M: nat] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06       => ( ( dvd_dvd_nat @ M @ N )
% 9.04/5.06         => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6586_dvd__minus__self,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 9.04/5.06        = ( ( ord_less_nat @ N @ M )
% 9.04/5.06          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6587_zdvd__antisym__nonneg,axiom,
% 9.04/5.06      ! [M: int,N: int] :
% 9.04/5.06        ( ( ord_less_eq_int @ zero_zero_int @ M )
% 9.04/5.06       => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 9.04/5.06         => ( ( dvd_dvd_int @ M @ N )
% 9.04/5.06           => ( ( dvd_dvd_int @ N @ M )
% 9.04/5.06             => ( M = N ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6588_zdvd__not__zless,axiom,
% 9.04/5.06      ! [M: int,N: int] :
% 9.04/5.06        ( ( ord_less_int @ zero_zero_int @ M )
% 9.04/5.06       => ( ( ord_less_int @ M @ N )
% 9.04/5.06         => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6589_dvd__diffD,axiom,
% 9.04/5.06      ! [K: nat,M: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.04/5.06       => ( ( dvd_dvd_nat @ K @ N )
% 9.04/5.06         => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.06           => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6590_dvd__diffD1,axiom,
% 9.04/5.06      ! [K: nat,M: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.04/5.06       => ( ( dvd_dvd_nat @ K @ M )
% 9.04/5.06         => ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.06           => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6591_less__eq__dvd__minus,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06       => ( ( dvd_dvd_nat @ M @ N )
% 9.04/5.06          = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6592_zdvd__mono,axiom,
% 9.04/5.06      ! [K: int,M: int,T: int] :
% 9.04/5.06        ( ( K != zero_zero_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ M @ T )
% 9.04/5.06          = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6593_zdvd__mult__cancel,axiom,
% 9.04/5.06      ! [K: int,M: int,N: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 9.04/5.06       => ( ( K != zero_zero_int )
% 9.04/5.06         => ( dvd_dvd_int @ M @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6594_bezout__add__nat,axiom,
% 9.04/5.06      ! [A: nat,B: nat] :
% 9.04/5.06      ? [D5: nat,X3: nat,Y4: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ D5 @ A )
% 9.04/5.06        & ( dvd_dvd_nat @ D5 @ B )
% 9.04/5.06        & ( ( ( times_times_nat @ A @ X3 )
% 9.04/5.06            = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D5 ) )
% 9.04/5.06          | ( ( times_times_nat @ B @ X3 )
% 9.04/5.06            = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D5 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6595_bezout__lemma__nat,axiom,
% 9.04/5.06      ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ D @ A )
% 9.04/5.06       => ( ( dvd_dvd_nat @ D @ B )
% 9.04/5.06         => ( ( ( ( times_times_nat @ A @ X )
% 9.04/5.06                = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
% 9.04/5.06              | ( ( times_times_nat @ B @ X )
% 9.04/5.06                = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
% 9.04/5.06           => ? [X3: nat,Y4: nat] :
% 9.04/5.06                ( ( dvd_dvd_nat @ D @ A )
% 9.04/5.06                & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 9.04/5.06                & ( ( ( times_times_nat @ A @ X3 )
% 9.04/5.06                    = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
% 9.04/5.06                  | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
% 9.04/5.06                    = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6596_zdvd__reduce,axiom,
% 9.04/5.06      ! [K: int,N: int,M: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 9.04/5.06        = ( dvd_dvd_int @ K @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6597_zdvd__period,axiom,
% 9.04/5.06      ! [A: int,D: int,X: int,T: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ A @ D )
% 9.04/5.06       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
% 9.04/5.06          = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6598_abs__div,axiom,
% 9.04/5.06      ! [Y: int,X: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ Y @ X )
% 9.04/5.06       => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
% 9.04/5.06          = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6599_even__of__int__iff,axiom,
% 9.04/5.06      ! [K: int] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 9.04/5.06        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6600_even__of__int__iff,axiom,
% 9.04/5.06      ! [K: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 9.04/5.06        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6601_div2__even__ext__nat,axiom,
% 9.04/5.06      ! [X: nat,Y: nat] :
% 9.04/5.06        ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06          = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.06       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
% 9.04/5.06            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
% 9.04/5.06         => ( X = Y ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6602_exp__gt__one,axiom,
% 9.04/5.06      ! [X: real] :
% 9.04/5.06        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.06       => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6603_exp__ge__add__one__self,axiom,
% 9.04/5.06      ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6604_unit__dvdE,axiom,
% 9.04/5.06      ! [A: code_integer,B: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.06       => ~ ( ( A != zero_z3403309356797280102nteger )
% 9.04/5.06           => ! [C4: code_integer] :
% 9.04/5.06                ( B
% 9.04/5.06               != ( times_3573771949741848930nteger @ A @ C4 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6605_unit__dvdE,axiom,
% 9.04/5.06      ! [A: nat,B: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.06       => ~ ( ( A != zero_zero_nat )
% 9.04/5.06           => ! [C4: nat] :
% 9.04/5.06                ( B
% 9.04/5.06               != ( times_times_nat @ A @ C4 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6606_unit__dvdE,axiom,
% 9.04/5.06      ! [A: int,B: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.06       => ~ ( ( A != zero_zero_int )
% 9.04/5.06           => ! [C4: int] :
% 9.04/5.06                ( B
% 9.04/5.06               != ( times_times_int @ A @ C4 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6607_unity__coeff__ex,axiom,
% 9.04/5.06      ! [P: code_integer > $o,L: code_integer] :
% 9.04/5.06        ( ( ? [X4: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X4 ) ) )
% 9.04/5.06        = ( ? [X4: code_integer] :
% 9.04/5.06              ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X4 @ zero_z3403309356797280102nteger ) )
% 9.04/5.06              & ( P @ X4 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6608_unity__coeff__ex,axiom,
% 9.04/5.06      ! [P: complex > $o,L: complex] :
% 9.04/5.06        ( ( ? [X4: complex] : ( P @ ( times_times_complex @ L @ X4 ) ) )
% 9.04/5.06        = ( ? [X4: complex] :
% 9.04/5.06              ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X4 @ zero_zero_complex ) )
% 9.04/5.06              & ( P @ X4 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6609_unity__coeff__ex,axiom,
% 9.04/5.06      ! [P: real > $o,L: real] :
% 9.04/5.06        ( ( ? [X4: real] : ( P @ ( times_times_real @ L @ X4 ) ) )
% 9.04/5.06        = ( ? [X4: real] :
% 9.04/5.06              ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X4 @ zero_zero_real ) )
% 9.04/5.06              & ( P @ X4 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6610_unity__coeff__ex,axiom,
% 9.04/5.06      ! [P: rat > $o,L: rat] :
% 9.04/5.06        ( ( ? [X4: rat] : ( P @ ( times_times_rat @ L @ X4 ) ) )
% 9.04/5.06        = ( ? [X4: rat] :
% 9.04/5.06              ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X4 @ zero_zero_rat ) )
% 9.04/5.06              & ( P @ X4 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6611_unity__coeff__ex,axiom,
% 9.04/5.06      ! [P: nat > $o,L: nat] :
% 9.04/5.06        ( ( ? [X4: nat] : ( P @ ( times_times_nat @ L @ X4 ) ) )
% 9.04/5.06        = ( ? [X4: nat] :
% 9.04/5.06              ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
% 9.04/5.06              & ( P @ X4 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6612_unity__coeff__ex,axiom,
% 9.04/5.06      ! [P: int > $o,L: int] :
% 9.04/5.06        ( ( ? [X4: int] : ( P @ ( times_times_int @ L @ X4 ) ) )
% 9.04/5.06        = ( ? [X4: int] :
% 9.04/5.06              ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
% 9.04/5.06              & ( P @ X4 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6613_dvd__div__eq__mult,axiom,
% 9.04/5.06      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.04/5.06        ( ( A != zero_z3403309356797280102nteger )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ A @ B )
% 9.04/5.06         => ( ( ( divide6298287555418463151nteger @ B @ A )
% 9.04/5.06              = C )
% 9.04/5.06            = ( B
% 9.04/5.06              = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6614_dvd__div__eq__mult,axiom,
% 9.04/5.06      ! [A: nat,B: nat,C: nat] :
% 9.04/5.06        ( ( A != zero_zero_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ A @ B )
% 9.04/5.06         => ( ( ( divide_divide_nat @ B @ A )
% 9.04/5.06              = C )
% 9.04/5.06            = ( B
% 9.04/5.06              = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6615_dvd__div__eq__mult,axiom,
% 9.04/5.06      ! [A: int,B: int,C: int] :
% 9.04/5.06        ( ( A != zero_zero_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ A @ B )
% 9.04/5.06         => ( ( ( divide_divide_int @ B @ A )
% 9.04/5.06              = C )
% 9.04/5.06            = ( B
% 9.04/5.06              = ( times_times_int @ C @ A ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6616_div__dvd__iff__mult,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.04/5.06        ( ( B != zero_z3403309356797280102nteger )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ B @ A )
% 9.04/5.06         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 9.04/5.06            = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6617_div__dvd__iff__mult,axiom,
% 9.04/5.06      ! [B: nat,A: nat,C: nat] :
% 9.04/5.06        ( ( B != zero_zero_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ B @ A )
% 9.04/5.06         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 9.04/5.06            = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6618_div__dvd__iff__mult,axiom,
% 9.04/5.06      ! [B: int,A: int,C: int] :
% 9.04/5.06        ( ( B != zero_zero_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ B @ A )
% 9.04/5.06         => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 9.04/5.06            = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6619_dvd__div__iff__mult,axiom,
% 9.04/5.06      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.04/5.06        ( ( C != zero_z3403309356797280102nteger )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.04/5.06         => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 9.04/5.06            = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6620_dvd__div__iff__mult,axiom,
% 9.04/5.06      ! [C: nat,B: nat,A: nat] :
% 9.04/5.06        ( ( C != zero_zero_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ C @ B )
% 9.04/5.06         => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 9.04/5.06            = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6621_dvd__div__iff__mult,axiom,
% 9.04/5.06      ! [C: int,B: int,A: int] :
% 9.04/5.06        ( ( C != zero_zero_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ C @ B )
% 9.04/5.06         => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 9.04/5.06            = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6622_dvd__div__div__eq__mult,axiom,
% 9.04/5.06      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 9.04/5.06        ( ( A != zero_z3403309356797280102nteger )
% 9.04/5.06       => ( ( C != zero_z3403309356797280102nteger )
% 9.04/5.06         => ( ( dvd_dvd_Code_integer @ A @ B )
% 9.04/5.06           => ( ( dvd_dvd_Code_integer @ C @ D )
% 9.04/5.06             => ( ( ( divide6298287555418463151nteger @ B @ A )
% 9.04/5.06                  = ( divide6298287555418463151nteger @ D @ C ) )
% 9.04/5.06                = ( ( times_3573771949741848930nteger @ B @ C )
% 9.04/5.06                  = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6623_dvd__div__div__eq__mult,axiom,
% 9.04/5.06      ! [A: nat,C: nat,B: nat,D: nat] :
% 9.04/5.06        ( ( A != zero_zero_nat )
% 9.04/5.06       => ( ( C != zero_zero_nat )
% 9.04/5.06         => ( ( dvd_dvd_nat @ A @ B )
% 9.04/5.06           => ( ( dvd_dvd_nat @ C @ D )
% 9.04/5.06             => ( ( ( divide_divide_nat @ B @ A )
% 9.04/5.06                  = ( divide_divide_nat @ D @ C ) )
% 9.04/5.06                = ( ( times_times_nat @ B @ C )
% 9.04/5.06                  = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6624_dvd__div__div__eq__mult,axiom,
% 9.04/5.06      ! [A: int,C: int,B: int,D: int] :
% 9.04/5.06        ( ( A != zero_zero_int )
% 9.04/5.06       => ( ( C != zero_zero_int )
% 9.04/5.06         => ( ( dvd_dvd_int @ A @ B )
% 9.04/5.06           => ( ( dvd_dvd_int @ C @ D )
% 9.04/5.06             => ( ( ( divide_divide_int @ B @ A )
% 9.04/5.06                  = ( divide_divide_int @ D @ C ) )
% 9.04/5.06                = ( ( times_times_int @ B @ C )
% 9.04/5.06                  = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6625_unit__div__eq__0__iff,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 9.04/5.06            = zero_z3403309356797280102nteger )
% 9.04/5.06          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6626_unit__div__eq__0__iff,axiom,
% 9.04/5.06      ! [B: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( ( divide_divide_nat @ A @ B )
% 9.04/5.06            = zero_zero_nat )
% 9.04/5.06          = ( A = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6627_unit__div__eq__0__iff,axiom,
% 9.04/5.06      ! [B: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( ( divide_divide_int @ A @ B )
% 9.04/5.06            = zero_zero_int )
% 9.04/5.06          = ( A = zero_zero_int ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6628_even__numeral,axiom,
% 9.04/5.06      ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6629_even__numeral,axiom,
% 9.04/5.06      ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6630_even__numeral,axiom,
% 9.04/5.06      ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6631_inf__period_I4_J,axiom,
% 9.04/5.06      ! [D: code_integer,D2: code_integer,T: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ D @ D2 )
% 9.04/5.06       => ! [X2: code_integer,K6: code_integer] :
% 9.04/5.06            ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T ) ) )
% 9.04/5.06            = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K6 @ D2 ) ) @ T ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6632_inf__period_I4_J,axiom,
% 9.04/5.06      ! [D: complex,D2: complex,T: complex] :
% 9.04/5.06        ( ( dvd_dvd_complex @ D @ D2 )
% 9.04/5.06       => ! [X2: complex,K6: complex] :
% 9.04/5.06            ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T ) ) )
% 9.04/5.06            = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K6 @ D2 ) ) @ T ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6633_inf__period_I4_J,axiom,
% 9.04/5.06      ! [D: real,D2: real,T: real] :
% 9.04/5.06        ( ( dvd_dvd_real @ D @ D2 )
% 9.04/5.06       => ! [X2: real,K6: real] :
% 9.04/5.06            ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) ) )
% 9.04/5.06            = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K6 @ D2 ) ) @ T ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6634_inf__period_I4_J,axiom,
% 9.04/5.06      ! [D: rat,D2: rat,T: rat] :
% 9.04/5.06        ( ( dvd_dvd_rat @ D @ D2 )
% 9.04/5.06       => ! [X2: rat,K6: rat] :
% 9.04/5.06            ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) ) )
% 9.04/5.06            = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K6 @ D2 ) ) @ T ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6635_inf__period_I4_J,axiom,
% 9.04/5.06      ! [D: int,D2: int,T: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ D @ D2 )
% 9.04/5.06       => ! [X2: int,K6: int] :
% 9.04/5.06            ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) ) )
% 9.04/5.06            = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K6 @ D2 ) ) @ T ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6636_inf__period_I3_J,axiom,
% 9.04/5.06      ! [D: code_integer,D2: code_integer,T: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ D @ D2 )
% 9.04/5.06       => ! [X2: code_integer,K6: code_integer] :
% 9.04/5.06            ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T ) )
% 9.04/5.06            = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K6 @ D2 ) ) @ T ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6637_inf__period_I3_J,axiom,
% 9.04/5.06      ! [D: complex,D2: complex,T: complex] :
% 9.04/5.06        ( ( dvd_dvd_complex @ D @ D2 )
% 9.04/5.06       => ! [X2: complex,K6: complex] :
% 9.04/5.06            ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T ) )
% 9.04/5.06            = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K6 @ D2 ) ) @ T ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6638_inf__period_I3_J,axiom,
% 9.04/5.06      ! [D: real,D2: real,T: real] :
% 9.04/5.06        ( ( dvd_dvd_real @ D @ D2 )
% 9.04/5.06       => ! [X2: real,K6: real] :
% 9.04/5.06            ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) )
% 9.04/5.06            = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K6 @ D2 ) ) @ T ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6639_inf__period_I3_J,axiom,
% 9.04/5.06      ! [D: rat,D2: rat,T: rat] :
% 9.04/5.06        ( ( dvd_dvd_rat @ D @ D2 )
% 9.04/5.06       => ! [X2: rat,K6: rat] :
% 9.04/5.06            ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) )
% 9.04/5.06            = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K6 @ D2 ) ) @ T ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6640_inf__period_I3_J,axiom,
% 9.04/5.06      ! [D: int,D2: int,T: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ D @ D2 )
% 9.04/5.06       => ! [X2: int,K6: int] :
% 9.04/5.06            ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 9.04/5.06            = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K6 @ D2 ) ) @ T ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6641_is__unit__div__mult2__eq,axiom,
% 9.04/5.06      ! [B: code_integer,C: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 9.04/5.06         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.04/5.06            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6642_is__unit__div__mult2__eq,axiom,
% 9.04/5.06      ! [B: nat,C: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 9.04/5.06         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.04/5.06            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6643_is__unit__div__mult2__eq,axiom,
% 9.04/5.06      ! [B: int,C: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ C @ one_one_int )
% 9.04/5.06         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 9.04/5.06            = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6644_unit__div__mult__swap,axiom,
% 9.04/5.06      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 9.04/5.06       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 9.04/5.06          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6645_unit__div__mult__swap,axiom,
% 9.04/5.06      ! [C: nat,A: nat,B: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 9.04/5.06       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 9.04/5.06          = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6646_unit__div__mult__swap,axiom,
% 9.04/5.06      ! [C: int,A: int,B: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ C @ one_one_int )
% 9.04/5.06       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 9.04/5.06          = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6647_unit__div__commute,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 9.04/5.06          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6648_unit__div__commute,axiom,
% 9.04/5.06      ! [B: nat,A: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 9.04/5.06          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6649_unit__div__commute,axiom,
% 9.04/5.06      ! [B: int,A: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 9.04/5.06          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6650_div__mult__unit2,axiom,
% 9.04/5.06      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ B @ A )
% 9.04/5.06         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.04/5.06            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6651_div__mult__unit2,axiom,
% 9.04/5.06      ! [C: nat,B: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ B @ A )
% 9.04/5.06         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.04/5.06            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6652_div__mult__unit2,axiom,
% 9.04/5.06      ! [C: int,B: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ C @ one_one_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ B @ A )
% 9.04/5.06         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 9.04/5.06            = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6653_unit__eq__div2,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( A
% 9.04/5.06            = ( divide6298287555418463151nteger @ C @ B ) )
% 9.04/5.06          = ( ( times_3573771949741848930nteger @ A @ B )
% 9.04/5.06            = C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6654_unit__eq__div2,axiom,
% 9.04/5.06      ! [B: nat,A: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( A
% 9.04/5.06            = ( divide_divide_nat @ C @ B ) )
% 9.04/5.06          = ( ( times_times_nat @ A @ B )
% 9.04/5.06            = C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6655_unit__eq__div2,axiom,
% 9.04/5.06      ! [B: int,A: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( A
% 9.04/5.06            = ( divide_divide_int @ C @ B ) )
% 9.04/5.06          = ( ( times_times_int @ A @ B )
% 9.04/5.06            = C ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6656_unit__eq__div1,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 9.04/5.06            = C )
% 9.04/5.06          = ( A
% 9.04/5.06            = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6657_unit__eq__div1,axiom,
% 9.04/5.06      ! [B: nat,A: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( ( divide_divide_nat @ A @ B )
% 9.04/5.06            = C )
% 9.04/5.06          = ( A
% 9.04/5.06            = ( times_times_nat @ C @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6658_unit__eq__div1,axiom,
% 9.04/5.06      ! [B: int,A: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( ( divide_divide_int @ A @ B )
% 9.04/5.06            = C )
% 9.04/5.06          = ( A
% 9.04/5.06            = ( times_times_int @ C @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6659_exp__minus__inverse,axiom,
% 9.04/5.06      ! [X: real] :
% 9.04/5.06        ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
% 9.04/5.06        = one_one_real ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6660_exp__minus__inverse,axiom,
% 9.04/5.06      ! [X: complex] :
% 9.04/5.06        ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
% 9.04/5.06        = one_one_complex ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6661_unit__imp__mod__eq__0,axiom,
% 9.04/5.06      ! [B: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06       => ( ( modulo_modulo_nat @ A @ B )
% 9.04/5.06          = zero_zero_nat ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6662_unit__imp__mod__eq__0,axiom,
% 9.04/5.06      ! [B: int,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06       => ( ( modulo_modulo_int @ A @ B )
% 9.04/5.06          = zero_zero_int ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6663_unit__imp__mod__eq__0,axiom,
% 9.04/5.06      ! [B: code_integer,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06       => ( ( modulo364778990260209775nteger @ A @ B )
% 9.04/5.06          = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6664_is__unit__power__iff,axiom,
% 9.04/5.06      ! [A: code_integer,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 9.04/5.06        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.06          | ( N = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6665_is__unit__power__iff,axiom,
% 9.04/5.06      ! [A: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 9.04/5.06        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.06          | ( N = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6666_is__unit__power__iff,axiom,
% 9.04/5.06      ! [A: int,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 9.04/5.06        = ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.06          | ( N = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6667_exp__of__nat2__mult,axiom,
% 9.04/5.06      ! [X: complex,N: nat] :
% 9.04/5.06        ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
% 9.04/5.06        = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6668_exp__of__nat2__mult,axiom,
% 9.04/5.06      ! [X: real,N: nat] :
% 9.04/5.06        ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.04/5.06        = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6669_exp__of__nat__mult,axiom,
% 9.04/5.06      ! [N: nat,X: complex] :
% 9.04/5.06        ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
% 9.04/5.06        = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6670_exp__of__nat__mult,axiom,
% 9.04/5.06      ! [N: nat,X: real] :
% 9.04/5.06        ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
% 9.04/5.06        = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6671_log__ln,axiom,
% 9.04/5.06      ( ln_ln_real
% 9.04/5.06      = ( log @ ( exp_real @ one_one_real ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6672_dvd__imp__le,axiom,
% 9.04/5.06      ! [K: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ K @ N )
% 9.04/5.06       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06         => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6673_nat__mult__dvd__cancel1,axiom,
% 9.04/5.06      ! [K: nat,M: nat,N: nat] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.04/5.06       => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.04/5.06          = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6674_dvd__mult__cancel,axiom,
% 9.04/5.06      ! [K: nat,M: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.04/5.06       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.04/5.06         => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6675_bezout__add__strong__nat,axiom,
% 9.04/5.06      ! [A: nat,B: nat] :
% 9.04/5.06        ( ( A != zero_zero_nat )
% 9.04/5.06       => ? [D5: nat,X3: nat,Y4: nat] :
% 9.04/5.06            ( ( dvd_dvd_nat @ D5 @ A )
% 9.04/5.06            & ( dvd_dvd_nat @ D5 @ B )
% 9.04/5.06            & ( ( times_times_nat @ A @ X3 )
% 9.04/5.06              = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D5 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6676_zdvd__imp__le,axiom,
% 9.04/5.06      ! [Z: int,N: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ Z @ N )
% 9.04/5.06       => ( ( ord_less_int @ zero_zero_int @ N )
% 9.04/5.06         => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6677_mod__greater__zero__iff__not__dvd,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 9.04/5.06        = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6678_dvd__imp__le__int,axiom,
% 9.04/5.06      ! [I: int,D: int] :
% 9.04/5.06        ( ( I != zero_zero_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ D @ I )
% 9.04/5.06         => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6679_mod__eq__dvd__iff__nat,axiom,
% 9.04/5.06      ! [N: nat,M: nat,Q3: nat] :
% 9.04/5.06        ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.06       => ( ( ( modulo_modulo_nat @ M @ Q3 )
% 9.04/5.06            = ( modulo_modulo_nat @ N @ Q3 ) )
% 9.04/5.06          = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6680_real__of__nat__div,axiom,
% 9.04/5.06      ! [D: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ D @ N )
% 9.04/5.06       => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 9.04/5.06          = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6681_real__of__int__div,axiom,
% 9.04/5.06      ! [D: int,N: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ D @ N )
% 9.04/5.06       => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 9.04/5.06          = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6682_exp__ge__add__one__self__aux,axiom,
% 9.04/5.06      ! [X: real] :
% 9.04/5.06        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.06       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6683_lemma__exp__total,axiom,
% 9.04/5.06      ! [Y: real] :
% 9.04/5.06        ( ( ord_less_eq_real @ one_one_real @ Y )
% 9.04/5.06       => ? [X3: real] :
% 9.04/5.06            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.04/5.06            & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y @ one_one_real ) )
% 9.04/5.06            & ( ( exp_real @ X3 )
% 9.04/5.06              = Y ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6684_ln__ge__iff,axiom,
% 9.04/5.06      ! [X: real,Y: real] :
% 9.04/5.06        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.06       => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
% 9.04/5.06          = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6685_even__zero,axiom,
% 9.04/5.06      dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6686_even__zero,axiom,
% 9.04/5.06      dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6687_even__zero,axiom,
% 9.04/5.06      dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6688_ln__x__over__x__mono,axiom,
% 9.04/5.06      ! [X: real,Y: real] :
% 9.04/5.06        ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
% 9.04/5.06       => ( ( ord_less_eq_real @ X @ Y )
% 9.04/5.06         => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6689_is__unit__div__mult__cancel__right,axiom,
% 9.04/5.06      ! [A: code_integer,B: code_integer] :
% 9.04/5.06        ( ( A != zero_z3403309356797280102nteger )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 9.04/5.06            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6690_is__unit__div__mult__cancel__right,axiom,
% 9.04/5.06      ! [A: nat,B: nat] :
% 9.04/5.06        ( ( A != zero_zero_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 9.04/5.06            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6691_is__unit__div__mult__cancel__right,axiom,
% 9.04/5.06      ! [A: int,B: int] :
% 9.04/5.06        ( ( A != zero_zero_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 9.04/5.06            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6692_is__unit__div__mult__cancel__left,axiom,
% 9.04/5.06      ! [A: code_integer,B: code_integer] :
% 9.04/5.06        ( ( A != zero_z3403309356797280102nteger )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.04/5.06         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 9.04/5.06            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6693_is__unit__div__mult__cancel__left,axiom,
% 9.04/5.06      ! [A: nat,B: nat] :
% 9.04/5.06        ( ( A != zero_zero_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.04/5.06         => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 9.04/5.06            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6694_is__unit__div__mult__cancel__left,axiom,
% 9.04/5.06      ! [A: int,B: int] :
% 9.04/5.06        ( ( A != zero_zero_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.04/5.06         => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 9.04/5.06            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6695_is__unitE,axiom,
% 9.04/5.06      ! [A: code_integer,C: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.04/5.06       => ~ ( ( A != zero_z3403309356797280102nteger )
% 9.04/5.06           => ! [B6: code_integer] :
% 9.04/5.06                ( ( B6 != zero_z3403309356797280102nteger )
% 9.04/5.06               => ( ( dvd_dvd_Code_integer @ B6 @ one_one_Code_integer )
% 9.04/5.06                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 9.04/5.06                      = B6 )
% 9.04/5.06                   => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B6 )
% 9.04/5.06                        = A )
% 9.04/5.06                     => ( ( ( times_3573771949741848930nteger @ A @ B6 )
% 9.04/5.06                          = one_one_Code_integer )
% 9.04/5.06                       => ( ( divide6298287555418463151nteger @ C @ A )
% 9.04/5.06                         != ( times_3573771949741848930nteger @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6696_is__unitE,axiom,
% 9.04/5.06      ! [A: nat,C: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.04/5.06       => ~ ( ( A != zero_zero_nat )
% 9.04/5.06           => ! [B6: nat] :
% 9.04/5.06                ( ( B6 != zero_zero_nat )
% 9.04/5.06               => ( ( dvd_dvd_nat @ B6 @ one_one_nat )
% 9.04/5.06                 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 9.04/5.06                      = B6 )
% 9.04/5.06                   => ( ( ( divide_divide_nat @ one_one_nat @ B6 )
% 9.04/5.06                        = A )
% 9.04/5.06                     => ( ( ( times_times_nat @ A @ B6 )
% 9.04/5.06                          = one_one_nat )
% 9.04/5.06                       => ( ( divide_divide_nat @ C @ A )
% 9.04/5.06                         != ( times_times_nat @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6697_is__unitE,axiom,
% 9.04/5.06      ! [A: int,C: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.04/5.06       => ~ ( ( A != zero_zero_int )
% 9.04/5.06           => ! [B6: int] :
% 9.04/5.06                ( ( B6 != zero_zero_int )
% 9.04/5.06               => ( ( dvd_dvd_int @ B6 @ one_one_int )
% 9.04/5.06                 => ( ( ( divide_divide_int @ one_one_int @ A )
% 9.04/5.06                      = B6 )
% 9.04/5.06                   => ( ( ( divide_divide_int @ one_one_int @ B6 )
% 9.04/5.06                        = A )
% 9.04/5.06                     => ( ( ( times_times_int @ A @ B6 )
% 9.04/5.06                          = one_one_int )
% 9.04/5.06                       => ( ( divide_divide_int @ C @ A )
% 9.04/5.06                         != ( times_times_int @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6698_evenE,axiom,
% 9.04/5.06      ! [A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ~ ! [B6: code_integer] :
% 9.04/5.06              ( A
% 9.04/5.06             != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6699_evenE,axiom,
% 9.04/5.06      ! [A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ~ ! [B6: nat] :
% 9.04/5.06              ( A
% 9.04/5.06             != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6700_evenE,axiom,
% 9.04/5.06      ! [A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ~ ! [B6: int] :
% 9.04/5.06              ( A
% 9.04/5.06             != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6701_odd__one,axiom,
% 9.04/5.06      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6702_odd__one,axiom,
% 9.04/5.06      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6703_odd__one,axiom,
% 9.04/5.06      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6704_odd__even__add,axiom,
% 9.04/5.06      ! [A: code_integer,B: code_integer] :
% 9.04/5.06        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 9.04/5.06         => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6705_odd__even__add,axiom,
% 9.04/5.06      ! [A: nat,B: nat] :
% 9.04/5.06        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.04/5.06         => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6706_odd__even__add,axiom,
% 9.04/5.06      ! [A: int,B: int] :
% 9.04/5.06        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 9.04/5.06         => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6707_bit__eq__rec,axiom,
% 9.04/5.06      ( ( ^ [Y3: code_integer,Z2: code_integer] : ( Y3 = Z2 ) )
% 9.04/5.06      = ( ^ [A3: code_integer,B4: code_integer] :
% 9.04/5.06            ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
% 9.04/5.06              = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) )
% 9.04/5.06            & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.06              = ( divide6298287555418463151nteger @ B4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6708_bit__eq__rec,axiom,
% 9.04/5.06      ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 9.04/5.06      = ( ^ [A3: nat,B4: nat] :
% 9.04/5.06            ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
% 9.04/5.06              = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
% 9.04/5.06            & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06              = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6709_bit__eq__rec,axiom,
% 9.04/5.06      ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 9.04/5.06      = ( ^ [A3: int,B4: int] :
% 9.04/5.06            ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
% 9.04/5.06              = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
% 9.04/5.06            & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.06              = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6710_even__minus,axiom,
% 9.04/5.06      ! [A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 9.04/5.06        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6711_even__minus,axiom,
% 9.04/5.06      ! [A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 9.04/5.06        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6712_dvd__power__iff,axiom,
% 9.04/5.06      ! [X: code_integer,M: nat,N: nat] :
% 9.04/5.06        ( ( X != zero_z3403309356797280102nteger )
% 9.04/5.06       => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
% 9.04/5.06          = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 9.04/5.06            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6713_dvd__power__iff,axiom,
% 9.04/5.06      ! [X: nat,M: nat,N: nat] :
% 9.04/5.06        ( ( X != zero_zero_nat )
% 9.04/5.06       => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
% 9.04/5.06          = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 9.04/5.06            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6714_dvd__power__iff,axiom,
% 9.04/5.06      ! [X: int,M: nat,N: nat] :
% 9.04/5.06        ( ( X != zero_zero_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
% 9.04/5.06          = ( ( dvd_dvd_int @ X @ one_one_int )
% 9.04/5.06            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6715_odd__numeral,axiom,
% 9.04/5.06      ! [N: num] :
% 9.04/5.06        ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6716_odd__numeral,axiom,
% 9.04/5.06      ! [N: num] :
% 9.04/5.06        ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6717_odd__numeral,axiom,
% 9.04/5.06      ! [N: num] :
% 9.04/5.06        ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6718_dvd__power,axiom,
% 9.04/5.06      ! [N: nat,X: code_integer] :
% 9.04/5.06        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06          | ( X = one_one_Code_integer ) )
% 9.04/5.06       => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6719_dvd__power,axiom,
% 9.04/5.06      ! [N: nat,X: rat] :
% 9.04/5.06        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06          | ( X = one_one_rat ) )
% 9.04/5.06       => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6720_dvd__power,axiom,
% 9.04/5.06      ! [N: nat,X: nat] :
% 9.04/5.06        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06          | ( X = one_one_nat ) )
% 9.04/5.06       => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6721_dvd__power,axiom,
% 9.04/5.06      ! [N: nat,X: real] :
% 9.04/5.06        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06          | ( X = one_one_real ) )
% 9.04/5.06       => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6722_dvd__power,axiom,
% 9.04/5.06      ! [N: nat,X: int] :
% 9.04/5.06        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06          | ( X = one_one_int ) )
% 9.04/5.06       => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6723_dvd__power,axiom,
% 9.04/5.06      ! [N: nat,X: complex] :
% 9.04/5.06        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06          | ( X = one_one_complex ) )
% 9.04/5.06       => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6724_even__signed__take__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 9.04/5.06        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6725_even__signed__take__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 9.04/5.06        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6726_even__even__mod__4__iff,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6727_dvd__mult__cancel2,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.06       => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 9.04/5.06          = ( N = one_one_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6728_dvd__mult__cancel1,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.06       => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 9.04/5.06          = ( N = one_one_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6729_powr__def,axiom,
% 9.04/5.06      ( powr_real
% 9.04/5.06      = ( ^ [X4: real,A3: real] : ( if_real @ ( X4 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X4 ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6730_dvd__minus__add,axiom,
% 9.04/5.06      ! [Q3: nat,N: nat,R2: nat,M: nat] :
% 9.04/5.06        ( ( ord_less_eq_nat @ Q3 @ N )
% 9.04/5.06       => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M ) )
% 9.04/5.06         => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q3 ) )
% 9.04/5.06            = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q3 ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6731_power__dvd__imp__le,axiom,
% 9.04/5.06      ! [I: nat,M: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 9.04/5.06       => ( ( ord_less_nat @ one_one_nat @ I )
% 9.04/5.06         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6732_mod__nat__eqI,axiom,
% 9.04/5.06      ! [R2: nat,N: nat,M: nat] :
% 9.04/5.06        ( ( ord_less_nat @ R2 @ N )
% 9.04/5.06       => ( ( ord_less_eq_nat @ R2 @ M )
% 9.04/5.06         => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 9.04/5.06           => ( ( modulo_modulo_nat @ M @ N )
% 9.04/5.06              = R2 ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6733_zdvd__mult__cancel1,axiom,
% 9.04/5.06      ! [M: int,N: int] :
% 9.04/5.06        ( ( M != zero_zero_int )
% 9.04/5.06       => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 9.04/5.06          = ( ( abs_abs_int @ N )
% 9.04/5.06            = one_one_int ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6734_mod__int__pos__iff,axiom,
% 9.04/5.06      ! [K: int,L: int] :
% 9.04/5.06        ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 9.04/5.06        = ( ( dvd_dvd_int @ L @ K )
% 9.04/5.06          | ( ( L = zero_zero_int )
% 9.04/5.06            & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 9.04/5.06          | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6735_exp__le,axiom,
% 9.04/5.06      ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6736_even__two__times__div__two,axiom,
% 9.04/5.06      ! [A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 9.04/5.06          = A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6737_even__two__times__div__two,axiom,
% 9.04/5.06      ! [A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.06          = A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6738_even__two__times__div__two,axiom,
% 9.04/5.06      ! [A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.04/5.06          = A ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6739_even__iff__mod__2__eq__zero,axiom,
% 9.04/5.06      ! [A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06          = zero_zero_nat ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6740_even__iff__mod__2__eq__zero,axiom,
% 9.04/5.06      ! [A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.06          = zero_zero_int ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6741_even__iff__mod__2__eq__zero,axiom,
% 9.04/5.06      ! [A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.06          = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6742_odd__iff__mod__2__eq__one,axiom,
% 9.04/5.06      ! [A: nat] :
% 9.04/5.06        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.06        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06          = one_one_nat ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6743_odd__iff__mod__2__eq__one,axiom,
% 9.04/5.06      ! [A: int] :
% 9.04/5.06        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.06        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.06          = one_one_int ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6744_odd__iff__mod__2__eq__one,axiom,
% 9.04/5.06      ! [A: code_integer] :
% 9.04/5.06        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.06        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.06          = one_one_Code_integer ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6745_power__mono__odd,axiom,
% 9.04/5.06      ! [N: nat,A: real,B: real] :
% 9.04/5.06        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( ord_less_eq_real @ A @ B )
% 9.04/5.06         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6746_power__mono__odd,axiom,
% 9.04/5.06      ! [N: nat,A: rat,B: rat] :
% 9.04/5.06        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( ord_less_eq_rat @ A @ B )
% 9.04/5.06         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6747_power__mono__odd,axiom,
% 9.04/5.06      ! [N: nat,A: int,B: int] :
% 9.04/5.06        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( ord_less_eq_int @ A @ B )
% 9.04/5.06         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6748_exp__divide__power__eq,axiom,
% 9.04/5.06      ! [N: nat,X: complex] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06       => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 9.04/5.06          = ( exp_complex @ X ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6749_exp__divide__power__eq,axiom,
% 9.04/5.06      ! [N: nat,X: real] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06       => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 9.04/5.06          = ( exp_real @ X ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6750_uminus__power__if,axiom,
% 9.04/5.06      ! [N: nat,A: real] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 9.04/5.06            = ( power_power_real @ A @ N ) ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 9.04/5.06            = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6751_uminus__power__if,axiom,
% 9.04/5.06      ! [N: nat,A: int] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 9.04/5.06            = ( power_power_int @ A @ N ) ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 9.04/5.06            = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6752_uminus__power__if,axiom,
% 9.04/5.06      ! [N: nat,A: complex] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 9.04/5.06            = ( power_power_complex @ A @ N ) ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 9.04/5.06            = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6753_uminus__power__if,axiom,
% 9.04/5.06      ! [N: nat,A: rat] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 9.04/5.06            = ( power_power_rat @ A @ N ) ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 9.04/5.06            = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6754_uminus__power__if,axiom,
% 9.04/5.06      ! [N: nat,A: code_integer] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 9.04/5.06            = ( power_8256067586552552935nteger @ A @ N ) ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 9.04/5.06            = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6755_odd__pos,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6756_power__even__abs,axiom,
% 9.04/5.06      ! [N: nat,A: rat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 9.04/5.06          = ( power_power_rat @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6757_power__even__abs,axiom,
% 9.04/5.06      ! [N: nat,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 9.04/5.06          = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6758_power__even__abs,axiom,
% 9.04/5.06      ! [N: nat,A: real] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 9.04/5.06          = ( power_power_real @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6759_power__even__abs,axiom,
% 9.04/5.06      ! [N: nat,A: int] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 9.04/5.06          = ( power_power_int @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6760_dvd__power__iff__le,axiom,
% 9.04/5.06      ! [K: nat,M: nat,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.04/5.06       => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 9.04/5.06          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6761_even__unset__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 9.04/5.06        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06          | ( M = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6762_even__unset__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 9.04/5.06        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06          | ( M = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6763_even__unset__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 9.04/5.06        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06          | ( M = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6764_even__set__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 9.04/5.06        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06          & ( M != zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6765_even__set__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 9.04/5.06        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06          & ( M != zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6766_even__set__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 9.04/5.06        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06          & ( M != zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6767_even__flip__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 9.04/5.06        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         != ( M = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6768_even__flip__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 9.04/5.06        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         != ( M = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6769_even__flip__bit__iff,axiom,
% 9.04/5.06      ! [M: nat,A: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 9.04/5.06        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         != ( M = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6770_tanh__altdef,axiom,
% 9.04/5.06      ( tanh_real
% 9.04/5.06      = ( ^ [X4: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6771_tanh__altdef,axiom,
% 9.04/5.06      ( tanh_complex
% 9.04/5.06      = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6772_even__diff__iff,axiom,
% 9.04/5.06      ! [K: int,L: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
% 9.04/5.06        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6773_even__add__abs__iff,axiom,
% 9.04/5.06      ! [K: int,L: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
% 9.04/5.06        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6774_even__abs__add__iff,axiom,
% 9.04/5.06      ! [K: int,L: int] :
% 9.04/5.06        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
% 9.04/5.06        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6775_exp__half__le2,axiom,
% 9.04/5.06      ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6776_oddE,axiom,
% 9.04/5.06      ! [A: code_integer] :
% 9.04/5.06        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ~ ! [B6: code_integer] :
% 9.04/5.06              ( A
% 9.04/5.06             != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B6 ) @ one_one_Code_integer ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6777_oddE,axiom,
% 9.04/5.06      ! [A: nat] :
% 9.04/5.06        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ~ ! [B6: nat] :
% 9.04/5.06              ( A
% 9.04/5.06             != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) @ one_one_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6778_oddE,axiom,
% 9.04/5.06      ! [A: int] :
% 9.04/5.06        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06       => ~ ! [B6: int] :
% 9.04/5.06              ( A
% 9.04/5.06             != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) @ one_one_int ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6779_parity__cases,axiom,
% 9.04/5.06      ! [A: nat] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06           != zero_zero_nat ) )
% 9.04/5.06       => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06           => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06             != one_one_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6780_parity__cases,axiom,
% 9.04/5.06      ! [A: int] :
% 9.04/5.06        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.06           != zero_zero_int ) )
% 9.04/5.06       => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06           => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.06             != one_one_int ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6781_parity__cases,axiom,
% 9.04/5.06      ! [A: code_integer] :
% 9.04/5.06        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.06           != zero_z3403309356797280102nteger ) )
% 9.04/5.06       => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06           => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.06             != one_one_Code_integer ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6782_mod2__eq__if,axiom,
% 9.04/5.06      ! [A: nat] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06            = zero_zero_nat ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06            = one_one_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6783_mod2__eq__if,axiom,
% 9.04/5.06      ! [A: int] :
% 9.04/5.06        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.06            = zero_zero_int ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.06            = one_one_int ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6784_mod2__eq__if,axiom,
% 9.04/5.06      ! [A: code_integer] :
% 9.04/5.06        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.06            = zero_z3403309356797280102nteger ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.06         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.06            = one_one_Code_integer ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6785_exp__double,axiom,
% 9.04/5.06      ! [Z: complex] :
% 9.04/5.06        ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 9.04/5.06        = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6786_exp__double,axiom,
% 9.04/5.06      ! [Z: real] :
% 9.04/5.06        ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 9.04/5.06        = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6787_zero__le__even__power,axiom,
% 9.04/5.06      ! [N: nat,A: real] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6788_zero__le__even__power,axiom,
% 9.04/5.06      ! [N: nat,A: rat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6789_zero__le__even__power,axiom,
% 9.04/5.06      ! [N: nat,A: int] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6790_zero__le__odd__power,axiom,
% 9.04/5.06      ! [N: nat,A: real] :
% 9.04/5.06        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 9.04/5.06          = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6791_zero__le__odd__power,axiom,
% 9.04/5.06      ! [N: nat,A: rat] :
% 9.04/5.06        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 9.04/5.06          = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6792_zero__le__odd__power,axiom,
% 9.04/5.06      ! [N: nat,A: int] :
% 9.04/5.06        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 9.04/5.06          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6793_zero__le__power__eq,axiom,
% 9.04/5.06      ! [A: real,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 9.04/5.06        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6794_zero__le__power__eq,axiom,
% 9.04/5.06      ! [A: rat,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 9.04/5.06        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6795_zero__le__power__eq,axiom,
% 9.04/5.06      ! [A: int,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 9.04/5.06        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6796_minus__one__power__iff,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 9.04/5.06            = one_one_real ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 9.04/5.06            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6797_minus__one__power__iff,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 9.04/5.06            = one_one_int ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 9.04/5.06            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6798_minus__one__power__iff,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 9.04/5.06            = one_one_complex ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 9.04/5.06            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6799_minus__one__power__iff,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 9.04/5.06            = one_one_rat ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 9.04/5.06            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6800_minus__one__power__iff,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 9.04/5.06            = one_one_Code_integer ) )
% 9.04/5.06        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 9.04/5.06            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6801_power__mono__even,axiom,
% 9.04/5.06      ! [N: nat,A: code_integer,B: code_integer] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 9.04/5.06         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6802_power__mono__even,axiom,
% 9.04/5.06      ! [N: nat,A: real,B: real] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 9.04/5.06         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6803_power__mono__even,axiom,
% 9.04/5.06      ! [N: nat,A: rat,B: rat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 9.04/5.06         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6804_power__mono__even,axiom,
% 9.04/5.06      ! [N: nat,A: int,B: int] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 9.04/5.06         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6805_central__binomial__odd,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.06          = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6806_add__0__iff,axiom,
% 9.04/5.06      ! [B: complex,A: complex] :
% 9.04/5.06        ( ( B
% 9.04/5.06          = ( plus_plus_complex @ B @ A ) )
% 9.04/5.06        = ( A = zero_zero_complex ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6807_add__0__iff,axiom,
% 9.04/5.06      ! [B: real,A: real] :
% 9.04/5.06        ( ( B
% 9.04/5.06          = ( plus_plus_real @ B @ A ) )
% 9.04/5.06        = ( A = zero_zero_real ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6808_add__0__iff,axiom,
% 9.04/5.06      ! [B: rat,A: rat] :
% 9.04/5.06        ( ( B
% 9.04/5.06          = ( plus_plus_rat @ B @ A ) )
% 9.04/5.06        = ( A = zero_zero_rat ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6809_add__0__iff,axiom,
% 9.04/5.06      ! [B: nat,A: nat] :
% 9.04/5.06        ( ( B
% 9.04/5.06          = ( plus_plus_nat @ B @ A ) )
% 9.04/5.06        = ( A = zero_zero_nat ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6810_add__0__iff,axiom,
% 9.04/5.06      ! [B: int,A: int] :
% 9.04/5.06        ( ( B
% 9.04/5.06          = ( plus_plus_int @ B @ A ) )
% 9.04/5.06        = ( A = zero_zero_int ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6811_crossproduct__noteq,axiom,
% 9.04/5.06      ! [A: complex,B: complex,C: complex,D: complex] :
% 9.04/5.06        ( ( ( A != B )
% 9.04/5.06          & ( C != D ) )
% 9.04/5.06        = ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) )
% 9.04/5.06         != ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6812_crossproduct__noteq,axiom,
% 9.04/5.06      ! [A: real,B: real,C: real,D: real] :
% 9.04/5.06        ( ( ( A != B )
% 9.04/5.06          & ( C != D ) )
% 9.04/5.06        = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 9.04/5.06         != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6813_crossproduct__noteq,axiom,
% 9.04/5.06      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.04/5.06        ( ( ( A != B )
% 9.04/5.06          & ( C != D ) )
% 9.04/5.06        = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 9.04/5.06         != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6814_crossproduct__noteq,axiom,
% 9.04/5.06      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.04/5.06        ( ( ( A != B )
% 9.04/5.06          & ( C != D ) )
% 9.04/5.06        = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 9.04/5.06         != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6815_crossproduct__noteq,axiom,
% 9.04/5.06      ! [A: int,B: int,C: int,D: int] :
% 9.04/5.06        ( ( ( A != B )
% 9.04/5.06          & ( C != D ) )
% 9.04/5.06        = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 9.04/5.06         != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6816_crossproduct__eq,axiom,
% 9.04/5.06      ! [W: complex,Y: complex,X: complex,Z: complex] :
% 9.04/5.06        ( ( ( plus_plus_complex @ ( times_times_complex @ W @ Y ) @ ( times_times_complex @ X @ Z ) )
% 9.04/5.06          = ( plus_plus_complex @ ( times_times_complex @ W @ Z ) @ ( times_times_complex @ X @ Y ) ) )
% 9.04/5.06        = ( ( W = X )
% 9.04/5.06          | ( Y = Z ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6817_crossproduct__eq,axiom,
% 9.04/5.06      ! [W: real,Y: real,X: real,Z: real] :
% 9.04/5.06        ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
% 9.04/5.06          = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
% 9.04/5.06        = ( ( W = X )
% 9.04/5.06          | ( Y = Z ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6818_crossproduct__eq,axiom,
% 9.04/5.06      ! [W: rat,Y: rat,X: rat,Z: rat] :
% 9.04/5.06        ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
% 9.04/5.06          = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
% 9.04/5.06        = ( ( W = X )
% 9.04/5.06          | ( Y = Z ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6819_crossproduct__eq,axiom,
% 9.04/5.06      ! [W: nat,Y: nat,X: nat,Z: nat] :
% 9.04/5.06        ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
% 9.04/5.06          = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
% 9.04/5.06        = ( ( W = X )
% 9.04/5.06          | ( Y = Z ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6820_crossproduct__eq,axiom,
% 9.04/5.06      ! [W: int,Y: int,X: int,Z: int] :
% 9.04/5.06        ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
% 9.04/5.06          = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
% 9.04/5.06        = ( ( W = X )
% 9.04/5.06          | ( Y = Z ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6821_list__decode_Ocases,axiom,
% 9.04/5.06      ! [X: nat] :
% 9.04/5.06        ( ( X != zero_zero_nat )
% 9.04/5.06       => ~ ! [N2: nat] :
% 9.04/5.06              ( X
% 9.04/5.06             != ( suc @ N2 ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6822_zero__less__power__eq,axiom,
% 9.04/5.06      ! [A: real,N: nat] :
% 9.04/5.06        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 9.04/5.06        = ( ( N = zero_zero_nat )
% 9.04/5.06          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            & ( A != zero_zero_real ) )
% 9.04/5.06          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6823_zero__less__power__eq,axiom,
% 9.04/5.06      ! [A: rat,N: nat] :
% 9.04/5.06        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 9.04/5.06        = ( ( N = zero_zero_nat )
% 9.04/5.06          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            & ( A != zero_zero_rat ) )
% 9.04/5.06          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6824_zero__less__power__eq,axiom,
% 9.04/5.06      ! [A: int,N: nat] :
% 9.04/5.06        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 9.04/5.06        = ( ( N = zero_zero_nat )
% 9.04/5.06          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            & ( A != zero_zero_int ) )
% 9.04/5.06          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6825_exp__bound__half,axiom,
% 9.04/5.06      ! [Z: real] :
% 9.04/5.06        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.06       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6826_exp__bound__half,axiom,
% 9.04/5.06      ! [Z: complex] :
% 9.04/5.06        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.06       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6827_exp__bound,axiom,
% 9.04/5.06      ! [X: real] :
% 9.04/5.06        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.06       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.06         => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6828_even__mask__div__iff_H,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( 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 ) ) )
% 9.04/5.06        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6829_even__mask__div__iff_H,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( 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 ) ) )
% 9.04/5.06        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6830_even__mask__div__iff_H,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( 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 ) ) )
% 9.04/5.06        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6831_power__le__zero__eq,axiom,
% 9.04/5.06      ! [A: real,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 9.04/5.06        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.04/5.06            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06              & ( A = zero_zero_real ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6832_power__le__zero__eq,axiom,
% 9.04/5.06      ! [A: rat,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 9.04/5.06        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.04/5.06            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06              & ( A = zero_zero_rat ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6833_power__le__zero__eq,axiom,
% 9.04/5.06      ! [A: int,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 9.04/5.06        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.04/5.06            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06              & ( A = zero_zero_int ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6834_even__mod__4__div__2,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.04/5.06          = ( suc @ zero_zero_nat ) )
% 9.04/5.06       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6835_real__exp__bound__lemma,axiom,
% 9.04/5.06      ! [X: real] :
% 9.04/5.06        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.06       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.06         => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6836_exp__ge__one__plus__x__over__n__power__n,axiom,
% 9.04/5.06      ! [N: nat,X: real] :
% 9.04/5.06        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
% 9.04/5.06       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06         => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6837_exp__ge__one__minus__x__over__n__power__n,axiom,
% 9.04/5.06      ! [X: real,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 9.04/5.06       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.06         => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6838_even__mask__div__iff,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( 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 ) ) )
% 9.04/5.06        = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            = zero_z3403309356797280102nteger )
% 9.04/5.06          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6839_even__mask__div__iff,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( 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 ) ) )
% 9.04/5.06        = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            = zero_zero_nat )
% 9.04/5.06          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6840_even__mask__div__iff,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( 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 ) ) )
% 9.04/5.06        = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            = zero_zero_int )
% 9.04/5.06          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6841_exp__bound__lemma,axiom,
% 9.04/5.06      ! [Z: real] :
% 9.04/5.06        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.06       => ( 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 ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6842_exp__bound__lemma,axiom,
% 9.04/5.06      ! [Z: complex] :
% 9.04/5.06        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.06       => ( 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 ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6843_odd__mod__4__div__2,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.04/5.06          = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.04/5.06       => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6844_Bernoulli__inequality__even,axiom,
% 9.04/5.06      ! [N: nat,X: real] :
% 9.04/5.06        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6845_exp__lower__Taylor__quadratic,axiom,
% 9.04/5.06      ! [X: real] :
% 9.04/5.06        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.06       => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6846_log__base__10__eq2,axiom,
% 9.04/5.06      ! [X: real] :
% 9.04/5.06        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.06       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.06          = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6847_tanh__real__altdef,axiom,
% 9.04/5.06      ( tanh_real
% 9.04/5.06      = ( ^ [X4: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6848_even__mult__exp__div__exp__iff,axiom,
% 9.04/5.06      ! [A: code_integer,M: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.06        = ( ( ord_less_nat @ N @ M )
% 9.04/5.06          | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            = zero_z3403309356797280102nteger )
% 9.04/5.06          | ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06            & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6849_even__mult__exp__div__exp__iff,axiom,
% 9.04/5.06      ! [A: nat,M: nat,N: nat] :
% 9.04/5.06        ( ( 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 ) ) )
% 9.04/5.06        = ( ( ord_less_nat @ N @ M )
% 9.04/5.06          | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            = zero_zero_nat )
% 9.04/5.06          | ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06            & ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6850_even__mult__exp__div__exp__iff,axiom,
% 9.04/5.06      ! [A: int,M: nat,N: nat] :
% 9.04/5.06        ( ( 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 ) ) )
% 9.04/5.06        = ( ( ord_less_nat @ N @ M )
% 9.04/5.06          | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.04/5.06            = zero_zero_int )
% 9.04/5.06          | ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06            & ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6851_triangle__def,axiom,
% 9.04/5.06      ( nat_triangle
% 9.04/5.06      = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6852_flip__bit__0,axiom,
% 9.04/5.06      ! [A: code_integer] :
% 9.04/5.06        ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 9.04/5.06        = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6853_flip__bit__0,axiom,
% 9.04/5.06      ! [A: int] :
% 9.04/5.06        ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 9.04/5.06        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6854_flip__bit__0,axiom,
% 9.04/5.06      ! [A: nat] :
% 9.04/5.06        ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 9.04/5.06        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6855_set__decode__Suc,axiom,
% 9.04/5.06      ! [N: nat,X: nat] :
% 9.04/5.06        ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
% 9.04/5.06        = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6856_VEBT__internal_Oheight_Opelims,axiom,
% 9.04/5.06      ! [X: vEBT_VEBT,Y: nat] :
% 9.04/5.06        ( ( ( vEBT_VEBT_height @ X )
% 9.04/5.06          = Y )
% 9.04/5.06       => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X )
% 9.04/5.06         => ( ! [A5: $o,B6: $o] :
% 9.04/5.06                ( ( X
% 9.04/5.06                  = ( vEBT_Leaf @ A5 @ B6 ) )
% 9.04/5.06               => ( ( Y = zero_zero_nat )
% 9.04/5.06                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A5 @ B6 ) ) ) )
% 9.04/5.06           => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.04/5.06                  ( ( X
% 9.04/5.06                    = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.06                 => ( ( Y
% 9.04/5.06                      = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList3 ) ) ) ) ) )
% 9.04/5.06                   => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6857_fact__double,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( 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 ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6858_fact__double,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( 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 ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6859_fact__double,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( 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 ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6860_mask__numeral,axiom,
% 9.04/5.06      ! [N: num] :
% 9.04/5.06        ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
% 9.04/5.06        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6861_mask__numeral,axiom,
% 9.04/5.06      ! [N: num] :
% 9.04/5.06        ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
% 9.04/5.06        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6862_mask__nat__positive__iff,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 9.04/5.06        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6863_of__bool__less__eq__iff,axiom,
% 9.04/5.06      ! [P: $o,Q: $o] :
% 9.04/5.06        ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 9.04/5.06        = ( P
% 9.04/5.06         => Q ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6864_of__bool__less__eq__iff,axiom,
% 9.04/5.06      ! [P: $o,Q: $o] :
% 9.04/5.06        ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 9.04/5.06        = ( P
% 9.04/5.06         => Q ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6865_of__bool__less__eq__iff,axiom,
% 9.04/5.06      ! [P: $o,Q: $o] :
% 9.04/5.06        ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 9.04/5.06        = ( P
% 9.04/5.06         => Q ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6866_of__bool__less__eq__iff,axiom,
% 9.04/5.06      ! [P: $o,Q: $o] :
% 9.04/5.06        ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 9.04/5.06        = ( P
% 9.04/5.06         => Q ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6867_of__bool__eq__0__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n1201886186963655149omplex @ P )
% 9.04/5.06          = zero_zero_complex )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6868_of__bool__eq__0__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n3304061248610475627l_real @ P )
% 9.04/5.06          = zero_zero_real )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6869_of__bool__eq__0__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 9.04/5.06          = zero_zero_rat )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6870_of__bool__eq__0__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 9.04/5.06          = zero_zero_nat )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6871_of__bool__eq__0__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n2684676970156552555ol_int @ P )
% 9.04/5.06          = zero_zero_int )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6872_of__bool__eq__0__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n356916108424825756nteger @ P )
% 9.04/5.06          = zero_z3403309356797280102nteger )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6873_of__bool__eq_I1_J,axiom,
% 9.04/5.06      ( ( zero_n1201886186963655149omplex @ $false )
% 9.04/5.06      = zero_zero_complex ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6874_of__bool__eq_I1_J,axiom,
% 9.04/5.06      ( ( zero_n3304061248610475627l_real @ $false )
% 9.04/5.06      = zero_zero_real ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6875_of__bool__eq_I1_J,axiom,
% 9.04/5.06      ( ( zero_n2052037380579107095ol_rat @ $false )
% 9.04/5.06      = zero_zero_rat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6876_of__bool__eq_I1_J,axiom,
% 9.04/5.06      ( ( zero_n2687167440665602831ol_nat @ $false )
% 9.04/5.06      = zero_zero_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6877_of__bool__eq_I1_J,axiom,
% 9.04/5.06      ( ( zero_n2684676970156552555ol_int @ $false )
% 9.04/5.06      = zero_zero_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6878_of__bool__eq_I1_J,axiom,
% 9.04/5.06      ( ( zero_n356916108424825756nteger @ $false )
% 9.04/5.06      = zero_z3403309356797280102nteger ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6879_of__bool__less__iff,axiom,
% 9.04/5.06      ! [P: $o,Q: $o] :
% 9.04/5.06        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 9.04/5.06        = ( ~ P
% 9.04/5.06          & Q ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6880_of__bool__less__iff,axiom,
% 9.04/5.06      ! [P: $o,Q: $o] :
% 9.04/5.06        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 9.04/5.06        = ( ~ P
% 9.04/5.06          & Q ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6881_of__bool__less__iff,axiom,
% 9.04/5.06      ! [P: $o,Q: $o] :
% 9.04/5.06        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 9.04/5.06        = ( ~ P
% 9.04/5.06          & Q ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6882_of__bool__less__iff,axiom,
% 9.04/5.06      ! [P: $o,Q: $o] :
% 9.04/5.06        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 9.04/5.06        = ( ~ P
% 9.04/5.06          & Q ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6883_of__bool__less__iff,axiom,
% 9.04/5.06      ! [P: $o,Q: $o] :
% 9.04/5.06        ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 9.04/5.06        = ( ~ P
% 9.04/5.06          & Q ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6884_of__bool__eq_I2_J,axiom,
% 9.04/5.06      ( ( zero_n1201886186963655149omplex @ $true )
% 9.04/5.06      = one_one_complex ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6885_of__bool__eq_I2_J,axiom,
% 9.04/5.06      ( ( zero_n3304061248610475627l_real @ $true )
% 9.04/5.06      = one_one_real ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6886_of__bool__eq_I2_J,axiom,
% 9.04/5.06      ( ( zero_n2052037380579107095ol_rat @ $true )
% 9.04/5.06      = one_one_rat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6887_of__bool__eq_I2_J,axiom,
% 9.04/5.06      ( ( zero_n2687167440665602831ol_nat @ $true )
% 9.04/5.06      = one_one_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6888_of__bool__eq_I2_J,axiom,
% 9.04/5.06      ( ( zero_n2684676970156552555ol_int @ $true )
% 9.04/5.06      = one_one_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6889_of__bool__eq_I2_J,axiom,
% 9.04/5.06      ( ( zero_n356916108424825756nteger @ $true )
% 9.04/5.06      = one_one_Code_integer ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6890_of__bool__eq__1__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n1201886186963655149omplex @ P )
% 9.04/5.06          = one_one_complex )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6891_of__bool__eq__1__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n3304061248610475627l_real @ P )
% 9.04/5.06          = one_one_real )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6892_of__bool__eq__1__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 9.04/5.06          = one_one_rat )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6893_of__bool__eq__1__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 9.04/5.06          = one_one_nat )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6894_of__bool__eq__1__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n2684676970156552555ol_int @ P )
% 9.04/5.06          = one_one_int )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6895_of__bool__eq__1__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ( zero_n356916108424825756nteger @ P )
% 9.04/5.06          = one_one_Code_integer )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6896_of__nat__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri681578069525770553at_rat @ ( semiri1408675320244567234ct_nat @ N ) )
% 9.04/5.06        = ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6897_of__nat__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri1314217659103216013at_int @ ( semiri1408675320244567234ct_nat @ N ) )
% 9.04/5.06        = ( semiri1406184849735516958ct_int @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6898_of__nat__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri1316708129612266289at_nat @ ( semiri1408675320244567234ct_nat @ N ) )
% 9.04/5.06        = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6899_of__nat__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri5074537144036343181t_real @ ( semiri1408675320244567234ct_nat @ N ) )
% 9.04/5.06        = ( semiri2265585572941072030t_real @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6900_of__nat__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri8010041392384452111omplex @ ( semiri1408675320244567234ct_nat @ N ) )
% 9.04/5.06        = ( semiri5044797733671781792omplex @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6901_of__nat__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.04/5.06        = ( zero_n1201886186963655149omplex @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6902_of__nat__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.04/5.06        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6903_of__nat__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.04/5.06        = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6904_of__nat__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.04/5.06        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6905_of__nat__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.04/5.06        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6906_of__nat__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.04/5.06        = ( zero_n356916108424825756nteger @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6907_norm__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( real_V7735802525324610683m_real @ ( semiri2265585572941072030t_real @ N ) )
% 9.04/5.06        = ( semiri2265585572941072030t_real @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6908_norm__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( real_V1022390504157884413omplex @ ( semiri5044797733671781792omplex @ N ) )
% 9.04/5.06        = ( semiri2265585572941072030t_real @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6909_of__int__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ring_1_of_int_rat @ ( semiri1406184849735516958ct_int @ N ) )
% 9.04/5.06        = ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6910_of__int__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ring_18347121197199848620nteger @ ( semiri1406184849735516958ct_int @ N ) )
% 9.04/5.06        = ( semiri3624122377584611663nteger @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6911_of__int__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ring_1_of_int_real @ ( semiri1406184849735516958ct_int @ N ) )
% 9.04/5.06        = ( semiri2265585572941072030t_real @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6912_of__int__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ring_17405671764205052669omplex @ ( semiri1406184849735516958ct_int @ N ) )
% 9.04/5.06        = ( semiri5044797733671781792omplex @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6913_of__int__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.04/5.06        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6914_of__int__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.04/5.06        = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6915_of__int__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ring_17405671764205052669omplex @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.04/5.06        = ( zero_n1201886186963655149omplex @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6916_of__int__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.04/5.06        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6917_of__int__of__bool,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.04/5.06        = ( zero_n356916108424825756nteger @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6918_of__real__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( real_V1803761363581548252l_real @ ( semiri2265585572941072030t_real @ N ) )
% 9.04/5.06        = ( semiri2265585572941072030t_real @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6919_of__real__fact,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( real_V4546457046886955230omplex @ ( semiri2265585572941072030t_real @ N ) )
% 9.04/5.06        = ( semiri5044797733671781792omplex @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6920_triangle__0,axiom,
% 9.04/5.06      ( ( nat_triangle @ zero_zero_nat )
% 9.04/5.06      = zero_zero_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6921_zero__less__of__bool__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6922_zero__less__of__bool__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6923_zero__less__of__bool__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6924_zero__less__of__bool__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6925_zero__less__of__bool__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 9.04/5.06        = P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6926_of__bool__less__one__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6927_of__bool__less__one__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6928_of__bool__less__one__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6929_of__bool__less__one__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6930_of__bool__less__one__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 9.04/5.06        = ~ P ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6931_fact__0,axiom,
% 9.04/5.06      ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 9.04/5.06      = one_one_rat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6932_fact__0,axiom,
% 9.04/5.06      ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 9.04/5.06      = one_one_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6933_fact__0,axiom,
% 9.04/5.06      ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 9.04/5.06      = one_one_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6934_fact__0,axiom,
% 9.04/5.06      ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 9.04/5.06      = one_one_real ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6935_fact__0,axiom,
% 9.04/5.06      ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 9.04/5.06      = one_one_complex ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6936_of__bool__not__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( zero_n1201886186963655149omplex @ ~ P )
% 9.04/5.06        = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6937_of__bool__not__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( zero_n3304061248610475627l_real @ ~ P )
% 9.04/5.06        = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6938_of__bool__not__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 9.04/5.06        = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6939_of__bool__not__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( zero_n2684676970156552555ol_int @ ~ P )
% 9.04/5.06        = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6940_of__bool__not__iff,axiom,
% 9.04/5.06      ! [P: $o] :
% 9.04/5.06        ( ( zero_n356916108424825756nteger @ ~ P )
% 9.04/5.06        = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6941_Suc__0__mod__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.04/5.06        = ( zero_n2687167440665602831ol_nat
% 9.04/5.06          @ ( N
% 9.04/5.06           != ( suc @ zero_zero_nat ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6942_mask__eq__0__iff,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ( bit_se2002935070580805687sk_nat @ N )
% 9.04/5.06          = zero_zero_nat )
% 9.04/5.06        = ( N = zero_zero_nat ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6943_mask__eq__0__iff,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ( bit_se2000444600071755411sk_int @ N )
% 9.04/5.06          = zero_zero_int )
% 9.04/5.06        = ( N = zero_zero_nat ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6944_mask__0,axiom,
% 9.04/5.06      ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
% 9.04/5.06      = zero_zero_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6945_mask__0,axiom,
% 9.04/5.06      ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
% 9.04/5.06      = zero_zero_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6946_fact__1,axiom,
% 9.04/5.06      ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 9.04/5.06      = one_one_rat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6947_fact__1,axiom,
% 9.04/5.06      ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 9.04/5.06      = one_one_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6948_fact__1,axiom,
% 9.04/5.06      ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 9.04/5.06      = one_one_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6949_fact__1,axiom,
% 9.04/5.06      ( ( semiri2265585572941072030t_real @ one_one_nat )
% 9.04/5.06      = one_one_real ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6950_fact__1,axiom,
% 9.04/5.06      ( ( semiri5044797733671781792omplex @ one_one_nat )
% 9.04/5.06      = one_one_complex ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6951_triangle__Suc,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( nat_triangle @ ( suc @ N ) )
% 9.04/5.06        = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6952_fact__Suc__0,axiom,
% 9.04/5.06      ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 9.04/5.06      = one_one_rat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6953_fact__Suc__0,axiom,
% 9.04/5.06      ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 9.04/5.06      = one_one_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6954_fact__Suc__0,axiom,
% 9.04/5.06      ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 9.04/5.06      = one_one_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6955_fact__Suc__0,axiom,
% 9.04/5.06      ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 9.04/5.06      = one_one_real ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6956_fact__Suc__0,axiom,
% 9.04/5.06      ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 9.04/5.06      = one_one_complex ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6957_fact__Suc,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 9.04/5.06        = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6958_fact__Suc,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 9.04/5.06        = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6959_fact__Suc,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 9.04/5.06        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6960_fact__Suc,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 9.04/5.06        = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6961_fact__Suc,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
% 9.04/5.06        = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6962_mask__Suc__0,axiom,
% 9.04/5.06      ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
% 9.04/5.06      = one_one_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6963_mask__Suc__0,axiom,
% 9.04/5.06      ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
% 9.04/5.06      = one_one_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6964_fact__2,axiom,
% 9.04/5.06      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6965_fact__2,axiom,
% 9.04/5.06      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6966_fact__2,axiom,
% 9.04/5.06      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6967_fact__2,axiom,
% 9.04/5.06      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6968_fact__2,axiom,
% 9.04/5.06      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6969_odd__of__bool__self,axiom,
% 9.04/5.06      ! [P5: $o] :
% 9.04/5.06        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P5 ) ) )
% 9.04/5.06        = P5 ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6970_odd__of__bool__self,axiom,
% 9.04/5.06      ! [P5: $o] :
% 9.04/5.06        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P5 ) ) )
% 9.04/5.06        = P5 ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6971_odd__of__bool__self,axiom,
% 9.04/5.06      ! [P5: $o] :
% 9.04/5.06        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P5 ) ) )
% 9.04/5.06        = P5 ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6972_of__bool__half__eq__0,axiom,
% 9.04/5.06      ! [B: $o] :
% 9.04/5.06        ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.06        = zero_zero_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6973_of__bool__half__eq__0,axiom,
% 9.04/5.06      ! [B: $o] :
% 9.04/5.06        ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.06        = zero_zero_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6974_of__bool__half__eq__0,axiom,
% 9.04/5.06      ! [B: $o] :
% 9.04/5.06        ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.06        = zero_z3403309356797280102nteger ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6975_set__decode__0,axiom,
% 9.04/5.06      ! [X: nat] :
% 9.04/5.06        ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 9.04/5.06        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6976_bits__1__div__exp,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6977_bits__1__div__exp,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6978_bits__1__div__exp,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6979_one__div__2__pow__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6980_one__div__2__pow__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6981_one__div__2__pow__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6982_one__mod__2__pow__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6983_one__mod__2__pow__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6984_one__mod__2__pow__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.06        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6985_dvd__antisym,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( dvd_dvd_nat @ M @ N )
% 9.04/5.06       => ( ( dvd_dvd_nat @ N @ M )
% 9.04/5.06         => ( M = N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6986_of__nat__mask__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 9.04/5.06        = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6987_of__nat__mask__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
% 9.04/5.06        = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6988_of__int__mask__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ring_18347121197199848620nteger @ ( bit_se2000444600071755411sk_int @ N ) )
% 9.04/5.06        = ( bit_se2119862282449309892nteger @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6989_of__int__mask__eq,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
% 9.04/5.06        = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6990_fact__ge__self,axiom,
% 9.04/5.06      ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6991_fact__mono__nat,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.06       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6992_fact__nonzero,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri773545260158071498ct_rat @ N )
% 9.04/5.06       != zero_zero_rat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6993_fact__nonzero,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri1406184849735516958ct_int @ N )
% 9.04/5.06       != zero_zero_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6994_fact__nonzero,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri1408675320244567234ct_nat @ N )
% 9.04/5.06       != zero_zero_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6995_fact__nonzero,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri2265585572941072030t_real @ N )
% 9.04/5.06       != zero_zero_real ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6996_fact__nonzero,axiom,
% 9.04/5.06      ! [N: nat] :
% 9.04/5.06        ( ( semiri5044797733671781792omplex @ N )
% 9.04/5.06       != zero_zero_complex ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6997_less__eq__mask,axiom,
% 9.04/5.06      ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6998_subset__decode__imp__le,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 9.04/5.06       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_6999_fact__less__mono__nat,axiom,
% 9.04/5.06      ! [M: nat,N: nat] :
% 9.04/5.06        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.06       => ( ( ord_less_nat @ M @ N )
% 9.04/5.06         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7000_zero__less__eq__of__bool,axiom,
% 9.04/5.06      ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7001_zero__less__eq__of__bool,axiom,
% 9.04/5.06      ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7002_zero__less__eq__of__bool,axiom,
% 9.04/5.06      ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7003_zero__less__eq__of__bool,axiom,
% 9.04/5.06      ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7004_zero__less__eq__of__bool,axiom,
% 9.04/5.06      ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7005_of__bool__less__eq__one,axiom,
% 9.04/5.06      ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7006_of__bool__less__eq__one,axiom,
% 9.04/5.06      ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7007_of__bool__less__eq__one,axiom,
% 9.04/5.06      ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7008_of__bool__less__eq__one,axiom,
% 9.04/5.06      ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7009_of__bool__less__eq__one,axiom,
% 9.04/5.06      ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7010_of__bool__def,axiom,
% 9.04/5.06      ( zero_n1201886186963655149omplex
% 9.04/5.06      = ( ^ [P4: $o] : ( if_complex @ P4 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7011_of__bool__def,axiom,
% 9.04/5.06      ( zero_n3304061248610475627l_real
% 9.04/5.06      = ( ^ [P4: $o] : ( if_real @ P4 @ one_one_real @ zero_zero_real ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7012_of__bool__def,axiom,
% 9.04/5.06      ( zero_n2052037380579107095ol_rat
% 9.04/5.06      = ( ^ [P4: $o] : ( if_rat @ P4 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7013_of__bool__def,axiom,
% 9.04/5.06      ( zero_n2687167440665602831ol_nat
% 9.04/5.06      = ( ^ [P4: $o] : ( if_nat @ P4 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 9.04/5.06  
% 9.04/5.06  thf(fact_7014_of__bool__def,axiom,
% 9.04/5.06      ( zero_n2684676970156552555ol_int
% 9.04/5.06      = ( ^ [P4: $o] : ( if_int @ P4 @ one_one_int @ zero_zero_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7015_of__bool__def,axiom,
% 9.04/5.07      ( zero_n356916108424825756nteger
% 9.04/5.07      = ( ^ [P4: $o] : ( if_Code_integer @ P4 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7016_split__of__bool,axiom,
% 9.04/5.07      ! [P: complex > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
% 9.04/5.07        = ( ( P5
% 9.04/5.07           => ( P @ one_one_complex ) )
% 9.04/5.07          & ( ~ P5
% 9.04/5.07           => ( P @ zero_zero_complex ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7017_split__of__bool,axiom,
% 9.04/5.07      ! [P: real > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
% 9.04/5.07        = ( ( P5
% 9.04/5.07           => ( P @ one_one_real ) )
% 9.04/5.07          & ( ~ P5
% 9.04/5.07           => ( P @ zero_zero_real ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7018_split__of__bool,axiom,
% 9.04/5.07      ! [P: rat > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
% 9.04/5.07        = ( ( P5
% 9.04/5.07           => ( P @ one_one_rat ) )
% 9.04/5.07          & ( ~ P5
% 9.04/5.07           => ( P @ zero_zero_rat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7019_split__of__bool,axiom,
% 9.04/5.07      ! [P: nat > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
% 9.04/5.07        = ( ( P5
% 9.04/5.07           => ( P @ one_one_nat ) )
% 9.04/5.07          & ( ~ P5
% 9.04/5.07           => ( P @ zero_zero_nat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7020_split__of__bool,axiom,
% 9.04/5.07      ! [P: int > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
% 9.04/5.07        = ( ( P5
% 9.04/5.07           => ( P @ one_one_int ) )
% 9.04/5.07          & ( ~ P5
% 9.04/5.07           => ( P @ zero_zero_int ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7021_split__of__bool,axiom,
% 9.04/5.07      ! [P: code_integer > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
% 9.04/5.07        = ( ( P5
% 9.04/5.07           => ( P @ one_one_Code_integer ) )
% 9.04/5.07          & ( ~ P5
% 9.04/5.07           => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7022_split__of__bool__asm,axiom,
% 9.04/5.07      ! [P: complex > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
% 9.04/5.07        = ( ~ ( ( P5
% 9.04/5.07                & ~ ( P @ one_one_complex ) )
% 9.04/5.07              | ( ~ P5
% 9.04/5.07                & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7023_split__of__bool__asm,axiom,
% 9.04/5.07      ! [P: real > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
% 9.04/5.07        = ( ~ ( ( P5
% 9.04/5.07                & ~ ( P @ one_one_real ) )
% 9.04/5.07              | ( ~ P5
% 9.04/5.07                & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7024_split__of__bool__asm,axiom,
% 9.04/5.07      ! [P: rat > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
% 9.04/5.07        = ( ~ ( ( P5
% 9.04/5.07                & ~ ( P @ one_one_rat ) )
% 9.04/5.07              | ( ~ P5
% 9.04/5.07                & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7025_split__of__bool__asm,axiom,
% 9.04/5.07      ! [P: nat > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
% 9.04/5.07        = ( ~ ( ( P5
% 9.04/5.07                & ~ ( P @ one_one_nat ) )
% 9.04/5.07              | ( ~ P5
% 9.04/5.07                & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7026_split__of__bool__asm,axiom,
% 9.04/5.07      ! [P: int > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
% 9.04/5.07        = ( ~ ( ( P5
% 9.04/5.07                & ~ ( P @ one_one_int ) )
% 9.04/5.07              | ( ~ P5
% 9.04/5.07                & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7027_split__of__bool__asm,axiom,
% 9.04/5.07      ! [P: code_integer > $o,P5: $o] :
% 9.04/5.07        ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
% 9.04/5.07        = ( ~ ( ( P5
% 9.04/5.07                & ~ ( P @ one_one_Code_integer ) )
% 9.04/5.07              | ( ~ P5
% 9.04/5.07                & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7028_dvd__fact,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 9.04/5.07       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07         => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7029_fact__ge__zero,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7030_fact__ge__zero,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7031_fact__ge__zero,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7032_fact__ge__zero,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7033_fact__not__neg,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7034_fact__not__neg,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7035_fact__not__neg,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7036_fact__not__neg,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7037_fact__gt__zero,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7038_fact__gt__zero,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7039_fact__gt__zero,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7040_fact__gt__zero,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7041_fact__ge__1,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7042_fact__ge__1,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7043_fact__ge__1,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7044_fact__ge__1,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7045_fact__mono,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07       => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7046_fact__mono,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07       => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7047_fact__mono,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7048_fact__mono,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07       => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7049_fact__dvd,axiom,
% 9.04/5.07      ! [N: nat,M: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.07       => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7050_fact__dvd,axiom,
% 9.04/5.07      ! [N: nat,M: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.07       => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7051_fact__dvd,axiom,
% 9.04/5.07      ! [N: nat,M: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.07       => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7052_fact__dvd,axiom,
% 9.04/5.07      ! [N: nat,M: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.07       => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7053_pochhammer__fact,axiom,
% 9.04/5.07      ( semiri773545260158071498ct_rat
% 9.04/5.07      = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7054_pochhammer__fact,axiom,
% 9.04/5.07      ( semiri1406184849735516958ct_int
% 9.04/5.07      = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7055_pochhammer__fact,axiom,
% 9.04/5.07      ( semiri1408675320244567234ct_nat
% 9.04/5.07      = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7056_pochhammer__fact,axiom,
% 9.04/5.07      ( semiri2265585572941072030t_real
% 9.04/5.07      = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7057_pochhammer__fact,axiom,
% 9.04/5.07      ( semiri5044797733671781792omplex
% 9.04/5.07      = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7058_mask__nonnegative__int,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7059_not__mask__negative__int,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7060_fact__ge__Suc__0__nat,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7061_fact__less__mono,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.07       => ( ( ord_less_nat @ M @ N )
% 9.04/5.07         => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7062_fact__less__mono,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.07       => ( ( ord_less_nat @ M @ N )
% 9.04/5.07         => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7063_fact__less__mono,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.07       => ( ( ord_less_nat @ M @ N )
% 9.04/5.07         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7064_fact__less__mono,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.07       => ( ( ord_less_nat @ M @ N )
% 9.04/5.07         => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7065_fact__fact__dvd__fact,axiom,
% 9.04/5.07      ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7066_fact__fact__dvd__fact,axiom,
% 9.04/5.07      ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7067_fact__fact__dvd__fact,axiom,
% 9.04/5.07      ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7068_fact__fact__dvd__fact,axiom,
% 9.04/5.07      ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7069_fact__fact__dvd__fact,axiom,
% 9.04/5.07      ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7070_fact__mod,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07       => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 9.04/5.07          = zero_zero_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7071_fact__mod,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07       => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 9.04/5.07          = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7072_fact__mod,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07       => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 9.04/5.07          = zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7073_fact__le__power,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7074_fact__le__power,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7075_fact__le__power,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7076_fact__le__power,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7077_less__mask,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.04/5.07       => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7078_fact__diff__Suc,axiom,
% 9.04/5.07      ! [N: nat,M: nat] :
% 9.04/5.07        ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.04/5.07       => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 9.04/5.07          = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7079_fact__div__fact__le__pow,axiom,
% 9.04/5.07      ! [R2: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ R2 @ N )
% 9.04/5.07       => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7080_binomial__fact__lemma,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 9.04/5.07          = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7081_choose__dvd,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7082_choose__dvd,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7083_choose__dvd,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7084_choose__dvd,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7085_choose__dvd,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7086_fact__numeral,axiom,
% 9.04/5.07      ! [K: num] :
% 9.04/5.07        ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 9.04/5.07        = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7087_fact__numeral,axiom,
% 9.04/5.07      ! [K: num] :
% 9.04/5.07        ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 9.04/5.07        = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7088_fact__numeral,axiom,
% 9.04/5.07      ! [K: num] :
% 9.04/5.07        ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 9.04/5.07        = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7089_fact__numeral,axiom,
% 9.04/5.07      ! [K: num] :
% 9.04/5.07        ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 9.04/5.07        = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7090_fact__numeral,axiom,
% 9.04/5.07      ! [K: num] :
% 9.04/5.07        ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 9.04/5.07        = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7091_of__bool__odd__eq__mod__2,axiom,
% 9.04/5.07      ! [A: nat] :
% 9.04/5.07        ( ( zero_n2687167440665602831ol_nat
% 9.04/5.07          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.07        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7092_of__bool__odd__eq__mod__2,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( zero_n2684676970156552555ol_int
% 9.04/5.07          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.07        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7093_of__bool__odd__eq__mod__2,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( zero_n356916108424825756nteger
% 9.04/5.07          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.07        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7094_binomial__altdef__nat,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( ( binomial @ N @ K )
% 9.04/5.07          = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7095_square__fact__le__2__fact,axiom,
% 9.04/5.07      ! [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 ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7096_Suc__mask__eq__exp,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 9.04/5.07        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7097_mask__nat__less__exp,axiom,
% 9.04/5.07      ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7098_bits__induct,axiom,
% 9.04/5.07      ! [P: nat > $o,A: nat] :
% 9.04/5.07        ( ! [A5: nat] :
% 9.04/5.07            ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.07              = A5 )
% 9.04/5.07           => ( P @ A5 ) )
% 9.04/5.07       => ( ! [A5: nat,B6: $o] :
% 9.04/5.07              ( ( P @ A5 )
% 9.04/5.07             => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B6 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.07                  = A5 )
% 9.04/5.07               => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B6 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 9.04/5.07         => ( P @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7099_bits__induct,axiom,
% 9.04/5.07      ! [P: int > $o,A: int] :
% 9.04/5.07        ( ! [A5: int] :
% 9.04/5.07            ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.07              = A5 )
% 9.04/5.07           => ( P @ A5 ) )
% 9.04/5.07       => ( ! [A5: int,B6: $o] :
% 9.04/5.07              ( ( P @ A5 )
% 9.04/5.07             => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B6 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.07                  = A5 )
% 9.04/5.07               => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B6 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 9.04/5.07         => ( P @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7100_bits__induct,axiom,
% 9.04/5.07      ! [P: code_integer > $o,A: code_integer] :
% 9.04/5.07        ( ! [A5: code_integer] :
% 9.04/5.07            ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.07              = A5 )
% 9.04/5.07           => ( P @ A5 ) )
% 9.04/5.07       => ( ! [A5: code_integer,B6: $o] :
% 9.04/5.07              ( ( P @ A5 )
% 9.04/5.07             => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B6 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.07                  = A5 )
% 9.04/5.07               => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B6 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 9.04/5.07         => ( P @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7101_fact__num__eq__if,axiom,
% 9.04/5.07      ( semiri773545260158071498ct_rat
% 9.04/5.07      = ( ^ [M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M6 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7102_fact__num__eq__if,axiom,
% 9.04/5.07      ( semiri1406184849735516958ct_int
% 9.04/5.07      = ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7103_fact__num__eq__if,axiom,
% 9.04/5.07      ( semiri1408675320244567234ct_nat
% 9.04/5.07      = ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7104_fact__num__eq__if,axiom,
% 9.04/5.07      ( semiri2265585572941072030t_real
% 9.04/5.07      = ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7105_fact__num__eq__if,axiom,
% 9.04/5.07      ( semiri5044797733671781792omplex
% 9.04/5.07      = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7106_fact__reduce,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.07       => ( ( semiri773545260158071498ct_rat @ N )
% 9.04/5.07          = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7107_fact__reduce,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.07       => ( ( semiri1406184849735516958ct_int @ N )
% 9.04/5.07          = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7108_fact__reduce,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.07       => ( ( semiri1408675320244567234ct_nat @ N )
% 9.04/5.07          = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7109_fact__reduce,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.07       => ( ( semiri2265585572941072030t_real @ N )
% 9.04/5.07          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7110_fact__reduce,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.07       => ( ( semiri5044797733671781792omplex @ N )
% 9.04/5.07          = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7111_pochhammer__same,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
% 9.04/5.07        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7112_pochhammer__same,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
% 9.04/5.07        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7113_pochhammer__same,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
% 9.04/5.07        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7114_pochhammer__same,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
% 9.04/5.07        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7115_pochhammer__same,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
% 9.04/5.07        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7116_binomial__fact,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 9.04/5.07          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7117_binomial__fact,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 9.04/5.07          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7118_binomial__fact,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 9.04/5.07          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7119_fact__binomial,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 9.04/5.07          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7120_fact__binomial,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 9.04/5.07          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7121_fact__binomial,axiom,
% 9.04/5.07      ! [K: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.07       => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 9.04/5.07          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7122_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
% 9.04/5.07        = ( N = zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7123_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
% 9.04/5.07        = ( N = zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7124_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
% 9.04/5.07        = ( N = zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7125_exp__mod__exp,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7126_exp__mod__exp,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7127_exp__mod__exp,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7128_mask__nat__def,axiom,
% 9.04/5.07      ( bit_se2002935070580805687sk_nat
% 9.04/5.07      = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7129_mask__half__int,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.07        = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7130_mask__int__def,axiom,
% 9.04/5.07      ( bit_se2000444600071755411sk_int
% 9.04/5.07      = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7131_mask__eq__exp__minus__1,axiom,
% 9.04/5.07      ( bit_se2002935070580805687sk_nat
% 9.04/5.07      = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7132_mask__eq__exp__minus__1,axiom,
% 9.04/5.07      ( bit_se2000444600071755411sk_int
% 9.04/5.07      = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7133_set__decode__plus__power__2,axiom,
% 9.04/5.07      ! [N: nat,Z: nat] :
% 9.04/5.07        ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 9.04/5.07       => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 9.04/5.07          = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7134_exp__div__exp__eq,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07        = ( times_times_nat
% 9.04/5.07          @ ( zero_n2687167440665602831ol_nat
% 9.04/5.07            @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.04/5.07               != zero_zero_nat )
% 9.04/5.07              & ( ord_less_eq_nat @ N @ M ) ) )
% 9.04/5.07          @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7135_exp__div__exp__eq,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07        = ( times_times_int
% 9.04/5.07          @ ( zero_n2684676970156552555ol_int
% 9.04/5.07            @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 9.04/5.07               != zero_zero_int )
% 9.04/5.07              & ( ord_less_eq_nat @ N @ M ) ) )
% 9.04/5.07          @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7136_exp__div__exp__eq,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07        = ( times_3573771949741848930nteger
% 9.04/5.07          @ ( zero_n356916108424825756nteger
% 9.04/5.07            @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 9.04/5.07               != zero_z3403309356797280102nteger )
% 9.04/5.07              & ( ord_less_eq_nat @ N @ M ) ) )
% 9.04/5.07          @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7137_sin__coeff__def,axiom,
% 9.04/5.07      ( sin_coeff
% 9.04/5.07      = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7138_binomial__code,axiom,
% 9.04/5.07      ( binomial
% 9.04/5.07      = ( ^ [N3: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K2 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7139_cos__coeff__def,axiom,
% 9.04/5.07      ( cos_coeff
% 9.04/5.07      = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7140_fact__code,axiom,
% 9.04/5.07      ( semiri773545260158071498ct_rat
% 9.04/5.07      = ( ^ [N3: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7141_fact__code,axiom,
% 9.04/5.07      ( semiri1406184849735516958ct_int
% 9.04/5.07      = ( ^ [N3: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7142_fact__code,axiom,
% 9.04/5.07      ( semiri1408675320244567234ct_nat
% 9.04/5.07      = ( ^ [N3: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7143_fact__code,axiom,
% 9.04/5.07      ( semiri2265585572941072030t_real
% 9.04/5.07      = ( ^ [N3: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7144_fact__code,axiom,
% 9.04/5.07      ( semiri5044797733671781792omplex
% 9.04/5.07      = ( ^ [N3: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7145_num_Osize__gen_I3_J,axiom,
% 9.04/5.07      ! [X32: num] :
% 9.04/5.07        ( ( size_num @ ( bit1 @ X32 ) )
% 9.04/5.07        = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7146_take__bit__rec,axiom,
% 9.04/5.07      ( bit_se1745604003318907178nteger
% 9.04/5.07      = ( ^ [N3: nat,A3: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7147_take__bit__rec,axiom,
% 9.04/5.07      ( bit_se2923211474154528505it_int
% 9.04/5.07      = ( ^ [N3: nat,A3: int] : ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7148_take__bit__rec,axiom,
% 9.04/5.07      ( bit_se2925701944663578781it_nat
% 9.04/5.07      = ( ^ [N3: nat,A3: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7149_take__bit__of__0,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 9.04/5.07        = zero_zero_int ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7150_take__bit__of__0,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 9.04/5.07        = zero_zero_nat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7151_concat__bit__of__zero__2,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 9.04/5.07        = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7152_take__bit__0,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 9.04/5.07        = zero_zero_int ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7153_take__bit__0,axiom,
% 9.04/5.07      ! [A: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 9.04/5.07        = zero_zero_nat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7154_take__bit__Suc__1,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
% 9.04/5.07        = one_one_int ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7155_take__bit__Suc__1,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
% 9.04/5.07        = one_one_nat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7156_take__bit__numeral__1,axiom,
% 9.04/5.07      ! [L: num] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
% 9.04/5.07        = one_one_int ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7157_take__bit__numeral__1,axiom,
% 9.04/5.07      ! [L: num] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
% 9.04/5.07        = one_one_nat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7158_sin__coeff__0,axiom,
% 9.04/5.07      ( ( sin_coeff @ zero_zero_nat )
% 9.04/5.07      = zero_zero_real ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7159_cos__coeff__0,axiom,
% 9.04/5.07      ( ( cos_coeff @ zero_zero_nat )
% 9.04/5.07      = one_one_real ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7160_take__bit__of__1__eq__0__iff,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 9.04/5.07          = zero_zero_int )
% 9.04/5.07        = ( N = zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7161_take__bit__of__1__eq__0__iff,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 9.04/5.07          = zero_zero_nat )
% 9.04/5.07        = ( N = zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7162_take__bit__minus__one__eq__mask,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.07        = ( bit_se2119862282449309892nteger @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7163_take__bit__minus__one__eq__mask,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.07        = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7164_take__bit__of__Suc__0,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.04/5.07        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7165_take__bit__of__1,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
% 9.04/5.07        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7166_take__bit__of__1,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 9.04/5.07        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7167_take__bit__of__1,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 9.04/5.07        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7168_even__take__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,A: code_integer] :
% 9.04/5.07        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 9.04/5.07        = ( ( N = zero_zero_nat )
% 9.04/5.07          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7169_even__take__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,A: int] :
% 9.04/5.07        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 9.04/5.07        = ( ( N = zero_zero_nat )
% 9.04/5.07          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7170_even__take__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,A: nat] :
% 9.04/5.07        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 9.04/5.07        = ( ( N = zero_zero_nat )
% 9.04/5.07          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7171_take__bit__Suc__0,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 9.04/5.07        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7172_take__bit__Suc__0,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 9.04/5.07        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7173_take__bit__Suc__0,axiom,
% 9.04/5.07      ! [A: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 9.04/5.07        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7174_take__bit__of__exp,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7175_take__bit__of__exp,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7176_take__bit__of__exp,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7177_take__bit__of__2,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.07        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7178_take__bit__of__2,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.07        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7179_take__bit__of__2,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.07        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7180_take__bit__add,axiom,
% 9.04/5.07      ! [N: nat,A: int,B: int] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
% 9.04/5.07        = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7181_take__bit__add,axiom,
% 9.04/5.07      ! [N: nat,A: nat,B: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
% 9.04/5.07        = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7182_take__bit__of__int,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( bit_se1745604003318907178nteger @ N @ ( ring_18347121197199848620nteger @ K ) )
% 9.04/5.07        = ( ring_18347121197199848620nteger @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7183_take__bit__of__int,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
% 9.04/5.07        = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7184_take__bit__of__nat,axiom,
% 9.04/5.07      ! [N: nat,M: nat] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 9.04/5.07        = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7185_take__bit__of__nat,axiom,
% 9.04/5.07      ! [N: nat,M: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 9.04/5.07        = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7186_take__bit__tightened,axiom,
% 9.04/5.07      ! [N: nat,A: int,B: int,M: nat] :
% 9.04/5.07        ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 9.04/5.07          = ( bit_se2923211474154528505it_int @ N @ B ) )
% 9.04/5.07       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07         => ( ( bit_se2923211474154528505it_int @ M @ A )
% 9.04/5.07            = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7187_take__bit__tightened,axiom,
% 9.04/5.07      ! [N: nat,A: nat,B: nat,M: nat] :
% 9.04/5.07        ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 9.04/5.07          = ( bit_se2925701944663578781it_nat @ N @ B ) )
% 9.04/5.07       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07         => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 9.04/5.07            = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7188_take__bit__nat__less__eq__self,axiom,
% 9.04/5.07      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7189_take__bit__tightened__less__eq__nat,axiom,
% 9.04/5.07      ! [M: nat,N: nat,Q3: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07       => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7190_take__bit__tightened__less__eq__int,axiom,
% 9.04/5.07      ! [M: nat,N: nat,K: int] :
% 9.04/5.07        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.07       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7191_take__bit__int__less__eq__self__iff,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 9.04/5.07        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7192_take__bit__nonnegative,axiom,
% 9.04/5.07      ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7193_take__bit__int__greater__self__iff,axiom,
% 9.04/5.07      ! [K: int,N: nat] :
% 9.04/5.07        ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.04/5.07        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7194_not__take__bit__negative,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7195_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,A: int,B: int] :
% 9.04/5.07        ( ( ( bit_ri631733984087533419it_int @ N @ A )
% 9.04/5.07          = ( bit_ri631733984087533419it_int @ N @ B ) )
% 9.04/5.07        = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 9.04/5.07          = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7196_signed__take__bit__take__bit,axiom,
% 9.04/5.07      ! [M: nat,N: nat,A: int] :
% 9.04/5.07        ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 9.04/5.07        = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7197_take__bit__unset__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,M: nat,A: int] :
% 9.04/5.07        ( ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 9.04/5.07            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 9.04/5.07        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 9.04/5.07            = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7198_take__bit__unset__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,M: nat,A: nat] :
% 9.04/5.07        ( ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 9.04/5.07            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 9.04/5.07        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 9.04/5.07            = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7199_take__bit__set__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,M: nat,A: int] :
% 9.04/5.07        ( ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 9.04/5.07            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 9.04/5.07        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 9.04/5.07            = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7200_take__bit__set__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,M: nat,A: nat] :
% 9.04/5.07        ( ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 9.04/5.07            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 9.04/5.07        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 9.04/5.07            = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7201_take__bit__flip__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,M: nat,A: int] :
% 9.04/5.07        ( ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 9.04/5.07            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 9.04/5.07        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 9.04/5.07            = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7202_take__bit__flip__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,M: nat,A: nat] :
% 9.04/5.07        ( ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 9.04/5.07            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 9.04/5.07        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.04/5.07         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 9.04/5.07            = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7203_sin__coeff__Suc,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( sin_coeff @ ( suc @ N ) )
% 9.04/5.07        = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7204_take__bit__signed__take__bit,axiom,
% 9.04/5.07      ! [M: nat,N: nat,A: int] :
% 9.04/5.07        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.04/5.07       => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 9.04/5.07          = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7205_take__bit__eq__mask__iff,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.04/5.07          = ( bit_se2000444600071755411sk_int @ N ) )
% 9.04/5.07        = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 9.04/5.07          = zero_zero_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7206_take__bit__decr__eq,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.04/5.07         != zero_zero_int )
% 9.04/5.07       => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 9.04/5.07          = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7207_cos__coeff__Suc,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( cos_coeff @ ( suc @ N ) )
% 9.04/5.07        = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7208_take__bit__Suc__bit0,axiom,
% 9.04/5.07      ! [N: nat,K: num] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 9.04/5.07        = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7209_take__bit__Suc__bit0,axiom,
% 9.04/5.07      ! [N: nat,K: num] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.04/5.07        = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7210_take__bit__eq__mod,axiom,
% 9.04/5.07      ( bit_se1745604003318907178nteger
% 9.04/5.07      = ( ^ [N3: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7211_take__bit__eq__mod,axiom,
% 9.04/5.07      ( bit_se2923211474154528505it_int
% 9.04/5.07      = ( ^ [N3: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7212_take__bit__eq__mod,axiom,
% 9.04/5.07      ( bit_se2925701944663578781it_nat
% 9.04/5.07      = ( ^ [N3: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7213_take__bit__nat__eq__self,axiom,
% 9.04/5.07      ! [M: nat,N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07       => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 9.04/5.07          = M ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7214_take__bit__nat__less__exp,axiom,
% 9.04/5.07      ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7215_take__bit__nat__eq__self__iff,axiom,
% 9.04/5.07      ! [N: nat,M: nat] :
% 9.04/5.07        ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 9.04/5.07          = M )
% 9.04/5.07        = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7216_take__bit__nat__def,axiom,
% 9.04/5.07      ( bit_se2925701944663578781it_nat
% 9.04/5.07      = ( ^ [N3: nat,M6: nat] : ( modulo_modulo_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7217_take__bit__int__less__exp,axiom,
% 9.04/5.07      ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7218_fold__atLeastAtMost__nat_Oelims,axiom,
% 9.04/5.07      ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
% 9.04/5.07        ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
% 9.04/5.07          = Y )
% 9.04/5.07       => ( ( ( ord_less_nat @ Xb @ Xa )
% 9.04/5.07           => ( Y = Xc ) )
% 9.04/5.07          & ( ~ ( ord_less_nat @ Xb @ Xa )
% 9.04/5.07           => ( Y
% 9.04/5.07              = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7219_fold__atLeastAtMost__nat_Osimps,axiom,
% 9.04/5.07      ( set_fo2584398358068434914at_nat
% 9.04/5.07      = ( ^ [F2: nat > nat > nat,A3: nat,B4: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A3 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B4 @ ( F2 @ A3 @ Acc ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7220_take__bit__int__def,axiom,
% 9.04/5.07      ( bit_se2923211474154528505it_int
% 9.04/5.07      = ( ^ [N3: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7221_num_Osize__gen_I1_J,axiom,
% 9.04/5.07      ( ( size_num @ one )
% 9.04/5.07      = zero_zero_nat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7222_take__bit__eq__0__iff,axiom,
% 9.04/5.07      ! [N: nat,A: code_integer] :
% 9.04/5.07        ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 9.04/5.07          = zero_z3403309356797280102nteger )
% 9.04/5.07        = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7223_take__bit__eq__0__iff,axiom,
% 9.04/5.07      ! [N: nat,A: int] :
% 9.04/5.07        ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 9.04/5.07          = zero_zero_int )
% 9.04/5.07        = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7224_take__bit__eq__0__iff,axiom,
% 9.04/5.07      ! [N: nat,A: nat] :
% 9.04/5.07        ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 9.04/5.07          = zero_zero_nat )
% 9.04/5.07        = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7225_take__bit__numeral__bit0,axiom,
% 9.04/5.07      ! [L: num,K: num] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 9.04/5.07        = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7226_take__bit__numeral__bit0,axiom,
% 9.04/5.07      ! [L: num,K: num] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.04/5.07        = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7227_take__bit__nat__less__self__iff,axiom,
% 9.04/5.07      ! [N: nat,M: nat] :
% 9.04/5.07        ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 9.04/5.07        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7228_take__bit__Suc__minus__bit0,axiom,
% 9.04/5.07      ! [N: nat,K: num] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.04/5.07        = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7229_take__bit__int__less__self__iff,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 9.04/5.07        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7230_take__bit__int__greater__eq__self__iff,axiom,
% 9.04/5.07      ! [K: int,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.04/5.07        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7231_take__bit__int__eq__self,axiom,
% 9.04/5.07      ! [K: int,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.07       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07         => ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.04/5.07            = K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7232_take__bit__int__eq__self__iff,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.04/5.07          = K )
% 9.04/5.07        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.07          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7233_take__bit__numeral__minus__bit0,axiom,
% 9.04/5.07      ! [L: num,K: num] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.04/5.07        = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7234_take__bit__incr__eq,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.04/5.07         != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 9.04/5.07       => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 9.04/5.07          = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7235_take__bit__eq__mask__iff__exp__dvd,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.04/5.07          = ( bit_se2000444600071755411sk_int @ N ) )
% 9.04/5.07        = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7236_take__bit__Suc__minus__1__eq,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.07        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7237_take__bit__Suc__minus__1__eq,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.07        = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7238_take__bit__Suc__bit1,axiom,
% 9.04/5.07      ! [N: nat,K: num] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 9.04/5.07        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7239_take__bit__Suc__bit1,axiom,
% 9.04/5.07      ! [N: nat,K: num] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.04/5.07        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7240_take__bit__numeral__minus__1__eq,axiom,
% 9.04/5.07      ! [K: num] :
% 9.04/5.07        ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.07        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7241_take__bit__numeral__minus__1__eq,axiom,
% 9.04/5.07      ! [K: num] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.07        = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7242_take__bit__Suc,axiom,
% 9.04/5.07      ! [N: nat,A: code_integer] :
% 9.04/5.07        ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
% 9.04/5.07        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7243_take__bit__Suc,axiom,
% 9.04/5.07      ! [N: nat,A: int] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 9.04/5.07        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7244_take__bit__Suc,axiom,
% 9.04/5.07      ! [N: nat,A: nat] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
% 9.04/5.07        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7245_take__bit__int__less__eq,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 9.04/5.07       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.07         => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7246_take__bit__int__greater__eq,axiom,
% 9.04/5.07      ! [K: int,N: nat] :
% 9.04/5.07        ( ( ord_less_int @ K @ zero_zero_int )
% 9.04/5.07       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7247_signed__take__bit__eq__take__bit__shift,axiom,
% 9.04/5.07      ( bit_ri631733984087533419it_int
% 9.04/5.07      = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7248_stable__imp__take__bit__eq,axiom,
% 9.04/5.07      ! [A: code_integer,N: nat] :
% 9.04/5.07        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.07          = A )
% 9.04/5.07       => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.07           => ( ( bit_se1745604003318907178nteger @ N @ A )
% 9.04/5.07              = zero_z3403309356797280102nteger ) )
% 9.04/5.07          & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.07           => ( ( bit_se1745604003318907178nteger @ N @ A )
% 9.04/5.07              = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7249_stable__imp__take__bit__eq,axiom,
% 9.04/5.07      ! [A: int,N: nat] :
% 9.04/5.07        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.07          = A )
% 9.04/5.07       => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.07           => ( ( bit_se2923211474154528505it_int @ N @ A )
% 9.04/5.07              = zero_zero_int ) )
% 9.04/5.07          & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.07           => ( ( bit_se2923211474154528505it_int @ N @ A )
% 9.04/5.07              = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7250_stable__imp__take__bit__eq,axiom,
% 9.04/5.07      ! [A: nat,N: nat] :
% 9.04/5.07        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.07          = A )
% 9.04/5.07       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.07           => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 9.04/5.07              = zero_zero_nat ) )
% 9.04/5.07          & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.07           => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 9.04/5.07              = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7251_take__bit__numeral__bit1,axiom,
% 9.04/5.07      ! [L: num,K: num] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 9.04/5.07        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7252_take__bit__numeral__bit1,axiom,
% 9.04/5.07      ! [L: num,K: num] :
% 9.04/5.07        ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.04/5.07        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7253_take__bit__minus__small__eq,axiom,
% 9.04/5.07      ! [K: int,N: nat] :
% 9.04/5.07        ( ( ord_less_int @ zero_zero_int @ K )
% 9.04/5.07       => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.07         => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 9.04/5.07            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7254_num_Osize__gen_I2_J,axiom,
% 9.04/5.07      ! [X22: num] :
% 9.04/5.07        ( ( size_num @ ( bit0 @ X22 ) )
% 9.04/5.07        = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7255_take__bit__numeral__minus__bit1,axiom,
% 9.04/5.07      ! [L: num,K: num] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.04/5.07        = ( 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 ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7256_take__bit__Suc__minus__bit1,axiom,
% 9.04/5.07      ! [N: nat,K: num] :
% 9.04/5.07        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.04/5.07        = ( 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 ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7257_modulo__int__unfold,axiom,
% 9.04/5.07      ! [L: int,K: int,N: nat,M: nat] :
% 9.04/5.07        ( ( ( ( ( sgn_sgn_int @ L )
% 9.04/5.07              = zero_zero_int )
% 9.04/5.07            | ( ( sgn_sgn_int @ K )
% 9.04/5.07              = zero_zero_int )
% 9.04/5.07            | ( N = zero_zero_nat ) )
% 9.04/5.07         => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.04/5.07            = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 9.04/5.07        & ( ~ ( ( ( sgn_sgn_int @ L )
% 9.04/5.07                = zero_zero_int )
% 9.04/5.07              | ( ( sgn_sgn_int @ K )
% 9.04/5.07                = zero_zero_int )
% 9.04/5.07              | ( N = zero_zero_nat ) )
% 9.04/5.07         => ( ( ( ( sgn_sgn_int @ K )
% 9.04/5.07                = ( sgn_sgn_int @ L ) )
% 9.04/5.07             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.04/5.07                = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 9.04/5.07            & ( ( ( sgn_sgn_int @ K )
% 9.04/5.07               != ( sgn_sgn_int @ L ) )
% 9.04/5.07             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.04/5.07                = ( times_times_int @ ( sgn_sgn_int @ L )
% 9.04/5.07                  @ ( minus_minus_int
% 9.04/5.07                    @ ( semiri1314217659103216013at_int
% 9.04/5.07                      @ ( times_times_nat @ N
% 9.04/5.07                        @ ( zero_n2687167440665602831ol_nat
% 9.04/5.07                          @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 9.04/5.07                    @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7258_powr__int,axiom,
% 9.04/5.07      ! [X: real,I: int] :
% 9.04/5.07        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.07       => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.04/5.07           => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 9.04/5.07              = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
% 9.04/5.07          & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 9.04/5.07           => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 9.04/5.07              = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7259_divide__int__unfold,axiom,
% 9.04/5.07      ! [L: int,K: int,N: nat,M: nat] :
% 9.04/5.07        ( ( ( ( ( sgn_sgn_int @ L )
% 9.04/5.07              = zero_zero_int )
% 9.04/5.07            | ( ( sgn_sgn_int @ K )
% 9.04/5.07              = zero_zero_int )
% 9.04/5.07            | ( N = zero_zero_nat ) )
% 9.04/5.07         => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.04/5.07            = zero_zero_int ) )
% 9.04/5.07        & ( ~ ( ( ( sgn_sgn_int @ L )
% 9.04/5.07                = zero_zero_int )
% 9.04/5.07              | ( ( sgn_sgn_int @ K )
% 9.04/5.07                = zero_zero_int )
% 9.04/5.07              | ( N = zero_zero_nat ) )
% 9.04/5.07         => ( ( ( ( sgn_sgn_int @ K )
% 9.04/5.07                = ( sgn_sgn_int @ L ) )
% 9.04/5.07             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.04/5.07                = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 9.04/5.07            & ( ( ( sgn_sgn_int @ K )
% 9.04/5.07               != ( sgn_sgn_int @ L ) )
% 9.04/5.07             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.04/5.07                = ( uminus_uminus_int
% 9.04/5.07                  @ ( semiri1314217659103216013at_int
% 9.04/5.07                    @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 9.04/5.07                      @ ( zero_n2687167440665602831ol_nat
% 9.04/5.07                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7260_invar__vebt_Ocases,axiom,
% 9.04/5.07      ! [A1: vEBT_VEBT,A22: nat] :
% 9.04/5.07        ( ( vEBT_invar_vebt @ A1 @ A22 )
% 9.04/5.07       => ( ( ? [A5: $o,B6: $o] :
% 9.04/5.07                ( A1
% 9.04/5.07                = ( vEBT_Leaf @ A5 @ B6 ) )
% 9.04/5.07           => ( A22
% 9.04/5.07             != ( suc @ zero_zero_nat ) ) )
% 9.04/5.07         => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
% 9.04/5.07                ( ( A1
% 9.04/5.07                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.07               => ( ( A22 = Deg2 )
% 9.04/5.07                 => ( ! [X2: vEBT_VEBT] :
% 9.04/5.07                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.07                       => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 9.04/5.07                   => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
% 9.04/5.07                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.04/5.07                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.04/5.07                       => ( ( M5 = N2 )
% 9.04/5.07                         => ( ( Deg2
% 9.04/5.07                              = ( plus_plus_nat @ N2 @ M5 ) )
% 9.04/5.07                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
% 9.04/5.07                             => ~ ! [X2: vEBT_VEBT] :
% 9.04/5.07                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.07                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
% 9.04/5.07           => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
% 9.04/5.07                  ( ( A1
% 9.04/5.07                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.07                 => ( ( A22 = Deg2 )
% 9.04/5.07                   => ( ! [X2: vEBT_VEBT] :
% 9.04/5.07                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.07                         => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 9.04/5.07                     => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
% 9.04/5.07                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.04/5.07                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.04/5.07                         => ( ( M5
% 9.04/5.07                              = ( suc @ N2 ) )
% 9.04/5.07                           => ( ( Deg2
% 9.04/5.07                                = ( plus_plus_nat @ N2 @ M5 ) )
% 9.04/5.07                             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
% 9.04/5.07                               => ~ ! [X2: vEBT_VEBT] :
% 9.04/5.07                                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.07                                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
% 9.04/5.07             => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi: nat,Ma2: nat] :
% 9.04/5.07                    ( ( A1
% 9.04/5.07                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.07                   => ( ( A22 = Deg2 )
% 9.04/5.07                     => ( ! [X2: vEBT_VEBT] :
% 9.04/5.07                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.07                           => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 9.04/5.07                       => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
% 9.04/5.07                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.04/5.07                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.04/5.07                           => ( ( M5 = N2 )
% 9.04/5.07                             => ( ( Deg2
% 9.04/5.07                                  = ( plus_plus_nat @ N2 @ M5 ) )
% 9.04/5.07                               => ( ! [I4: nat] :
% 9.04/5.07                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.04/5.07                                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 9.04/5.07                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.04/5.07                                 => ( ( ( Mi = Ma2 )
% 9.04/5.07                                     => ! [X2: vEBT_VEBT] :
% 9.04/5.07                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.07                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
% 9.04/5.07                                   => ( ( ord_less_eq_nat @ Mi @ Ma2 )
% 9.04/5.07                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.07                                       => ~ ( ( Mi != Ma2 )
% 9.04/5.07                                           => ! [I4: nat] :
% 9.04/5.07                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.04/5.07                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 9.04/5.07                                                      = I4 )
% 9.04/5.07                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 9.04/5.07                                                  & ! [X2: nat] :
% 9.04/5.07                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
% 9.04/5.07                                                          = I4 )
% 9.04/5.07                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
% 9.04/5.07                                                     => ( ( ord_less_nat @ Mi @ X2 )
% 9.04/5.07                                                        & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 9.04/5.07               => ~ ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi: nat,Ma2: nat] :
% 9.04/5.07                      ( ( A1
% 9.04/5.07                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.07                     => ( ( A22 = Deg2 )
% 9.04/5.07                       => ( ! [X2: vEBT_VEBT] :
% 9.04/5.07                              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.07                             => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 9.04/5.07                         => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
% 9.04/5.07                           => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.04/5.07                                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.04/5.07                             => ( ( M5
% 9.04/5.07                                  = ( suc @ N2 ) )
% 9.04/5.07                               => ( ( Deg2
% 9.04/5.07                                    = ( plus_plus_nat @ N2 @ M5 ) )
% 9.04/5.07                                 => ( ! [I4: nat] :
% 9.04/5.07                                        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.04/5.07                                       => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 9.04/5.07                                          = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.04/5.07                                   => ( ( ( Mi = Ma2 )
% 9.04/5.07                                       => ! [X2: vEBT_VEBT] :
% 9.04/5.07                                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.07                                           => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
% 9.04/5.07                                     => ( ( ord_less_eq_nat @ Mi @ Ma2 )
% 9.04/5.07                                       => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.07                                         => ~ ( ( Mi != Ma2 )
% 9.04/5.07                                             => ! [I4: nat] :
% 9.04/5.07                                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.04/5.07                                                 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 9.04/5.07                                                        = I4 )
% 9.04/5.07                                                     => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 9.04/5.07                                                    & ! [X2: nat] :
% 9.04/5.07                                                        ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
% 9.04/5.07                                                            = I4 )
% 9.04/5.07                                                          & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
% 9.04/5.07                                                       => ( ( ord_less_nat @ Mi @ X2 )
% 9.04/5.07                                                          & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7261_sgn__zero,axiom,
% 9.04/5.07      ( ( sgn_sgn_complex @ zero_zero_complex )
% 9.04/5.07      = zero_zero_complex ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7262_sgn__zero,axiom,
% 9.04/5.07      ( ( sgn_sgn_real @ zero_zero_real )
% 9.04/5.07      = zero_zero_real ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7263_sgn__0,axiom,
% 9.04/5.07      ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 9.04/5.07      = zero_z3403309356797280102nteger ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7264_sgn__0,axiom,
% 9.04/5.07      ( ( sgn_sgn_complex @ zero_zero_complex )
% 9.04/5.07      = zero_zero_complex ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7265_sgn__0,axiom,
% 9.04/5.07      ( ( sgn_sgn_real @ zero_zero_real )
% 9.04/5.07      = zero_zero_real ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7266_sgn__0,axiom,
% 9.04/5.07      ( ( sgn_sgn_rat @ zero_zero_rat )
% 9.04/5.07      = zero_zero_rat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7267_sgn__0,axiom,
% 9.04/5.07      ( ( sgn_sgn_int @ zero_zero_int )
% 9.04/5.07      = zero_zero_int ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7268_sgn__one,axiom,
% 9.04/5.07      ( ( sgn_sgn_real @ one_one_real )
% 9.04/5.07      = one_one_real ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7269_sgn__one,axiom,
% 9.04/5.07      ( ( sgn_sgn_complex @ one_one_complex )
% 9.04/5.07      = one_one_complex ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7270_sgn__1,axiom,
% 9.04/5.07      ( ( sgn_sgn_int @ one_one_int )
% 9.04/5.07      = one_one_int ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7271_sgn__1,axiom,
% 9.04/5.07      ( ( sgn_sgn_real @ one_one_real )
% 9.04/5.07      = one_one_real ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7272_sgn__1,axiom,
% 9.04/5.07      ( ( sgn_sgn_complex @ one_one_complex )
% 9.04/5.07      = one_one_complex ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7273_sgn__1,axiom,
% 9.04/5.07      ( ( sgn_sgn_rat @ one_one_rat )
% 9.04/5.07      = one_one_rat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7274_sgn__1,axiom,
% 9.04/5.07      ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
% 9.04/5.07      = one_one_Code_integer ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7275_sgn__divide,axiom,
% 9.04/5.07      ! [A: complex,B: complex] :
% 9.04/5.07        ( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.04/5.07        = ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7276_sgn__divide,axiom,
% 9.04/5.07      ! [A: real,B: real] :
% 9.04/5.07        ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B ) )
% 9.04/5.07        = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7277_sgn__divide,axiom,
% 9.04/5.07      ! [A: rat,B: rat] :
% 9.04/5.07        ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B ) )
% 9.04/5.07        = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7278_power__sgn,axiom,
% 9.04/5.07      ! [A: rat,N: nat] :
% 9.04/5.07        ( ( sgn_sgn_rat @ ( power_power_rat @ A @ N ) )
% 9.04/5.07        = ( power_power_rat @ ( sgn_sgn_rat @ A ) @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7279_power__sgn,axiom,
% 9.04/5.07      ! [A: code_integer,N: nat] :
% 9.04/5.07        ( ( sgn_sgn_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 9.04/5.07        = ( power_8256067586552552935nteger @ ( sgn_sgn_Code_integer @ A ) @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7280_power__sgn,axiom,
% 9.04/5.07      ! [A: real,N: nat] :
% 9.04/5.07        ( ( sgn_sgn_real @ ( power_power_real @ A @ N ) )
% 9.04/5.07        = ( power_power_real @ ( sgn_sgn_real @ A ) @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7281_power__sgn,axiom,
% 9.04/5.07      ! [A: int,N: nat] :
% 9.04/5.07        ( ( sgn_sgn_int @ ( power_power_int @ A @ N ) )
% 9.04/5.07        = ( power_power_int @ ( sgn_sgn_int @ A ) @ N ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7282_not__Some__eq,axiom,
% 9.04/5.07      ! [X: option4927543243414619207at_nat] :
% 9.04/5.07        ( ( ! [Y6: product_prod_nat_nat] :
% 9.04/5.07              ( X
% 9.04/5.07             != ( some_P7363390416028606310at_nat @ Y6 ) ) )
% 9.04/5.07        = ( X = none_P5556105721700978146at_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7283_not__Some__eq,axiom,
% 9.04/5.07      ! [X: option_num] :
% 9.04/5.07        ( ( ! [Y6: num] :
% 9.04/5.07              ( X
% 9.04/5.07             != ( some_num @ Y6 ) ) )
% 9.04/5.07        = ( X = none_num ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7284_not__None__eq,axiom,
% 9.04/5.07      ! [X: option4927543243414619207at_nat] :
% 9.04/5.07        ( ( X != none_P5556105721700978146at_nat )
% 9.04/5.07        = ( ? [Y6: product_prod_nat_nat] :
% 9.04/5.07              ( X
% 9.04/5.07              = ( some_P7363390416028606310at_nat @ Y6 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7285_not__None__eq,axiom,
% 9.04/5.07      ! [X: option_num] :
% 9.04/5.07        ( ( X != none_num )
% 9.04/5.07        = ( ? [Y6: num] :
% 9.04/5.07              ( X
% 9.04/5.07              = ( some_num @ Y6 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7286_nat__int,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
% 9.04/5.07        = N ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7287_mi__ma__2__deg,axiom,
% 9.04/5.07      ! [Mi2: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.04/5.07        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 9.04/5.07       => ( ( ord_less_eq_nat @ Mi2 @ Ma )
% 9.04/5.07          & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7288_sgn__greater,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
% 9.04/5.07        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7289_sgn__greater,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 9.04/5.07        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7290_sgn__greater,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
% 9.04/5.07        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7291_sgn__greater,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
% 9.04/5.07        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7292_sgn__less,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 9.04/5.07        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7293_sgn__less,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 9.04/5.07        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7294_sgn__less,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
% 9.04/5.07        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7295_sgn__less,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
% 9.04/5.07        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7296_divide__sgn,axiom,
% 9.04/5.07      ! [A: real,B: real] :
% 9.04/5.07        ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B ) )
% 9.04/5.07        = ( times_times_real @ A @ ( sgn_sgn_real @ B ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7297_divide__sgn,axiom,
% 9.04/5.07      ! [A: rat,B: rat] :
% 9.04/5.07        ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
% 9.04/5.07        = ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7298_nat__numeral,axiom,
% 9.04/5.07      ! [K: num] :
% 9.04/5.07        ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 9.04/5.07        = ( numeral_numeral_nat @ K ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7299_pred__numeral__inc,axiom,
% 9.04/5.07      ! [K: num] :
% 9.04/5.07        ( ( pred_numeral @ ( inc @ K ) )
% 9.04/5.07        = ( numeral_numeral_nat @ K ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7300_both__member__options__from__complete__tree__to__child,axiom,
% 9.04/5.07      ! [Deg: nat,Mi2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 9.04/5.07       => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.04/5.07         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.07            | ( X = Mi2 )
% 9.04/5.07            | ( X = Ma ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7301_nat__of__bool,axiom,
% 9.04/5.07      ! [P: $o] :
% 9.04/5.07        ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.04/5.07        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7302_sgn__pos,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 9.04/5.07       => ( ( sgn_sgn_Code_integer @ A )
% 9.04/5.07          = one_one_Code_integer ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7303_sgn__pos,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.07       => ( ( sgn_sgn_real @ A )
% 9.04/5.07          = one_one_real ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7304_sgn__pos,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.04/5.07       => ( ( sgn_sgn_rat @ A )
% 9.04/5.07          = one_one_rat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7305_sgn__pos,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( ord_less_int @ zero_zero_int @ A )
% 9.04/5.07       => ( ( sgn_sgn_int @ A )
% 9.04/5.07          = one_one_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7306_abs__sgn__eq__1,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( A != zero_z3403309356797280102nteger )
% 9.04/5.07       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 9.04/5.07          = one_one_Code_integer ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7307_abs__sgn__eq__1,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( A != zero_zero_real )
% 9.04/5.07       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 9.04/5.07          = one_one_real ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7308_abs__sgn__eq__1,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( A != zero_zero_rat )
% 9.04/5.07       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 9.04/5.07          = one_one_rat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7309_abs__sgn__eq__1,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( A != zero_zero_int )
% 9.04/5.07       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 9.04/5.07          = one_one_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7310_nat__1,axiom,
% 9.04/5.07      ( ( nat2 @ one_one_int )
% 9.04/5.07      = ( suc @ zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7311_sgn__mult__self__eq,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
% 9.04/5.07        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7312_sgn__mult__self__eq,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 9.04/5.07        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7313_sgn__mult__self__eq,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
% 9.04/5.07        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7314_sgn__mult__self__eq,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 9.04/5.07        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7315_nat__le__0,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 9.04/5.07       => ( ( nat2 @ Z )
% 9.04/5.07          = zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7316_nat__0__iff,axiom,
% 9.04/5.07      ! [I: int] :
% 9.04/5.07        ( ( ( nat2 @ I )
% 9.04/5.07          = zero_zero_nat )
% 9.04/5.07        = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7317_zless__nat__conj,axiom,
% 9.04/5.07      ! [W: int,Z: int] :
% 9.04/5.07        ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 9.04/5.07        = ( ( ord_less_int @ zero_zero_int @ Z )
% 9.04/5.07          & ( ord_less_int @ W @ Z ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7318_nat__neg__numeral,axiom,
% 9.04/5.07      ! [K: num] :
% 9.04/5.07        ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.04/5.07        = zero_zero_nat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7319_both__member__options__from__chilf__to__complete__tree,axiom,
% 9.04/5.07      ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi2: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.04/5.07        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.04/5.07       => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 9.04/5.07         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.07           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7320_idom__abs__sgn__class_Oabs__sgn,axiom,
% 9.04/5.07      ! [A: complex] :
% 9.04/5.07        ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
% 9.04/5.07        = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7321_idom__abs__sgn__class_Oabs__sgn,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
% 9.04/5.07        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7322_idom__abs__sgn__class_Oabs__sgn,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
% 9.04/5.07        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7323_idom__abs__sgn__class_Oabs__sgn,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
% 9.04/5.07        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7324_idom__abs__sgn__class_Oabs__sgn,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 9.04/5.07        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7325_sgn__abs,axiom,
% 9.04/5.07      ! [A: complex] :
% 9.04/5.07        ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
% 9.04/5.07        = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7326_sgn__abs,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 9.04/5.07        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7327_sgn__abs,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 9.04/5.07        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7328_sgn__abs,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 9.04/5.07        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7329_sgn__abs,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 9.04/5.07        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7330_nat__zminus__int,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.04/5.07        = zero_zero_nat ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7331_int__nat__eq,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 9.04/5.07            = Z ) )
% 9.04/5.07        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 9.04/5.07            = zero_zero_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7332_of__nat__nat__take__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( semiri4939895301339042750nteger @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 9.04/5.07        = ( ring_18347121197199848620nteger @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7333_of__nat__nat__take__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 9.04/5.07        = ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7334_of__nat__nat__take__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 9.04/5.07        = ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7335_of__nat__nat__take__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 9.04/5.07        = ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7336_of__nat__nat__take__bit__eq,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 9.04/5.07        = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7337_dvd__mult__sgn__iff,axiom,
% 9.04/5.07      ! [L: int,K: int,R2: int] :
% 9.04/5.07        ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
% 9.04/5.07        = ( ( dvd_dvd_int @ L @ K )
% 9.04/5.07          | ( R2 = zero_zero_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7338_dvd__sgn__mult__iff,axiom,
% 9.04/5.07      ! [L: int,R2: int,K: int] :
% 9.04/5.07        ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
% 9.04/5.07        = ( ( dvd_dvd_int @ L @ K )
% 9.04/5.07          | ( R2 = zero_zero_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7339_mult__sgn__dvd__iff,axiom,
% 9.04/5.07      ! [L: int,R2: int,K: int] :
% 9.04/5.07        ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
% 9.04/5.07        = ( ( dvd_dvd_int @ L @ K )
% 9.04/5.07          & ( ( R2 = zero_zero_int )
% 9.04/5.07           => ( K = zero_zero_int ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7340_sgn__mult__dvd__iff,axiom,
% 9.04/5.07      ! [R2: int,L: int,K: int] :
% 9.04/5.07        ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
% 9.04/5.07        = ( ( dvd_dvd_int @ L @ K )
% 9.04/5.07          & ( ( R2 = zero_zero_int )
% 9.04/5.07           => ( K = zero_zero_int ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7341_sgn__neg,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( ord_less_real @ A @ zero_zero_real )
% 9.04/5.07       => ( ( sgn_sgn_real @ A )
% 9.04/5.07          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7342_sgn__neg,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( ord_less_int @ A @ zero_zero_int )
% 9.04/5.07       => ( ( sgn_sgn_int @ A )
% 9.04/5.07          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7343_sgn__neg,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.04/5.07       => ( ( sgn_sgn_rat @ A )
% 9.04/5.07          = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7344_sgn__neg,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 9.04/5.07       => ( ( sgn_sgn_Code_integer @ A )
% 9.04/5.07          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7345_zero__less__nat__eq,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 9.04/5.07        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7346_add__neg__numeral__special_I6_J,axiom,
% 9.04/5.07      ! [M: num] :
% 9.04/5.07        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.04/5.07        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7347_add__neg__numeral__special_I6_J,axiom,
% 9.04/5.07      ! [M: num] :
% 9.04/5.07        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.07        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7348_add__neg__numeral__special_I6_J,axiom,
% 9.04/5.07      ! [M: num] :
% 9.04/5.07        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.04/5.07        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7349_add__neg__numeral__special_I6_J,axiom,
% 9.04/5.07      ! [M: num] :
% 9.04/5.07        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.04/5.07        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7350_add__neg__numeral__special_I6_J,axiom,
% 9.04/5.07      ! [M: num] :
% 9.04/5.07        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.07        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7351_add__neg__numeral__special_I5_J,axiom,
% 9.04/5.07      ! [N: num] :
% 9.04/5.07        ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.04/5.07        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7352_add__neg__numeral__special_I5_J,axiom,
% 9.04/5.07      ! [N: num] :
% 9.04/5.07        ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.07        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7353_add__neg__numeral__special_I5_J,axiom,
% 9.04/5.07      ! [N: num] :
% 9.04/5.07        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.04/5.07        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7354_add__neg__numeral__special_I5_J,axiom,
% 9.04/5.07      ! [N: num] :
% 9.04/5.07        ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.04/5.07        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7355_add__neg__numeral__special_I5_J,axiom,
% 9.04/5.07      ! [N: num] :
% 9.04/5.07        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.04/5.07        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7356_diff__numeral__special_I6_J,axiom,
% 9.04/5.07      ! [M: num] :
% 9.04/5.07        ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.04/5.07        = ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7357_diff__numeral__special_I6_J,axiom,
% 9.04/5.07      ! [M: num] :
% 9.04/5.07        ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.07        = ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7358_diff__numeral__special_I6_J,axiom,
% 9.04/5.07      ! [M: num] :
% 9.04/5.07        ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.04/5.07        = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7359_diff__numeral__special_I6_J,axiom,
% 9.04/5.07      ! [M: num] :
% 9.04/5.07        ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.04/5.07        = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7360_diff__numeral__special_I6_J,axiom,
% 9.04/5.07      ! [M: num] :
% 9.04/5.07        ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.07        = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7361_diff__numeral__special_I5_J,axiom,
% 9.04/5.07      ! [N: num] :
% 9.04/5.07        ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 9.04/5.07        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7362_diff__numeral__special_I5_J,axiom,
% 9.04/5.07      ! [N: num] :
% 9.04/5.07        ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.07        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7363_diff__numeral__special_I5_J,axiom,
% 9.04/5.07      ! [N: num] :
% 9.04/5.07        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 9.04/5.07        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7364_diff__numeral__special_I5_J,axiom,
% 9.04/5.07      ! [N: num] :
% 9.04/5.07        ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 9.04/5.07        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7365_diff__numeral__special_I5_J,axiom,
% 9.04/5.07      ! [N: num] :
% 9.04/5.07        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 9.04/5.07        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7366_of__nat__nat,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( semiri4939895301339042750nteger @ ( nat2 @ Z ) )
% 9.04/5.07          = ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7367_of__nat__nat,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( semiri8010041392384452111omplex @ ( nat2 @ Z ) )
% 9.04/5.07          = ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7368_of__nat__nat,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
% 9.04/5.07          = ( ring_1_of_int_real @ Z ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7369_of__nat__nat,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
% 9.04/5.07          = ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7370_of__nat__nat,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 9.04/5.07          = ( ring_1_of_int_int @ Z ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7371_sgn__of__nat,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.04/5.07        = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7372_sgn__of__nat,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.04/5.07        = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7373_sgn__of__nat,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.04/5.07        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7374_sgn__of__nat,axiom,
% 9.04/5.07      ! [N: nat] :
% 9.04/5.07        ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 9.04/5.07        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7375_diff__nat__numeral,axiom,
% 9.04/5.07      ! [V: num,V2: num] :
% 9.04/5.07        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
% 9.04/5.07        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7376_nat__eq__numeral__power__cancel__iff,axiom,
% 9.04/5.07      ! [Y: int,X: num,N: nat] :
% 9.04/5.07        ( ( ( nat2 @ Y )
% 9.04/5.07          = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 9.04/5.07        = ( Y
% 9.04/5.07          = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7377_numeral__power__eq__nat__cancel__iff,axiom,
% 9.04/5.07      ! [X: num,N: nat,Y: int] :
% 9.04/5.07        ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 9.04/5.07          = ( nat2 @ Y ) )
% 9.04/5.07        = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 9.04/5.07          = Y ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7378_nat__abs__dvd__iff,axiom,
% 9.04/5.07      ! [K: int,N: nat] :
% 9.04/5.07        ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 9.04/5.07        = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7379_dvd__nat__abs__iff,axiom,
% 9.04/5.07      ! [N: nat,K: int] :
% 9.04/5.07        ( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 9.04/5.07        = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7380_nat__ceiling__le__eq,axiom,
% 9.04/5.07      ! [X: real,A: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 9.04/5.07        = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7381_one__less__nat__eq,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 9.04/5.07        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7382_nat__numeral__diff__1,axiom,
% 9.04/5.07      ! [V: num] :
% 9.04/5.07        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 9.04/5.07        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7383_numeral__power__less__nat__cancel__iff,axiom,
% 9.04/5.07      ! [X: num,N: nat,A: int] :
% 9.04/5.07        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 9.04/5.07        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7384_nat__less__numeral__power__cancel__iff,axiom,
% 9.04/5.07      ! [A: int,X: num,N: nat] :
% 9.04/5.07        ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 9.04/5.07        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7385_numeral__power__le__nat__cancel__iff,axiom,
% 9.04/5.07      ! [X: num,N: nat,A: int] :
% 9.04/5.07        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 9.04/5.07        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7386_nat__le__numeral__power__cancel__iff,axiom,
% 9.04/5.07      ! [A: int,X: num,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 9.04/5.07        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7387_sgn__zero__iff,axiom,
% 9.04/5.07      ! [X: complex] :
% 9.04/5.07        ( ( ( sgn_sgn_complex @ X )
% 9.04/5.07          = zero_zero_complex )
% 9.04/5.07        = ( X = zero_zero_complex ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7388_sgn__zero__iff,axiom,
% 9.04/5.07      ! [X: real] :
% 9.04/5.07        ( ( ( sgn_sgn_real @ X )
% 9.04/5.07          = zero_zero_real )
% 9.04/5.07        = ( X = zero_zero_real ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7389_sgn__eq__0__iff,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( ( sgn_sgn_Code_integer @ A )
% 9.04/5.07          = zero_z3403309356797280102nteger )
% 9.04/5.07        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7390_sgn__eq__0__iff,axiom,
% 9.04/5.07      ! [A: complex] :
% 9.04/5.07        ( ( ( sgn_sgn_complex @ A )
% 9.04/5.07          = zero_zero_complex )
% 9.04/5.07        = ( A = zero_zero_complex ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7391_sgn__eq__0__iff,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( ( sgn_sgn_real @ A )
% 9.04/5.07          = zero_zero_real )
% 9.04/5.07        = ( A = zero_zero_real ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7392_sgn__eq__0__iff,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( ( sgn_sgn_rat @ A )
% 9.04/5.07          = zero_zero_rat )
% 9.04/5.07        = ( A = zero_zero_rat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7393_sgn__eq__0__iff,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( ( sgn_sgn_int @ A )
% 9.04/5.07          = zero_zero_int )
% 9.04/5.07        = ( A = zero_zero_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7394_sgn__0__0,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( ( sgn_sgn_Code_integer @ A )
% 9.04/5.07          = zero_z3403309356797280102nteger )
% 9.04/5.07        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7395_sgn__0__0,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( ( sgn_sgn_real @ A )
% 9.04/5.07          = zero_zero_real )
% 9.04/5.07        = ( A = zero_zero_real ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7396_sgn__0__0,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( ( sgn_sgn_rat @ A )
% 9.04/5.07          = zero_zero_rat )
% 9.04/5.07        = ( A = zero_zero_rat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7397_sgn__0__0,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( ( sgn_sgn_int @ A )
% 9.04/5.07          = zero_zero_int )
% 9.04/5.07        = ( A = zero_zero_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7398_same__sgn__sgn__add,axiom,
% 9.04/5.07      ! [B: code_integer,A: code_integer] :
% 9.04/5.07        ( ( ( sgn_sgn_Code_integer @ B )
% 9.04/5.07          = ( sgn_sgn_Code_integer @ A ) )
% 9.04/5.07       => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.04/5.07          = ( sgn_sgn_Code_integer @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7399_same__sgn__sgn__add,axiom,
% 9.04/5.07      ! [B: real,A: real] :
% 9.04/5.07        ( ( ( sgn_sgn_real @ B )
% 9.04/5.07          = ( sgn_sgn_real @ A ) )
% 9.04/5.07       => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
% 9.04/5.07          = ( sgn_sgn_real @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7400_same__sgn__sgn__add,axiom,
% 9.04/5.07      ! [B: rat,A: rat] :
% 9.04/5.07        ( ( ( sgn_sgn_rat @ B )
% 9.04/5.07          = ( sgn_sgn_rat @ A ) )
% 9.04/5.07       => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
% 9.04/5.07          = ( sgn_sgn_rat @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7401_same__sgn__sgn__add,axiom,
% 9.04/5.07      ! [B: int,A: int] :
% 9.04/5.07        ( ( ( sgn_sgn_int @ B )
% 9.04/5.07          = ( sgn_sgn_int @ A ) )
% 9.04/5.07       => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
% 9.04/5.07          = ( sgn_sgn_int @ A ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7402_combine__options__cases,axiom,
% 9.04/5.07      ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 9.04/5.07        ( ( ( X = none_P5556105721700978146at_nat )
% 9.04/5.07         => ( P @ X @ Y ) )
% 9.04/5.07       => ( ( ( Y = none_P5556105721700978146at_nat )
% 9.04/5.07           => ( P @ X @ Y ) )
% 9.04/5.07         => ( ! [A5: product_prod_nat_nat,B6: product_prod_nat_nat] :
% 9.04/5.07                ( ( X
% 9.04/5.07                  = ( some_P7363390416028606310at_nat @ A5 ) )
% 9.04/5.07               => ( ( Y
% 9.04/5.07                    = ( some_P7363390416028606310at_nat @ B6 ) )
% 9.04/5.07                 => ( P @ X @ Y ) ) )
% 9.04/5.07           => ( P @ X @ Y ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7403_combine__options__cases,axiom,
% 9.04/5.07      ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
% 9.04/5.07        ( ( ( X = none_P5556105721700978146at_nat )
% 9.04/5.07         => ( P @ X @ Y ) )
% 9.04/5.07       => ( ( ( Y = none_num )
% 9.04/5.07           => ( P @ X @ Y ) )
% 9.04/5.07         => ( ! [A5: product_prod_nat_nat,B6: num] :
% 9.04/5.07                ( ( X
% 9.04/5.07                  = ( some_P7363390416028606310at_nat @ A5 ) )
% 9.04/5.07               => ( ( Y
% 9.04/5.07                    = ( some_num @ B6 ) )
% 9.04/5.07                 => ( P @ X @ Y ) ) )
% 9.04/5.07           => ( P @ X @ Y ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7404_combine__options__cases,axiom,
% 9.04/5.07      ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 9.04/5.07        ( ( ( X = none_num )
% 9.04/5.07         => ( P @ X @ Y ) )
% 9.04/5.07       => ( ( ( Y = none_P5556105721700978146at_nat )
% 9.04/5.07           => ( P @ X @ Y ) )
% 9.04/5.07         => ( ! [A5: num,B6: product_prod_nat_nat] :
% 9.04/5.07                ( ( X
% 9.04/5.07                  = ( some_num @ A5 ) )
% 9.04/5.07               => ( ( Y
% 9.04/5.07                    = ( some_P7363390416028606310at_nat @ B6 ) )
% 9.04/5.07                 => ( P @ X @ Y ) ) )
% 9.04/5.07           => ( P @ X @ Y ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7405_combine__options__cases,axiom,
% 9.04/5.07      ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
% 9.04/5.07        ( ( ( X = none_num )
% 9.04/5.07         => ( P @ X @ Y ) )
% 9.04/5.07       => ( ( ( Y = none_num )
% 9.04/5.07           => ( P @ X @ Y ) )
% 9.04/5.07         => ( ! [A5: num,B6: num] :
% 9.04/5.07                ( ( X
% 9.04/5.07                  = ( some_num @ A5 ) )
% 9.04/5.07               => ( ( Y
% 9.04/5.07                    = ( some_num @ B6 ) )
% 9.04/5.07                 => ( P @ X @ Y ) ) )
% 9.04/5.07           => ( P @ X @ Y ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7406_split__option__all,axiom,
% 9.04/5.07      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 9.04/5.07          ! [X5: option4927543243414619207at_nat] : ( P2 @ X5 ) )
% 9.04/5.07      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 9.04/5.07            ( ( P3 @ none_P5556105721700978146at_nat )
% 9.04/5.07            & ! [X4: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X4 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7407_split__option__all,axiom,
% 9.04/5.07      ( ( ^ [P2: option_num > $o] :
% 9.04/5.07          ! [X5: option_num] : ( P2 @ X5 ) )
% 9.04/5.07      = ( ^ [P3: option_num > $o] :
% 9.04/5.07            ( ( P3 @ none_num )
% 9.04/5.07            & ! [X4: num] : ( P3 @ ( some_num @ X4 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7408_split__option__ex,axiom,
% 9.04/5.07      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 9.04/5.07          ? [X5: option4927543243414619207at_nat] : ( P2 @ X5 ) )
% 9.04/5.07      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 9.04/5.07            ( ( P3 @ none_P5556105721700978146at_nat )
% 9.04/5.07            | ? [X4: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X4 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7409_split__option__ex,axiom,
% 9.04/5.07      ( ( ^ [P2: option_num > $o] :
% 9.04/5.07          ? [X5: option_num] : ( P2 @ X5 ) )
% 9.04/5.07      = ( ^ [P3: option_num > $o] :
% 9.04/5.07            ( ( P3 @ none_num )
% 9.04/5.07            | ? [X4: num] : ( P3 @ ( some_num @ X4 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7410_option_Oexhaust,axiom,
% 9.04/5.07      ! [Y: option4927543243414619207at_nat] :
% 9.04/5.07        ( ( Y != none_P5556105721700978146at_nat )
% 9.04/5.07       => ~ ! [X23: product_prod_nat_nat] :
% 9.04/5.07              ( Y
% 9.04/5.07             != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7411_option_Oexhaust,axiom,
% 9.04/5.07      ! [Y: option_num] :
% 9.04/5.07        ( ( Y != none_num )
% 9.04/5.07       => ~ ! [X23: num] :
% 9.04/5.07              ( Y
% 9.04/5.07             != ( some_num @ X23 ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7412_option_OdiscI,axiom,
% 9.04/5.07      ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 9.04/5.07        ( ( Option
% 9.04/5.07          = ( some_P7363390416028606310at_nat @ X22 ) )
% 9.04/5.07       => ( Option != none_P5556105721700978146at_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7413_option_OdiscI,axiom,
% 9.04/5.07      ! [Option: option_num,X22: num] :
% 9.04/5.07        ( ( Option
% 9.04/5.07          = ( some_num @ X22 ) )
% 9.04/5.07       => ( Option != none_num ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7414_option_Odistinct_I1_J,axiom,
% 9.04/5.07      ! [X22: product_prod_nat_nat] :
% 9.04/5.07        ( none_P5556105721700978146at_nat
% 9.04/5.07       != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7415_option_Odistinct_I1_J,axiom,
% 9.04/5.07      ! [X22: num] :
% 9.04/5.07        ( none_num
% 9.04/5.07       != ( some_num @ X22 ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7416_None__notin__image__Some,axiom,
% 9.04/5.07      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.04/5.07        ~ ( member3954567711264315760at_nat @ none_P5556105721700978146at_nat @ ( image_4198897800814241419at_nat @ some_P7363390416028606310at_nat @ A2 ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7417_None__notin__image__Some,axiom,
% 9.04/5.07      ! [A2: set_num] :
% 9.04/5.07        ~ ( member_option_num @ none_num @ ( image_num_option_num @ some_num @ A2 ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7418_num__induct,axiom,
% 9.04/5.07      ! [P: num > $o,X: num] :
% 9.04/5.07        ( ( P @ one )
% 9.04/5.07       => ( ! [X3: num] :
% 9.04/5.07              ( ( P @ X3 )
% 9.04/5.07             => ( P @ ( inc @ X3 ) ) )
% 9.04/5.07         => ( P @ X ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7419_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 9.04/5.07      ! [Mi2: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
% 9.04/5.07        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X )
% 9.04/5.07        = ( ( X = Mi2 )
% 9.04/5.07          | ( X = Ma ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7420_add__inc,axiom,
% 9.04/5.07      ! [X: num,Y: num] :
% 9.04/5.07        ( ( plus_plus_num @ X @ ( inc @ Y ) )
% 9.04/5.07        = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7421_sgn__not__eq__imp,axiom,
% 9.04/5.07      ! [B: real,A: real] :
% 9.04/5.07        ( ( ( sgn_sgn_real @ B )
% 9.04/5.07         != ( sgn_sgn_real @ A ) )
% 9.04/5.07       => ( ( ( sgn_sgn_real @ A )
% 9.04/5.07           != zero_zero_real )
% 9.04/5.07         => ( ( ( sgn_sgn_real @ B )
% 9.04/5.07             != zero_zero_real )
% 9.04/5.07           => ( ( sgn_sgn_real @ A )
% 9.04/5.07              = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7422_sgn__not__eq__imp,axiom,
% 9.04/5.07      ! [B: int,A: int] :
% 9.04/5.07        ( ( ( sgn_sgn_int @ B )
% 9.04/5.07         != ( sgn_sgn_int @ A ) )
% 9.04/5.07       => ( ( ( sgn_sgn_int @ A )
% 9.04/5.07           != zero_zero_int )
% 9.04/5.07         => ( ( ( sgn_sgn_int @ B )
% 9.04/5.07             != zero_zero_int )
% 9.04/5.07           => ( ( sgn_sgn_int @ A )
% 9.04/5.07              = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7423_sgn__not__eq__imp,axiom,
% 9.04/5.07      ! [B: rat,A: rat] :
% 9.04/5.07        ( ( ( sgn_sgn_rat @ B )
% 9.04/5.07         != ( sgn_sgn_rat @ A ) )
% 9.04/5.07       => ( ( ( sgn_sgn_rat @ A )
% 9.04/5.07           != zero_zero_rat )
% 9.04/5.07         => ( ( ( sgn_sgn_rat @ B )
% 9.04/5.07             != zero_zero_rat )
% 9.04/5.07           => ( ( sgn_sgn_rat @ A )
% 9.04/5.07              = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7424_sgn__not__eq__imp,axiom,
% 9.04/5.07      ! [B: code_integer,A: code_integer] :
% 9.04/5.07        ( ( ( sgn_sgn_Code_integer @ B )
% 9.04/5.07         != ( sgn_sgn_Code_integer @ A ) )
% 9.04/5.07       => ( ( ( sgn_sgn_Code_integer @ A )
% 9.04/5.07           != zero_z3403309356797280102nteger )
% 9.04/5.07         => ( ( ( sgn_sgn_Code_integer @ B )
% 9.04/5.07             != zero_z3403309356797280102nteger )
% 9.04/5.07           => ( ( sgn_sgn_Code_integer @ A )
% 9.04/5.07              = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7425_sgn__minus__1,axiom,
% 9.04/5.07      ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.04/5.07      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7426_sgn__minus__1,axiom,
% 9.04/5.07      ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.07      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7427_sgn__minus__1,axiom,
% 9.04/5.07      ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.04/5.07      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7428_sgn__minus__1,axiom,
% 9.04/5.07      ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.04/5.07      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7429_sgn__minus__1,axiom,
% 9.04/5.07      ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.07      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7430_nat__zero__as__int,axiom,
% 9.04/5.07      ( zero_zero_nat
% 9.04/5.07      = ( nat2 @ zero_zero_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7431_nat__numeral__as__int,axiom,
% 9.04/5.07      ( numeral_numeral_nat
% 9.04/5.07      = ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7432_int__sgnE,axiom,
% 9.04/5.07      ! [K: int] :
% 9.04/5.07        ~ ! [N2: nat,L2: int] :
% 9.04/5.07            ( K
% 9.04/5.07           != ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7433_nat__mono,axiom,
% 9.04/5.07      ! [X: int,Y: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ X @ Y )
% 9.04/5.07       => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7434_same__sgn__abs__add,axiom,
% 9.04/5.07      ! [B: code_integer,A: code_integer] :
% 9.04/5.07        ( ( ( sgn_sgn_Code_integer @ B )
% 9.04/5.07          = ( sgn_sgn_Code_integer @ A ) )
% 9.04/5.07       => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.04/5.07          = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7435_same__sgn__abs__add,axiom,
% 9.04/5.07      ! [B: real,A: real] :
% 9.04/5.07        ( ( ( sgn_sgn_real @ B )
% 9.04/5.07          = ( sgn_sgn_real @ A ) )
% 9.04/5.07       => ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
% 9.04/5.07          = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7436_same__sgn__abs__add,axiom,
% 9.04/5.07      ! [B: rat,A: rat] :
% 9.04/5.07        ( ( ( sgn_sgn_rat @ B )
% 9.04/5.07          = ( sgn_sgn_rat @ A ) )
% 9.04/5.07       => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
% 9.04/5.07          = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7437_same__sgn__abs__add,axiom,
% 9.04/5.07      ! [B: int,A: int] :
% 9.04/5.07        ( ( ( sgn_sgn_int @ B )
% 9.04/5.07          = ( sgn_sgn_int @ A ) )
% 9.04/5.07       => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
% 9.04/5.07          = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7438_ex__nat,axiom,
% 9.04/5.07      ( ( ^ [P2: nat > $o] :
% 9.04/5.07          ? [X5: nat] : ( P2 @ X5 ) )
% 9.04/5.07      = ( ^ [P3: nat > $o] :
% 9.04/5.07          ? [X4: int] :
% 9.04/5.07            ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 9.04/5.07            & ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7439_all__nat,axiom,
% 9.04/5.07      ( ( ^ [P2: nat > $o] :
% 9.04/5.07          ! [X5: nat] : ( P2 @ X5 ) )
% 9.04/5.07      = ( ^ [P3: nat > $o] :
% 9.04/5.07          ! [X4: int] :
% 9.04/5.07            ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 9.04/5.07           => ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7440_eq__nat__nat__iff,axiom,
% 9.04/5.07      ! [Z: int,Z6: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 9.04/5.07         => ( ( ( nat2 @ Z )
% 9.04/5.07              = ( nat2 @ Z6 ) )
% 9.04/5.07            = ( Z = Z6 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7441_take__bit__nat__eq,axiom,
% 9.04/5.07      ! [K: int,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.07       => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 9.04/5.07          = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7442_nat__take__bit__eq,axiom,
% 9.04/5.07      ! [K: int,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.07       => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.04/5.07          = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7443_nat__one__as__int,axiom,
% 9.04/5.07      ( one_one_nat
% 9.04/5.07      = ( nat2 @ one_one_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7444_div__eq__sgn__abs,axiom,
% 9.04/5.07      ! [K: int,L: int] :
% 9.04/5.07        ( ( ( sgn_sgn_int @ K )
% 9.04/5.07          = ( sgn_sgn_int @ L ) )
% 9.04/5.07       => ( ( divide_divide_int @ K @ L )
% 9.04/5.07          = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7445_unset__bit__nat__def,axiom,
% 9.04/5.07      ( bit_se4205575877204974255it_nat
% 9.04/5.07      = ( ^ [M6: nat,N3: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M6 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7446_inc_Osimps_I1_J,axiom,
% 9.04/5.07      ( ( inc @ one )
% 9.04/5.07      = ( bit0 @ one ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7447_inc_Osimps_I3_J,axiom,
% 9.04/5.07      ! [X: num] :
% 9.04/5.07        ( ( inc @ ( bit1 @ X ) )
% 9.04/5.07        = ( bit0 @ ( inc @ X ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7448_inc_Osimps_I2_J,axiom,
% 9.04/5.07      ! [X: num] :
% 9.04/5.07        ( ( inc @ ( bit0 @ X ) )
% 9.04/5.07        = ( bit1 @ X ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7449_sgn__1__pos,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( ( sgn_sgn_Code_integer @ A )
% 9.04/5.07          = one_one_Code_integer )
% 9.04/5.07        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7450_sgn__1__pos,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( ( sgn_sgn_real @ A )
% 9.04/5.07          = one_one_real )
% 9.04/5.07        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7451_sgn__1__pos,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( ( sgn_sgn_rat @ A )
% 9.04/5.07          = one_one_rat )
% 9.04/5.07        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7452_sgn__1__pos,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( ( sgn_sgn_int @ A )
% 9.04/5.07          = one_one_int )
% 9.04/5.07        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7453_add__One,axiom,
% 9.04/5.07      ! [X: num] :
% 9.04/5.07        ( ( plus_plus_num @ X @ one )
% 9.04/5.07        = ( inc @ X ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7454_abs__sgn__eq,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( ( A = zero_z3403309356797280102nteger )
% 9.04/5.07         => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 9.04/5.07            = zero_z3403309356797280102nteger ) )
% 9.04/5.07        & ( ( A != zero_z3403309356797280102nteger )
% 9.04/5.07         => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 9.04/5.07            = one_one_Code_integer ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7455_abs__sgn__eq,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( ( A = zero_zero_real )
% 9.04/5.07         => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 9.04/5.07            = zero_zero_real ) )
% 9.04/5.07        & ( ( A != zero_zero_real )
% 9.04/5.07         => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 9.04/5.07            = one_one_real ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7456_abs__sgn__eq,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( ( A = zero_zero_rat )
% 9.04/5.07         => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 9.04/5.07            = zero_zero_rat ) )
% 9.04/5.07        & ( ( A != zero_zero_rat )
% 9.04/5.07         => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 9.04/5.07            = one_one_rat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7457_abs__sgn__eq,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( ( A = zero_zero_int )
% 9.04/5.07         => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 9.04/5.07            = zero_zero_int ) )
% 9.04/5.07        & ( ( A != zero_zero_int )
% 9.04/5.07         => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 9.04/5.07            = one_one_int ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7458_nat__mono__iff,axiom,
% 9.04/5.07      ! [Z: int,W: int] :
% 9.04/5.07        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 9.04/5.07          = ( ord_less_int @ W @ Z ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7459_of__nat__ceiling,axiom,
% 9.04/5.07      ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7460_of__nat__ceiling,axiom,
% 9.04/5.07      ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7461_zless__nat__eq__int__zless,axiom,
% 9.04/5.07      ! [M: nat,Z: int] :
% 9.04/5.07        ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 9.04/5.07        = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7462_nat__le__iff,axiom,
% 9.04/5.07      ! [X: int,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 9.04/5.07        = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7463_int__eq__iff,axiom,
% 9.04/5.07      ! [M: nat,Z: int] :
% 9.04/5.07        ( ( ( semiri1314217659103216013at_int @ M )
% 9.04/5.07          = Z )
% 9.04/5.07        = ( ( M
% 9.04/5.07            = ( nat2 @ Z ) )
% 9.04/5.07          & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7464_nat__0__le,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 9.04/5.07          = Z ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7465_nat__int__add,axiom,
% 9.04/5.07      ! [A: nat,B: nat] :
% 9.04/5.07        ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 9.04/5.07        = ( plus_plus_nat @ A @ B ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7466_sgn__mod,axiom,
% 9.04/5.07      ! [L: int,K: int] :
% 9.04/5.07        ( ( L != zero_zero_int )
% 9.04/5.07       => ( ~ ( dvd_dvd_int @ L @ K )
% 9.04/5.07         => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
% 9.04/5.07            = ( sgn_sgn_int @ L ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7467_int__minus,axiom,
% 9.04/5.07      ! [N: nat,M: nat] :
% 9.04/5.07        ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 9.04/5.07        = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7468_nat__abs__mult__distrib,axiom,
% 9.04/5.07      ! [W: int,Z: int] :
% 9.04/5.07        ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 9.04/5.07        = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7469_mult__inc,axiom,
% 9.04/5.07      ! [X: num,Y: num] :
% 9.04/5.07        ( ( times_times_num @ X @ ( inc @ Y ) )
% 9.04/5.07        = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7470_real__nat__ceiling__ge,axiom,
% 9.04/5.07      ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7471_option_Osize_I4_J,axiom,
% 9.04/5.07      ! [X22: product_prod_nat_nat] :
% 9.04/5.07        ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.04/5.07        = ( suc @ zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7472_option_Osize_I4_J,axiom,
% 9.04/5.07      ! [X22: num] :
% 9.04/5.07        ( ( size_size_option_num @ ( some_num @ X22 ) )
% 9.04/5.07        = ( suc @ zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7473_sgn__if,axiom,
% 9.04/5.07      ( sgn_sgn_real
% 9.04/5.07      = ( ^ [X4: real] : ( if_real @ ( X4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7474_sgn__if,axiom,
% 9.04/5.07      ( sgn_sgn_int
% 9.04/5.07      = ( ^ [X4: int] : ( if_int @ ( X4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7475_sgn__if,axiom,
% 9.04/5.07      ( sgn_sgn_rat
% 9.04/5.07      = ( ^ [X4: rat] : ( if_rat @ ( X4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7476_sgn__if,axiom,
% 9.04/5.07      ( sgn_sgn_Code_integer
% 9.04/5.07      = ( ^ [X4: code_integer] : ( if_Code_integer @ ( X4 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X4 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7477_sgn__1__neg,axiom,
% 9.04/5.07      ! [A: real] :
% 9.04/5.07        ( ( ( sgn_sgn_real @ A )
% 9.04/5.07          = ( uminus_uminus_real @ one_one_real ) )
% 9.04/5.07        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7478_sgn__1__neg,axiom,
% 9.04/5.07      ! [A: int] :
% 9.04/5.07        ( ( ( sgn_sgn_int @ A )
% 9.04/5.07          = ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.07        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7479_sgn__1__neg,axiom,
% 9.04/5.07      ! [A: rat] :
% 9.04/5.07        ( ( ( sgn_sgn_rat @ A )
% 9.04/5.07          = ( uminus_uminus_rat @ one_one_rat ) )
% 9.04/5.07        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7480_sgn__1__neg,axiom,
% 9.04/5.07      ! [A: code_integer] :
% 9.04/5.07        ( ( ( sgn_sgn_Code_integer @ A )
% 9.04/5.07          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.07        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7481_of__nat__floor,axiom,
% 9.04/5.07      ! [R2: real] :
% 9.04/5.07        ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 9.04/5.07       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7482_of__nat__floor,axiom,
% 9.04/5.07      ! [R2: rat] :
% 9.04/5.07        ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 9.04/5.07       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7483_zsgn__def,axiom,
% 9.04/5.07      ( sgn_sgn_int
% 9.04/5.07      = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7484_numeral__inc,axiom,
% 9.04/5.07      ! [X: num] :
% 9.04/5.07        ( ( numera6690914467698888265omplex @ ( inc @ X ) )
% 9.04/5.07        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7485_numeral__inc,axiom,
% 9.04/5.07      ! [X: num] :
% 9.04/5.07        ( ( numeral_numeral_real @ ( inc @ X ) )
% 9.04/5.07        = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7486_numeral__inc,axiom,
% 9.04/5.07      ! [X: num] :
% 9.04/5.07        ( ( numeral_numeral_rat @ ( inc @ X ) )
% 9.04/5.07        = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7487_numeral__inc,axiom,
% 9.04/5.07      ! [X: num] :
% 9.04/5.07        ( ( numeral_numeral_nat @ ( inc @ X ) )
% 9.04/5.07        = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7488_numeral__inc,axiom,
% 9.04/5.07      ! [X: num] :
% 9.04/5.07        ( ( numeral_numeral_int @ ( inc @ X ) )
% 9.04/5.07        = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7489_nat__less__eq__zless,axiom,
% 9.04/5.07      ! [W: int,Z: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ W )
% 9.04/5.07       => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 9.04/5.07          = ( ord_less_int @ W @ Z ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7490_norm__sgn,axiom,
% 9.04/5.07      ! [X: real] :
% 9.04/5.07        ( ( ( X = zero_zero_real )
% 9.04/5.07         => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 9.04/5.07            = zero_zero_real ) )
% 9.04/5.07        & ( ( X != zero_zero_real )
% 9.04/5.07         => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 9.04/5.07            = one_one_real ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7491_norm__sgn,axiom,
% 9.04/5.07      ! [X: complex] :
% 9.04/5.07        ( ( ( X = zero_zero_complex )
% 9.04/5.07         => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 9.04/5.07            = zero_zero_real ) )
% 9.04/5.07        & ( ( X != zero_zero_complex )
% 9.04/5.07         => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 9.04/5.07            = one_one_real ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7492_nat__le__eq__zle,axiom,
% 9.04/5.07      ! [W: int,Z: int] :
% 9.04/5.07        ( ( ( ord_less_int @ zero_zero_int @ W )
% 9.04/5.07          | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 9.04/5.07       => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 9.04/5.07          = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7493_nat__eq__iff2,axiom,
% 9.04/5.07      ! [M: nat,W: int] :
% 9.04/5.07        ( ( M
% 9.04/5.07          = ( nat2 @ W ) )
% 9.04/5.07        = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 9.04/5.07           => ( W
% 9.04/5.07              = ( semiri1314217659103216013at_int @ M ) ) )
% 9.04/5.07          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 9.04/5.07           => ( M = zero_zero_nat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7494_nat__eq__iff,axiom,
% 9.04/5.07      ! [W: int,M: nat] :
% 9.04/5.07        ( ( ( nat2 @ W )
% 9.04/5.07          = M )
% 9.04/5.07        = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 9.04/5.07           => ( W
% 9.04/5.07              = ( semiri1314217659103216013at_int @ M ) ) )
% 9.04/5.07          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 9.04/5.07           => ( M = zero_zero_nat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7495_split__nat,axiom,
% 9.04/5.07      ! [P: nat > $o,I: int] :
% 9.04/5.07        ( ( P @ ( nat2 @ I ) )
% 9.04/5.07        = ( ! [N3: nat] :
% 9.04/5.07              ( ( I
% 9.04/5.07                = ( semiri1314217659103216013at_int @ N3 ) )
% 9.04/5.07             => ( P @ N3 ) )
% 9.04/5.07          & ( ( ord_less_int @ I @ zero_zero_int )
% 9.04/5.07           => ( P @ zero_zero_nat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7496_le__mult__nat__floor,axiom,
% 9.04/5.07      ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7497_le__mult__nat__floor,axiom,
% 9.04/5.07      ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7498_le__nat__iff,axiom,
% 9.04/5.07      ! [K: int,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.07       => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 9.04/5.07          = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7499_nat__add__distrib,axiom,
% 9.04/5.07      ! [Z: int,Z6: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 9.04/5.07         => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
% 9.04/5.07            = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7500_div__sgn__abs__cancel,axiom,
% 9.04/5.07      ! [V: int,K: int,L: int] :
% 9.04/5.07        ( ( V != zero_zero_int )
% 9.04/5.07       => ( ( 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 ) ) )
% 9.04/5.07          = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7501_nat__mult__distrib,axiom,
% 9.04/5.07      ! [Z: int,Z6: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 9.04/5.07          = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7502_Suc__as__int,axiom,
% 9.04/5.07      ( suc
% 9.04/5.07      = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7503_nat__diff__distrib_H,axiom,
% 9.04/5.07      ! [X: int,Y: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.07       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.07         => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
% 9.04/5.07            = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7504_nat__diff__distrib,axiom,
% 9.04/5.07      ! [Z6: int,Z: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 9.04/5.07       => ( ( ord_less_eq_int @ Z6 @ Z )
% 9.04/5.07         => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
% 9.04/5.07            = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7505_nat__abs__triangle__ineq,axiom,
% 9.04/5.07      ! [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 ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7506_nat__div__distrib_H,axiom,
% 9.04/5.07      ! [Y: int,X: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.07       => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 9.04/5.07          = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7507_nat__div__distrib,axiom,
% 9.04/5.07      ! [X: int,Y: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.07       => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 9.04/5.07          = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7508_div__dvd__sgn__abs,axiom,
% 9.04/5.07      ! [L: int,K: int] :
% 9.04/5.07        ( ( dvd_dvd_int @ L @ K )
% 9.04/5.07       => ( ( divide_divide_int @ K @ L )
% 9.04/5.07          = ( 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 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7509_nat__floor__neg,axiom,
% 9.04/5.07      ! [X: real] :
% 9.04/5.07        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.04/5.07       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 9.04/5.07          = zero_zero_nat ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7510_nat__power__eq,axiom,
% 9.04/5.07      ! [Z: int,N: nat] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 9.04/5.07          = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7511_nat__mod__distrib,axiom,
% 9.04/5.07      ! [X: int,Y: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.07       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.07         => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
% 9.04/5.07            = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7512_div__abs__eq__div__nat,axiom,
% 9.04/5.07      ! [K: int,L: int] :
% 9.04/5.07        ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 9.04/5.07        = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7513_floor__eq3,axiom,
% 9.04/5.07      ! [N: nat,X: real] :
% 9.04/5.07        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 9.04/5.07       => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 9.04/5.07         => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 9.04/5.07            = N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7514_le__nat__floor,axiom,
% 9.04/5.07      ! [X: nat,A: real] :
% 9.04/5.07        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 9.04/5.07       => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7515_mod__abs__eq__div__nat,axiom,
% 9.04/5.07      ! [K: int,L: int] :
% 9.04/5.07        ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 9.04/5.07        = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7516_divide__int__def,axiom,
% 9.04/5.07      ( divide_divide_int
% 9.04/5.07      = ( ^ [K2: int,L3: int] :
% 9.04/5.07            ( if_int @ ( L3 = zero_zero_int ) @ zero_zero_int
% 9.04/5.07            @ ( if_int
% 9.04/5.07              @ ( ( sgn_sgn_int @ K2 )
% 9.04/5.07                = ( sgn_sgn_int @ L3 ) )
% 9.04/5.07              @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) )
% 9.04/5.07              @ ( uminus_uminus_int
% 9.04/5.07                @ ( semiri1314217659103216013at_int
% 9.04/5.07                  @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) )
% 9.04/5.07                    @ ( zero_n2687167440665602831ol_nat
% 9.04/5.07                      @ ~ ( dvd_dvd_int @ L3 @ K2 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7517_modulo__int__def,axiom,
% 9.04/5.07      ( modulo_modulo_int
% 9.04/5.07      = ( ^ [K2: int,L3: int] :
% 9.04/5.07            ( if_int @ ( L3 = zero_zero_int ) @ K2
% 9.04/5.07            @ ( if_int
% 9.04/5.07              @ ( ( sgn_sgn_int @ K2 )
% 9.04/5.07                = ( sgn_sgn_int @ L3 ) )
% 9.04/5.07              @ ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) )
% 9.04/5.07              @ ( times_times_int @ ( sgn_sgn_int @ L3 )
% 9.04/5.07                @ ( minus_minus_int
% 9.04/5.07                  @ ( times_times_int @ ( abs_abs_int @ L3 )
% 9.04/5.07                    @ ( zero_n2684676970156552555ol_int
% 9.04/5.07                      @ ~ ( dvd_dvd_int @ L3 @ K2 ) ) )
% 9.04/5.07                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7518_nat__2,axiom,
% 9.04/5.07      ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.07      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7519_Suc__nat__eq__nat__zadd1,axiom,
% 9.04/5.07      ! [Z: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07       => ( ( suc @ ( nat2 @ Z ) )
% 9.04/5.07          = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7520_nat__less__iff,axiom,
% 9.04/5.07      ! [W: int,M: nat] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ W )
% 9.04/5.07       => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 9.04/5.07          = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7521_nat__mult__distrib__neg,axiom,
% 9.04/5.07      ! [Z: int,Z6: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 9.04/5.07       => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 9.04/5.07          = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7522_nat__abs__int__diff,axiom,
% 9.04/5.07      ! [A: nat,B: nat] :
% 9.04/5.07        ( ( ( ord_less_eq_nat @ A @ B )
% 9.04/5.07         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 9.04/5.07            = ( minus_minus_nat @ B @ A ) ) )
% 9.04/5.07        & ( ~ ( ord_less_eq_nat @ A @ B )
% 9.04/5.07         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 9.04/5.07            = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7523_floor__eq4,axiom,
% 9.04/5.07      ! [N: nat,X: real] :
% 9.04/5.07        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 9.04/5.07       => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 9.04/5.07         => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 9.04/5.07            = N ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7524_of__int__of__nat,axiom,
% 9.04/5.07      ( ring_18347121197199848620nteger
% 9.04/5.07      = ( ^ [K2: int] : ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K2 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7525_of__int__of__nat,axiom,
% 9.04/5.07      ( ring_17405671764205052669omplex
% 9.04/5.07      = ( ^ [K2: int] : ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K2 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7526_of__int__of__nat,axiom,
% 9.04/5.07      ( ring_1_of_int_real
% 9.04/5.07      = ( ^ [K2: int] : ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K2 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7527_of__int__of__nat,axiom,
% 9.04/5.07      ( ring_1_of_int_rat
% 9.04/5.07      = ( ^ [K2: int] : ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K2 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7528_of__int__of__nat,axiom,
% 9.04/5.07      ( ring_1_of_int_int
% 9.04/5.07      = ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7529_nat__dvd__iff,axiom,
% 9.04/5.07      ! [Z: int,M: nat] :
% 9.04/5.07        ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 9.04/5.07        = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07           => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 9.04/5.07          & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.04/5.07           => ( M = zero_zero_nat ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7530_div__noneq__sgn__abs,axiom,
% 9.04/5.07      ! [L: int,K: int] :
% 9.04/5.07        ( ( L != zero_zero_int )
% 9.04/5.07       => ( ( ( sgn_sgn_int @ K )
% 9.04/5.07           != ( sgn_sgn_int @ L ) )
% 9.04/5.07         => ( ( divide_divide_int @ K @ L )
% 9.04/5.07            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
% 9.04/5.07              @ ( zero_n2684676970156552555ol_int
% 9.04/5.07                @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7531_even__nat__iff,axiom,
% 9.04/5.07      ! [K: int] :
% 9.04/5.07        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.07       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 9.04/5.07          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 9.04/5.07  
% 9.04/5.07  thf(fact_7532_invar__vebt_Ointros_I4_J,axiom,
% 9.04/5.07      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma: nat] :
% 9.04/5.08        ( ! [X3: vEBT_VEBT] :
% 9.04/5.08            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.04/5.08           => ( vEBT_invar_vebt @ X3 @ N ) )
% 9.04/5.08       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.04/5.08         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.04/5.08              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.08           => ( ( M = N )
% 9.04/5.08             => ( ( Deg
% 9.04/5.08                  = ( plus_plus_nat @ N @ M ) )
% 9.04/5.08               => ( ! [I2: nat] :
% 9.04/5.08                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.08                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 9.04/5.08                        = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 9.04/5.08                 => ( ( ( Mi2 = Ma )
% 9.04/5.08                     => ! [X3: vEBT_VEBT] :
% 9.04/5.08                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.04/5.08                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
% 9.04/5.08                   => ( ( ord_less_eq_nat @ Mi2 @ Ma )
% 9.04/5.08                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.04/5.08                       => ( ( ( Mi2 != Ma )
% 9.04/5.08                           => ! [I2: nat] :
% 9.04/5.08                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.08                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 9.04/5.08                                      = I2 )
% 9.04/5.08                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 9.04/5.08                                  & ! [X3: nat] :
% 9.04/5.08                                      ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 9.04/5.08                                          = I2 )
% 9.04/5.08                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 9.04/5.08                                     => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.04/5.08                                        & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 9.04/5.08                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7533_invar__vebt_Ointros_I5_J,axiom,
% 9.04/5.08      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma: nat] :
% 9.04/5.08        ( ! [X3: vEBT_VEBT] :
% 9.04/5.08            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.04/5.08           => ( vEBT_invar_vebt @ X3 @ N ) )
% 9.04/5.08       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.04/5.08         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.04/5.08              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.08           => ( ( M
% 9.04/5.08                = ( suc @ N ) )
% 9.04/5.08             => ( ( Deg
% 9.04/5.08                  = ( plus_plus_nat @ N @ M ) )
% 9.04/5.08               => ( ! [I2: nat] :
% 9.04/5.08                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.08                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 9.04/5.08                        = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 9.04/5.08                 => ( ( ( Mi2 = Ma )
% 9.04/5.08                     => ! [X3: vEBT_VEBT] :
% 9.04/5.08                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.04/5.08                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
% 9.04/5.08                   => ( ( ord_less_eq_nat @ Mi2 @ Ma )
% 9.04/5.08                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.04/5.08                       => ( ( ( Mi2 != Ma )
% 9.04/5.08                           => ! [I2: nat] :
% 9.04/5.08                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.04/5.08                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 9.04/5.08                                      = I2 )
% 9.04/5.08                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 9.04/5.08                                  & ! [X3: nat] :
% 9.04/5.08                                      ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 9.04/5.08                                          = I2 )
% 9.04/5.08                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 9.04/5.08                                     => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.04/5.08                                        & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 9.04/5.08                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7534_invar__vebt_Osimps,axiom,
% 9.04/5.08      ( vEBT_invar_vebt
% 9.04/5.08      = ( ^ [A12: vEBT_VEBT,A23: nat] :
% 9.04/5.08            ( ( ? [A3: $o,B4: $o] :
% 9.04/5.08                  ( A12
% 9.04/5.08                  = ( vEBT_Leaf @ A3 @ B4 ) )
% 9.04/5.08              & ( A23
% 9.04/5.08                = ( suc @ zero_zero_nat ) ) )
% 9.04/5.08            | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
% 9.04/5.08                ( ( A12
% 9.04/5.08                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList2 @ Summary3 ) )
% 9.04/5.08                & ! [X4: vEBT_VEBT] :
% 9.04/5.08                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.04/5.08                   => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 9.04/5.08                & ( vEBT_invar_vebt @ Summary3 @ N3 )
% 9.04/5.08                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.04/5.08                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 9.04/5.08                & ( A23
% 9.04/5.08                  = ( plus_plus_nat @ N3 @ N3 ) )
% 9.04/5.08                & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 9.04/5.08                & ! [X4: vEBT_VEBT] :
% 9.04/5.08                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.04/5.08                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.08            | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
% 9.04/5.08                ( ( A12
% 9.04/5.08                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList2 @ Summary3 ) )
% 9.04/5.08                & ! [X4: vEBT_VEBT] :
% 9.04/5.08                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.04/5.08                   => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 9.04/5.08                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
% 9.04/5.08                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.04/5.08                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 9.04/5.08                & ( A23
% 9.04/5.08                  = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 9.04/5.08                & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 9.04/5.08                & ! [X4: vEBT_VEBT] :
% 9.04/5.08                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.04/5.08                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.08            | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 9.04/5.08                ( ( A12
% 9.04/5.08                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList2 @ Summary3 ) )
% 9.04/5.08                & ! [X4: vEBT_VEBT] :
% 9.04/5.08                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.04/5.08                   => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 9.04/5.08                & ( vEBT_invar_vebt @ Summary3 @ N3 )
% 9.04/5.08                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.04/5.08                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 9.04/5.08                & ( A23
% 9.04/5.08                  = ( plus_plus_nat @ N3 @ N3 ) )
% 9.04/5.08                & ! [I3: nat] :
% 9.04/5.08                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 9.04/5.08                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
% 9.04/5.08                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 9.04/5.08                & ( ( Mi3 = Ma3 )
% 9.04/5.08                 => ! [X4: vEBT_VEBT] :
% 9.04/5.08                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.04/5.08                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.08                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.04/5.08                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 9.04/5.08                & ( ( Mi3 != Ma3 )
% 9.04/5.08                 => ! [I3: nat] :
% 9.04/5.08                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 9.04/5.08                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 9.04/5.08                            = I3 )
% 9.04/5.08                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 9.04/5.08                        & ! [X4: nat] :
% 9.04/5.08                            ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
% 9.04/5.08                                = I3 )
% 9.04/5.08                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
% 9.04/5.08                           => ( ( ord_less_nat @ Mi3 @ X4 )
% 9.04/5.08                              & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) )
% 9.04/5.08            | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 9.04/5.08                ( ( A12
% 9.04/5.08                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList2 @ Summary3 ) )
% 9.04/5.08                & ! [X4: vEBT_VEBT] :
% 9.04/5.08                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.04/5.08                   => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 9.04/5.08                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
% 9.04/5.08                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.04/5.08                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 9.04/5.08                & ( A23
% 9.04/5.08                  = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 9.04/5.08                & ! [I3: nat] :
% 9.04/5.08                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 9.04/5.08                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
% 9.04/5.08                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 9.04/5.08                & ( ( Mi3 = Ma3 )
% 9.04/5.08                 => ! [X4: vEBT_VEBT] :
% 9.04/5.08                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.04/5.08                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.08                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.04/5.08                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 9.04/5.08                & ( ( Mi3 != Ma3 )
% 9.04/5.08                 => ! [I3: nat] :
% 9.04/5.08                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 9.04/5.08                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 9.04/5.08                            = I3 )
% 9.04/5.08                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 9.04/5.08                        & ! [X4: nat] :
% 9.04/5.08                            ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
% 9.04/5.08                                = I3 )
% 9.04/5.08                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
% 9.04/5.08                           => ( ( ord_less_nat @ Mi3 @ X4 )
% 9.04/5.08                              & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7535_divmod__step__eq,axiom,
% 9.04/5.08      ! [L: num,R2: nat,Q3: nat] :
% 9.04/5.08        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 9.04/5.08         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 9.04/5.08            = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
% 9.04/5.08        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 9.04/5.08         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 9.04/5.08            = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7536_divmod__step__eq,axiom,
% 9.04/5.08      ! [L: num,R2: int,Q3: int] :
% 9.04/5.08        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 9.04/5.08         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.04/5.08            = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
% 9.04/5.08        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 9.04/5.08         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.04/5.08            = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7537_divmod__step__eq,axiom,
% 9.04/5.08      ! [L: num,R2: code_integer,Q3: code_integer] :
% 9.04/5.08        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 9.04/5.08         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 9.04/5.08            = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 9.04/5.08        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 9.04/5.08         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 9.04/5.08            = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7538_divides__aux__eq,axiom,
% 9.04/5.08      ! [Q3: nat,R2: nat] :
% 9.04/5.08        ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 9.04/5.08        = ( R2 = zero_zero_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7539_divides__aux__eq,axiom,
% 9.04/5.08      ! [Q3: int,R2: int] :
% 9.04/5.08        ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.04/5.08        = ( R2 = zero_zero_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7540_product__nth,axiom,
% 9.04/5.08      ! [N: nat,Xs: list_num,Ys2: list_num] :
% 9.04/5.08        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs ) @ ( size_size_list_num @ Ys2 ) ) )
% 9.04/5.08       => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs @ Ys2 ) @ N )
% 9.04/5.08          = ( product_Pair_num_num @ ( nth_num @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7541_product__nth,axiom,
% 9.04/5.08      ! [N: nat,Xs: list_nat,Ys2: list_num] :
% 9.04/5.08        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_num @ Ys2 ) ) )
% 9.04/5.08       => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs @ Ys2 ) @ N )
% 9.04/5.08          = ( product_Pair_nat_num @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7542_product__nth,axiom,
% 9.04/5.08      ! [N: nat,Xs: list_nat,Ys2: list_nat] :
% 9.04/5.08        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
% 9.04/5.08       => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys2 ) @ N )
% 9.04/5.08          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7543_product__nth,axiom,
% 9.04/5.08      ! [N: nat,Xs: list_nat,Ys2: list_VEBT_VEBT] :
% 9.04/5.08        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 9.04/5.08       => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys2 ) @ N )
% 9.04/5.08          = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7544_product__nth,axiom,
% 9.04/5.08      ! [N: nat,Xs: list_nat,Ys2: list_o] :
% 9.04/5.08        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
% 9.04/5.08       => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys2 ) @ N )
% 9.04/5.08          = ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7545_product__nth,axiom,
% 9.04/5.08      ! [N: nat,Xs: list_Code_integer,Ys2: list_o] :
% 9.04/5.08        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
% 9.04/5.08       => ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs @ Ys2 ) @ N )
% 9.04/5.08          = ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7546_product__nth,axiom,
% 9.04/5.08      ! [N: nat,Xs: list_nat,Ys2: list_int] :
% 9.04/5.08        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
% 9.04/5.08       => ( ( nth_Pr3440142176431000676at_int @ ( product_nat_int @ Xs @ Ys2 ) @ N )
% 9.04/5.08          = ( product_Pair_nat_int @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7547_product__nth,axiom,
% 9.04/5.08      ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
% 9.04/5.08        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
% 9.04/5.08       => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys2 ) @ N )
% 9.04/5.08          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7548_product__nth,axiom,
% 9.04/5.08      ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 9.04/5.08        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 9.04/5.08       => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) @ N )
% 9.04/5.08          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7549_product__nth,axiom,
% 9.04/5.08      ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
% 9.04/5.08        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
% 9.04/5.08       => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys2 ) @ N )
% 9.04/5.08          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7550_option_Osize__gen_I2_J,axiom,
% 9.04/5.08      ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 9.04/5.08        ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.04/5.08        = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7551_option_Osize__gen_I2_J,axiom,
% 9.04/5.08      ! [X: num > nat,X22: num] :
% 9.04/5.08        ( ( size_option_num @ X @ ( some_num @ X22 ) )
% 9.04/5.08        = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7552_signed__take__bit__eq__take__bit__minus,axiom,
% 9.04/5.08      ( bit_ri631733984087533419it_int
% 9.04/5.08      = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7553_zero__le__sgn__iff,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
% 9.04/5.08        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7554_sgn__le__0__iff,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
% 9.04/5.08        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7555_length__product,axiom,
% 9.04/5.08      ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 9.04/5.08        ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) )
% 9.04/5.08        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7556_length__product,axiom,
% 9.04/5.08      ! [Xs: list_VEBT_VEBT,Ys2: list_o] :
% 9.04/5.08        ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys2 ) )
% 9.04/5.08        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7557_length__product,axiom,
% 9.04/5.08      ! [Xs: list_VEBT_VEBT,Ys2: list_int] :
% 9.04/5.08        ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys2 ) )
% 9.04/5.08        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7558_length__product,axiom,
% 9.04/5.08      ! [Xs: list_o,Ys2: list_VEBT_VEBT] :
% 9.04/5.08        ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys2 ) )
% 9.04/5.08        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7559_length__product,axiom,
% 9.04/5.08      ! [Xs: list_o,Ys2: list_o] :
% 9.04/5.08        ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys2 ) )
% 9.04/5.08        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7560_length__product,axiom,
% 9.04/5.08      ! [Xs: list_o,Ys2: list_int] :
% 9.04/5.08        ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys2 ) )
% 9.04/5.08        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7561_length__product,axiom,
% 9.04/5.08      ! [Xs: list_int,Ys2: list_VEBT_VEBT] :
% 9.04/5.08        ( ( size_s6639371672096860321T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys2 ) )
% 9.04/5.08        = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7562_length__product,axiom,
% 9.04/5.08      ! [Xs: list_int,Ys2: list_o] :
% 9.04/5.08        ( ( size_s4246224855604898693_int_o @ ( product_int_o @ Xs @ Ys2 ) )
% 9.04/5.08        = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7563_length__product,axiom,
% 9.04/5.08      ! [Xs: list_int,Ys2: list_int] :
% 9.04/5.08        ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys2 ) )
% 9.04/5.08        = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7564_bit__numeral__Bit0__Suc__iff,axiom,
% 9.04/5.08      ! [M: num,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 9.04/5.08        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7565_bit__numeral__Bit0__Suc__iff,axiom,
% 9.04/5.08      ! [M: num,N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 9.04/5.08        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7566_bit__numeral__Bit1__Suc__iff,axiom,
% 9.04/5.08      ! [M: num,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 9.04/5.08        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7567_bit__numeral__Bit1__Suc__iff,axiom,
% 9.04/5.08      ! [M: num,N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 9.04/5.08        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7568_signed__take__bit__nonnegative__iff,axiom,
% 9.04/5.08      ! [N: nat,K: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.04/5.08        = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7569_signed__take__bit__negative__iff,axiom,
% 9.04/5.08      ! [N: nat,K: int] :
% 9.04/5.08        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 9.04/5.08        = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7570_bit__numeral__simps_I2_J,axiom,
% 9.04/5.08      ! [W: num,N: num] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 9.04/5.08        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7571_bit__numeral__simps_I2_J,axiom,
% 9.04/5.08      ! [W: num,N: num] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 9.04/5.08        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7572_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 9.04/5.08      ! [W: num,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 9.04/5.08        = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7573_bit__numeral__simps_I3_J,axiom,
% 9.04/5.08      ! [W: num,N: num] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 9.04/5.08        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7574_bit__numeral__simps_I3_J,axiom,
% 9.04/5.08      ! [W: num,N: num] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 9.04/5.08        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7575_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 9.04/5.08      ! [W: num,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 9.04/5.08        = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7576_bit__0,axiom,
% 9.04/5.08      ! [A: code_integer] :
% 9.04/5.08        ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 9.04/5.08        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7577_bit__0,axiom,
% 9.04/5.08      ! [A: int] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 9.04/5.08        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7578_bit__0,axiom,
% 9.04/5.08      ! [A: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 9.04/5.08        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7579_bit__minus__numeral__int_I1_J,axiom,
% 9.04/5.08      ! [W: num,N: num] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 9.04/5.08        = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7580_bit__minus__numeral__int_I2_J,axiom,
% 9.04/5.08      ! [W: num,N: num] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 9.04/5.08        = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7581_bit__mod__2__iff,axiom,
% 9.04/5.08      ! [A: code_integer,N: nat] :
% 9.04/5.08        ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
% 9.04/5.08        = ( ( N = zero_zero_nat )
% 9.04/5.08          & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7582_bit__mod__2__iff,axiom,
% 9.04/5.08      ! [A: int,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
% 9.04/5.08        = ( ( N = zero_zero_nat )
% 9.04/5.08          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7583_bit__mod__2__iff,axiom,
% 9.04/5.08      ! [A: nat,N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 9.04/5.08        = ( ( N = zero_zero_nat )
% 9.04/5.08          & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7584_bit__disjunctive__add__iff,axiom,
% 9.04/5.08      ! [A: int,B: int,N: nat] :
% 9.04/5.08        ( ! [N2: nat] :
% 9.04/5.08            ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
% 9.04/5.08            | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
% 9.04/5.08       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N )
% 9.04/5.08          = ( ( bit_se1146084159140164899it_int @ A @ N )
% 9.04/5.08            | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7585_bit__disjunctive__add__iff,axiom,
% 9.04/5.08      ! [A: nat,B: nat,N: nat] :
% 9.04/5.08        ( ! [N2: nat] :
% 9.04/5.08            ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
% 9.04/5.08            | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
% 9.04/5.08       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 9.04/5.08          = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 9.04/5.08            | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7586_bit__of__nat__iff__bit,axiom,
% 9.04/5.08      ! [M: nat,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
% 9.04/5.08        = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7587_bit__of__nat__iff__bit,axiom,
% 9.04/5.08      ! [M: nat,N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
% 9.04/5.08        = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7588_bit__numeral__iff,axiom,
% 9.04/5.08      ! [M: num,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
% 9.04/5.08        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7589_bit__numeral__iff,axiom,
% 9.04/5.08      ! [M: num,N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
% 9.04/5.08        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7590_sgn__eq,axiom,
% 9.04/5.08      ( sgn_sgn_complex
% 9.04/5.08      = ( ^ [Z4: complex] : ( divide1717551699836669952omplex @ Z4 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z4 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7591_not__bit__1__Suc,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7592_not__bit__1__Suc,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7593_bit__1__iff,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
% 9.04/5.08        = ( N = zero_zero_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7594_bit__1__iff,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
% 9.04/5.08        = ( N = zero_zero_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7595_bit__numeral__simps_I1_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7596_bit__numeral__simps_I1_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7597_bit__take__bit__iff,axiom,
% 9.04/5.08      ! [M: nat,A: int,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
% 9.04/5.08        = ( ( ord_less_nat @ N @ M )
% 9.04/5.08          & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7598_bit__take__bit__iff,axiom,
% 9.04/5.08      ! [M: nat,A: nat,N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
% 9.04/5.08        = ( ( ord_less_nat @ N @ M )
% 9.04/5.08          & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7599_bit__of__bool__iff,axiom,
% 9.04/5.08      ! [B: $o,N: nat] :
% 9.04/5.08        ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N )
% 9.04/5.08        = ( B
% 9.04/5.08          & ( N = zero_zero_nat ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7600_bit__of__bool__iff,axiom,
% 9.04/5.08      ! [B: $o,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N )
% 9.04/5.08        = ( B
% 9.04/5.08          & ( N = zero_zero_nat ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7601_bit__of__bool__iff,axiom,
% 9.04/5.08      ! [B: $o,N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N )
% 9.04/5.08        = ( B
% 9.04/5.08          & ( N = zero_zero_nat ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7602_real__sgn__eq,axiom,
% 9.04/5.08      ( sgn_sgn_real
% 9.04/5.08      = ( ^ [X4: real] : ( divide_divide_real @ X4 @ ( abs_abs_real @ X4 ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7603_VEBT__internal_Ovalid_H_Ocases,axiom,
% 9.04/5.08      ! [X: produc9072475918466114483BT_nat] :
% 9.04/5.08        ( ! [Uu2: $o,Uv2: $o,D5: nat] :
% 9.04/5.08            ( X
% 9.04/5.08           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D5 ) )
% 9.04/5.08       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 9.04/5.08              ( X
% 9.04/5.08             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7604_bit__not__int__iff_H,axiom,
% 9.04/5.08      ! [K: int,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 9.04/5.08        = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7605_xor__num_Ocases,axiom,
% 9.04/5.08      ! [X: product_prod_num_num] :
% 9.04/5.08        ( ( X
% 9.04/5.08         != ( product_Pair_num_num @ one @ one ) )
% 9.04/5.08       => ( ! [N2: num] :
% 9.04/5.08              ( X
% 9.04/5.08             != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
% 9.04/5.08         => ( ! [N2: num] :
% 9.04/5.08                ( X
% 9.04/5.08               != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
% 9.04/5.08           => ( ! [M5: num] :
% 9.04/5.08                  ( X
% 9.04/5.08                 != ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
% 9.04/5.08             => ( ! [M5: num,N2: num] :
% 9.04/5.08                    ( X
% 9.04/5.08                   != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N2 ) ) )
% 9.04/5.08               => ( ! [M5: num,N2: num] :
% 9.04/5.08                      ( X
% 9.04/5.08                     != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N2 ) ) )
% 9.04/5.08                 => ( ! [M5: num] :
% 9.04/5.08                        ( X
% 9.04/5.08                       != ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
% 9.04/5.08                   => ( ! [M5: num,N2: num] :
% 9.04/5.08                          ( X
% 9.04/5.08                         != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N2 ) ) )
% 9.04/5.08                     => ~ ! [M5: num,N2: num] :
% 9.04/5.08                            ( X
% 9.04/5.08                           != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7606_sgn__real__def,axiom,
% 9.04/5.08      ( sgn_sgn_real
% 9.04/5.08      = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7607_bit__imp__take__bit__positive,axiom,
% 9.04/5.08      ! [N: nat,M: nat,K: int] :
% 9.04/5.08        ( ( ord_less_nat @ N @ M )
% 9.04/5.08       => ( ( bit_se1146084159140164899it_int @ K @ N )
% 9.04/5.08         => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7608_bit__concat__bit__iff,axiom,
% 9.04/5.08      ! [M: nat,K: int,L: int,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 9.04/5.08        = ( ( ( ord_less_nat @ N @ M )
% 9.04/5.08            & ( bit_se1146084159140164899it_int @ K @ N ) )
% 9.04/5.08          | ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.08            & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7609_VEBT__internal_Onaive__member_Ocases,axiom,
% 9.04/5.08      ! [X: produc9072475918466114483BT_nat] :
% 9.04/5.08        ( ! [A5: $o,B6: $o,X3: nat] :
% 9.04/5.08            ( X
% 9.04/5.08           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ X3 ) )
% 9.04/5.08       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 9.04/5.08              ( X
% 9.04/5.08             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 9.04/5.08         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
% 9.04/5.08                ( X
% 9.04/5.08               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ X3 ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7610_exp__eq__0__imp__not__bit,axiom,
% 9.04/5.08      ! [N: nat,A: int] :
% 9.04/5.08        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.04/5.08          = zero_zero_int )
% 9.04/5.08       => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7611_exp__eq__0__imp__not__bit,axiom,
% 9.04/5.08      ! [N: nat,A: nat] :
% 9.04/5.08        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.08          = zero_zero_nat )
% 9.04/5.08       => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7612_bit__Suc,axiom,
% 9.04/5.08      ! [A: int,N: nat] :
% 9.04/5.08        ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
% 9.04/5.08        = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7613_bit__Suc,axiom,
% 9.04/5.08      ! [A: nat,N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
% 9.04/5.08        = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7614_stable__imp__bit__iff__odd,axiom,
% 9.04/5.08      ! [A: code_integer,N: nat] :
% 9.04/5.08        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.08          = A )
% 9.04/5.08       => ( ( bit_se9216721137139052372nteger @ A @ N )
% 9.04/5.08          = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7615_stable__imp__bit__iff__odd,axiom,
% 9.04/5.08      ! [A: int,N: nat] :
% 9.04/5.08        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.08          = A )
% 9.04/5.08       => ( ( bit_se1146084159140164899it_int @ A @ N )
% 9.04/5.08          = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7616_stable__imp__bit__iff__odd,axiom,
% 9.04/5.08      ! [A: nat,N: nat] :
% 9.04/5.08        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.08          = A )
% 9.04/5.08       => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 9.04/5.08          = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7617_bit__iff__idd__imp__stable,axiom,
% 9.04/5.08      ! [A: code_integer] :
% 9.04/5.08        ( ! [N2: nat] :
% 9.04/5.08            ( ( bit_se9216721137139052372nteger @ A @ N2 )
% 9.04/5.08            = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 9.04/5.08       => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.08          = A ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7618_bit__iff__idd__imp__stable,axiom,
% 9.04/5.08      ! [A: int] :
% 9.04/5.08        ( ! [N2: nat] :
% 9.04/5.08            ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 9.04/5.08            = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 9.04/5.08       => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.08          = A ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7619_bit__iff__idd__imp__stable,axiom,
% 9.04/5.08      ! [A: nat] :
% 9.04/5.08        ( ! [N2: nat] :
% 9.04/5.08            ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 9.04/5.08            = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 9.04/5.08       => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.08          = A ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7620_sgn__power__injE,axiom,
% 9.04/5.08      ! [A: real,N: nat,X: real,B: real] :
% 9.04/5.08        ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 9.04/5.08          = X )
% 9.04/5.08       => ( ( X
% 9.04/5.08            = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 9.04/5.08         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.08           => ( A = B ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7621_int__bit__bound,axiom,
% 9.04/5.08      ! [K: int] :
% 9.04/5.08        ~ ! [N2: nat] :
% 9.04/5.08            ( ! [M3: nat] :
% 9.04/5.08                ( ( ord_less_eq_nat @ N2 @ M3 )
% 9.04/5.08               => ( ( bit_se1146084159140164899it_int @ K @ M3 )
% 9.04/5.08                  = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
% 9.04/5.08           => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.04/5.08               => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 9.04/5.08                  = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7622_bit__iff__odd,axiom,
% 9.04/5.08      ( bit_se9216721137139052372nteger
% 9.04/5.08      = ( ^ [A3: code_integer,N3: nat] :
% 9.04/5.08            ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7623_bit__iff__odd,axiom,
% 9.04/5.08      ( bit_se1146084159140164899it_int
% 9.04/5.08      = ( ^ [A3: int,N3: nat] :
% 9.04/5.08            ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7624_bit__iff__odd,axiom,
% 9.04/5.08      ( bit_se1148574629649215175it_nat
% 9.04/5.08      = ( ^ [A3: nat,N3: nat] :
% 9.04/5.08            ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7625_bit__int__def,axiom,
% 9.04/5.08      ( bit_se1146084159140164899it_int
% 9.04/5.08      = ( ^ [K2: int,N3: nat] :
% 9.04/5.08            ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7626_even__bit__succ__iff,axiom,
% 9.04/5.08      ! [A: code_integer,N: nat] :
% 9.04/5.08        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.08       => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
% 9.04/5.08          = ( ( bit_se9216721137139052372nteger @ A @ N )
% 9.04/5.08            | ( N = zero_zero_nat ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7627_even__bit__succ__iff,axiom,
% 9.04/5.08      ! [A: int,N: nat] :
% 9.04/5.08        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.08       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
% 9.04/5.08          = ( ( bit_se1146084159140164899it_int @ A @ N )
% 9.04/5.08            | ( N = zero_zero_nat ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7628_even__bit__succ__iff,axiom,
% 9.04/5.08      ! [A: nat,N: nat] :
% 9.04/5.08        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.08       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
% 9.04/5.08          = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 9.04/5.08            | ( N = zero_zero_nat ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7629_odd__bit__iff__bit__pred,axiom,
% 9.04/5.08      ! [A: code_integer,N: nat] :
% 9.04/5.08        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.08       => ( ( bit_se9216721137139052372nteger @ A @ N )
% 9.04/5.08          = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
% 9.04/5.08            | ( N = zero_zero_nat ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7630_odd__bit__iff__bit__pred,axiom,
% 9.04/5.08      ! [A: int,N: nat] :
% 9.04/5.08        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.08       => ( ( bit_se1146084159140164899it_int @ A @ N )
% 9.04/5.08          = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
% 9.04/5.08            | ( N = zero_zero_nat ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7631_odd__bit__iff__bit__pred,axiom,
% 9.04/5.08      ! [A: nat,N: nat] :
% 9.04/5.08        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.08       => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 9.04/5.08          = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
% 9.04/5.08            | ( N = zero_zero_nat ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7632_bit__sum__mult__2__cases,axiom,
% 9.04/5.08      ! [A: code_integer,B: code_integer,N: nat] :
% 9.04/5.08        ( ! [J3: nat] :
% 9.04/5.08            ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J3 ) )
% 9.04/5.08       => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
% 9.04/5.08          = ( ( ( N = zero_zero_nat )
% 9.04/5.08             => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.08            & ( ( N != zero_zero_nat )
% 9.04/5.08             => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7633_bit__sum__mult__2__cases,axiom,
% 9.04/5.08      ! [A: int,B: int,N: nat] :
% 9.04/5.08        ( ! [J3: nat] :
% 9.04/5.08            ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J3 ) )
% 9.04/5.08       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N )
% 9.04/5.08          = ( ( ( N = zero_zero_nat )
% 9.04/5.08             => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.08            & ( ( N != zero_zero_nat )
% 9.04/5.08             => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7634_bit__sum__mult__2__cases,axiom,
% 9.04/5.08      ! [A: nat,B: nat,N: nat] :
% 9.04/5.08        ( ! [J3: nat] :
% 9.04/5.08            ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J3 ) )
% 9.04/5.08       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N )
% 9.04/5.08          = ( ( ( N = zero_zero_nat )
% 9.04/5.08             => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.04/5.08            & ( ( N != zero_zero_nat )
% 9.04/5.08             => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7635_bit__rec,axiom,
% 9.04/5.08      ( bit_se9216721137139052372nteger
% 9.04/5.08      = ( ^ [A3: code_integer,N3: nat] :
% 9.04/5.08            ( ( ( N3 = zero_zero_nat )
% 9.04/5.08             => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
% 9.04/5.08            & ( ( N3 != zero_zero_nat )
% 9.04/5.08             => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7636_bit__rec,axiom,
% 9.04/5.08      ( bit_se1146084159140164899it_int
% 9.04/5.08      = ( ^ [A3: int,N3: nat] :
% 9.04/5.08            ( ( ( N3 = zero_zero_nat )
% 9.04/5.08             => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
% 9.04/5.08            & ( ( N3 != zero_zero_nat )
% 9.04/5.08             => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7637_bit__rec,axiom,
% 9.04/5.08      ( bit_se1148574629649215175it_nat
% 9.04/5.08      = ( ^ [A3: nat,N3: nat] :
% 9.04/5.08            ( ( ( N3 = zero_zero_nat )
% 9.04/5.08             => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
% 9.04/5.08            & ( ( N3 != zero_zero_nat )
% 9.04/5.08             => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7638_VEBT__internal_Omembermima_Ocases,axiom,
% 9.04/5.08      ! [X: produc9072475918466114483BT_nat] :
% 9.04/5.08        ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 9.04/5.08            ( X
% 9.04/5.08           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 9.04/5.08       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 9.04/5.08              ( X
% 9.04/5.08             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 9.04/5.08         => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
% 9.04/5.08                ( X
% 9.04/5.08               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X3 ) )
% 9.04/5.08           => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
% 9.04/5.08                  ( X
% 9.04/5.08                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) @ X3 ) )
% 9.04/5.08             => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
% 9.04/5.08                    ( X
% 9.04/5.08                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ X3 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7639_option_Osize__gen_I1_J,axiom,
% 9.04/5.08      ! [X: product_prod_nat_nat > nat] :
% 9.04/5.08        ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
% 9.04/5.08        = ( suc @ zero_zero_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7640_option_Osize__gen_I1_J,axiom,
% 9.04/5.08      ! [X: num > nat] :
% 9.04/5.08        ( ( size_option_num @ X @ none_num )
% 9.04/5.08        = ( suc @ zero_zero_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7641_set__bit__eq,axiom,
% 9.04/5.08      ( bit_se7879613467334960850it_int
% 9.04/5.08      = ( ^ [N3: nat,K2: int] :
% 9.04/5.08            ( plus_plus_int @ K2
% 9.04/5.08            @ ( times_times_int
% 9.04/5.08              @ ( zero_n2684676970156552555ol_int
% 9.04/5.08                @ ~ ( bit_se1146084159140164899it_int @ K2 @ N3 ) )
% 9.04/5.08              @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7642_unset__bit__eq,axiom,
% 9.04/5.08      ( bit_se4203085406695923979it_int
% 9.04/5.08      = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7643_take__bit__Suc__from__most,axiom,
% 9.04/5.08      ! [N: nat,K: int] :
% 9.04/5.08        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 9.04/5.08        = ( 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 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7644_divmod__algorithm__code_I7_J,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( ( ord_less_eq_num @ M @ N )
% 9.04/5.08         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 9.04/5.08        & ( ~ ( ord_less_eq_num @ M @ N )
% 9.04/5.08         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7645_divmod__algorithm__code_I7_J,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( ( ord_less_eq_num @ M @ N )
% 9.04/5.08         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 9.04/5.08        & ( ~ ( ord_less_eq_num @ M @ N )
% 9.04/5.08         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7646_divmod__algorithm__code_I7_J,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( ( ord_less_eq_num @ M @ N )
% 9.04/5.08         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 9.04/5.08        & ( ~ ( ord_less_eq_num @ M @ N )
% 9.04/5.08         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7647_divmod__algorithm__code_I8_J,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( ( ord_less_num @ M @ N )
% 9.04/5.08         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 9.04/5.08        & ( ~ ( ord_less_num @ M @ N )
% 9.04/5.08         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7648_divmod__algorithm__code_I8_J,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( ( ord_less_num @ M @ N )
% 9.04/5.08         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 9.04/5.08        & ( ~ ( ord_less_num @ M @ N )
% 9.04/5.08         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7649_divmod__algorithm__code_I8_J,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( ( ord_less_num @ M @ N )
% 9.04/5.08         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 9.04/5.08        & ( ~ ( ord_less_num @ M @ N )
% 9.04/5.08         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.08            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7650_neg__eucl__rel__int__mult__2,axiom,
% 9.04/5.08      ! [B: int,A: int,Q3: int,R2: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.04/5.08       => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.04/5.08         => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7651_Divides_Oadjust__div__eq,axiom,
% 9.04/5.08      ! [Q3: int,R2: int] :
% 9.04/5.08        ( ( adjust_div @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.04/5.08        = ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7652_pos__eucl__rel__int__mult__2,axiom,
% 9.04/5.08      ! [B: int,A: int,Q3: int,R2: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.04/5.08       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.04/5.08         => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7653_numeral__div__minus__numeral,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.08        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7654_minus__numeral__div__numeral,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.08        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7655_dvd__numeral__simp,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.08        = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7656_dvd__numeral__simp,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.04/5.08        = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7657_dvd__numeral__simp,axiom,
% 9.04/5.08      ! [M: num,N: num] :
% 9.04/5.08        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 9.04/5.08        = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7658_divmod__algorithm__code_I2_J,axiom,
% 9.04/5.08      ! [M: num] :
% 9.04/5.08        ( ( unique5052692396658037445od_int @ M @ one )
% 9.04/5.08        = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7659_divmod__algorithm__code_I2_J,axiom,
% 9.04/5.08      ! [M: num] :
% 9.04/5.08        ( ( unique5055182867167087721od_nat @ M @ one )
% 9.04/5.08        = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7660_divmod__algorithm__code_I2_J,axiom,
% 9.04/5.08      ! [M: num] :
% 9.04/5.08        ( ( unique3479559517661332726nteger @ M @ one )
% 9.04/5.08        = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7661_divmod__algorithm__code_I3_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 9.04/5.08        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7662_divmod__algorithm__code_I3_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 9.04/5.08        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7663_divmod__algorithm__code_I3_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 9.04/5.08        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7664_divmod__algorithm__code_I4_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 9.04/5.08        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7665_divmod__algorithm__code_I4_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 9.04/5.08        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7666_divmod__algorithm__code_I4_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 9.04/5.08        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7667_one__div__minus__numeral,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.08        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7668_minus__one__div__numeral,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.08        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7669_eucl__rel__int__by0,axiom,
% 9.04/5.08      ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7670_div__int__unique,axiom,
% 9.04/5.08      ! [K: int,L: int,Q3: int,R2: int] :
% 9.04/5.08        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.04/5.08       => ( ( divide_divide_int @ K @ L )
% 9.04/5.08          = Q3 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7671_bit__Suc__0__iff,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.04/5.08        = ( N = zero_zero_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7672_not__bit__Suc__0__Suc,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7673_not__bit__Suc__0__numeral,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7674_eucl__rel__int__dividesI,axiom,
% 9.04/5.08      ! [L: int,K: int,Q3: int] :
% 9.04/5.08        ( ( L != zero_zero_int )
% 9.04/5.08       => ( ( K
% 9.04/5.08            = ( times_times_int @ Q3 @ L ) )
% 9.04/5.08         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7675_eucl__rel__int,axiom,
% 9.04/5.08      ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7676_bit__nat__iff,axiom,
% 9.04/5.08      ! [K: int,N: nat] :
% 9.04/5.08        ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 9.04/5.08        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.08          & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7677_divmod__int__def,axiom,
% 9.04/5.08      ( unique5052692396658037445od_int
% 9.04/5.08      = ( ^ [M6: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7678_divmod__def,axiom,
% 9.04/5.08      ( unique5052692396658037445od_int
% 9.04/5.08      = ( ^ [M6: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7679_divmod__def,axiom,
% 9.04/5.08      ( unique5055182867167087721od_nat
% 9.04/5.08      = ( ^ [M6: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7680_divmod__def,axiom,
% 9.04/5.08      ( unique3479559517661332726nteger
% 9.04/5.08      = ( ^ [M6: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7681_bit__nat__def,axiom,
% 9.04/5.08      ( bit_se1148574629649215175it_nat
% 9.04/5.08      = ( ^ [M6: nat,N3: nat] :
% 9.04/5.08            ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7682_divmod_H__nat__def,axiom,
% 9.04/5.08      ( unique5055182867167087721od_nat
% 9.04/5.08      = ( ^ [M6: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7683_zminus1__lemma,axiom,
% 9.04/5.08      ! [A: int,B: int,Q3: int,R2: int] :
% 9.04/5.08        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.04/5.08       => ( ( B != zero_zero_int )
% 9.04/5.08         => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7684_eucl__rel__int__iff,axiom,
% 9.04/5.08      ! [K: int,L: int,Q3: int,R2: int] :
% 9.04/5.08        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.04/5.08        = ( ( K
% 9.04/5.08            = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
% 9.04/5.08          & ( ( ord_less_int @ zero_zero_int @ L )
% 9.04/5.08           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.04/5.08              & ( ord_less_int @ R2 @ L ) ) )
% 9.04/5.08          & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 9.04/5.08           => ( ( ( ord_less_int @ L @ zero_zero_int )
% 9.04/5.08               => ( ( ord_less_int @ L @ R2 )
% 9.04/5.08                  & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 9.04/5.08              & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 9.04/5.08               => ( Q3 = zero_zero_int ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7685_eucl__rel__int__remainderI,axiom,
% 9.04/5.08      ! [R2: int,L: int,K: int,Q3: int] :
% 9.04/5.08        ( ( ( sgn_sgn_int @ R2 )
% 9.04/5.08          = ( sgn_sgn_int @ L ) )
% 9.04/5.08       => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 9.04/5.08         => ( ( K
% 9.04/5.08              = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
% 9.04/5.08           => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7686_eucl__rel__int_Ocases,axiom,
% 9.04/5.08      ! [A1: int,A22: int,A32: product_prod_int_int] :
% 9.04/5.08        ( ( eucl_rel_int @ A1 @ A22 @ A32 )
% 9.04/5.08       => ( ( ( A22 = zero_zero_int )
% 9.04/5.08           => ( A32
% 9.04/5.08             != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
% 9.04/5.08         => ( ! [Q2: int] :
% 9.04/5.08                ( ( A32
% 9.04/5.08                  = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
% 9.04/5.08               => ( ( A22 != zero_zero_int )
% 9.04/5.08                 => ( A1
% 9.04/5.08                   != ( times_times_int @ Q2 @ A22 ) ) ) )
% 9.04/5.08           => ~ ! [R3: int,Q2: int] :
% 9.04/5.08                  ( ( A32
% 9.04/5.08                    = ( product_Pair_int_int @ Q2 @ R3 ) )
% 9.04/5.08                 => ( ( ( sgn_sgn_int @ R3 )
% 9.04/5.08                      = ( sgn_sgn_int @ A22 ) )
% 9.04/5.08                   => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A22 ) )
% 9.04/5.08                     => ( A1
% 9.04/5.08                       != ( plus_plus_int @ ( times_times_int @ Q2 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7687_eucl__rel__int_Osimps,axiom,
% 9.04/5.08      ( eucl_rel_int
% 9.04/5.08      = ( ^ [A12: int,A23: int,A33: product_prod_int_int] :
% 9.04/5.08            ( ? [K2: int] :
% 9.04/5.08                ( ( A12 = K2 )
% 9.04/5.08                & ( A23 = zero_zero_int )
% 9.04/5.08                & ( A33
% 9.04/5.08                  = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
% 9.04/5.08            | ? [L3: int,K2: int,Q4: int] :
% 9.04/5.08                ( ( A12 = K2 )
% 9.04/5.08                & ( A23 = L3 )
% 9.04/5.08                & ( A33
% 9.04/5.08                  = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 9.04/5.08                & ( L3 != zero_zero_int )
% 9.04/5.08                & ( K2
% 9.04/5.08                  = ( times_times_int @ Q4 @ L3 ) ) )
% 9.04/5.08            | ? [R5: int,L3: int,K2: int,Q4: int] :
% 9.04/5.08                ( ( A12 = K2 )
% 9.04/5.08                & ( A23 = L3 )
% 9.04/5.08                & ( A33
% 9.04/5.08                  = ( product_Pair_int_int @ Q4 @ R5 ) )
% 9.04/5.08                & ( ( sgn_sgn_int @ R5 )
% 9.04/5.08                  = ( sgn_sgn_int @ L3 ) )
% 9.04/5.08                & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L3 ) )
% 9.04/5.08                & ( K2
% 9.04/5.08                  = ( plus_plus_int @ ( times_times_int @ Q4 @ L3 ) @ R5 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7688_divmod__divmod__step,axiom,
% 9.04/5.08      ( unique5055182867167087721od_nat
% 9.04/5.08      = ( ^ [M6: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7689_divmod__divmod__step,axiom,
% 9.04/5.08      ( unique5052692396658037445od_int
% 9.04/5.08      = ( ^ [M6: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7690_divmod__divmod__step,axiom,
% 9.04/5.08      ( unique3479559517661332726nteger
% 9.04/5.08      = ( ^ [M6: num,N3: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N3 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N3 @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7691_arctan__inverse,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( X != zero_zero_real )
% 9.04/5.08       => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
% 9.04/5.08          = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7692_and__int__unfold,axiom,
% 9.04/5.08      ( bit_se725231765392027082nd_int
% 9.04/5.08      = ( ^ [K2: int,L3: int] :
% 9.04/5.08            ( if_int
% 9.04/5.08            @ ( ( K2 = zero_zero_int )
% 9.04/5.08              | ( L3 = zero_zero_int ) )
% 9.04/5.08            @ zero_zero_int
% 9.04/5.08            @ ( if_int
% 9.04/5.08              @ ( K2
% 9.04/5.08                = ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.08              @ L3
% 9.04/5.08              @ ( if_int
% 9.04/5.08                @ ( L3
% 9.04/5.08                  = ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.08                @ K2
% 9.04/5.08                @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7693_arctan__half,axiom,
% 9.04/5.08      ( arctan
% 9.04/5.08      = ( ^ [X4: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X4 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7694_machin,axiom,
% 9.04/5.08      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.04/5.08      = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7695_machin__Euler,axiom,
% 9.04/5.08      ( ( 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 ) ) ) ) ) ) ) ) ) ) )
% 9.04/5.08      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7696_and__zero__eq,axiom,
% 9.04/5.08      ! [A: int] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 9.04/5.08        = zero_zero_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7697_and__zero__eq,axiom,
% 9.04/5.08      ! [A: nat] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 9.04/5.08        = zero_zero_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7698_zero__and__eq,axiom,
% 9.04/5.08      ! [A: int] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 9.04/5.08        = zero_zero_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7699_zero__and__eq,axiom,
% 9.04/5.08      ! [A: nat] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 9.04/5.08        = zero_zero_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7700_bit_Oconj__zero__left,axiom,
% 9.04/5.08      ! [X: int] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
% 9.04/5.08        = zero_zero_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7701_bit_Oconj__zero__right,axiom,
% 9.04/5.08      ! [X: int] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
% 9.04/5.08        = zero_zero_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7702_real__sqrt__zero,axiom,
% 9.04/5.08      ( ( sqrt @ zero_zero_real )
% 9.04/5.08      = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7703_real__sqrt__eq__zero__cancel__iff,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ( sqrt @ X )
% 9.04/5.08          = zero_zero_real )
% 9.04/5.08        = ( X = zero_zero_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7704_real__sqrt__less__iff,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 9.04/5.08        = ( ord_less_real @ X @ Y ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7705_real__sqrt__le__iff,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 9.04/5.08        = ( ord_less_eq_real @ X @ Y ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7706_real__sqrt__eq__1__iff,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ( sqrt @ X )
% 9.04/5.08          = one_one_real )
% 9.04/5.08        = ( X = one_one_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7707_real__sqrt__one,axiom,
% 9.04/5.08      ( ( sqrt @ one_one_real )
% 9.04/5.08      = one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7708_and_Oleft__neutral,axiom,
% 9.04/5.08      ! [A: code_integer] :
% 9.04/5.08        ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
% 9.04/5.08        = A ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7709_and_Oleft__neutral,axiom,
% 9.04/5.08      ! [A: int] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
% 9.04/5.08        = A ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7710_and_Oright__neutral,axiom,
% 9.04/5.08      ! [A: code_integer] :
% 9.04/5.08        ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.08        = A ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7711_and_Oright__neutral,axiom,
% 9.04/5.08      ! [A: int] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.08        = A ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7712_bit_Oconj__one__right,axiom,
% 9.04/5.08      ! [X: code_integer] :
% 9.04/5.08        ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.08        = X ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7713_bit_Oconj__one__right,axiom,
% 9.04/5.08      ! [X: int] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.08        = X ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7714_real__sqrt__gt__0__iff,axiom,
% 9.04/5.08      ! [Y: real] :
% 9.04/5.08        ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
% 9.04/5.08        = ( ord_less_real @ zero_zero_real @ Y ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7715_real__sqrt__lt__0__iff,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 9.04/5.08        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7716_real__sqrt__le__0__iff,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
% 9.04/5.08        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7717_real__sqrt__ge__0__iff,axiom,
% 9.04/5.08      ! [Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
% 9.04/5.08        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7718_real__sqrt__gt__1__iff,axiom,
% 9.04/5.08      ! [Y: real] :
% 9.04/5.08        ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
% 9.04/5.08        = ( ord_less_real @ one_one_real @ Y ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7719_real__sqrt__lt__1__iff,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 9.04/5.08        = ( ord_less_real @ X @ one_one_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7720_real__sqrt__le__1__iff,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
% 9.04/5.08        = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7721_real__sqrt__ge__1__iff,axiom,
% 9.04/5.08      ! [Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
% 9.04/5.08        = ( ord_less_eq_real @ one_one_real @ Y ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7722_and__nonnegative__int__iff,axiom,
% 9.04/5.08      ! [K: int,L: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 9.04/5.08        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.08          | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7723_and__negative__int__iff,axiom,
% 9.04/5.08      ! [K: int,L: int] :
% 9.04/5.08        ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 9.04/5.08        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.04/5.08          & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7724_and__numerals_I2_J,axiom,
% 9.04/5.08      ! [Y: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 9.04/5.08        = one_one_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7725_and__numerals_I2_J,axiom,
% 9.04/5.08      ! [Y: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.04/5.08        = one_one_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7726_and__numerals_I8_J,axiom,
% 9.04/5.08      ! [X: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 9.04/5.08        = one_one_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7727_and__numerals_I8_J,axiom,
% 9.04/5.08      ! [X: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 9.04/5.08        = one_one_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7728_real__sqrt__four,axiom,
% 9.04/5.08      ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.04/5.08      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7729_and__numerals_I1_J,axiom,
% 9.04/5.08      ! [Y: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 9.04/5.08        = zero_zero_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7730_and__numerals_I1_J,axiom,
% 9.04/5.08      ! [Y: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.04/5.08        = zero_zero_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7731_and__numerals_I5_J,axiom,
% 9.04/5.08      ! [X: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 9.04/5.08        = zero_zero_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7732_and__numerals_I5_J,axiom,
% 9.04/5.08      ! [X: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 9.04/5.08        = zero_zero_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7733_and__numerals_I3_J,axiom,
% 9.04/5.08      ! [X: num,Y: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 9.04/5.08        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7734_and__numerals_I3_J,axiom,
% 9.04/5.08      ! [X: num,Y: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.04/5.08        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7735_and__minus__numerals_I6_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 9.04/5.08        = one_one_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7736_and__minus__numerals_I2_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.04/5.08        = one_one_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7737_and__numerals_I4_J,axiom,
% 9.04/5.08      ! [X: num,Y: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 9.04/5.08        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7738_and__numerals_I4_J,axiom,
% 9.04/5.08      ! [X: num,Y: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.04/5.08        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7739_and__numerals_I6_J,axiom,
% 9.04/5.08      ! [X: num,Y: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 9.04/5.08        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7740_and__numerals_I6_J,axiom,
% 9.04/5.08      ! [X: num,Y: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.04/5.08        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7741_real__sqrt__abs,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.08        = ( abs_abs_real @ X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7742_and__minus__numerals_I5_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 9.04/5.08        = zero_zero_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7743_and__minus__numerals_I1_J,axiom,
% 9.04/5.08      ! [N: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.04/5.08        = zero_zero_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7744_real__sqrt__pow2__iff,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.08          = X )
% 9.04/5.08        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7745_real__sqrt__pow2,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.08          = X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7746_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 9.04/5.08      ! [X: real,Y: real,Xa: real,Ya: real] :
% 9.04/5.08        ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.08        = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7747_and__numerals_I7_J,axiom,
% 9.04/5.08      ! [X: num,Y: num] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 9.04/5.08        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7748_and__numerals_I7_J,axiom,
% 9.04/5.08      ! [X: num,Y: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.04/5.08        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7749_real__sqrt__divide,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
% 9.04/5.08        = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7750_real__sqrt__less__mono,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_real @ X @ Y )
% 9.04/5.08       => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7751_real__sqrt__le__mono,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ X @ Y )
% 9.04/5.08       => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7752_of__int__and__eq,axiom,
% 9.04/5.08      ! [K: int,L: int] :
% 9.04/5.08        ( ( ring_18347121197199848620nteger @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 9.04/5.08        = ( bit_se3949692690581998587nteger @ ( ring_18347121197199848620nteger @ K ) @ ( ring_18347121197199848620nteger @ L ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7753_of__int__and__eq,axiom,
% 9.04/5.08      ! [K: int,L: int] :
% 9.04/5.08        ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 9.04/5.08        = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7754_of__nat__and__eq,axiom,
% 9.04/5.08      ! [M: nat,N: nat] :
% 9.04/5.08        ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 9.04/5.08        = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7755_of__nat__and__eq,axiom,
% 9.04/5.08      ! [M: nat,N: nat] :
% 9.04/5.08        ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 9.04/5.08        = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7756_real__sqrt__power,axiom,
% 9.04/5.08      ! [X: real,K: nat] :
% 9.04/5.08        ( ( sqrt @ ( power_power_real @ X @ K ) )
% 9.04/5.08        = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7757_pi__neq__zero,axiom,
% 9.04/5.08      pi != zero_zero_real ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7758_and__eq__minus__1__iff,axiom,
% 9.04/5.08      ! [A: code_integer,B: code_integer] :
% 9.04/5.08        ( ( ( bit_se3949692690581998587nteger @ A @ B )
% 9.04/5.08          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.08        = ( ( A
% 9.04/5.08            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.08          & ( B
% 9.04/5.08            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7759_and__eq__minus__1__iff,axiom,
% 9.04/5.08      ! [A: int,B: int] :
% 9.04/5.08        ( ( ( bit_se725231765392027082nd_int @ A @ B )
% 9.04/5.08          = ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.08        = ( ( A
% 9.04/5.08            = ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.08          & ( B
% 9.04/5.08            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7760_real__sqrt__gt__zero,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7761_real__sqrt__eq__zero__cancel,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ( ( sqrt @ X )
% 9.04/5.08            = zero_zero_real )
% 9.04/5.08         => ( X = zero_zero_real ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7762_real__sqrt__ge__zero,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7763_real__sqrt__ge__one,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ one_one_real @ X )
% 9.04/5.08       => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7764_AND__upper2_H,axiom,
% 9.04/5.08      ! [Y: int,Z: int,X: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.08       => ( ( ord_less_eq_int @ Y @ Z )
% 9.04/5.08         => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7765_AND__upper1_H,axiom,
% 9.04/5.08      ! [Y: int,Z: int,Ya: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.08       => ( ( ord_less_eq_int @ Y @ Z )
% 9.04/5.08         => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7766_AND__upper2,axiom,
% 9.04/5.08      ! [Y: int,X: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.08       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7767_AND__upper1,axiom,
% 9.04/5.08      ! [X: int,Y: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.08       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7768_AND__lower,axiom,
% 9.04/5.08      ! [X: int,Y: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.08       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7769_pi__not__less__zero,axiom,
% 9.04/5.08      ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7770_pi__gt__zero,axiom,
% 9.04/5.08      ord_less_real @ zero_zero_real @ pi ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7771_pi__ge__zero,axiom,
% 9.04/5.08      ord_less_eq_real @ zero_zero_real @ pi ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7772_real__div__sqrt,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
% 9.04/5.08          = ( sqrt @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7773_sqrt__add__le__add__sqrt,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.08         => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7774_le__real__sqrt__sumsq,axiom,
% 9.04/5.08      ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7775_AND__upper2_H_H,axiom,
% 9.04/5.08      ! [Y: int,Z: int,X: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.08       => ( ( ord_less_int @ Y @ Z )
% 9.04/5.08         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7776_AND__upper1_H_H,axiom,
% 9.04/5.08      ! [Y: int,Z: int,Ya: int] :
% 9.04/5.08        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.08       => ( ( ord_less_int @ Y @ Z )
% 9.04/5.08         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7777_and__less__eq,axiom,
% 9.04/5.08      ! [L: int,K: int] :
% 9.04/5.08        ( ( ord_less_int @ L @ zero_zero_int )
% 9.04/5.08       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7778_even__and__iff,axiom,
% 9.04/5.08      ! [A: code_integer,B: code_integer] :
% 9.04/5.08        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
% 9.04/5.08        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.08          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7779_even__and__iff,axiom,
% 9.04/5.08      ! [A: int,B: int] :
% 9.04/5.08        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 9.04/5.08        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.08          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7780_even__and__iff,axiom,
% 9.04/5.08      ! [A: nat,B: nat] :
% 9.04/5.08        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 9.04/5.08        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.08          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7781_sqrt2__less__2,axiom,
% 9.04/5.08      ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7782_even__and__iff__int,axiom,
% 9.04/5.08      ! [K: int,L: int] :
% 9.04/5.08        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 9.04/5.08        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.04/5.08          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7783_pi__less__4,axiom,
% 9.04/5.08      ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7784_pi__ge__two,axiom,
% 9.04/5.08      ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7785_pi__half__neq__two,axiom,
% 9.04/5.08      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.08     != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7786_one__and__eq,axiom,
% 9.04/5.08      ! [A: code_integer] :
% 9.04/5.08        ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
% 9.04/5.08        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7787_one__and__eq,axiom,
% 9.04/5.08      ! [A: int] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
% 9.04/5.08        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7788_one__and__eq,axiom,
% 9.04/5.08      ! [A: nat] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
% 9.04/5.08        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7789_and__one__eq,axiom,
% 9.04/5.08      ! [A: code_integer] :
% 9.04/5.08        ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
% 9.04/5.08        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7790_and__one__eq,axiom,
% 9.04/5.08      ! [A: int] :
% 9.04/5.08        ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
% 9.04/5.08        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7791_and__one__eq,axiom,
% 9.04/5.08      ! [A: nat] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
% 9.04/5.08        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7792_real__less__rsqrt,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 9.04/5.08       => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7793_sqrt__le__D,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
% 9.04/5.08       => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7794_real__le__rsqrt,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 9.04/5.08       => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7795_pi__half__neq__zero,axiom,
% 9.04/5.08      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.08     != zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7796_pi__half__less__two,axiom,
% 9.04/5.08      ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7797_pi__half__le__two,axiom,
% 9.04/5.08      ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7798_and__exp__eq__0__iff__not__bit,axiom,
% 9.04/5.08      ! [A: int,N: nat] :
% 9.04/5.08        ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.08          = zero_zero_int )
% 9.04/5.08        = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7799_and__exp__eq__0__iff__not__bit,axiom,
% 9.04/5.08      ! [A: nat,N: nat] :
% 9.04/5.08        ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.08          = zero_zero_nat )
% 9.04/5.08        = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7800_real__sqrt__unique,axiom,
% 9.04/5.08      ! [Y: real,X: real] :
% 9.04/5.08        ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.08          = X )
% 9.04/5.08       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.08         => ( ( sqrt @ X )
% 9.04/5.08            = Y ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7801_real__le__lsqrt,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.08         => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.08           => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7802_lemma__real__divide__sqrt__less,axiom,
% 9.04/5.08      ! [U: real] :
% 9.04/5.08        ( ( ord_less_real @ zero_zero_real @ U )
% 9.04/5.08       => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7803_real__sqrt__sum__squares__eq__cancel,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.08          = X )
% 9.04/5.08       => ( Y = zero_zero_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7804_real__sqrt__sum__squares__eq__cancel2,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.08          = Y )
% 9.04/5.08       => ( X = zero_zero_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7805_real__sqrt__sum__squares__ge1,axiom,
% 9.04/5.08      ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7806_real__sqrt__sum__squares__ge2,axiom,
% 9.04/5.08      ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7807_real__sqrt__sum__squares__triangle__ineq,axiom,
% 9.04/5.08      ! [A: real,C: real,B: 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 @ B @ 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 @ B @ ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7808_sqrt__ge__absD,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
% 9.04/5.08       => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7809_pi__half__gt__zero,axiom,
% 9.04/5.08      ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7810_pi__half__ge__zero,axiom,
% 9.04/5.08      ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7811_m2pi__less__pi,axiom,
% 9.04/5.08      ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7812_arctan__ubound,axiom,
% 9.04/5.08      ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7813_arctan__one,axiom,
% 9.04/5.08      ( ( arctan @ one_one_real )
% 9.04/5.08      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7814_real__less__lsqrt,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.08         => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.08           => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7815_sqrt__sum__squares__le__sum,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.08         => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7816_ln__sqrt,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ( ln_ln_real @ ( sqrt @ X ) )
% 9.04/5.08          = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7817_sqrt__sum__squares__le__sum__abs,axiom,
% 9.04/5.08      ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7818_real__sqrt__ge__abs2,axiom,
% 9.04/5.08      ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7819_real__sqrt__ge__abs1,axiom,
% 9.04/5.08      ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7820_sqrt__even__pow2,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.08       => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.08          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7821_minus__pi__half__less__zero,axiom,
% 9.04/5.08      ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7822_arctan__bounded,axiom,
% 9.04/5.08      ! [Y: real] :
% 9.04/5.08        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 9.04/5.08        & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7823_arctan__lbound,axiom,
% 9.04/5.08      ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7824_arsinh__real__aux,axiom,
% 9.04/5.08      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7825_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 9.04/5.08      ! [X: real,Y: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7826_real__sqrt__power__even,axiom,
% 9.04/5.08      ! [N: nat,X: real] :
% 9.04/5.08        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.08       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08         => ( ( power_power_real @ ( sqrt @ X ) @ N )
% 9.04/5.08            = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7827_arith__geo__mean__sqrt,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.08         => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7828_powr__half__sqrt,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08       => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.08          = ( sqrt @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7829_arsinh__real__def,axiom,
% 9.04/5.08      ( arsinh_real
% 9.04/5.08      = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7830_and__int__rec,axiom,
% 9.04/5.08      ( bit_se725231765392027082nd_int
% 9.04/5.08      = ( ^ [K2: int,L3: int] :
% 9.04/5.08            ( plus_plus_int
% 9.04/5.08            @ ( zero_n2684676970156552555ol_int
% 9.04/5.08              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 9.04/5.08                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 9.04/5.08            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7831_cos__x__y__le__one,axiom,
% 9.04/5.08      ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7832_real__sqrt__sum__squares__less,axiom,
% 9.04/5.08      ! [X: real,U: real,Y: real] :
% 9.04/5.08        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.04/5.08       => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.04/5.08         => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7833_arcosh__real__def,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ord_less_eq_real @ one_one_real @ X )
% 9.04/5.08       => ( ( arcosh_real @ X )
% 9.04/5.08          = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7834_sqrt__sum__squares__half__less,axiom,
% 9.04/5.08      ! [X: real,U: real,Y: real] :
% 9.04/5.08        ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.08       => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.08         => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.08           => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.08             => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7835_sin__cos__npi,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( 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 ) ) ) )
% 9.04/5.08        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7836_and__int_Osimps,axiom,
% 9.04/5.08      ( bit_se725231765392027082nd_int
% 9.04/5.08      = ( ^ [K2: int,L3: int] :
% 9.04/5.08            ( if_int
% 9.04/5.08            @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.04/5.08              & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.04/5.08            @ ( uminus_uminus_int
% 9.04/5.08              @ ( zero_n2684676970156552555ol_int
% 9.04/5.08                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 9.04/5.08                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
% 9.04/5.08            @ ( plus_plus_int
% 9.04/5.08              @ ( zero_n2684676970156552555ol_int
% 9.04/5.08                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 9.04/5.08                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 9.04/5.08              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7837_and__int_Oelims,axiom,
% 9.04/5.08      ! [X: int,Xa: int,Y: int] :
% 9.04/5.08        ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 9.04/5.08          = Y )
% 9.04/5.08       => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.04/5.08              & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.04/5.08           => ( Y
% 9.04/5.08              = ( uminus_uminus_int
% 9.04/5.08                @ ( zero_n2684676970156552555ol_int
% 9.04/5.08                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 9.04/5.08                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 9.04/5.08          & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.04/5.08                & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.04/5.08           => ( Y
% 9.04/5.08              = ( plus_plus_int
% 9.04/5.08                @ ( zero_n2684676970156552555ol_int
% 9.04/5.08                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 9.04/5.08                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 9.04/5.08                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7838_cos__pi__eq__zero,axiom,
% 9.04/5.08      ! [M: nat] :
% 9.04/5.08        ( ( 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 ) ) ) )
% 9.04/5.08        = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7839_divmod__BitM__2__eq,axiom,
% 9.04/5.08      ! [M: num] :
% 9.04/5.08        ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 9.04/5.08        = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7840_empty__iff,axiom,
% 9.04/5.08      ! [C: complex] :
% 9.04/5.08        ~ ( member_complex @ C @ bot_bot_set_complex ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7841_empty__iff,axiom,
% 9.04/5.08      ! [C: set_nat] :
% 9.04/5.08        ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7842_empty__iff,axiom,
% 9.04/5.08      ! [C: real] :
% 9.04/5.08        ~ ( member_real @ C @ bot_bot_set_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7843_empty__iff,axiom,
% 9.04/5.08      ! [C: $o] :
% 9.04/5.08        ~ ( member_o @ C @ bot_bot_set_o ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7844_empty__iff,axiom,
% 9.04/5.08      ! [C: nat] :
% 9.04/5.08        ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7845_empty__iff,axiom,
% 9.04/5.08      ! [C: int] :
% 9.04/5.08        ~ ( member_int @ C @ bot_bot_set_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7846_all__not__in__conv,axiom,
% 9.04/5.08      ! [A2: set_complex] :
% 9.04/5.08        ( ( ! [X4: complex] :
% 9.04/5.08              ~ ( member_complex @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_complex ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7847_all__not__in__conv,axiom,
% 9.04/5.08      ! [A2: set_set_nat] :
% 9.04/5.08        ( ( ! [X4: set_nat] :
% 9.04/5.08              ~ ( member_set_nat @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7848_all__not__in__conv,axiom,
% 9.04/5.08      ! [A2: set_real] :
% 9.04/5.08        ( ( ! [X4: real] :
% 9.04/5.08              ~ ( member_real @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7849_all__not__in__conv,axiom,
% 9.04/5.08      ! [A2: set_o] :
% 9.04/5.08        ( ( ! [X4: $o] :
% 9.04/5.08              ~ ( member_o @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7850_all__not__in__conv,axiom,
% 9.04/5.08      ! [A2: set_nat] :
% 9.04/5.08        ( ( ! [X4: nat] :
% 9.04/5.08              ~ ( member_nat @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7851_all__not__in__conv,axiom,
% 9.04/5.08      ! [A2: set_int] :
% 9.04/5.08        ( ( ! [X4: int] :
% 9.04/5.08              ~ ( member_int @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7852_Collect__empty__eq,axiom,
% 9.04/5.08      ! [P: complex > $o] :
% 9.04/5.08        ( ( ( collect_complex @ P )
% 9.04/5.08          = bot_bot_set_complex )
% 9.04/5.08        = ( ! [X4: complex] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7853_Collect__empty__eq,axiom,
% 9.04/5.08      ! [P: set_nat > $o] :
% 9.04/5.08        ( ( ( collect_set_nat @ P )
% 9.04/5.08          = bot_bot_set_set_nat )
% 9.04/5.08        = ( ! [X4: set_nat] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7854_Collect__empty__eq,axiom,
% 9.04/5.08      ! [P: real > $o] :
% 9.04/5.08        ( ( ( collect_real @ P )
% 9.04/5.08          = bot_bot_set_real )
% 9.04/5.08        = ( ! [X4: real] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7855_Collect__empty__eq,axiom,
% 9.04/5.08      ! [P: $o > $o] :
% 9.04/5.08        ( ( ( collect_o @ P )
% 9.04/5.08          = bot_bot_set_o )
% 9.04/5.08        = ( ! [X4: $o] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7856_Collect__empty__eq,axiom,
% 9.04/5.08      ! [P: nat > $o] :
% 9.04/5.08        ( ( ( collect_nat @ P )
% 9.04/5.08          = bot_bot_set_nat )
% 9.04/5.08        = ( ! [X4: nat] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7857_Collect__empty__eq,axiom,
% 9.04/5.08      ! [P: int > $o] :
% 9.04/5.08        ( ( ( collect_int @ P )
% 9.04/5.08          = bot_bot_set_int )
% 9.04/5.08        = ( ! [X4: int] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7858_empty__Collect__eq,axiom,
% 9.04/5.08      ! [P: complex > $o] :
% 9.04/5.08        ( ( bot_bot_set_complex
% 9.04/5.08          = ( collect_complex @ P ) )
% 9.04/5.08        = ( ! [X4: complex] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7859_empty__Collect__eq,axiom,
% 9.04/5.08      ! [P: set_nat > $o] :
% 9.04/5.08        ( ( bot_bot_set_set_nat
% 9.04/5.08          = ( collect_set_nat @ P ) )
% 9.04/5.08        = ( ! [X4: set_nat] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7860_empty__Collect__eq,axiom,
% 9.04/5.08      ! [P: real > $o] :
% 9.04/5.08        ( ( bot_bot_set_real
% 9.04/5.08          = ( collect_real @ P ) )
% 9.04/5.08        = ( ! [X4: real] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7861_empty__Collect__eq,axiom,
% 9.04/5.08      ! [P: $o > $o] :
% 9.04/5.08        ( ( bot_bot_set_o
% 9.04/5.08          = ( collect_o @ P ) )
% 9.04/5.08        = ( ! [X4: $o] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7862_empty__Collect__eq,axiom,
% 9.04/5.08      ! [P: nat > $o] :
% 9.04/5.08        ( ( bot_bot_set_nat
% 9.04/5.08          = ( collect_nat @ P ) )
% 9.04/5.08        = ( ! [X4: nat] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7863_empty__Collect__eq,axiom,
% 9.04/5.08      ! [P: int > $o] :
% 9.04/5.08        ( ( bot_bot_set_int
% 9.04/5.08          = ( collect_int @ P ) )
% 9.04/5.08        = ( ! [X4: int] :
% 9.04/5.08              ~ ( P @ X4 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7864_image__is__empty,axiom,
% 9.04/5.08      ! [F: real > real,A2: set_real] :
% 9.04/5.08        ( ( ( image_real_real @ F @ A2 )
% 9.04/5.08          = bot_bot_set_real )
% 9.04/5.08        = ( A2 = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7865_image__is__empty,axiom,
% 9.04/5.08      ! [F: $o > real,A2: set_o] :
% 9.04/5.08        ( ( ( image_o_real @ F @ A2 )
% 9.04/5.08          = bot_bot_set_real )
% 9.04/5.08        = ( A2 = bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7866_image__is__empty,axiom,
% 9.04/5.08      ! [F: nat > real,A2: set_nat] :
% 9.04/5.08        ( ( ( image_nat_real @ F @ A2 )
% 9.04/5.08          = bot_bot_set_real )
% 9.04/5.08        = ( A2 = bot_bot_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7867_image__is__empty,axiom,
% 9.04/5.08      ! [F: int > real,A2: set_int] :
% 9.04/5.08        ( ( ( image_int_real @ F @ A2 )
% 9.04/5.08          = bot_bot_set_real )
% 9.04/5.08        = ( A2 = bot_bot_set_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7868_image__is__empty,axiom,
% 9.04/5.08      ! [F: real > $o,A2: set_real] :
% 9.04/5.08        ( ( ( image_real_o @ F @ A2 )
% 9.04/5.08          = bot_bot_set_o )
% 9.04/5.08        = ( A2 = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7869_image__is__empty,axiom,
% 9.04/5.08      ! [F: $o > $o,A2: set_o] :
% 9.04/5.08        ( ( ( image_o_o @ F @ A2 )
% 9.04/5.08          = bot_bot_set_o )
% 9.04/5.08        = ( A2 = bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7870_image__is__empty,axiom,
% 9.04/5.08      ! [F: nat > $o,A2: set_nat] :
% 9.04/5.08        ( ( ( image_nat_o @ F @ A2 )
% 9.04/5.08          = bot_bot_set_o )
% 9.04/5.08        = ( A2 = bot_bot_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7871_image__is__empty,axiom,
% 9.04/5.08      ! [F: int > $o,A2: set_int] :
% 9.04/5.08        ( ( ( image_int_o @ F @ A2 )
% 9.04/5.08          = bot_bot_set_o )
% 9.04/5.08        = ( A2 = bot_bot_set_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7872_image__is__empty,axiom,
% 9.04/5.08      ! [F: real > nat,A2: set_real] :
% 9.04/5.08        ( ( ( image_real_nat @ F @ A2 )
% 9.04/5.08          = bot_bot_set_nat )
% 9.04/5.08        = ( A2 = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7873_image__is__empty,axiom,
% 9.04/5.08      ! [F: $o > nat,A2: set_o] :
% 9.04/5.08        ( ( ( image_o_nat @ F @ A2 )
% 9.04/5.08          = bot_bot_set_nat )
% 9.04/5.08        = ( A2 = bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7874_empty__is__image,axiom,
% 9.04/5.08      ! [F: real > real,A2: set_real] :
% 9.04/5.08        ( ( bot_bot_set_real
% 9.04/5.08          = ( image_real_real @ F @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7875_empty__is__image,axiom,
% 9.04/5.08      ! [F: $o > real,A2: set_o] :
% 9.04/5.08        ( ( bot_bot_set_real
% 9.04/5.08          = ( image_o_real @ F @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7876_empty__is__image,axiom,
% 9.04/5.08      ! [F: nat > real,A2: set_nat] :
% 9.04/5.08        ( ( bot_bot_set_real
% 9.04/5.08          = ( image_nat_real @ F @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7877_empty__is__image,axiom,
% 9.04/5.08      ! [F: int > real,A2: set_int] :
% 9.04/5.08        ( ( bot_bot_set_real
% 9.04/5.08          = ( image_int_real @ F @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7878_empty__is__image,axiom,
% 9.04/5.08      ! [F: real > $o,A2: set_real] :
% 9.04/5.08        ( ( bot_bot_set_o
% 9.04/5.08          = ( image_real_o @ F @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7879_empty__is__image,axiom,
% 9.04/5.08      ! [F: $o > $o,A2: set_o] :
% 9.04/5.08        ( ( bot_bot_set_o
% 9.04/5.08          = ( image_o_o @ F @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7880_empty__is__image,axiom,
% 9.04/5.08      ! [F: nat > $o,A2: set_nat] :
% 9.04/5.08        ( ( bot_bot_set_o
% 9.04/5.08          = ( image_nat_o @ F @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7881_empty__is__image,axiom,
% 9.04/5.08      ! [F: int > $o,A2: set_int] :
% 9.04/5.08        ( ( bot_bot_set_o
% 9.04/5.08          = ( image_int_o @ F @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7882_empty__is__image,axiom,
% 9.04/5.08      ! [F: real > nat,A2: set_real] :
% 9.04/5.08        ( ( bot_bot_set_nat
% 9.04/5.08          = ( image_real_nat @ F @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7883_empty__is__image,axiom,
% 9.04/5.08      ! [F: $o > nat,A2: set_o] :
% 9.04/5.08        ( ( bot_bot_set_nat
% 9.04/5.08          = ( image_o_nat @ F @ A2 ) )
% 9.04/5.08        = ( A2 = bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7884_image__empty,axiom,
% 9.04/5.08      ! [F: real > real] :
% 9.04/5.08        ( ( image_real_real @ F @ bot_bot_set_real )
% 9.04/5.08        = bot_bot_set_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7885_image__empty,axiom,
% 9.04/5.08      ! [F: real > $o] :
% 9.04/5.08        ( ( image_real_o @ F @ bot_bot_set_real )
% 9.04/5.08        = bot_bot_set_o ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7886_image__empty,axiom,
% 9.04/5.08      ! [F: real > nat] :
% 9.04/5.08        ( ( image_real_nat @ F @ bot_bot_set_real )
% 9.04/5.08        = bot_bot_set_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7887_image__empty,axiom,
% 9.04/5.08      ! [F: real > int] :
% 9.04/5.08        ( ( image_real_int @ F @ bot_bot_set_real )
% 9.04/5.08        = bot_bot_set_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7888_image__empty,axiom,
% 9.04/5.08      ! [F: $o > real] :
% 9.04/5.08        ( ( image_o_real @ F @ bot_bot_set_o )
% 9.04/5.08        = bot_bot_set_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7889_image__empty,axiom,
% 9.04/5.08      ! [F: $o > $o] :
% 9.04/5.08        ( ( image_o_o @ F @ bot_bot_set_o )
% 9.04/5.08        = bot_bot_set_o ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7890_image__empty,axiom,
% 9.04/5.08      ! [F: $o > nat] :
% 9.04/5.08        ( ( image_o_nat @ F @ bot_bot_set_o )
% 9.04/5.08        = bot_bot_set_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7891_image__empty,axiom,
% 9.04/5.08      ! [F: $o > int] :
% 9.04/5.08        ( ( image_o_int @ F @ bot_bot_set_o )
% 9.04/5.08        = bot_bot_set_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7892_image__empty,axiom,
% 9.04/5.08      ! [F: nat > real] :
% 9.04/5.08        ( ( image_nat_real @ F @ bot_bot_set_nat )
% 9.04/5.08        = bot_bot_set_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7893_image__empty,axiom,
% 9.04/5.08      ! [F: nat > $o] :
% 9.04/5.08        ( ( image_nat_o @ F @ bot_bot_set_nat )
% 9.04/5.08        = bot_bot_set_o ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7894_subset__empty,axiom,
% 9.04/5.08      ! [A2: set_real] :
% 9.04/5.08        ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 9.04/5.08        = ( A2 = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7895_subset__empty,axiom,
% 9.04/5.08      ! [A2: set_o] :
% 9.04/5.08        ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
% 9.04/5.08        = ( A2 = bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7896_subset__empty,axiom,
% 9.04/5.08      ! [A2: set_nat] :
% 9.04/5.08        ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 9.04/5.08        = ( A2 = bot_bot_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7897_subset__empty,axiom,
% 9.04/5.08      ! [A2: set_int] :
% 9.04/5.08        ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 9.04/5.08        = ( A2 = bot_bot_set_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7898_empty__subsetI,axiom,
% 9.04/5.08      ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7899_empty__subsetI,axiom,
% 9.04/5.08      ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7900_empty__subsetI,axiom,
% 9.04/5.08      ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7901_empty__subsetI,axiom,
% 9.04/5.08      ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7902_singletonI,axiom,
% 9.04/5.08      ! [A: vEBT_VEBT] : ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7903_singletonI,axiom,
% 9.04/5.08      ! [A: complex] : ( member_complex @ A @ ( insert_complex @ A @ bot_bot_set_complex ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7904_singletonI,axiom,
% 9.04/5.08      ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7905_singletonI,axiom,
% 9.04/5.08      ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7906_singletonI,axiom,
% 9.04/5.08      ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7907_singletonI,axiom,
% 9.04/5.08      ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7908_singletonI,axiom,
% 9.04/5.08      ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7909_Diff__empty,axiom,
% 9.04/5.08      ! [A2: set_real] :
% 9.04/5.08        ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 9.04/5.08        = A2 ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7910_Diff__empty,axiom,
% 9.04/5.08      ! [A2: set_o] :
% 9.04/5.08        ( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
% 9.04/5.08        = A2 ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7911_Diff__empty,axiom,
% 9.04/5.08      ! [A2: set_int] :
% 9.04/5.08        ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 9.04/5.08        = A2 ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7912_Diff__empty,axiom,
% 9.04/5.08      ! [A2: set_nat] :
% 9.04/5.08        ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 9.04/5.08        = A2 ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7913_empty__Diff,axiom,
% 9.04/5.08      ! [A2: set_real] :
% 9.04/5.08        ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 9.04/5.08        = bot_bot_set_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7914_empty__Diff,axiom,
% 9.04/5.08      ! [A2: set_o] :
% 9.04/5.08        ( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
% 9.04/5.08        = bot_bot_set_o ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7915_empty__Diff,axiom,
% 9.04/5.08      ! [A2: set_int] :
% 9.04/5.08        ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 9.04/5.08        = bot_bot_set_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7916_empty__Diff,axiom,
% 9.04/5.08      ! [A2: set_nat] :
% 9.04/5.08        ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 9.04/5.08        = bot_bot_set_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7917_Diff__cancel,axiom,
% 9.04/5.08      ! [A2: set_real] :
% 9.04/5.08        ( ( minus_minus_set_real @ A2 @ A2 )
% 9.04/5.08        = bot_bot_set_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7918_Diff__cancel,axiom,
% 9.04/5.08      ! [A2: set_o] :
% 9.04/5.08        ( ( minus_minus_set_o @ A2 @ A2 )
% 9.04/5.08        = bot_bot_set_o ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7919_Diff__cancel,axiom,
% 9.04/5.08      ! [A2: set_int] :
% 9.04/5.08        ( ( minus_minus_set_int @ A2 @ A2 )
% 9.04/5.08        = bot_bot_set_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7920_Diff__cancel,axiom,
% 9.04/5.08      ! [A2: set_nat] :
% 9.04/5.08        ( ( minus_minus_set_nat @ A2 @ A2 )
% 9.04/5.08        = bot_bot_set_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7921_sin__zero,axiom,
% 9.04/5.08      ( ( sin_complex @ zero_zero_complex )
% 9.04/5.08      = zero_zero_complex ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7922_sin__zero,axiom,
% 9.04/5.08      ( ( sin_real @ zero_zero_real )
% 9.04/5.08      = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7923_cos__minus,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( cos_real @ ( uminus_uminus_real @ X ) )
% 9.04/5.08        = ( cos_real @ X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7924_cos__minus,axiom,
% 9.04/5.08      ! [X: complex] :
% 9.04/5.08        ( ( cos_complex @ ( uminus1482373934393186551omplex @ X ) )
% 9.04/5.08        = ( cos_complex @ X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7925_sin__minus,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( sin_real @ ( uminus_uminus_real @ X ) )
% 9.04/5.08        = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7926_sin__minus,axiom,
% 9.04/5.08      ! [X: complex] :
% 9.04/5.08        ( ( sin_complex @ ( uminus1482373934393186551omplex @ X ) )
% 9.04/5.08        = ( uminus1482373934393186551omplex @ ( sin_complex @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7927_singleton__insert__inj__eq,axiom,
% 9.04/5.08      ! [B: vEBT_VEBT,A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.04/5.08        ( ( ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT )
% 9.04/5.08          = ( insert_VEBT_VEBT @ A @ A2 ) )
% 9.04/5.08        = ( ( A = B )
% 9.04/5.08          & ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7928_singleton__insert__inj__eq,axiom,
% 9.04/5.08      ! [B: real,A: real,A2: set_real] :
% 9.04/5.08        ( ( ( insert_real @ B @ bot_bot_set_real )
% 9.04/5.08          = ( insert_real @ A @ A2 ) )
% 9.04/5.08        = ( ( A = B )
% 9.04/5.08          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7929_singleton__insert__inj__eq,axiom,
% 9.04/5.08      ! [B: $o,A: $o,A2: set_o] :
% 9.04/5.08        ( ( ( insert_o @ B @ bot_bot_set_o )
% 9.04/5.08          = ( insert_o @ A @ A2 ) )
% 9.04/5.08        = ( ( A = B )
% 9.04/5.08          & ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7930_singleton__insert__inj__eq,axiom,
% 9.04/5.08      ! [B: nat,A: nat,A2: set_nat] :
% 9.04/5.08        ( ( ( insert_nat @ B @ bot_bot_set_nat )
% 9.04/5.08          = ( insert_nat @ A @ A2 ) )
% 9.04/5.08        = ( ( A = B )
% 9.04/5.08          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7931_singleton__insert__inj__eq,axiom,
% 9.04/5.08      ! [B: int,A: int,A2: set_int] :
% 9.04/5.08        ( ( ( insert_int @ B @ bot_bot_set_int )
% 9.04/5.08          = ( insert_int @ A @ A2 ) )
% 9.04/5.08        = ( ( A = B )
% 9.04/5.08          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7932_singleton__insert__inj__eq_H,axiom,
% 9.04/5.08      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT] :
% 9.04/5.08        ( ( ( insert_VEBT_VEBT @ A @ A2 )
% 9.04/5.08          = ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
% 9.04/5.08        = ( ( A = B )
% 9.04/5.08          & ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7933_singleton__insert__inj__eq_H,axiom,
% 9.04/5.08      ! [A: real,A2: set_real,B: real] :
% 9.04/5.08        ( ( ( insert_real @ A @ A2 )
% 9.04/5.08          = ( insert_real @ B @ bot_bot_set_real ) )
% 9.04/5.08        = ( ( A = B )
% 9.04/5.08          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7934_singleton__insert__inj__eq_H,axiom,
% 9.04/5.08      ! [A: $o,A2: set_o,B: $o] :
% 9.04/5.08        ( ( ( insert_o @ A @ A2 )
% 9.04/5.08          = ( insert_o @ B @ bot_bot_set_o ) )
% 9.04/5.08        = ( ( A = B )
% 9.04/5.08          & ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7935_singleton__insert__inj__eq_H,axiom,
% 9.04/5.08      ! [A: nat,A2: set_nat,B: nat] :
% 9.04/5.08        ( ( ( insert_nat @ A @ A2 )
% 9.04/5.08          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 9.04/5.08        = ( ( A = B )
% 9.04/5.08          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7936_singleton__insert__inj__eq_H,axiom,
% 9.04/5.08      ! [A: int,A2: set_int,B: int] :
% 9.04/5.08        ( ( ( insert_int @ A @ A2 )
% 9.04/5.08          = ( insert_int @ B @ bot_bot_set_int ) )
% 9.04/5.08        = ( ( A = B )
% 9.04/5.08          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7937_cos__zero,axiom,
% 9.04/5.08      ( ( cos_complex @ zero_zero_complex )
% 9.04/5.08      = one_one_complex ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7938_cos__zero,axiom,
% 9.04/5.08      ( ( cos_real @ zero_zero_real )
% 9.04/5.08      = one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7939_Diff__eq__empty__iff,axiom,
% 9.04/5.08      ! [A2: set_real,B2: set_real] :
% 9.04/5.08        ( ( ( minus_minus_set_real @ A2 @ B2 )
% 9.04/5.08          = bot_bot_set_real )
% 9.04/5.08        = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7940_Diff__eq__empty__iff,axiom,
% 9.04/5.08      ! [A2: set_o,B2: set_o] :
% 9.04/5.08        ( ( ( minus_minus_set_o @ A2 @ B2 )
% 9.04/5.08          = bot_bot_set_o )
% 9.04/5.08        = ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7941_Diff__eq__empty__iff,axiom,
% 9.04/5.08      ! [A2: set_nat,B2: set_nat] :
% 9.04/5.08        ( ( ( minus_minus_set_nat @ A2 @ B2 )
% 9.04/5.08          = bot_bot_set_nat )
% 9.04/5.08        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7942_Diff__eq__empty__iff,axiom,
% 9.04/5.08      ! [A2: set_int,B2: set_int] :
% 9.04/5.08        ( ( ( minus_minus_set_int @ A2 @ B2 )
% 9.04/5.08          = bot_bot_set_int )
% 9.04/5.08        = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7943_insert__Diff__single,axiom,
% 9.04/5.08      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.04/5.08        ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
% 9.04/5.08        = ( insert_VEBT_VEBT @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7944_insert__Diff__single,axiom,
% 9.04/5.08      ! [A: real,A2: set_real] :
% 9.04/5.08        ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.04/5.08        = ( insert_real @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7945_insert__Diff__single,axiom,
% 9.04/5.08      ! [A: $o,A2: set_o] :
% 9.04/5.08        ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
% 9.04/5.08        = ( insert_o @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7946_insert__Diff__single,axiom,
% 9.04/5.08      ! [A: int,A2: set_int] :
% 9.04/5.08        ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.04/5.08        = ( insert_int @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7947_insert__Diff__single,axiom,
% 9.04/5.08      ! [A: nat,A2: set_nat] :
% 9.04/5.08        ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.04/5.08        = ( insert_nat @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7948_Max__singleton,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( lattic4275903605611617917x_real @ ( insert_real @ X @ bot_bot_set_real ) )
% 9.04/5.08        = X ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7949_Max__singleton,axiom,
% 9.04/5.08      ! [X: $o] :
% 9.04/5.08        ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 9.04/5.08        = X ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7950_Max__singleton,axiom,
% 9.04/5.08      ! [X: nat] :
% 9.04/5.08        ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 9.04/5.08        = X ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7951_Max__singleton,axiom,
% 9.04/5.08      ! [X: int] :
% 9.04/5.08        ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 9.04/5.08        = X ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7952_sin__pi,axiom,
% 9.04/5.08      ( ( sin_real @ pi )
% 9.04/5.08      = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7953_sin__pi__minus,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
% 9.04/5.08        = ( sin_real @ X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7954_dbl__dec__simps_I5_J,axiom,
% 9.04/5.08      ! [K: num] :
% 9.04/5.08        ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
% 9.04/5.08        = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7955_dbl__dec__simps_I5_J,axiom,
% 9.04/5.08      ! [K: num] :
% 9.04/5.08        ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
% 9.04/5.08        = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7956_dbl__dec__simps_I5_J,axiom,
% 9.04/5.08      ! [K: num] :
% 9.04/5.08        ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
% 9.04/5.08        = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7957_dbl__dec__simps_I5_J,axiom,
% 9.04/5.08      ! [K: num] :
% 9.04/5.08        ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
% 9.04/5.08        = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7958_subset__Compl__singleton,axiom,
% 9.04/5.08      ! [A2: set_VEBT_VEBT,B: vEBT_VEBT] :
% 9.04/5.08        ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) )
% 9.04/5.08        = ( ~ ( member_VEBT_VEBT @ B @ A2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7959_subset__Compl__singleton,axiom,
% 9.04/5.08      ! [A2: set_complex,B: complex] :
% 9.04/5.08        ( ( ord_le211207098394363844omplex @ A2 @ ( uminus8566677241136511917omplex @ ( insert_complex @ B @ bot_bot_set_complex ) ) )
% 9.04/5.08        = ( ~ ( member_complex @ B @ A2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7960_subset__Compl__singleton,axiom,
% 9.04/5.08      ! [A2: set_set_nat,B: set_nat] :
% 9.04/5.08        ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
% 9.04/5.08        = ( ~ ( member_set_nat @ B @ A2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7961_subset__Compl__singleton,axiom,
% 9.04/5.08      ! [A2: set_real,B: real] :
% 9.04/5.08        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
% 9.04/5.08        = ( ~ ( member_real @ B @ A2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7962_subset__Compl__singleton,axiom,
% 9.04/5.08      ! [A2: set_o,B: $o] :
% 9.04/5.08        ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 9.04/5.08        = ( ~ ( member_o @ B @ A2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7963_subset__Compl__singleton,axiom,
% 9.04/5.08      ! [A2: set_nat,B: nat] :
% 9.04/5.08        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 9.04/5.08        = ( ~ ( member_nat @ B @ A2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7964_subset__Compl__singleton,axiom,
% 9.04/5.08      ! [A2: set_int,B: int] :
% 9.04/5.08        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 9.04/5.08        = ( ~ ( member_int @ B @ A2 ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7965_sin__of__real__pi,axiom,
% 9.04/5.08      ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
% 9.04/5.08      = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7966_sin__of__real__pi,axiom,
% 9.04/5.08      ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 9.04/5.08      = zero_zero_complex ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7967_cos__pi,axiom,
% 9.04/5.08      ( ( cos_real @ pi )
% 9.04/5.08      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7968_cos__periodic__pi,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
% 9.04/5.08        = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7969_cos__periodic__pi2,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
% 9.04/5.08        = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7970_sin__periodic__pi,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
% 9.04/5.08        = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7971_sin__periodic__pi2,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
% 9.04/5.08        = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7972_cos__minus__pi,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
% 9.04/5.08        = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7973_cos__pi__minus,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
% 9.04/5.08        = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7974_sin__minus__pi,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
% 9.04/5.08        = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7975_pred__numeral__simps_I2_J,axiom,
% 9.04/5.08      ! [K: num] :
% 9.04/5.08        ( ( pred_numeral @ ( bit0 @ K ) )
% 9.04/5.08        = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7976_sin__cos__squared__add3,axiom,
% 9.04/5.08      ! [X: complex] :
% 9.04/5.08        ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
% 9.04/5.08        = one_one_complex ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7977_sin__cos__squared__add3,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
% 9.04/5.08        = one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7978_cos__of__real__pi,axiom,
% 9.04/5.08      ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
% 9.04/5.08      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7979_cos__of__real__pi,axiom,
% 9.04/5.08      ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 9.04/5.08      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7980_sin__npi2,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.04/5.08        = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7981_sin__npi,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 9.04/5.08        = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7982_and__nat__numerals_I3_J,axiom,
% 9.04/5.08      ! [X: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.04/5.08        = zero_zero_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7983_and__nat__numerals_I1_J,axiom,
% 9.04/5.08      ! [Y: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.04/5.08        = zero_zero_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7984_sin__npi__int,axiom,
% 9.04/5.08      ! [N: int] :
% 9.04/5.08        ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 9.04/5.08        = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7985_and__nat__numerals_I4_J,axiom,
% 9.04/5.08      ! [X: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.04/5.08        = one_one_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7986_and__nat__numerals_I2_J,axiom,
% 9.04/5.08      ! [Y: num] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.04/5.08        = one_one_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7987_cos__pi__half,axiom,
% 9.04/5.08      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.08      = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7988_sin__two__pi,axiom,
% 9.04/5.08      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.04/5.08      = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7989_sin__pi__half,axiom,
% 9.04/5.08      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.08      = one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7990_cos__two__pi,axiom,
% 9.04/5.08      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.04/5.08      = one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7991_cos__periodic,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.04/5.08        = ( cos_real @ X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7992_sin__periodic,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.04/5.08        = ( sin_real @ X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7993_cos__2pi__minus,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 9.04/5.08        = ( cos_real @ X ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7994_Suc__0__and__eq,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.04/5.08        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7995_and__Suc__0__eq,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.04/5.08        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7996_cos__npi,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 9.04/5.08        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7997_cos__npi2,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.04/5.08        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7998_sin__cos__squared__add,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.08        = one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_7999_sin__cos__squared__add,axiom,
% 9.04/5.08      ! [X: complex] :
% 9.04/5.08        ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.08        = one_one_complex ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8000_sin__cos__squared__add2,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.08        = one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8001_sin__cos__squared__add2,axiom,
% 9.04/5.08      ! [X: complex] :
% 9.04/5.08        ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.08        = one_one_complex ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8002_cos__of__real__pi__half,axiom,
% 9.04/5.08      ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.08      = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8003_cos__of__real__pi__half,axiom,
% 9.04/5.08      ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 9.04/5.08      = zero_zero_complex ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8004_sin__of__real__pi__half,axiom,
% 9.04/5.08      ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.08      = one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8005_sin__of__real__pi__half,axiom,
% 9.04/5.08      ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 9.04/5.08      = one_one_complex ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8006_sin__2npi,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 9.04/5.08        = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8007_cos__2npi,axiom,
% 9.04/5.08      ! [N: nat] :
% 9.04/5.08        ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 9.04/5.08        = one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8008_sin__2pi__minus,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 9.04/5.08        = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8009_sin__int__2pin,axiom,
% 9.04/5.08      ! [N: int] :
% 9.04/5.08        ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 9.04/5.08        = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8010_cos__int__2pin,axiom,
% 9.04/5.08      ! [N: int] :
% 9.04/5.08        ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 9.04/5.08        = one_one_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8011_cos__3over2__pi,axiom,
% 9.04/5.08      ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 9.04/5.08      = zero_zero_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8012_sin__3over2__pi,axiom,
% 9.04/5.08      ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 9.04/5.08      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8013_cos__npi__int,axiom,
% 9.04/5.08      ! [N: int] :
% 9.04/5.08        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.04/5.08         => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 9.04/5.08            = one_one_real ) )
% 9.04/5.08        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.04/5.08         => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 9.04/5.08            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8014_cos__one__sin__zero,axiom,
% 9.04/5.08      ! [X: complex] :
% 9.04/5.08        ( ( ( cos_complex @ X )
% 9.04/5.08          = one_one_complex )
% 9.04/5.08       => ( ( sin_complex @ X )
% 9.04/5.08          = zero_zero_complex ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8015_cos__one__sin__zero,axiom,
% 9.04/5.08      ! [X: real] :
% 9.04/5.08        ( ( ( cos_real @ X )
% 9.04/5.08          = one_one_real )
% 9.04/5.08       => ( ( sin_real @ X )
% 9.04/5.08          = zero_zero_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8016_emptyE,axiom,
% 9.04/5.08      ! [A: complex] :
% 9.04/5.08        ~ ( member_complex @ A @ bot_bot_set_complex ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8017_emptyE,axiom,
% 9.04/5.08      ! [A: set_nat] :
% 9.04/5.08        ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8018_emptyE,axiom,
% 9.04/5.08      ! [A: real] :
% 9.04/5.08        ~ ( member_real @ A @ bot_bot_set_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8019_emptyE,axiom,
% 9.04/5.08      ! [A: $o] :
% 9.04/5.08        ~ ( member_o @ A @ bot_bot_set_o ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8020_emptyE,axiom,
% 9.04/5.08      ! [A: nat] :
% 9.04/5.08        ~ ( member_nat @ A @ bot_bot_set_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8021_emptyE,axiom,
% 9.04/5.08      ! [A: int] :
% 9.04/5.08        ~ ( member_int @ A @ bot_bot_set_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8022_equals0D,axiom,
% 9.04/5.08      ! [A2: set_complex,A: complex] :
% 9.04/5.08        ( ( A2 = bot_bot_set_complex )
% 9.04/5.08       => ~ ( member_complex @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8023_equals0D,axiom,
% 9.04/5.08      ! [A2: set_set_nat,A: set_nat] :
% 9.04/5.08        ( ( A2 = bot_bot_set_set_nat )
% 9.04/5.08       => ~ ( member_set_nat @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8024_equals0D,axiom,
% 9.04/5.08      ! [A2: set_real,A: real] :
% 9.04/5.08        ( ( A2 = bot_bot_set_real )
% 9.04/5.08       => ~ ( member_real @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8025_equals0D,axiom,
% 9.04/5.08      ! [A2: set_o,A: $o] :
% 9.04/5.08        ( ( A2 = bot_bot_set_o )
% 9.04/5.08       => ~ ( member_o @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8026_equals0D,axiom,
% 9.04/5.08      ! [A2: set_nat,A: nat] :
% 9.04/5.08        ( ( A2 = bot_bot_set_nat )
% 9.04/5.08       => ~ ( member_nat @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8027_equals0D,axiom,
% 9.04/5.08      ! [A2: set_int,A: int] :
% 9.04/5.08        ( ( A2 = bot_bot_set_int )
% 9.04/5.08       => ~ ( member_int @ A @ A2 ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8028_equals0I,axiom,
% 9.04/5.08      ! [A2: set_complex] :
% 9.04/5.08        ( ! [Y4: complex] :
% 9.04/5.08            ~ ( member_complex @ Y4 @ A2 )
% 9.04/5.08       => ( A2 = bot_bot_set_complex ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8029_equals0I,axiom,
% 9.04/5.08      ! [A2: set_set_nat] :
% 9.04/5.08        ( ! [Y4: set_nat] :
% 9.04/5.08            ~ ( member_set_nat @ Y4 @ A2 )
% 9.04/5.08       => ( A2 = bot_bot_set_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8030_equals0I,axiom,
% 9.04/5.08      ! [A2: set_real] :
% 9.04/5.08        ( ! [Y4: real] :
% 9.04/5.08            ~ ( member_real @ Y4 @ A2 )
% 9.04/5.08       => ( A2 = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8031_equals0I,axiom,
% 9.04/5.08      ! [A2: set_o] :
% 9.04/5.08        ( ! [Y4: $o] :
% 9.04/5.08            ~ ( member_o @ Y4 @ A2 )
% 9.04/5.08       => ( A2 = bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8032_equals0I,axiom,
% 9.04/5.08      ! [A2: set_nat] :
% 9.04/5.08        ( ! [Y4: nat] :
% 9.04/5.08            ~ ( member_nat @ Y4 @ A2 )
% 9.04/5.08       => ( A2 = bot_bot_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8033_equals0I,axiom,
% 9.04/5.08      ! [A2: set_int] :
% 9.04/5.08        ( ! [Y4: int] :
% 9.04/5.08            ~ ( member_int @ Y4 @ A2 )
% 9.04/5.08       => ( A2 = bot_bot_set_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8034_ex__in__conv,axiom,
% 9.04/5.08      ! [A2: set_complex] :
% 9.04/5.08        ( ( ? [X4: complex] : ( member_complex @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 != bot_bot_set_complex ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8035_ex__in__conv,axiom,
% 9.04/5.08      ! [A2: set_set_nat] :
% 9.04/5.08        ( ( ? [X4: set_nat] : ( member_set_nat @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 != bot_bot_set_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8036_ex__in__conv,axiom,
% 9.04/5.08      ! [A2: set_real] :
% 9.04/5.08        ( ( ? [X4: real] : ( member_real @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 != bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8037_ex__in__conv,axiom,
% 9.04/5.08      ! [A2: set_o] :
% 9.04/5.08        ( ( ? [X4: $o] : ( member_o @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 != bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8038_ex__in__conv,axiom,
% 9.04/5.08      ! [A2: set_nat] :
% 9.04/5.08        ( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 != bot_bot_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8039_ex__in__conv,axiom,
% 9.04/5.08      ! [A2: set_int] :
% 9.04/5.08        ( ( ? [X4: int] : ( member_int @ X4 @ A2 ) )
% 9.04/5.08        = ( A2 != bot_bot_set_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8040_polar__Ex,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08      ? [R3: real,A5: real] :
% 9.04/5.08        ( ( X
% 9.04/5.08          = ( times_times_real @ R3 @ ( cos_real @ A5 ) ) )
% 9.04/5.08        & ( Y
% 9.04/5.08          = ( times_times_real @ R3 @ ( sin_real @ A5 ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8041_sin__diff,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( sin_real @ ( minus_minus_real @ X @ Y ) )
% 9.04/5.08        = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8042_sin__add,axiom,
% 9.04/5.08      ! [X: complex,Y: complex] :
% 9.04/5.08        ( ( sin_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.04/5.08        = ( plus_plus_complex @ ( times_times_complex @ ( sin_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8043_sin__add,axiom,
% 9.04/5.08      ! [X: real,Y: real] :
% 9.04/5.08        ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.08        = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8044_bot_Oextremum__strict,axiom,
% 9.04/5.08      ! [A: set_real] :
% 9.04/5.08        ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8045_bot_Oextremum__strict,axiom,
% 9.04/5.08      ! [A: set_o] :
% 9.04/5.08        ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8046_bot_Oextremum__strict,axiom,
% 9.04/5.08      ! [A: set_nat] :
% 9.04/5.08        ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8047_bot_Oextremum__strict,axiom,
% 9.04/5.08      ! [A: set_int] :
% 9.04/5.08        ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8048_bot_Oextremum__strict,axiom,
% 9.04/5.08      ! [A: nat] :
% 9.04/5.08        ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8049_bot_Onot__eq__extremum,axiom,
% 9.04/5.08      ! [A: set_real] :
% 9.04/5.08        ( ( A != bot_bot_set_real )
% 9.04/5.08        = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8050_bot_Onot__eq__extremum,axiom,
% 9.04/5.08      ! [A: set_o] :
% 9.04/5.08        ( ( A != bot_bot_set_o )
% 9.04/5.08        = ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8051_bot_Onot__eq__extremum,axiom,
% 9.04/5.08      ! [A: set_nat] :
% 9.04/5.08        ( ( A != bot_bot_set_nat )
% 9.04/5.08        = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8052_bot_Onot__eq__extremum,axiom,
% 9.04/5.08      ! [A: set_int] :
% 9.04/5.08        ( ( A != bot_bot_set_int )
% 9.04/5.08        = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8053_bot_Onot__eq__extremum,axiom,
% 9.04/5.08      ! [A: nat] :
% 9.04/5.08        ( ( A != bot_bot_nat )
% 9.04/5.08        = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8054_bot_Oextremum,axiom,
% 9.04/5.08      ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8055_bot_Oextremum,axiom,
% 9.04/5.08      ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8056_bot_Oextremum,axiom,
% 9.04/5.08      ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8057_bot_Oextremum,axiom,
% 9.04/5.08      ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8058_bot_Oextremum,axiom,
% 9.04/5.08      ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8059_bot_Oextremum__unique,axiom,
% 9.04/5.08      ! [A: set_real] :
% 9.04/5.08        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 9.04/5.08        = ( A = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8060_bot_Oextremum__unique,axiom,
% 9.04/5.08      ! [A: set_o] :
% 9.04/5.08        ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
% 9.04/5.08        = ( A = bot_bot_set_o ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8061_bot_Oextremum__unique,axiom,
% 9.04/5.08      ! [A: set_nat] :
% 9.04/5.08        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 9.04/5.08        = ( A = bot_bot_set_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8062_bot_Oextremum__unique,axiom,
% 9.04/5.08      ! [A: set_int] :
% 9.04/5.08        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 9.04/5.08        = ( A = bot_bot_set_int ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8063_bot_Oextremum__unique,axiom,
% 9.04/5.08      ! [A: nat] :
% 9.04/5.08        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 9.04/5.08        = ( A = bot_bot_nat ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8064_bot_Oextremum__uniqueI,axiom,
% 9.04/5.08      ! [A: set_real] :
% 9.04/5.08        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 9.04/5.08       => ( A = bot_bot_set_real ) ) ).
% 9.04/5.08  
% 9.04/5.08  thf(fact_8065_bot_Oextremum__uniqueI,axiom,
% 9.04/5.08      ! [A: set_o] :
% 9.04/5.08        ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
% 9.04/5.09       => ( A = bot_bot_set_o ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8066_bot_Oextremum__uniqueI,axiom,
% 9.04/5.09      ! [A: set_nat] :
% 9.04/5.09        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 9.04/5.09       => ( A = bot_bot_set_nat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8067_bot_Oextremum__uniqueI,axiom,
% 9.04/5.09      ! [A: set_int] :
% 9.04/5.09        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 9.04/5.09       => ( A = bot_bot_set_int ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8068_bot_Oextremum__uniqueI,axiom,
% 9.04/5.09      ! [A: nat] :
% 9.04/5.09        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 9.04/5.09       => ( A = bot_bot_nat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8069_cos__of__real,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( cos_real @ ( real_V1803761363581548252l_real @ X ) )
% 9.04/5.09        = ( real_V1803761363581548252l_real @ ( cos_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8070_cos__of__real,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( cos_complex @ ( real_V4546457046886955230omplex @ X ) )
% 9.04/5.09        = ( real_V4546457046886955230omplex @ ( cos_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8071_sin__of__real,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( sin_real @ ( real_V1803761363581548252l_real @ X ) )
% 9.04/5.09        = ( real_V1803761363581548252l_real @ ( sin_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8072_sin__of__real,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( sin_complex @ ( real_V4546457046886955230omplex @ X ) )
% 9.04/5.09        = ( real_V4546457046886955230omplex @ ( sin_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8073_singleton__inject,axiom,
% 9.04/5.09      ! [A: vEBT_VEBT,B: vEBT_VEBT] :
% 9.04/5.09        ( ( ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT )
% 9.04/5.09          = ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
% 9.04/5.09       => ( A = B ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8074_singleton__inject,axiom,
% 9.04/5.09      ! [A: real,B: real] :
% 9.04/5.09        ( ( ( insert_real @ A @ bot_bot_set_real )
% 9.04/5.09          = ( insert_real @ B @ bot_bot_set_real ) )
% 9.04/5.09       => ( A = B ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8075_singleton__inject,axiom,
% 9.04/5.09      ! [A: $o,B: $o] :
% 9.04/5.09        ( ( ( insert_o @ A @ bot_bot_set_o )
% 9.04/5.09          = ( insert_o @ B @ bot_bot_set_o ) )
% 9.04/5.09       => ( A = B ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8076_singleton__inject,axiom,
% 9.04/5.09      ! [A: nat,B: nat] :
% 9.04/5.09        ( ( ( insert_nat @ A @ bot_bot_set_nat )
% 9.04/5.09          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 9.04/5.09       => ( A = B ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8077_singleton__inject,axiom,
% 9.04/5.09      ! [A: int,B: int] :
% 9.04/5.09        ( ( ( insert_int @ A @ bot_bot_set_int )
% 9.04/5.09          = ( insert_int @ B @ bot_bot_set_int ) )
% 9.04/5.09       => ( A = B ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8078_insert__not__empty,axiom,
% 9.04/5.09      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.04/5.09        ( ( insert_VEBT_VEBT @ A @ A2 )
% 9.04/5.09       != bot_bo8194388402131092736T_VEBT ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8079_insert__not__empty,axiom,
% 9.04/5.09      ! [A: real,A2: set_real] :
% 9.04/5.09        ( ( insert_real @ A @ A2 )
% 9.04/5.09       != bot_bot_set_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8080_insert__not__empty,axiom,
% 9.04/5.09      ! [A: $o,A2: set_o] :
% 9.04/5.09        ( ( insert_o @ A @ A2 )
% 9.04/5.09       != bot_bot_set_o ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8081_insert__not__empty,axiom,
% 9.04/5.09      ! [A: nat,A2: set_nat] :
% 9.04/5.09        ( ( insert_nat @ A @ A2 )
% 9.04/5.09       != bot_bot_set_nat ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8082_insert__not__empty,axiom,
% 9.04/5.09      ! [A: int,A2: set_int] :
% 9.04/5.09        ( ( insert_int @ A @ A2 )
% 9.04/5.09       != bot_bot_set_int ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8083_doubleton__eq__iff,axiom,
% 9.04/5.09      ! [A: vEBT_VEBT,B: vEBT_VEBT,C: vEBT_VEBT,D: vEBT_VEBT] :
% 9.04/5.09        ( ( ( insert_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
% 9.04/5.09          = ( insert_VEBT_VEBT @ C @ ( insert_VEBT_VEBT @ D @ bot_bo8194388402131092736T_VEBT ) ) )
% 9.04/5.09        = ( ( ( A = C )
% 9.04/5.09            & ( B = D ) )
% 9.04/5.09          | ( ( A = D )
% 9.04/5.09            & ( B = C ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8084_doubleton__eq__iff,axiom,
% 9.04/5.09      ! [A: real,B: real,C: real,D: real] :
% 9.04/5.09        ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
% 9.04/5.09          = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
% 9.04/5.09        = ( ( ( A = C )
% 9.04/5.09            & ( B = D ) )
% 9.04/5.09          | ( ( A = D )
% 9.04/5.09            & ( B = C ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8085_doubleton__eq__iff,axiom,
% 9.04/5.09      ! [A: $o,B: $o,C: $o,D: $o] :
% 9.04/5.09        ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
% 9.04/5.09          = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
% 9.04/5.09        = ( ( ( A = C )
% 9.04/5.09            & ( B = D ) )
% 9.04/5.09          | ( ( A = D )
% 9.04/5.09            & ( B = C ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8086_doubleton__eq__iff,axiom,
% 9.04/5.09      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.04/5.09        ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
% 9.04/5.09          = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
% 9.04/5.09        = ( ( ( A = C )
% 9.04/5.09            & ( B = D ) )
% 9.04/5.09          | ( ( A = D )
% 9.04/5.09            & ( B = C ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8087_doubleton__eq__iff,axiom,
% 9.04/5.09      ! [A: int,B: int,C: int,D: int] :
% 9.04/5.09        ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
% 9.04/5.09          = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
% 9.04/5.09        = ( ( ( A = C )
% 9.04/5.09            & ( B = D ) )
% 9.04/5.09          | ( ( A = D )
% 9.04/5.09            & ( B = C ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8088_singleton__iff,axiom,
% 9.04/5.09      ! [B: vEBT_VEBT,A: vEBT_VEBT] :
% 9.04/5.09        ( ( member_VEBT_VEBT @ B @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
% 9.04/5.09        = ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8089_singleton__iff,axiom,
% 9.04/5.09      ! [B: complex,A: complex] :
% 9.04/5.09        ( ( member_complex @ B @ ( insert_complex @ A @ bot_bot_set_complex ) )
% 9.04/5.09        = ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8090_singleton__iff,axiom,
% 9.04/5.09      ! [B: set_nat,A: set_nat] :
% 9.04/5.09        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 9.04/5.09        = ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8091_singleton__iff,axiom,
% 9.04/5.09      ! [B: real,A: real] :
% 9.04/5.09        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 9.04/5.09        = ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8092_singleton__iff,axiom,
% 9.04/5.09      ! [B: $o,A: $o] :
% 9.04/5.09        ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
% 9.04/5.09        = ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8093_singleton__iff,axiom,
% 9.04/5.09      ! [B: nat,A: nat] :
% 9.04/5.09        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 9.04/5.09        = ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8094_singleton__iff,axiom,
% 9.04/5.09      ! [B: int,A: int] :
% 9.04/5.09        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 9.04/5.09        = ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8095_singletonD,axiom,
% 9.04/5.09      ! [B: vEBT_VEBT,A: vEBT_VEBT] :
% 9.04/5.09        ( ( member_VEBT_VEBT @ B @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
% 9.04/5.09       => ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8096_singletonD,axiom,
% 9.04/5.09      ! [B: complex,A: complex] :
% 9.04/5.09        ( ( member_complex @ B @ ( insert_complex @ A @ bot_bot_set_complex ) )
% 9.04/5.09       => ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8097_singletonD,axiom,
% 9.04/5.09      ! [B: set_nat,A: set_nat] :
% 9.04/5.09        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 9.04/5.09       => ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8098_singletonD,axiom,
% 9.04/5.09      ! [B: real,A: real] :
% 9.04/5.09        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 9.04/5.09       => ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8099_singletonD,axiom,
% 9.04/5.09      ! [B: $o,A: $o] :
% 9.04/5.09        ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
% 9.04/5.09       => ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8100_singletonD,axiom,
% 9.04/5.09      ! [B: nat,A: nat] :
% 9.04/5.09        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 9.04/5.09       => ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8101_singletonD,axiom,
% 9.04/5.09      ! [B: int,A: int] :
% 9.04/5.09        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 9.04/5.09       => ( B = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8102_cos__add,axiom,
% 9.04/5.09      ! [X: complex,Y: complex] :
% 9.04/5.09        ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.04/5.09        = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8103_cos__add,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.09        = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8104_cos__diff,axiom,
% 9.04/5.09      ! [X: complex,Y: complex] :
% 9.04/5.09        ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
% 9.04/5.09        = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8105_cos__diff,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 9.04/5.09        = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8106_not__psubset__empty,axiom,
% 9.04/5.09      ! [A2: set_real] :
% 9.04/5.09        ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8107_not__psubset__empty,axiom,
% 9.04/5.09      ! [A2: set_o] :
% 9.04/5.09        ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8108_not__psubset__empty,axiom,
% 9.04/5.09      ! [A2: set_nat] :
% 9.04/5.09        ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8109_not__psubset__empty,axiom,
% 9.04/5.09      ! [A2: set_int] :
% 9.04/5.09        ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8110_semiring__norm_I26_J,axiom,
% 9.04/5.09      ( ( bitM @ one )
% 9.04/5.09      = one ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8111_sin__zero__norm__cos__one,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( sin_real @ X )
% 9.04/5.09          = zero_zero_real )
% 9.04/5.09       => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 9.04/5.09          = one_one_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8112_sin__zero__norm__cos__one,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( ( sin_complex @ X )
% 9.04/5.09          = zero_zero_complex )
% 9.04/5.09       => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 9.04/5.09          = one_one_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8113_sin__zero__abs__cos__one,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( sin_real @ X )
% 9.04/5.09          = zero_zero_real )
% 9.04/5.09       => ( ( abs_abs_real @ ( cos_real @ X ) )
% 9.04/5.09          = one_one_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8114_sin__double,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.09        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8115_sin__double,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.09        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8116_sincos__principal__value,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09      ? [Y4: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 9.04/5.09        & ( ord_less_eq_real @ Y4 @ pi )
% 9.04/5.09        & ( ( sin_real @ Y4 )
% 9.04/5.09          = ( sin_real @ X ) )
% 9.04/5.09        & ( ( cos_real @ Y4 )
% 9.04/5.09          = ( cos_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8117_diff__shunt__var,axiom,
% 9.04/5.09      ! [X: set_real,Y: set_real] :
% 9.04/5.09        ( ( ( minus_minus_set_real @ X @ Y )
% 9.04/5.09          = bot_bot_set_real )
% 9.04/5.09        = ( ord_less_eq_set_real @ X @ Y ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8118_diff__shunt__var,axiom,
% 9.04/5.09      ! [X: set_o,Y: set_o] :
% 9.04/5.09        ( ( ( minus_minus_set_o @ X @ Y )
% 9.04/5.09          = bot_bot_set_o )
% 9.04/5.09        = ( ord_less_eq_set_o @ X @ Y ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8119_diff__shunt__var,axiom,
% 9.04/5.09      ! [X: set_nat,Y: set_nat] :
% 9.04/5.09        ( ( ( minus_minus_set_nat @ X @ Y )
% 9.04/5.09          = bot_bot_set_nat )
% 9.04/5.09        = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8120_diff__shunt__var,axiom,
% 9.04/5.09      ! [X: set_int,Y: set_int] :
% 9.04/5.09        ( ( ( minus_minus_set_int @ X @ Y )
% 9.04/5.09          = bot_bot_set_int )
% 9.04/5.09        = ( ord_less_eq_set_int @ X @ Y ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8121_sin__x__le__x,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8122_sin__le__one,axiom,
% 9.04/5.09      ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8123_cos__le__one,axiom,
% 9.04/5.09      ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8124_abs__sin__x__le__abs__x,axiom,
% 9.04/5.09      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8125_cos__arctan__not__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( cos_real @ ( arctan @ X ) )
% 9.04/5.09       != zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8126_subset__singletonD,axiom,
% 9.04/5.09      ! [A2: set_VEBT_VEBT,X: vEBT_VEBT] :
% 9.04/5.09        ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
% 9.04/5.09       => ( ( A2 = bot_bo8194388402131092736T_VEBT )
% 9.04/5.09          | ( A2
% 9.04/5.09            = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8127_subset__singletonD,axiom,
% 9.04/5.09      ! [A2: set_real,X: real] :
% 9.04/5.09        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) )
% 9.04/5.09       => ( ( A2 = bot_bot_set_real )
% 9.04/5.09          | ( A2
% 9.04/5.09            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8128_subset__singletonD,axiom,
% 9.04/5.09      ! [A2: set_o,X: $o] :
% 9.04/5.09        ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) )
% 9.04/5.09       => ( ( A2 = bot_bot_set_o )
% 9.04/5.09          | ( A2
% 9.04/5.09            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8129_subset__singletonD,axiom,
% 9.04/5.09      ! [A2: set_nat,X: nat] :
% 9.04/5.09        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 9.04/5.09       => ( ( A2 = bot_bot_set_nat )
% 9.04/5.09          | ( A2
% 9.04/5.09            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8130_subset__singletonD,axiom,
% 9.04/5.09      ! [A2: set_int,X: int] :
% 9.04/5.09        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
% 9.04/5.09       => ( ( A2 = bot_bot_set_int )
% 9.04/5.09          | ( A2
% 9.04/5.09            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8131_subset__singleton__iff,axiom,
% 9.04/5.09      ! [X8: set_VEBT_VEBT,A: vEBT_VEBT] :
% 9.04/5.09        ( ( ord_le4337996190870823476T_VEBT @ X8 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
% 9.04/5.09        = ( ( X8 = bot_bo8194388402131092736T_VEBT )
% 9.04/5.09          | ( X8
% 9.04/5.09            = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8132_subset__singleton__iff,axiom,
% 9.04/5.09      ! [X8: set_real,A: real] :
% 9.04/5.09        ( ( ord_less_eq_set_real @ X8 @ ( insert_real @ A @ bot_bot_set_real ) )
% 9.04/5.09        = ( ( X8 = bot_bot_set_real )
% 9.04/5.09          | ( X8
% 9.04/5.09            = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8133_subset__singleton__iff,axiom,
% 9.04/5.09      ! [X8: set_o,A: $o] :
% 9.04/5.09        ( ( ord_less_eq_set_o @ X8 @ ( insert_o @ A @ bot_bot_set_o ) )
% 9.04/5.09        = ( ( X8 = bot_bot_set_o )
% 9.04/5.09          | ( X8
% 9.04/5.09            = ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8134_subset__singleton__iff,axiom,
% 9.04/5.09      ! [X8: set_nat,A: nat] :
% 9.04/5.09        ( ( ord_less_eq_set_nat @ X8 @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 9.04/5.09        = ( ( X8 = bot_bot_set_nat )
% 9.04/5.09          | ( X8
% 9.04/5.09            = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8135_subset__singleton__iff,axiom,
% 9.04/5.09      ! [X8: set_int,A: int] :
% 9.04/5.09        ( ( ord_less_eq_set_int @ X8 @ ( insert_int @ A @ bot_bot_set_int ) )
% 9.04/5.09        = ( ( X8 = bot_bot_set_int )
% 9.04/5.09          | ( X8
% 9.04/5.09            = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8136_Diff__insert,axiom,
% 9.04/5.09      ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.04/5.09        ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 9.04/5.09        = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ B2 ) @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8137_Diff__insert,axiom,
% 9.04/5.09      ! [A2: set_real,A: real,B2: set_real] :
% 9.04/5.09        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 9.04/5.09        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8138_Diff__insert,axiom,
% 9.04/5.09      ! [A2: set_o,A: $o,B2: set_o] :
% 9.04/5.09        ( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
% 9.04/5.09        = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B2 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8139_Diff__insert,axiom,
% 9.04/5.09      ! [A2: set_int,A: int,B2: set_int] :
% 9.04/5.09        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 9.04/5.09        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8140_Diff__insert,axiom,
% 9.04/5.09      ! [A2: set_nat,A: nat,B2: set_nat] :
% 9.04/5.09        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 9.04/5.09        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8141_insert__Diff,axiom,
% 9.04/5.09      ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.04/5.09        ( ( member_VEBT_VEBT @ A @ A2 )
% 9.04/5.09       => ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8142_insert__Diff,axiom,
% 9.04/5.09      ! [A: complex,A2: set_complex] :
% 9.04/5.09        ( ( member_complex @ A @ A2 )
% 9.04/5.09       => ( ( insert_complex @ A @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8143_insert__Diff,axiom,
% 9.04/5.09      ! [A: set_nat,A2: set_set_nat] :
% 9.04/5.09        ( ( member_set_nat @ A @ A2 )
% 9.04/5.09       => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8144_insert__Diff,axiom,
% 9.04/5.09      ! [A: real,A2: set_real] :
% 9.04/5.09        ( ( member_real @ A @ A2 )
% 9.04/5.09       => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8145_insert__Diff,axiom,
% 9.04/5.09      ! [A: $o,A2: set_o] :
% 9.04/5.09        ( ( member_o @ A @ A2 )
% 9.04/5.09       => ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8146_insert__Diff,axiom,
% 9.04/5.09      ! [A: int,A2: set_int] :
% 9.04/5.09        ( ( member_int @ A @ A2 )
% 9.04/5.09       => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8147_insert__Diff,axiom,
% 9.04/5.09      ! [A: nat,A2: set_nat] :
% 9.04/5.09        ( ( member_nat @ A @ A2 )
% 9.04/5.09       => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8148_Diff__insert2,axiom,
% 9.04/5.09      ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.04/5.09        ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ B2 ) )
% 9.04/5.09        = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) @ B2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8149_Diff__insert2,axiom,
% 9.04/5.09      ! [A2: set_real,A: real,B2: set_real] :
% 9.04/5.09        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 9.04/5.09        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8150_Diff__insert2,axiom,
% 9.04/5.09      ! [A2: set_o,A: $o,B2: set_o] :
% 9.04/5.09        ( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
% 9.04/5.09        = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8151_Diff__insert2,axiom,
% 9.04/5.09      ! [A2: set_int,A: int,B2: set_int] :
% 9.04/5.09        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 9.04/5.09        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8152_Diff__insert2,axiom,
% 9.04/5.09      ! [A2: set_nat,A: nat,B2: set_nat] :
% 9.04/5.09        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 9.04/5.09        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8153_Diff__insert__absorb,axiom,
% 9.04/5.09      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.04/5.09        ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 9.04/5.09       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8154_Diff__insert__absorb,axiom,
% 9.04/5.09      ! [X: complex,A2: set_complex] :
% 9.04/5.09        ( ~ ( member_complex @ X @ A2 )
% 9.04/5.09       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ ( insert_complex @ X @ bot_bot_set_complex ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8155_Diff__insert__absorb,axiom,
% 9.04/5.09      ! [X: set_nat,A2: set_set_nat] :
% 9.04/5.09        ( ~ ( member_set_nat @ X @ A2 )
% 9.04/5.09       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8156_Diff__insert__absorb,axiom,
% 9.04/5.09      ! [X: real,A2: set_real] :
% 9.04/5.09        ( ~ ( member_real @ X @ A2 )
% 9.04/5.09       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8157_Diff__insert__absorb,axiom,
% 9.04/5.09      ! [X: $o,A2: set_o] :
% 9.04/5.09        ( ~ ( member_o @ X @ A2 )
% 9.04/5.09       => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8158_Diff__insert__absorb,axiom,
% 9.04/5.09      ! [X: int,A2: set_int] :
% 9.04/5.09        ( ~ ( member_int @ X @ A2 )
% 9.04/5.09       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8159_Diff__insert__absorb,axiom,
% 9.04/5.09      ! [X: nat,A2: set_nat] :
% 9.04/5.09        ( ~ ( member_nat @ X @ A2 )
% 9.04/5.09       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 9.04/5.09          = A2 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8160_in__image__insert__iff,axiom,
% 9.04/5.09      ! [B2: set_set_VEBT_VEBT,X: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.04/5.09        ( ! [C5: set_VEBT_VEBT] :
% 9.04/5.09            ( ( member_set_VEBT_VEBT @ C5 @ B2 )
% 9.04/5.09           => ~ ( member_VEBT_VEBT @ X @ C5 ) )
% 9.04/5.09       => ( ( member_set_VEBT_VEBT @ A2 @ ( image_1661326939266726661T_VEBT @ ( insert_VEBT_VEBT @ X ) @ B2 ) )
% 9.04/5.09          = ( ( member_VEBT_VEBT @ X @ A2 )
% 9.04/5.09            & ( member_set_VEBT_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8161_in__image__insert__iff,axiom,
% 9.04/5.09      ! [B2: set_set_complex,X: complex,A2: set_complex] :
% 9.04/5.09        ( ! [C5: set_complex] :
% 9.04/5.09            ( ( member_set_complex @ C5 @ B2 )
% 9.04/5.09           => ~ ( member_complex @ X @ C5 ) )
% 9.04/5.09       => ( ( member_set_complex @ A2 @ ( image_7998606247489673935omplex @ ( insert_complex @ X ) @ B2 ) )
% 9.04/5.09          = ( ( member_complex @ X @ A2 )
% 9.04/5.09            & ( member_set_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8162_in__image__insert__iff,axiom,
% 9.04/5.09      ! [B2: set_set_set_nat,X: set_nat,A2: set_set_nat] :
% 9.04/5.09        ( ! [C5: set_set_nat] :
% 9.04/5.09            ( ( member_set_set_nat @ C5 @ B2 )
% 9.04/5.09           => ~ ( member_set_nat @ X @ C5 ) )
% 9.04/5.09       => ( ( member_set_set_nat @ A2 @ ( image_7884819252390400639et_nat @ ( insert_set_nat @ X ) @ B2 ) )
% 9.04/5.09          = ( ( member_set_nat @ X @ A2 )
% 9.04/5.09            & ( member_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8163_in__image__insert__iff,axiom,
% 9.04/5.09      ! [B2: set_set_real,X: real,A2: set_real] :
% 9.04/5.09        ( ! [C5: set_real] :
% 9.04/5.09            ( ( member_set_real @ C5 @ B2 )
% 9.04/5.09           => ~ ( member_real @ X @ C5 ) )
% 9.04/5.09       => ( ( member_set_real @ A2 @ ( image_2436557299294012491t_real @ ( insert_real @ X ) @ B2 ) )
% 9.04/5.09          = ( ( member_real @ X @ A2 )
% 9.04/5.09            & ( member_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8164_in__image__insert__iff,axiom,
% 9.04/5.09      ! [B2: set_set_o,X: $o,A2: set_o] :
% 9.04/5.09        ( ! [C5: set_o] :
% 9.04/5.09            ( ( member_set_o @ C5 @ B2 )
% 9.04/5.09           => ~ ( member_o @ X @ C5 ) )
% 9.04/5.09       => ( ( member_set_o @ A2 @ ( image_set_o_set_o @ ( insert_o @ X ) @ B2 ) )
% 9.04/5.09          = ( ( member_o @ X @ A2 )
% 9.04/5.09            & ( member_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8165_in__image__insert__iff,axiom,
% 9.04/5.09      ! [B2: set_set_int,X: int,A2: set_int] :
% 9.04/5.09        ( ! [C5: set_int] :
% 9.04/5.09            ( ( member_set_int @ C5 @ B2 )
% 9.04/5.09           => ~ ( member_int @ X @ C5 ) )
% 9.04/5.09       => ( ( member_set_int @ A2 @ ( image_524474410958335435et_int @ ( insert_int @ X ) @ B2 ) )
% 9.04/5.09          = ( ( member_int @ X @ A2 )
% 9.04/5.09            & ( member_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8166_in__image__insert__iff,axiom,
% 9.04/5.09      ! [B2: set_set_nat,X: nat,A2: set_nat] :
% 9.04/5.09        ( ! [C5: set_nat] :
% 9.04/5.09            ( ( member_set_nat @ C5 @ B2 )
% 9.04/5.09           => ~ ( member_nat @ X @ C5 ) )
% 9.04/5.09       => ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B2 ) )
% 9.04/5.09          = ( ( member_nat @ X @ A2 )
% 9.04/5.09            & ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8167_cos__int__times__real,axiom,
% 9.04/5.09      ! [M: int,X: real] :
% 9.04/5.09        ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 9.04/5.09        = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8168_cos__int__times__real,axiom,
% 9.04/5.09      ! [M: int,X: real] :
% 9.04/5.09        ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 9.04/5.09        = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8169_sin__cos__le1,axiom,
% 9.04/5.09      ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8170_subset__Compl__self__eq,axiom,
% 9.04/5.09      ! [A2: set_real] :
% 9.04/5.09        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 9.04/5.09        = ( A2 = bot_bot_set_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8171_subset__Compl__self__eq,axiom,
% 9.04/5.09      ! [A2: set_o] :
% 9.04/5.09        ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
% 9.04/5.09        = ( A2 = bot_bot_set_o ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8172_subset__Compl__self__eq,axiom,
% 9.04/5.09      ! [A2: set_nat] :
% 9.04/5.09        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 9.04/5.09        = ( A2 = bot_bot_set_nat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8173_subset__Compl__self__eq,axiom,
% 9.04/5.09      ! [A2: set_int] :
% 9.04/5.09        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 9.04/5.09        = ( A2 = bot_bot_set_int ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8174_sin__int__times__real,axiom,
% 9.04/5.09      ! [M: int,X: real] :
% 9.04/5.09        ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 9.04/5.09        = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8175_sin__int__times__real,axiom,
% 9.04/5.09      ! [M: int,X: real] :
% 9.04/5.09        ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 9.04/5.09        = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8176_semiring__norm_I28_J,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( bitM @ ( bit1 @ N ) )
% 9.04/5.09        = ( bit1 @ ( bit0 @ N ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8177_semiring__norm_I27_J,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( bitM @ ( bit0 @ N ) )
% 9.04/5.09        = ( bit1 @ ( bitM @ N ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8178_sin__squared__eq,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.09        = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8179_sin__squared__eq,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.09        = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8180_cos__squared__eq,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.09        = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8181_cos__squared__eq,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.09        = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8182_inc__BitM__eq,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( inc @ ( bitM @ N ) )
% 9.04/5.09        = ( bit0 @ N ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8183_BitM__inc__eq,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( bitM @ ( inc @ N ) )
% 9.04/5.09        = ( bit1 @ N ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8184_sin__gt__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ pi )
% 9.04/5.09         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8185_sin__x__ge__neg__x,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8186_sin__ge__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ pi )
% 9.04/5.09         => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8187_sin__ge__minus__one,axiom,
% 9.04/5.09      ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8188_cos__monotone__0__pi__le,axiom,
% 9.04/5.09      ! [Y: real,X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ X )
% 9.04/5.09         => ( ( ord_less_eq_real @ X @ pi )
% 9.04/5.09           => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8189_cos__mono__le__eq,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ pi )
% 9.04/5.09         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.09           => ( ( ord_less_eq_real @ Y @ pi )
% 9.04/5.09             => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
% 9.04/5.09                = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8190_cos__inj__pi,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ pi )
% 9.04/5.09         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.09           => ( ( ord_less_eq_real @ Y @ pi )
% 9.04/5.09             => ( ( ( cos_real @ X )
% 9.04/5.09                  = ( cos_real @ Y ) )
% 9.04/5.09               => ( X = Y ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8191_cos__ge__minus__one,axiom,
% 9.04/5.09      ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8192_abs__sin__le__one,axiom,
% 9.04/5.09      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8193_abs__cos__le__one,axiom,
% 9.04/5.09      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8194_and__nat__def,axiom,
% 9.04/5.09      ( bit_se727722235901077358nd_nat
% 9.04/5.09      = ( ^ [M6: nat,N3: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8195_cos__diff__cos,axiom,
% 9.04/5.09      ! [W: complex,Z: complex] :
% 9.04/5.09        ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8196_cos__diff__cos,axiom,
% 9.04/5.09      ! [W: real,Z: real] :
% 9.04/5.09        ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8197_sin__diff__sin,axiom,
% 9.04/5.09      ! [W: complex,Z: complex] :
% 9.04/5.09        ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8198_sin__diff__sin,axiom,
% 9.04/5.09      ! [W: real,Z: real] :
% 9.04/5.09        ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8199_sin__plus__sin,axiom,
% 9.04/5.09      ! [W: complex,Z: complex] :
% 9.04/5.09        ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8200_sin__plus__sin,axiom,
% 9.04/5.09      ! [W: real,Z: real] :
% 9.04/5.09        ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8201_cos__times__sin,axiom,
% 9.04/5.09      ! [W: complex,Z: complex] :
% 9.04/5.09        ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 9.04/5.09        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8202_cos__times__sin,axiom,
% 9.04/5.09      ! [W: real,Z: real] :
% 9.04/5.09        ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8203_sin__times__cos,axiom,
% 9.04/5.09      ! [W: complex,Z: complex] :
% 9.04/5.09        ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 9.04/5.09        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8204_sin__times__cos,axiom,
% 9.04/5.09      ! [W: real,Z: real] :
% 9.04/5.09        ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8205_sin__times__sin,axiom,
% 9.04/5.09      ! [W: complex,Z: complex] :
% 9.04/5.09        ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 9.04/5.09        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8206_sin__times__sin,axiom,
% 9.04/5.09      ! [W: real,Z: real] :
% 9.04/5.09        ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8207_cos__double,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.09        = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8208_cos__double,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.09        = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8209_cos__sin__eq,axiom,
% 9.04/5.09      ( cos_real
% 9.04/5.09      = ( ^ [X4: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8210_cos__sin__eq,axiom,
% 9.04/5.09      ( cos_complex
% 9.04/5.09      = ( ^ [X4: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8211_sin__cos__eq,axiom,
% 9.04/5.09      ( sin_real
% 9.04/5.09      = ( ^ [X4: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8212_sin__cos__eq,axiom,
% 9.04/5.09      ( sin_complex
% 9.04/5.09      = ( ^ [X4: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8213_subset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.04/5.09        ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_VEBT_VEBT @ X @ A2 )
% 9.04/5.09           => ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B2 ) )
% 9.04/5.09          & ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 9.04/5.09           => ( ord_le4337996190870823476T_VEBT @ A2 @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8214_subset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_complex,X: complex,B2: set_complex] :
% 9.04/5.09        ( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_complex @ X @ A2 )
% 9.04/5.09           => ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) )
% 9.04/5.09          & ( ~ ( member_complex @ X @ A2 )
% 9.04/5.09           => ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8215_subset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
% 9.04/5.09        ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_set_nat @ X @ A2 )
% 9.04/5.09           => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
% 9.04/5.09          & ( ~ ( member_set_nat @ X @ A2 )
% 9.04/5.09           => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8216_subset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_real,X: real,B2: set_real] :
% 9.04/5.09        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_real @ X @ A2 )
% 9.04/5.09           => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
% 9.04/5.09          & ( ~ ( member_real @ X @ A2 )
% 9.04/5.09           => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8217_subset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_o,X: $o,B2: set_o] :
% 9.04/5.09        ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_o @ X @ A2 )
% 9.04/5.09           => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
% 9.04/5.09          & ( ~ ( member_o @ X @ A2 )
% 9.04/5.09           => ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8218_subset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_nat,X: nat,B2: set_nat] :
% 9.04/5.09        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_nat @ X @ A2 )
% 9.04/5.09           => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
% 9.04/5.09          & ( ~ ( member_nat @ X @ A2 )
% 9.04/5.09           => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8219_subset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_int,X: int,B2: set_int] :
% 9.04/5.09        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_int @ X @ A2 )
% 9.04/5.09           => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
% 9.04/5.09          & ( ~ ( member_int @ X @ A2 )
% 9.04/5.09           => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8220_Diff__single__insert,axiom,
% 9.04/5.09      ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.04/5.09        ( ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B2 )
% 9.04/5.09       => ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B2 ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8221_Diff__single__insert,axiom,
% 9.04/5.09      ! [A2: set_real,X: real,B2: set_real] :
% 9.04/5.09        ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 )
% 9.04/5.09       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8222_Diff__single__insert,axiom,
% 9.04/5.09      ! [A2: set_o,X: $o,B2: set_o] :
% 9.04/5.09        ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 )
% 9.04/5.09       => ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8223_Diff__single__insert,axiom,
% 9.04/5.09      ! [A2: set_nat,X: nat,B2: set_nat] :
% 9.04/5.09        ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 )
% 9.04/5.09       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8224_Diff__single__insert,axiom,
% 9.04/5.09      ! [A2: set_int,X: int,B2: set_int] :
% 9.04/5.09        ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 )
% 9.04/5.09       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8225_Compl__insert,axiom,
% 9.04/5.09      ! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
% 9.04/5.09        ( ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) )
% 9.04/5.09        = ( minus_5127226145743854075T_VEBT @ ( uminus8041839845116263051T_VEBT @ A2 ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8226_Compl__insert,axiom,
% 9.04/5.09      ! [X: real,A2: set_real] :
% 9.04/5.09        ( ( uminus612125837232591019t_real @ ( insert_real @ X @ A2 ) )
% 9.04/5.09        = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8227_Compl__insert,axiom,
% 9.04/5.09      ! [X: $o,A2: set_o] :
% 9.04/5.09        ( ( uminus_uminus_set_o @ ( insert_o @ X @ A2 ) )
% 9.04/5.09        = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8228_Compl__insert,axiom,
% 9.04/5.09      ! [X: int,A2: set_int] :
% 9.04/5.09        ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
% 9.04/5.09        = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8229_Compl__insert,axiom,
% 9.04/5.09      ! [X: nat,A2: set_nat] :
% 9.04/5.09        ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
% 9.04/5.09        = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8230_cos__double__sin,axiom,
% 9.04/5.09      ! [W: complex] :
% 9.04/5.09        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 9.04/5.09        = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8231_cos__double__sin,axiom,
% 9.04/5.09      ! [W: real] :
% 9.04/5.09        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 9.04/5.09        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8232_eval__nat__numeral_I2_J,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 9.04/5.09        = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8233_minus__sin__cos__eq,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( uminus_uminus_real @ ( sin_real @ X ) )
% 9.04/5.09        = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8234_minus__sin__cos__eq,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
% 9.04/5.09        = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8235_BitM__plus__one,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 9.04/5.09        = ( bit0 @ N ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8236_one__plus__BitM,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 9.04/5.09        = ( bit0 @ N ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8237_cos__two__neq__zero,axiom,
% 9.04/5.09      ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.09     != zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8238_cos__monotone__0__pi,axiom,
% 9.04/5.09      ! [Y: real,X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.09       => ( ( ord_less_real @ Y @ X )
% 9.04/5.09         => ( ( ord_less_eq_real @ X @ pi )
% 9.04/5.09           => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8239_cos__mono__less__eq,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ pi )
% 9.04/5.09         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.09           => ( ( ord_less_eq_real @ Y @ pi )
% 9.04/5.09             => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
% 9.04/5.09                = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8240_sin__eq__0__pi,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ pi )
% 9.04/5.09         => ( ( ( sin_real @ X )
% 9.04/5.09              = zero_zero_real )
% 9.04/5.09           => ( X = zero_zero_real ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8241_sin__zero__pi__iff,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
% 9.04/5.09       => ( ( ( sin_real @ X )
% 9.04/5.09            = zero_zero_real )
% 9.04/5.09          = ( X = zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8242_cos__monotone__minus__pi__0_H,axiom,
% 9.04/5.09      ! [Y: real,X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ X )
% 9.04/5.09         => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.04/5.09           => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8243_sin__zero__iff__int2,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( sin_real @ X )
% 9.04/5.09          = zero_zero_real )
% 9.04/5.09        = ( ? [I3: int] :
% 9.04/5.09              ( X
% 9.04/5.09              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8244_sincos__total__pi,axiom,
% 9.04/5.09      ! [Y: real,X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.09       => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.09            = one_one_real )
% 9.04/5.09         => ? [T3: real] :
% 9.04/5.09              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 9.04/5.09              & ( ord_less_eq_real @ T3 @ pi )
% 9.04/5.09              & ( X
% 9.04/5.09                = ( cos_real @ T3 ) )
% 9.04/5.09              & ( Y
% 9.04/5.09                = ( sin_real @ T3 ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8245_sin__cos__sqrt,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
% 9.04/5.09       => ( ( sin_real @ X )
% 9.04/5.09          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8246_sin__expansion__lemma,axiom,
% 9.04/5.09      ! [X: real,M: nat] :
% 9.04/5.09        ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.04/5.09        = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8247_psubset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.04/5.09        ( ( ord_le3480810397992357184T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_VEBT_VEBT @ X @ B2 )
% 9.04/5.09           => ( ord_le3480810397992357184T_VEBT @ A2 @ B2 ) )
% 9.04/5.09          & ( ~ ( member_VEBT_VEBT @ X @ B2 )
% 9.04/5.09           => ( ( ( member_VEBT_VEBT @ X @ A2 )
% 9.04/5.09               => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B2 ) )
% 9.04/5.09              & ( ~ ( member_VEBT_VEBT @ X @ A2 )
% 9.04/5.09               => ( ord_le4337996190870823476T_VEBT @ A2 @ B2 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8248_psubset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_complex,X: complex,B2: set_complex] :
% 9.04/5.09        ( ( ord_less_set_complex @ A2 @ ( insert_complex @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_complex @ X @ B2 )
% 9.04/5.09           => ( ord_less_set_complex @ A2 @ B2 ) )
% 9.04/5.09          & ( ~ ( member_complex @ X @ B2 )
% 9.04/5.09           => ( ( ( member_complex @ X @ A2 )
% 9.04/5.09               => ( ord_less_set_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) )
% 9.04/5.09              & ( ~ ( member_complex @ X @ A2 )
% 9.04/5.09               => ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8249_psubset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
% 9.04/5.09        ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_set_nat @ X @ B2 )
% 9.04/5.09           => ( ord_less_set_set_nat @ A2 @ B2 ) )
% 9.04/5.09          & ( ~ ( member_set_nat @ X @ B2 )
% 9.04/5.09           => ( ( ( member_set_nat @ X @ A2 )
% 9.04/5.09               => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
% 9.04/5.09              & ( ~ ( member_set_nat @ X @ A2 )
% 9.04/5.09               => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8250_psubset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_real,X: real,B2: set_real] :
% 9.04/5.09        ( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_real @ X @ B2 )
% 9.04/5.09           => ( ord_less_set_real @ A2 @ B2 ) )
% 9.04/5.09          & ( ~ ( member_real @ X @ B2 )
% 9.04/5.09           => ( ( ( member_real @ X @ A2 )
% 9.04/5.09               => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
% 9.04/5.09              & ( ~ ( member_real @ X @ A2 )
% 9.04/5.09               => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8251_psubset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_o,X: $o,B2: set_o] :
% 9.04/5.09        ( ( ord_less_set_o @ A2 @ ( insert_o @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_o @ X @ B2 )
% 9.04/5.09           => ( ord_less_set_o @ A2 @ B2 ) )
% 9.04/5.09          & ( ~ ( member_o @ X @ B2 )
% 9.04/5.09           => ( ( ( member_o @ X @ A2 )
% 9.04/5.09               => ( ord_less_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
% 9.04/5.09              & ( ~ ( member_o @ X @ A2 )
% 9.04/5.09               => ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8252_psubset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_nat,X: nat,B2: set_nat] :
% 9.04/5.09        ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_nat @ X @ B2 )
% 9.04/5.09           => ( ord_less_set_nat @ A2 @ B2 ) )
% 9.04/5.09          & ( ~ ( member_nat @ X @ B2 )
% 9.04/5.09           => ( ( ( member_nat @ X @ A2 )
% 9.04/5.09               => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
% 9.04/5.09              & ( ~ ( member_nat @ X @ A2 )
% 9.04/5.09               => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8253_psubset__insert__iff,axiom,
% 9.04/5.09      ! [A2: set_int,X: int,B2: set_int] :
% 9.04/5.09        ( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 9.04/5.09        = ( ( ( member_int @ X @ B2 )
% 9.04/5.09           => ( ord_less_set_int @ A2 @ B2 ) )
% 9.04/5.09          & ( ~ ( member_int @ X @ B2 )
% 9.04/5.09           => ( ( ( member_int @ X @ A2 )
% 9.04/5.09               => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
% 9.04/5.09              & ( ~ ( member_int @ X @ A2 )
% 9.04/5.09               => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8254_numeral__BitM,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
% 9.04/5.09        = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8255_numeral__BitM,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( numeral_numeral_real @ ( bitM @ N ) )
% 9.04/5.09        = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8256_numeral__BitM,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( numeral_numeral_rat @ ( bitM @ N ) )
% 9.04/5.09        = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8257_numeral__BitM,axiom,
% 9.04/5.09      ! [N: num] :
% 9.04/5.09        ( ( numeral_numeral_int @ ( bitM @ N ) )
% 9.04/5.09        = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8258_odd__numeral__BitM,axiom,
% 9.04/5.09      ! [W: num] :
% 9.04/5.09        ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8259_odd__numeral__BitM,axiom,
% 9.04/5.09      ! [W: num] :
% 9.04/5.09        ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8260_odd__numeral__BitM,axiom,
% 9.04/5.09      ! [W: num] :
% 9.04/5.09        ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8261_cos__expansion__lemma,axiom,
% 9.04/5.09      ! [X: real,M: nat] :
% 9.04/5.09        ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.04/5.09        = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8262_sin__gt__zero__02,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.09         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8263_cos__two__less__zero,axiom,
% 9.04/5.09      ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8264_cos__is__zero,axiom,
% 9.04/5.09      ? [X3: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.04/5.09        & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.09        & ( ( cos_real @ X3 )
% 9.04/5.09          = zero_zero_real )
% 9.04/5.09        & ! [Y5: real] :
% 9.04/5.09            ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 9.04/5.09              & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.09              & ( ( cos_real @ Y5 )
% 9.04/5.09                = zero_zero_real ) )
% 9.04/5.09           => ( Y5 = X3 ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8265_cos__two__le__zero,axiom,
% 9.04/5.09      ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8266_cos__monotone__minus__pi__0,axiom,
% 9.04/5.09      ! [Y: real,X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
% 9.04/5.09       => ( ( ord_less_real @ Y @ X )
% 9.04/5.09         => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.04/5.09           => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8267_cos__total,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ? [X3: real] :
% 9.04/5.09              ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.04/5.09              & ( ord_less_eq_real @ X3 @ pi )
% 9.04/5.09              & ( ( cos_real @ X3 )
% 9.04/5.09                = Y )
% 9.04/5.09              & ! [Y5: real] :
% 9.04/5.09                  ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 9.04/5.09                    & ( ord_less_eq_real @ Y5 @ pi )
% 9.04/5.09                    & ( ( cos_real @ Y5 )
% 9.04/5.09                      = Y ) )
% 9.04/5.09                 => ( Y5 = X3 ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8268_sincos__total__pi__half,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.09         => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.09              = one_one_real )
% 9.04/5.09           => ? [T3: real] :
% 9.04/5.09                ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 9.04/5.09                & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09                & ( X
% 9.04/5.09                  = ( cos_real @ T3 ) )
% 9.04/5.09                & ( Y
% 9.04/5.09                  = ( sin_real @ T3 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8269_sincos__total__2pi__le,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.09          = one_one_real )
% 9.04/5.09       => ? [T3: real] :
% 9.04/5.09            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 9.04/5.09            & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.04/5.09            & ( X
% 9.04/5.09              = ( cos_real @ T3 ) )
% 9.04/5.09            & ( Y
% 9.04/5.09              = ( sin_real @ T3 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8270_sincos__total__2pi,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.09          = one_one_real )
% 9.04/5.09       => ~ ! [T3: real] :
% 9.04/5.09              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 9.04/5.09             => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.04/5.09               => ( ( X
% 9.04/5.09                    = ( cos_real @ T3 ) )
% 9.04/5.09                 => ( Y
% 9.04/5.09                   != ( sin_real @ T3 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8271_sin__pi__divide__n__ge__0,axiom,
% 9.04/5.09      ! [N: nat] :
% 9.04/5.09        ( ( N != zero_zero_nat )
% 9.04/5.09       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8272_sin__45,axiom,
% 9.04/5.09      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.04/5.09      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8273_cos__45,axiom,
% 9.04/5.09      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.04/5.09      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8274_cos__plus__cos,axiom,
% 9.04/5.09      ! [W: complex,Z: complex] :
% 9.04/5.09        ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8275_cos__plus__cos,axiom,
% 9.04/5.09      ! [W: real,Z: real] :
% 9.04/5.09        ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8276_cos__times__cos,axiom,
% 9.04/5.09      ! [W: complex,Z: complex] :
% 9.04/5.09        ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 9.04/5.09        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8277_cos__times__cos,axiom,
% 9.04/5.09      ! [W: real,Z: real] :
% 9.04/5.09        ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 9.04/5.09        = ( 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 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8278_sin__gt__zero2,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8279_sin__lt__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ pi @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.04/5.09         => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8280_cos__double__less__one,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.09         => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8281_sin__30,axiom,
% 9.04/5.09      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 9.04/5.09      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8282_cos__gt__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8283_sin__monotone__2pi__le,axiom,
% 9.04/5.09      ! [Y: real,X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ X )
% 9.04/5.09         => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09           => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8284_sin__mono__le__eq,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.04/5.09           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09             => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 9.04/5.09                = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8285_sin__inj__pi,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.04/5.09           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09             => ( ( ( sin_real @ X )
% 9.04/5.09                  = ( sin_real @ Y ) )
% 9.04/5.09               => ( X = Y ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8286_cos__60,axiom,
% 9.04/5.09      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 9.04/5.09      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8287_sin__60,axiom,
% 9.04/5.09      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 9.04/5.09      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8288_cos__30,axiom,
% 9.04/5.09      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 9.04/5.09      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8289_cos__one__2pi__int,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( cos_real @ X )
% 9.04/5.09          = one_one_real )
% 9.04/5.09        = ( ? [X4: int] :
% 9.04/5.09              ( X
% 9.04/5.09              = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8290_cos__double__cos,axiom,
% 9.04/5.09      ! [W: complex] :
% 9.04/5.09        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 9.04/5.09        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8291_cos__double__cos,axiom,
% 9.04/5.09      ! [W: real] :
% 9.04/5.09        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 9.04/5.09        = ( 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 ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8292_cos__treble__cos,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
% 9.04/5.09        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8293_cos__treble__cos,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
% 9.04/5.09        = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8294_and__nat__unfold,axiom,
% 9.04/5.09      ( bit_se727722235901077358nd_nat
% 9.04/5.09      = ( ^ [M6: nat,N3: nat] :
% 9.04/5.09            ( if_nat
% 9.04/5.09            @ ( ( M6 = zero_zero_nat )
% 9.04/5.09              | ( N3 = zero_zero_nat ) )
% 9.04/5.09            @ zero_zero_nat
% 9.04/5.09            @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8295_sin__le__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ pi @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.04/5.09         => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8296_sin__less__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ zero_zero_real )
% 9.04/5.09         => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8297_sin__monotone__2pi,axiom,
% 9.04/5.09      ! [Y: real,X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.04/5.09       => ( ( ord_less_real @ Y @ X )
% 9.04/5.09         => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09           => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8298_sin__mono__less__eq,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.04/5.09           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09             => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 9.04/5.09                = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8299_sin__total,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ? [X3: real] :
% 9.04/5.09              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.04/5.09              & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09              & ( ( sin_real @ X3 )
% 9.04/5.09                = Y )
% 9.04/5.09              & ! [Y5: real] :
% 9.04/5.09                  ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 9.04/5.09                    & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09                    & ( ( sin_real @ Y5 )
% 9.04/5.09                      = Y ) )
% 9.04/5.09                 => ( Y5 = X3 ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8300_and__nat__rec,axiom,
% 9.04/5.09      ( bit_se727722235901077358nd_nat
% 9.04/5.09      = ( ^ [M6: nat,N3: nat] :
% 9.04/5.09            ( plus_plus_nat
% 9.04/5.09            @ ( zero_n2687167440665602831ol_nat
% 9.04/5.09              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 9.04/5.09                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 9.04/5.09            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8301_cos__gt__zero__pi,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8302_cos__ge__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8303_cos__one__2pi,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( cos_real @ X )
% 9.04/5.09          = one_one_real )
% 9.04/5.09        = ( ? [X4: nat] :
% 9.04/5.09              ( X
% 9.04/5.09              = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 9.04/5.09          | ? [X4: nat] :
% 9.04/5.09              ( X
% 9.04/5.09              = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8304_sin__pi__divide__n__gt__0,axiom,
% 9.04/5.09      ! [N: nat] :
% 9.04/5.09        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.09       => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8305_sin__arctan,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( sin_real @ ( arctan @ X ) )
% 9.04/5.09        = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8306_cos__arctan,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( cos_real @ ( arctan @ X ) )
% 9.04/5.09        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8307_sin__zero__iff__int,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( sin_real @ X )
% 9.04/5.09          = zero_zero_real )
% 9.04/5.09        = ( ? [I3: int] :
% 9.04/5.09              ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
% 9.04/5.09              & ( X
% 9.04/5.09                = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8308_cos__zero__iff__int,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( cos_real @ X )
% 9.04/5.09          = zero_zero_real )
% 9.04/5.09        = ( ? [I3: int] :
% 9.04/5.09              ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
% 9.04/5.09              & ( X
% 9.04/5.09                = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8309_sin__zero__lemma,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ( sin_real @ X )
% 9.04/5.09            = zero_zero_real )
% 9.04/5.09         => ? [N2: nat] :
% 9.04/5.09              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.04/5.09              & ( X
% 9.04/5.09                = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8310_sin__zero__iff,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( sin_real @ X )
% 9.04/5.09          = zero_zero_real )
% 9.04/5.09        = ( ? [N3: nat] :
% 9.04/5.09              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 9.04/5.09              & ( X
% 9.04/5.09                = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.09          | ? [N3: nat] :
% 9.04/5.09              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 9.04/5.09              & ( X
% 9.04/5.09                = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8311_cos__zero__lemma,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ( cos_real @ X )
% 9.04/5.09            = zero_zero_real )
% 9.04/5.09         => ? [N2: nat] :
% 9.04/5.09              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.04/5.09              & ( X
% 9.04/5.09                = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8312_cos__zero__iff,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( cos_real @ X )
% 9.04/5.09          = zero_zero_real )
% 9.04/5.09        = ( ? [N3: nat] :
% 9.04/5.09              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 9.04/5.09              & ( X
% 9.04/5.09                = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.09          | ? [N3: nat] :
% 9.04/5.09              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 9.04/5.09              & ( X
% 9.04/5.09                = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8313_tan__double,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( ( cos_complex @ X )
% 9.04/5.09         != zero_zero_complex )
% 9.04/5.09       => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.09           != zero_zero_complex )
% 9.04/5.09         => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.09            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8314_tan__double,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( cos_real @ X )
% 9.04/5.09         != zero_zero_real )
% 9.04/5.09       => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.09           != zero_zero_real )
% 9.04/5.09         => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.09            = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8315_sin__tan,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09       => ( ( sin_real @ X )
% 9.04/5.09          = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8316_cos__tan,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09       => ( ( cos_real @ X )
% 9.04/5.09          = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8317_complex__unimodular__polar,axiom,
% 9.04/5.09      ! [Z: complex] :
% 9.04/5.09        ( ( ( real_V1022390504157884413omplex @ Z )
% 9.04/5.09          = one_one_real )
% 9.04/5.09       => ~ ! [T3: real] :
% 9.04/5.09              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 9.04/5.09             => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.04/5.09               => ( Z
% 9.04/5.09                 != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8318_and__int_Opelims,axiom,
% 9.04/5.09      ! [X: int,Xa: int,Y: int] :
% 9.04/5.09        ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 9.04/5.09          = Y )
% 9.04/5.09       => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 9.04/5.09         => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.04/5.09                    & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.04/5.09                 => ( Y
% 9.04/5.09                    = ( uminus_uminus_int
% 9.04/5.09                      @ ( zero_n2684676970156552555ol_int
% 9.04/5.09                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 9.04/5.09                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 9.04/5.09                & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.04/5.09                      & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.04/5.09                 => ( Y
% 9.04/5.09                    = ( plus_plus_int
% 9.04/5.09                      @ ( zero_n2684676970156552555ol_int
% 9.04/5.09                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 9.04/5.09                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 9.04/5.09                      @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 9.04/5.09             => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8319_bit__0__eq,axiom,
% 9.04/5.09      ( ( bit_se1146084159140164899it_int @ zero_zero_int )
% 9.04/5.09      = bot_bot_nat_o ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8320_bit__0__eq,axiom,
% 9.04/5.09      ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
% 9.04/5.09      = bot_bot_nat_o ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8321_tan__zero,axiom,
% 9.04/5.09      ( ( tan_complex @ zero_zero_complex )
% 9.04/5.09      = zero_zero_complex ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8322_tan__zero,axiom,
% 9.04/5.09      ( ( tan_real @ zero_zero_real )
% 9.04/5.09      = zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8323_tan__minus,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( tan_real @ ( uminus_uminus_real @ X ) )
% 9.04/5.09        = ( uminus_uminus_real @ ( tan_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8324_tan__minus,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( tan_complex @ ( uminus1482373934393186551omplex @ X ) )
% 9.04/5.09        = ( uminus1482373934393186551omplex @ ( tan_complex @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8325_set__decode__zero,axiom,
% 9.04/5.09      ( ( nat_set_decode @ zero_zero_nat )
% 9.04/5.09      = bot_bot_set_nat ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8326_tan__pi,axiom,
% 9.04/5.09      ( ( tan_real @ pi )
% 9.04/5.09      = zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8327_tan__periodic__pi,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
% 9.04/5.09        = ( tan_real @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8328_tan__npi,axiom,
% 9.04/5.09      ! [N: nat] :
% 9.04/5.09        ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 9.04/5.09        = zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8329_tan__periodic__n,axiom,
% 9.04/5.09      ! [X: real,N: num] :
% 9.04/5.09        ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 9.04/5.09        = ( tan_real @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8330_tan__periodic__nat,axiom,
% 9.04/5.09      ! [X: real,N: nat] :
% 9.04/5.09        ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 9.04/5.09        = ( tan_real @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8331_tan__periodic__int,axiom,
% 9.04/5.09      ! [X: real,I: int] :
% 9.04/5.09        ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 9.04/5.09        = ( tan_real @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8332_norm__cos__sin,axiom,
% 9.04/5.09      ! [T: real] :
% 9.04/5.09        ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 9.04/5.09        = one_one_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8333_tan__periodic,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.04/5.09        = ( tan_real @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8334_bot__nat__def,axiom,
% 9.04/5.09      bot_bot_nat = zero_zero_nat ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8335_bot__set__def,axiom,
% 9.04/5.09      ( bot_bot_set_complex
% 9.04/5.09      = ( collect_complex @ bot_bot_complex_o ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8336_bot__set__def,axiom,
% 9.04/5.09      ( bot_bot_set_set_nat
% 9.04/5.09      = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8337_bot__set__def,axiom,
% 9.04/5.09      ( bot_bot_set_real
% 9.04/5.09      = ( collect_real @ bot_bot_real_o ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8338_bot__set__def,axiom,
% 9.04/5.09      ( bot_bot_set_o
% 9.04/5.09      = ( collect_o @ bot_bot_o_o ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8339_bot__set__def,axiom,
% 9.04/5.09      ( bot_bot_set_nat
% 9.04/5.09      = ( collect_nat @ bot_bot_nat_o ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8340_bot__set__def,axiom,
% 9.04/5.09      ( bot_bot_set_int
% 9.04/5.09      = ( collect_int @ bot_bot_int_o ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8341_tan__of__real,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( real_V1803761363581548252l_real @ ( tan_real @ X ) )
% 9.04/5.09        = ( tan_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8342_tan__of__real,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( real_V4546457046886955230omplex @ ( tan_real @ X ) )
% 9.04/5.09        = ( tan_complex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8343_zero__complex_Ocode,axiom,
% 9.04/5.09      ( zero_zero_complex
% 9.04/5.09      = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8344_Complex__eq__0,axiom,
% 9.04/5.09      ! [A: real,B: real] :
% 9.04/5.09        ( ( ( complex2 @ A @ B )
% 9.04/5.09          = zero_zero_complex )
% 9.04/5.09        = ( ( A = zero_zero_real )
% 9.04/5.09          & ( B = zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8345_tan__arctan,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( tan_real @ ( arctan @ Y ) )
% 9.04/5.09        = Y ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8346_bot__enat__def,axiom,
% 9.04/5.09      bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8347_one__complex_Ocode,axiom,
% 9.04/5.09      ( one_one_complex
% 9.04/5.09      = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8348_Complex__eq__1,axiom,
% 9.04/5.09      ! [A: real,B: real] :
% 9.04/5.09        ( ( ( complex2 @ A @ B )
% 9.04/5.09          = one_one_complex )
% 9.04/5.09        = ( ( A = one_one_real )
% 9.04/5.09          & ( B = zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8349_Complex__eq__numeral,axiom,
% 9.04/5.09      ! [A: real,B: real,W: num] :
% 9.04/5.09        ( ( ( complex2 @ A @ B )
% 9.04/5.09          = ( numera6690914467698888265omplex @ W ) )
% 9.04/5.09        = ( ( A
% 9.04/5.09            = ( numeral_numeral_real @ W ) )
% 9.04/5.09          & ( B = zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8350_complex__eq__cancel__iff2,axiom,
% 9.04/5.09      ! [X: real,Y: real,Xa: real] :
% 9.04/5.09        ( ( ( complex2 @ X @ Y )
% 9.04/5.09          = ( real_V4546457046886955230omplex @ Xa ) )
% 9.04/5.09        = ( ( X = Xa )
% 9.04/5.09          & ( Y = zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8351_complex__of__real__code,axiom,
% 9.04/5.09      ( real_V4546457046886955230omplex
% 9.04/5.09      = ( ^ [X4: real] : ( complex2 @ X4 @ zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8352_complex__of__real__def,axiom,
% 9.04/5.09      ( real_V4546457046886955230omplex
% 9.04/5.09      = ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8353_Complex__eq__neg__1,axiom,
% 9.04/5.09      ! [A: real,B: real] :
% 9.04/5.09        ( ( ( complex2 @ A @ B )
% 9.04/5.09          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.04/5.09        = ( ( A
% 9.04/5.09            = ( uminus_uminus_real @ one_one_real ) )
% 9.04/5.09          & ( B = zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8354_Complex__eq__neg__numeral,axiom,
% 9.04/5.09      ! [A: real,B: real,W: num] :
% 9.04/5.09        ( ( ( complex2 @ A @ B )
% 9.04/5.09          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.04/5.09        = ( ( A
% 9.04/5.09            = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.04/5.09          & ( B = zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8355_tan__def,axiom,
% 9.04/5.09      ( tan_complex
% 9.04/5.09      = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X4 ) @ ( cos_complex @ X4 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8356_tan__def,axiom,
% 9.04/5.09      ( tan_real
% 9.04/5.09      = ( ^ [X4: real] : ( divide_divide_real @ ( sin_real @ X4 ) @ ( cos_real @ X4 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8357_tan__45,axiom,
% 9.04/5.09      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.04/5.09      = one_one_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8358_tan__60,axiom,
% 9.04/5.09      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 9.04/5.09      = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8359_tan__gt__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8360_lemma__tan__total,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.04/5.09       => ? [X3: real] :
% 9.04/5.09            ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.04/5.09            & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09            & ( ord_less_real @ Y @ ( tan_real @ X3 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8361_tan__total,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09      ? [X3: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.04/5.09        & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09        & ( ( tan_real @ X3 )
% 9.04/5.09          = Y )
% 9.04/5.09        & ! [Y5: real] :
% 9.04/5.09            ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 9.04/5.09              & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09              & ( ( tan_real @ Y5 )
% 9.04/5.09                = Y ) )
% 9.04/5.09           => ( Y5 = X3 ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8362_tan__monotone,axiom,
% 9.04/5.09      ! [Y: real,X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.04/5.09       => ( ( ord_less_real @ Y @ X )
% 9.04/5.09         => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09           => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8363_tan__monotone_H,axiom,
% 9.04/5.09      ! [Y: real,X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.04/5.09       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09           => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09             => ( ( ord_less_real @ Y @ X )
% 9.04/5.09                = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8364_tan__mono__lt__eq,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.04/5.09           => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09             => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 9.04/5.09                = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8365_lemma__tan__total1,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09      ? [X3: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.04/5.09        & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09        & ( ( tan_real @ X3 )
% 9.04/5.09          = Y ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8366_tan__minus__45,axiom,
% 9.04/5.09      ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.09      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8367_tan__inverse,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
% 9.04/5.09        = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8368_complex__norm,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
% 9.04/5.09        = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8369_add__tan__eq,axiom,
% 9.04/5.09      ! [X: complex,Y: complex] :
% 9.04/5.09        ( ( ( cos_complex @ X )
% 9.04/5.09         != zero_zero_complex )
% 9.04/5.09       => ( ( ( cos_complex @ Y )
% 9.04/5.09           != zero_zero_complex )
% 9.04/5.09         => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
% 9.04/5.09            = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8370_add__tan__eq,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ( cos_real @ X )
% 9.04/5.09         != zero_zero_real )
% 9.04/5.09       => ( ( ( cos_real @ Y )
% 9.04/5.09           != zero_zero_real )
% 9.04/5.09         => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 9.04/5.09            = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8371_tan__total__pos,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.09       => ? [X3: real] :
% 9.04/5.09            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.04/5.09            & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09            & ( ( tan_real @ X3 )
% 9.04/5.09              = Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8372_tan__pos__pi2__le,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8373_tan__less__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ zero_zero_real )
% 9.04/5.09         => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8374_tan__mono__le,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ Y )
% 9.04/5.09         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09           => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8375_tan__mono__le__eq,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.04/5.09           => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09             => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 9.04/5.09                = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8376_tan__bound__pi2,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.04/5.09       => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8377_tan__30,axiom,
% 9.04/5.09      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 9.04/5.09      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8378_arctan,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 9.04/5.09        & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09        & ( ( tan_real @ ( arctan @ Y ) )
% 9.04/5.09          = Y ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8379_arctan__tan,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ( arctan @ ( tan_real @ X ) )
% 9.04/5.09            = X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8380_arctan__unique,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ( ( tan_real @ X )
% 9.04/5.09              = Y )
% 9.04/5.09           => ( ( arctan @ Y )
% 9.04/5.09              = X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8381_and__int_Opinduct,axiom,
% 9.04/5.09      ! [A0: int,A1: int,P: int > int > $o] :
% 9.04/5.09        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 9.04/5.09       => ( ! [K3: int,L2: int] :
% 9.04/5.09              ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L2 ) )
% 9.04/5.09             => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.04/5.09                      & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.04/5.09                 => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.04/5.09               => ( P @ K3 @ L2 ) ) )
% 9.04/5.09         => ( P @ A0 @ A1 ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8382_tan__add,axiom,
% 9.04/5.09      ! [X: complex,Y: complex] :
% 9.04/5.09        ( ( ( cos_complex @ X )
% 9.04/5.09         != zero_zero_complex )
% 9.04/5.09       => ( ( ( cos_complex @ Y )
% 9.04/5.09           != zero_zero_complex )
% 9.04/5.09         => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.04/5.09             != zero_zero_complex )
% 9.04/5.09           => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.04/5.09              = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8383_tan__add,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ( cos_real @ X )
% 9.04/5.09         != zero_zero_real )
% 9.04/5.09       => ( ( ( cos_real @ Y )
% 9.04/5.09           != zero_zero_real )
% 9.04/5.09         => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.09             != zero_zero_real )
% 9.04/5.09           => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.09              = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8384_tan__diff,axiom,
% 9.04/5.09      ! [X: complex,Y: complex] :
% 9.04/5.09        ( ( ( cos_complex @ X )
% 9.04/5.09         != zero_zero_complex )
% 9.04/5.09       => ( ( ( cos_complex @ Y )
% 9.04/5.09           != zero_zero_complex )
% 9.04/5.09         => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
% 9.04/5.09             != zero_zero_complex )
% 9.04/5.09           => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
% 9.04/5.09              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8385_tan__diff,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ( cos_real @ X )
% 9.04/5.09         != zero_zero_real )
% 9.04/5.09       => ( ( ( cos_real @ Y )
% 9.04/5.09           != zero_zero_real )
% 9.04/5.09         => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 9.04/5.09             != zero_zero_real )
% 9.04/5.09           => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
% 9.04/5.09              = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8386_lemma__tan__add1,axiom,
% 9.04/5.09      ! [X: complex,Y: complex] :
% 9.04/5.09        ( ( ( cos_complex @ X )
% 9.04/5.09         != zero_zero_complex )
% 9.04/5.09       => ( ( ( cos_complex @ Y )
% 9.04/5.09           != zero_zero_complex )
% 9.04/5.09         => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
% 9.04/5.09            = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8387_lemma__tan__add1,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ( cos_real @ X )
% 9.04/5.09         != zero_zero_real )
% 9.04/5.09       => ( ( ( cos_real @ Y )
% 9.04/5.09           != zero_zero_real )
% 9.04/5.09         => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
% 9.04/5.09            = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8388_tan__total__pi4,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.09       => ? [Z3: real] :
% 9.04/5.09            ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
% 9.04/5.09            & ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.04/5.09            & ( ( tan_real @ Z3 )
% 9.04/5.09              = X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8389_tan__half,axiom,
% 9.04/5.09      ( tan_complex
% 9.04/5.09      = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_complex ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8390_tan__half,axiom,
% 9.04/5.09      ( tan_real
% 9.04/5.09      = ( ^ [X4: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8391_and__int_Opsimps,axiom,
% 9.04/5.09      ! [K: int,L: int] :
% 9.04/5.09        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 9.04/5.09       => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.04/5.09              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.04/5.09           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 9.04/5.09              = ( uminus_uminus_int
% 9.04/5.09                @ ( zero_n2684676970156552555ol_int
% 9.04/5.09                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.04/5.09                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 9.04/5.09          & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.04/5.09                & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.04/5.09           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 9.04/5.09              = ( plus_plus_int
% 9.04/5.09                @ ( zero_n2684676970156552555ol_int
% 9.04/5.09                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.04/5.09                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 9.04/5.09                @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8392_upto_Opinduct,axiom,
% 9.04/5.09      ! [A0: int,A1: int,P: int > int > $o] :
% 9.04/5.09        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 9.04/5.09       => ( ! [I2: int,J3: int] :
% 9.04/5.09              ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J3 ) )
% 9.04/5.09             => ( ( ( ord_less_eq_int @ I2 @ J3 )
% 9.04/5.09                 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) )
% 9.04/5.09               => ( P @ I2 @ J3 ) ) )
% 9.04/5.09         => ( P @ A0 @ A1 ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8393_cos__arcsin,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.09         => ( ( cos_real @ ( arcsin @ X ) )
% 9.04/5.09            = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8394_sin__arccos__abs,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.04/5.09       => ( ( sin_real @ ( arccos @ Y ) )
% 9.04/5.09          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8395_sin__arccos,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.09         => ( ( sin_real @ ( arccos @ X ) )
% 9.04/5.09            = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8396_cot__less__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ zero_zero_real )
% 9.04/5.09         => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8397_arcsin__0,axiom,
% 9.04/5.09      ( ( arcsin @ zero_zero_real )
% 9.04/5.09      = zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8398_cot__zero,axiom,
% 9.04/5.09      ( ( cot_complex @ zero_zero_complex )
% 9.04/5.09      = zero_zero_complex ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8399_cot__zero,axiom,
% 9.04/5.09      ( ( cot_real @ zero_zero_real )
% 9.04/5.09      = zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8400_cot__minus,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( cot_real @ ( uminus_uminus_real @ X ) )
% 9.04/5.09        = ( uminus_uminus_real @ ( cot_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8401_cot__minus,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( cot_complex @ ( uminus1482373934393186551omplex @ X ) )
% 9.04/5.09        = ( uminus1482373934393186551omplex @ ( cot_complex @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8402_arccos__1,axiom,
% 9.04/5.09      ( ( arccos @ one_one_real )
% 9.04/5.09      = zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8403_cot__pi,axiom,
% 9.04/5.09      ( ( cot_real @ pi )
% 9.04/5.09      = zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8404_arccos__minus__1,axiom,
% 9.04/5.09      ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 9.04/5.09      = pi ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8405_cos__arccos,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ( cos_real @ ( arccos @ Y ) )
% 9.04/5.09            = Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8406_sin__arcsin,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ( sin_real @ ( arcsin @ Y ) )
% 9.04/5.09            = Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8407_cot__npi,axiom,
% 9.04/5.09      ! [N: nat] :
% 9.04/5.09        ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 9.04/5.09        = zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8408_arccos__0,axiom,
% 9.04/5.09      ( ( arccos @ zero_zero_real )
% 9.04/5.09      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8409_arcsin__1,axiom,
% 9.04/5.09      ( ( arcsin @ one_one_real )
% 9.04/5.09      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8410_cot__periodic,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.04/5.09        = ( cot_real @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8411_arcsin__minus__1,axiom,
% 9.04/5.09      ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 9.04/5.09      = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8412_cot__of__real,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( real_V1803761363581548252l_real @ ( cot_real @ X ) )
% 9.04/5.09        = ( cot_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8413_cot__of__real,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( real_V4546457046886955230omplex @ ( cot_real @ X ) )
% 9.04/5.09        = ( cot_complex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8414_arccos__le__arccos,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ Y )
% 9.04/5.09         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09           => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8415_arccos__le__mono,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.09       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 9.04/5.09            = ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8416_arccos__eq__iff,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.09          & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
% 9.04/5.09       => ( ( ( arccos @ X )
% 9.04/5.09            = ( arccos @ Y ) )
% 9.04/5.09          = ( X = Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8417_arcsin__le__arcsin,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ Y )
% 9.04/5.09         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09           => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8418_arcsin__minus,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.09         => ( ( arcsin @ ( uminus_uminus_real @ X ) )
% 9.04/5.09            = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8419_arcsin__le__mono,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.09       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 9.04/5.09            = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8420_arcsin__eq__iff,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.09       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.04/5.09         => ( ( ( arcsin @ X )
% 9.04/5.09              = ( arcsin @ Y ) )
% 9.04/5.09            = ( X = Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8421_arccos__lbound,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8422_arccos__less__arccos,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ Y )
% 9.04/5.09         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09           => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8423_arccos__less__mono,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.09       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.04/5.09         => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 9.04/5.09            = ( ord_less_real @ Y @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8424_arccos__ubound,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8425_arccos__cos,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ pi )
% 9.04/5.09         => ( ( arccos @ ( cos_real @ X ) )
% 9.04/5.09            = X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8426_arcsin__less__arcsin,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ Y )
% 9.04/5.09         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09           => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8427_arcsin__less__mono,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.09       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.04/5.09         => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 9.04/5.09            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8428_cos__arccos__abs,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.04/5.09       => ( ( cos_real @ ( arccos @ Y ) )
% 9.04/5.09          = Y ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8429_arccos__cos__eq__abs,axiom,
% 9.04/5.09      ! [Theta: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 9.04/5.09       => ( ( arccos @ ( cos_real @ Theta ) )
% 9.04/5.09          = ( abs_abs_real @ Theta ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8430_cot__def,axiom,
% 9.04/5.09      ( cot_complex
% 9.04/5.09      = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X4 ) @ ( sin_complex @ X4 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8431_cot__def,axiom,
% 9.04/5.09      ( cot_real
% 9.04/5.09      = ( ^ [X4: real] : ( divide_divide_real @ ( cos_real @ X4 ) @ ( sin_real @ X4 ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8432_arccos__lt__bounded,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_real @ Y @ one_one_real )
% 9.04/5.09         => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
% 9.04/5.09            & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8433_arccos__bounded,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 9.04/5.09            & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8434_sin__arccos__nonzero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ one_one_real )
% 9.04/5.09         => ( ( sin_real @ ( arccos @ X ) )
% 9.04/5.09           != zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8435_arccos__cos2,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.04/5.09       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
% 9.04/5.09         => ( ( arccos @ ( cos_real @ X ) )
% 9.04/5.09            = ( uminus_uminus_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8436_arccos__minus,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.09         => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 9.04/5.09            = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8437_cos__arcsin__nonzero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ one_one_real )
% 9.04/5.09         => ( ( cos_real @ ( arcsin @ X ) )
% 9.04/5.09           != zero_zero_real ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8438_arccos,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 9.04/5.09            & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
% 9.04/5.09            & ( ( cos_real @ ( arccos @ Y ) )
% 9.04/5.09              = Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8439_arccos__minus__abs,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.09       => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 9.04/5.09          = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8440_arccos__le__pi2,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8441_arcsin__lt__bounded,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_real @ Y @ one_one_real )
% 9.04/5.09         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 9.04/5.09            & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8442_arcsin__bounded,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 9.04/5.09            & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8443_arcsin__ubound,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8444_arcsin__lbound,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8445_cot__gt__zero,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8446_arcsin__sin,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09         => ( ( arcsin @ ( sin_real @ X ) )
% 9.04/5.09            = X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8447_tan__cot_H,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 9.04/5.09        = ( cot_real @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8448_le__arcsin__iff,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 9.04/5.09           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09             => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
% 9.04/5.09                = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8449_arcsin__le__iff,axiom,
% 9.04/5.09      ! [X: real,Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.09       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 9.04/5.09           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09             => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
% 9.04/5.09                = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8450_arcsin__pi,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 9.04/5.09            & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
% 9.04/5.09            & ( ( sin_real @ ( arcsin @ Y ) )
% 9.04/5.09              = Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8451_arcsin,axiom,
% 9.04/5.09      ! [Y: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.04/5.09       => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 9.04/5.09            & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09            & ( ( sin_real @ ( arcsin @ Y ) )
% 9.04/5.09              = Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8452_arccos__cos__eq__abs__2pi,axiom,
% 9.04/5.09      ! [Theta: real] :
% 9.04/5.09        ~ ! [K3: int] :
% 9.04/5.09            ( ( arccos @ ( cos_real @ Theta ) )
% 9.04/5.09           != ( 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 ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8453_exp__two__pi__i_H,axiom,
% 9.04/5.09      ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 9.04/5.09      = one_one_complex ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8454_exp__two__pi__i,axiom,
% 9.04/5.09      ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 9.04/5.09      = one_one_complex ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8455_powr__real__of__int,axiom,
% 9.04/5.09      ! [X: real,N: int] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.09       => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 9.04/5.09           => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 9.04/5.09              = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
% 9.04/5.09          & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 9.04/5.09           => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 9.04/5.09              = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8456_cis__2pi,axiom,
% 9.04/5.09      ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.04/5.09      = one_one_complex ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8457_Suc__0__xor__eq,axiom,
% 9.04/5.09      ! [N: nat] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.04/5.09        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.09          @ ( zero_n2687167440665602831ol_nat
% 9.04/5.09            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8458_inverse__nonzero__iff__nonzero,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( ( inverse_inverse_real @ A )
% 9.04/5.09          = zero_zero_real )
% 9.04/5.09        = ( A = zero_zero_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8459_inverse__nonzero__iff__nonzero,axiom,
% 9.04/5.09      ! [A: complex] :
% 9.04/5.09        ( ( ( invers8013647133539491842omplex @ A )
% 9.04/5.09          = zero_zero_complex )
% 9.04/5.09        = ( A = zero_zero_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8460_inverse__nonzero__iff__nonzero,axiom,
% 9.04/5.09      ! [A: rat] :
% 9.04/5.09        ( ( ( inverse_inverse_rat @ A )
% 9.04/5.09          = zero_zero_rat )
% 9.04/5.09        = ( A = zero_zero_rat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8461_inverse__zero,axiom,
% 9.04/5.09      ( ( inverse_inverse_real @ zero_zero_real )
% 9.04/5.09      = zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8462_inverse__zero,axiom,
% 9.04/5.09      ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 9.04/5.09      = zero_zero_complex ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8463_inverse__zero,axiom,
% 9.04/5.09      ( ( inverse_inverse_rat @ zero_zero_rat )
% 9.04/5.09      = zero_zero_rat ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8464_inverse__1,axiom,
% 9.04/5.09      ( ( inverse_inverse_real @ one_one_real )
% 9.04/5.09      = one_one_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8465_inverse__1,axiom,
% 9.04/5.09      ( ( invers8013647133539491842omplex @ one_one_complex )
% 9.04/5.09      = one_one_complex ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8466_inverse__1,axiom,
% 9.04/5.09      ( ( inverse_inverse_rat @ one_one_rat )
% 9.04/5.09      = one_one_rat ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8467_inverse__eq__1__iff,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( ( inverse_inverse_real @ X )
% 9.04/5.09          = one_one_real )
% 9.04/5.09        = ( X = one_one_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8468_inverse__eq__1__iff,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( ( invers8013647133539491842omplex @ X )
% 9.04/5.09          = one_one_complex )
% 9.04/5.09        = ( X = one_one_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8469_inverse__eq__1__iff,axiom,
% 9.04/5.09      ! [X: rat] :
% 9.04/5.09        ( ( ( inverse_inverse_rat @ X )
% 9.04/5.09          = one_one_rat )
% 9.04/5.09        = ( X = one_one_rat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8470_inverse__divide,axiom,
% 9.04/5.09      ! [A: real,B: real] :
% 9.04/5.09        ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
% 9.04/5.09        = ( divide_divide_real @ B @ A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8471_inverse__divide,axiom,
% 9.04/5.09      ! [A: complex,B: complex] :
% 9.04/5.09        ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.04/5.09        = ( divide1717551699836669952omplex @ B @ A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8472_inverse__divide,axiom,
% 9.04/5.09      ! [A: rat,B: rat] :
% 9.04/5.09        ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
% 9.04/5.09        = ( divide_divide_rat @ B @ A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8473_bit_Oxor__self,axiom,
% 9.04/5.09      ! [X: int] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ X @ X )
% 9.04/5.09        = zero_zero_int ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8474_xor__self__eq,axiom,
% 9.04/5.09      ! [A: nat] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ A @ A )
% 9.04/5.09        = zero_zero_nat ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8475_xor__self__eq,axiom,
% 9.04/5.09      ! [A: int] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ A @ A )
% 9.04/5.09        = zero_zero_int ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8476_xor_Oleft__neutral,axiom,
% 9.04/5.09      ! [A: nat] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 9.04/5.09        = A ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8477_xor_Oleft__neutral,axiom,
% 9.04/5.09      ! [A: int] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 9.04/5.09        = A ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8478_xor_Oright__neutral,axiom,
% 9.04/5.09      ! [A: nat] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 9.04/5.09        = A ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8479_xor_Oright__neutral,axiom,
% 9.04/5.09      ! [A: int] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 9.04/5.09        = A ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8480_inverse__nonnegative__iff__nonnegative,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 9.04/5.09        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8481_inverse__nonnegative__iff__nonnegative,axiom,
% 9.04/5.09      ! [A: rat] :
% 9.04/5.09        ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 9.04/5.09        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8482_inverse__nonpositive__iff__nonpositive,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 9.04/5.09        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8483_inverse__nonpositive__iff__nonpositive,axiom,
% 9.04/5.09      ! [A: rat] :
% 9.04/5.09        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 9.04/5.09        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8484_inverse__less__iff__less,axiom,
% 9.04/5.09      ! [A: real,B: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.09       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.04/5.09         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.09            = ( ord_less_real @ B @ A ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8485_inverse__less__iff__less,axiom,
% 9.04/5.09      ! [A: rat,B: rat] :
% 9.04/5.09        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.04/5.09       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.04/5.09         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.09            = ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8486_inverse__less__iff__less__neg,axiom,
% 9.04/5.09      ! [A: real,B: real] :
% 9.04/5.09        ( ( ord_less_real @ A @ zero_zero_real )
% 9.04/5.09       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.04/5.09         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.09            = ( ord_less_real @ B @ A ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8487_inverse__less__iff__less__neg,axiom,
% 9.04/5.09      ! [A: rat,B: rat] :
% 9.04/5.09        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.04/5.09       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.04/5.09         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.09            = ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8488_inverse__negative__iff__negative,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 9.04/5.09        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8489_inverse__negative__iff__negative,axiom,
% 9.04/5.09      ! [A: rat] :
% 9.04/5.09        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 9.04/5.09        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8490_inverse__positive__iff__positive,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 9.04/5.09        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8491_inverse__positive__iff__positive,axiom,
% 9.04/5.09      ! [A: rat] :
% 9.04/5.09        ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 9.04/5.09        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8492_norm__ii,axiom,
% 9.04/5.09      ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 9.04/5.09      = one_one_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8493_norm__cis,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 9.04/5.09        = one_one_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8494_cis__zero,axiom,
% 9.04/5.09      ( ( cis @ zero_zero_real )
% 9.04/5.09      = one_one_complex ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8495_inverse__le__iff__le,axiom,
% 9.04/5.09      ! [A: real,B: real] :
% 9.04/5.09        ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.09       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.09            = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8496_inverse__le__iff__le,axiom,
% 9.04/5.09      ! [A: rat,B: rat] :
% 9.04/5.09        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.04/5.09       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.04/5.09         => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.09            = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8497_inverse__le__iff__le__neg,axiom,
% 9.04/5.09      ! [A: real,B: real] :
% 9.04/5.09        ( ( ord_less_real @ A @ zero_zero_real )
% 9.04/5.09       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.04/5.09         => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.09            = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8498_inverse__le__iff__le__neg,axiom,
% 9.04/5.09      ! [A: rat,B: rat] :
% 9.04/5.09        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.04/5.09       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.04/5.09         => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.09            = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8499_right__inverse,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( A != zero_zero_real )
% 9.04/5.09       => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
% 9.04/5.09          = one_one_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8500_right__inverse,axiom,
% 9.04/5.09      ! [A: complex] :
% 9.04/5.09        ( ( A != zero_zero_complex )
% 9.04/5.09       => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
% 9.04/5.09          = one_one_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8501_right__inverse,axiom,
% 9.04/5.09      ! [A: rat] :
% 9.04/5.09        ( ( A != zero_zero_rat )
% 9.04/5.09       => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
% 9.04/5.09          = one_one_rat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8502_left__inverse,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( A != zero_zero_real )
% 9.04/5.09       => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 9.04/5.09          = one_one_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8503_left__inverse,axiom,
% 9.04/5.09      ! [A: complex] :
% 9.04/5.09        ( ( A != zero_zero_complex )
% 9.04/5.09       => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 9.04/5.09          = one_one_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8504_left__inverse,axiom,
% 9.04/5.09      ! [A: rat] :
% 9.04/5.09        ( ( A != zero_zero_rat )
% 9.04/5.09       => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 9.04/5.09          = one_one_rat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8505_inverse__eq__divide__numeral,axiom,
% 9.04/5.09      ! [W: num] :
% 9.04/5.09        ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
% 9.04/5.09        = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8506_inverse__eq__divide__numeral,axiom,
% 9.04/5.09      ! [W: num] :
% 9.04/5.09        ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.04/5.09        = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8507_inverse__eq__divide__numeral,axiom,
% 9.04/5.09      ! [W: num] :
% 9.04/5.09        ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
% 9.04/5.09        = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8508_cis__pi,axiom,
% 9.04/5.09      ( ( cis @ pi )
% 9.04/5.09      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8509_divide__i,axiom,
% 9.04/5.09      ! [X: complex] :
% 9.04/5.09        ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
% 9.04/5.09        = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8510_i__squared,axiom,
% 9.04/5.09      ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 9.04/5.09      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8511_inverse__eq__divide__neg__numeral,axiom,
% 9.04/5.09      ! [W: num] :
% 9.04/5.09        ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.04/5.09        = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8512_inverse__eq__divide__neg__numeral,axiom,
% 9.04/5.09      ! [W: num] :
% 9.04/5.09        ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.04/5.09        = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8513_inverse__eq__divide__neg__numeral,axiom,
% 9.04/5.09      ! [W: num] :
% 9.04/5.09        ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.04/5.09        = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8514_xor__numerals_I3_J,axiom,
% 9.04/5.09      ! [X: num,Y: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.04/5.09        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8515_xor__numerals_I3_J,axiom,
% 9.04/5.09      ! [X: num,Y: num] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 9.04/5.09        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8516_xor__numerals_I1_J,axiom,
% 9.04/5.09      ! [Y: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.04/5.09        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8517_xor__numerals_I1_J,axiom,
% 9.04/5.09      ! [Y: num] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 9.04/5.09        = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8518_xor__numerals_I2_J,axiom,
% 9.04/5.09      ! [Y: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.04/5.09        = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8519_xor__numerals_I2_J,axiom,
% 9.04/5.09      ! [Y: num] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 9.04/5.09        = ( numeral_numeral_int @ ( bit0 @ Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8520_xor__numerals_I5_J,axiom,
% 9.04/5.09      ! [X: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 9.04/5.09        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8521_xor__numerals_I5_J,axiom,
% 9.04/5.09      ! [X: num] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 9.04/5.09        = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8522_xor__numerals_I8_J,axiom,
% 9.04/5.09      ! [X: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 9.04/5.09        = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8523_xor__numerals_I8_J,axiom,
% 9.04/5.09      ! [X: num] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 9.04/5.09        = ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8524_xor__numerals_I7_J,axiom,
% 9.04/5.09      ! [X: num,Y: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.04/5.09        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8525_xor__numerals_I7_J,axiom,
% 9.04/5.09      ! [X: num,Y: num] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 9.04/5.09        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8526_divide__numeral__i,axiom,
% 9.04/5.09      ! [Z: complex,N: num] :
% 9.04/5.09        ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 9.04/5.09        = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8527_xor__nat__numerals_I1_J,axiom,
% 9.04/5.09      ! [Y: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.04/5.09        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8528_xor__nat__numerals_I2_J,axiom,
% 9.04/5.09      ! [Y: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.04/5.09        = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8529_xor__nat__numerals_I3_J,axiom,
% 9.04/5.09      ! [X: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.04/5.09        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8530_xor__nat__numerals_I4_J,axiom,
% 9.04/5.09      ! [X: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.04/5.09        = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8531_power2__i,axiom,
% 9.04/5.09      ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.09      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8532_cis__pi__half,axiom,
% 9.04/5.09      ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.09      = imaginary_unit ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8533_i__even__power,axiom,
% 9.04/5.09      ! [N: nat] :
% 9.04/5.09        ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.09        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8534_xor__numerals_I4_J,axiom,
% 9.04/5.09      ! [X: num,Y: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.04/5.09        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8535_xor__numerals_I4_J,axiom,
% 9.04/5.09      ! [X: num,Y: num] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 9.04/5.09        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8536_xor__numerals_I6_J,axiom,
% 9.04/5.09      ! [X: num,Y: num] :
% 9.04/5.09        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.04/5.09        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8537_xor__numerals_I6_J,axiom,
% 9.04/5.09      ! [X: num,Y: num] :
% 9.04/5.09        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 9.04/5.09        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8538_exp__pi__i_H,axiom,
% 9.04/5.09      ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 9.04/5.09      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8539_exp__pi__i,axiom,
% 9.04/5.09      ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 9.04/5.09      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8540_cis__minus__pi__half,axiom,
% 9.04/5.09      ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.04/5.09      = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8541_field__class_Ofield__inverse__zero,axiom,
% 9.04/5.09      ( ( inverse_inverse_real @ zero_zero_real )
% 9.04/5.09      = zero_zero_real ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8542_field__class_Ofield__inverse__zero,axiom,
% 9.04/5.09      ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 9.04/5.09      = zero_zero_complex ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8543_field__class_Ofield__inverse__zero,axiom,
% 9.04/5.09      ( ( inverse_inverse_rat @ zero_zero_rat )
% 9.04/5.09      = zero_zero_rat ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8544_inverse__zero__imp__zero,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( ( inverse_inverse_real @ A )
% 9.04/5.09          = zero_zero_real )
% 9.04/5.09       => ( A = zero_zero_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8545_inverse__zero__imp__zero,axiom,
% 9.04/5.09      ! [A: complex] :
% 9.04/5.09        ( ( ( invers8013647133539491842omplex @ A )
% 9.04/5.09          = zero_zero_complex )
% 9.04/5.09       => ( A = zero_zero_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8546_inverse__zero__imp__zero,axiom,
% 9.04/5.09      ! [A: rat] :
% 9.04/5.09        ( ( ( inverse_inverse_rat @ A )
% 9.04/5.09          = zero_zero_rat )
% 9.04/5.09       => ( A = zero_zero_rat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8547_nonzero__inverse__eq__imp__eq,axiom,
% 9.04/5.09      ! [A: real,B: real] :
% 9.04/5.09        ( ( ( inverse_inverse_real @ A )
% 9.04/5.09          = ( inverse_inverse_real @ B ) )
% 9.04/5.09       => ( ( A != zero_zero_real )
% 9.04/5.09         => ( ( B != zero_zero_real )
% 9.04/5.09           => ( A = B ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8548_nonzero__inverse__eq__imp__eq,axiom,
% 9.04/5.09      ! [A: complex,B: complex] :
% 9.04/5.09        ( ( ( invers8013647133539491842omplex @ A )
% 9.04/5.09          = ( invers8013647133539491842omplex @ B ) )
% 9.04/5.09       => ( ( A != zero_zero_complex )
% 9.04/5.09         => ( ( B != zero_zero_complex )
% 9.04/5.09           => ( A = B ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8549_nonzero__inverse__eq__imp__eq,axiom,
% 9.04/5.09      ! [A: rat,B: rat] :
% 9.04/5.09        ( ( ( inverse_inverse_rat @ A )
% 9.04/5.09          = ( inverse_inverse_rat @ B ) )
% 9.04/5.09       => ( ( A != zero_zero_rat )
% 9.04/5.09         => ( ( B != zero_zero_rat )
% 9.04/5.09           => ( A = B ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8550_nonzero__inverse__inverse__eq,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( A != zero_zero_real )
% 9.04/5.09       => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 9.04/5.09          = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8551_nonzero__inverse__inverse__eq,axiom,
% 9.04/5.09      ! [A: complex] :
% 9.04/5.09        ( ( A != zero_zero_complex )
% 9.04/5.09       => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 9.04/5.09          = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8552_nonzero__inverse__inverse__eq,axiom,
% 9.04/5.09      ! [A: rat] :
% 9.04/5.09        ( ( A != zero_zero_rat )
% 9.04/5.09       => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 9.04/5.09          = A ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8553_nonzero__imp__inverse__nonzero,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( A != zero_zero_real )
% 9.04/5.09       => ( ( inverse_inverse_real @ A )
% 9.04/5.09         != zero_zero_real ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8554_nonzero__imp__inverse__nonzero,axiom,
% 9.04/5.09      ! [A: complex] :
% 9.04/5.09        ( ( A != zero_zero_complex )
% 9.04/5.09       => ( ( invers8013647133539491842omplex @ A )
% 9.04/5.09         != zero_zero_complex ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8555_nonzero__imp__inverse__nonzero,axiom,
% 9.04/5.09      ! [A: rat] :
% 9.04/5.09        ( ( A != zero_zero_rat )
% 9.04/5.09       => ( ( inverse_inverse_rat @ A )
% 9.04/5.09         != zero_zero_rat ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8556_nonzero__norm__inverse,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( A != zero_zero_real )
% 9.04/5.09       => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
% 9.04/5.09          = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8557_nonzero__norm__inverse,axiom,
% 9.04/5.09      ! [A: complex] :
% 9.04/5.09        ( ( A != zero_zero_complex )
% 9.04/5.09       => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
% 9.04/5.09          = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8558_nonzero__of__real__inverse,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( X != zero_zero_real )
% 9.04/5.09       => ( ( real_V1803761363581548252l_real @ ( inverse_inverse_real @ X ) )
% 9.04/5.09          = ( inverse_inverse_real @ ( real_V1803761363581548252l_real @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8559_nonzero__of__real__inverse,axiom,
% 9.04/5.09      ! [X: real] :
% 9.04/5.09        ( ( X != zero_zero_real )
% 9.04/5.09       => ( ( real_V4546457046886955230omplex @ ( inverse_inverse_real @ X ) )
% 9.04/5.09          = ( invers8013647133539491842omplex @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8560_power__inverse,axiom,
% 9.04/5.09      ! [A: real,N: nat] :
% 9.04/5.09        ( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
% 9.04/5.09        = ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8561_power__inverse,axiom,
% 9.04/5.09      ! [A: complex,N: nat] :
% 9.04/5.09        ( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N )
% 9.04/5.09        = ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8562_power__inverse,axiom,
% 9.04/5.09      ! [A: rat,N: nat] :
% 9.04/5.09        ( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N )
% 9.04/5.09        = ( inverse_inverse_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8563_of__int__xor__eq,axiom,
% 9.04/5.09      ! [K: int,L: int] :
% 9.04/5.09        ( ( ring_18347121197199848620nteger @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 9.04/5.09        = ( bit_se3222712562003087583nteger @ ( ring_18347121197199848620nteger @ K ) @ ( ring_18347121197199848620nteger @ L ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8564_of__int__xor__eq,axiom,
% 9.04/5.09      ! [K: int,L: int] :
% 9.04/5.09        ( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 9.04/5.09        = ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8565_of__nat__xor__eq,axiom,
% 9.04/5.09      ! [M: nat,N: nat] :
% 9.04/5.09        ( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 9.04/5.09        = ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8566_of__nat__xor__eq,axiom,
% 9.04/5.09      ! [M: nat,N: nat] :
% 9.04/5.09        ( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 9.04/5.09        = ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8567_complex__i__not__zero,axiom,
% 9.04/5.09      imaginary_unit != zero_zero_complex ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8568_complex__i__not__one,axiom,
% 9.04/5.09      imaginary_unit != one_one_complex ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8569_cis__neq__zero,axiom,
% 9.04/5.09      ! [A: real] :
% 9.04/5.09        ( ( cis @ A )
% 9.04/5.09       != zero_zero_complex ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8570_complex__i__not__numeral,axiom,
% 9.04/5.09      ! [W: num] :
% 9.04/5.09        ( imaginary_unit
% 9.04/5.09       != ( numera6690914467698888265omplex @ W ) ) ).
% 9.04/5.09  
% 9.04/5.09  thf(fact_8571_norm__inverse__le__norm,axiom,
% 9.04/5.09      ! [R2: real,X: real] :
% 9.04/5.09        ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X ) )
% 9.04/5.10       => ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.04/5.10         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8572_norm__inverse__le__norm,axiom,
% 9.04/5.10      ! [R2: real,X: complex] :
% 9.04/5.10        ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X ) )
% 9.04/5.10       => ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.04/5.10         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8573_inverse__less__imp__less,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.10       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.10         => ( ord_less_real @ B @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8574_inverse__less__imp__less,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.10       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.04/5.10         => ( ord_less_rat @ B @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8575_less__imp__inverse__less,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_real @ A @ B )
% 9.04/5.10       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.10         => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8576_less__imp__inverse__less,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_rat @ A @ B )
% 9.04/5.10       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.04/5.10         => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8577_inverse__less__imp__less__neg,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.10       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.04/5.10         => ( ord_less_real @ B @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8578_inverse__less__imp__less__neg,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.10       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.04/5.10         => ( ord_less_rat @ B @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8579_less__imp__inverse__less__neg,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_real @ A @ B )
% 9.04/5.10       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.04/5.10         => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8580_less__imp__inverse__less__neg,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_rat @ A @ B )
% 9.04/5.10       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.04/5.10         => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8581_inverse__negative__imp__negative,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 9.04/5.10       => ( ( A != zero_zero_real )
% 9.04/5.10         => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8582_inverse__negative__imp__negative,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 9.04/5.10       => ( ( A != zero_zero_rat )
% 9.04/5.10         => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8583_inverse__positive__imp__positive,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 9.04/5.10       => ( ( A != zero_zero_real )
% 9.04/5.10         => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8584_inverse__positive__imp__positive,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 9.04/5.10       => ( ( A != zero_zero_rat )
% 9.04/5.10         => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8585_negative__imp__inverse__negative,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( ord_less_real @ A @ zero_zero_real )
% 9.04/5.10       => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8586_negative__imp__inverse__negative,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.04/5.10       => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8587_positive__imp__inverse__positive,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8588_positive__imp__inverse__positive,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8589_nonzero__inverse__mult__distrib,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( A != zero_zero_real )
% 9.04/5.10       => ( ( B != zero_zero_real )
% 9.04/5.10         => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 9.04/5.10            = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8590_nonzero__inverse__mult__distrib,axiom,
% 9.04/5.10      ! [A: complex,B: complex] :
% 9.04/5.10        ( ( A != zero_zero_complex )
% 9.04/5.10       => ( ( B != zero_zero_complex )
% 9.04/5.10         => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 9.04/5.10            = ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8591_nonzero__inverse__mult__distrib,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( A != zero_zero_rat )
% 9.04/5.10       => ( ( B != zero_zero_rat )
% 9.04/5.10         => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 9.04/5.10            = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8592_nonzero__inverse__minus__eq,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( A != zero_zero_real )
% 9.04/5.10       => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 9.04/5.10          = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8593_nonzero__inverse__minus__eq,axiom,
% 9.04/5.10      ! [A: complex] :
% 9.04/5.10        ( ( A != zero_zero_complex )
% 9.04/5.10       => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 9.04/5.10          = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8594_nonzero__inverse__minus__eq,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( A != zero_zero_rat )
% 9.04/5.10       => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 9.04/5.10          = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8595_inverse__unique,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ( times_times_real @ A @ B )
% 9.04/5.10          = one_one_real )
% 9.04/5.10       => ( ( inverse_inverse_real @ A )
% 9.04/5.10          = B ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8596_inverse__unique,axiom,
% 9.04/5.10      ! [A: complex,B: complex] :
% 9.04/5.10        ( ( ( times_times_complex @ A @ B )
% 9.04/5.10          = one_one_complex )
% 9.04/5.10       => ( ( invers8013647133539491842omplex @ A )
% 9.04/5.10          = B ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8597_inverse__unique,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ( times_times_rat @ A @ B )
% 9.04/5.10          = one_one_rat )
% 9.04/5.10       => ( ( inverse_inverse_rat @ A )
% 9.04/5.10          = B ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8598_inverse__numeral__1,axiom,
% 9.04/5.10      ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
% 9.04/5.10      = ( numeral_numeral_real @ one ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8599_inverse__numeral__1,axiom,
% 9.04/5.10      ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
% 9.04/5.10      = ( numera6690914467698888265omplex @ one ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8600_inverse__numeral__1,axiom,
% 9.04/5.10      ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
% 9.04/5.10      = ( numeral_numeral_rat @ one ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8601_divide__inverse__commute,axiom,
% 9.04/5.10      ( divide_divide_real
% 9.04/5.10      = ( ^ [A3: real,B4: real] : ( times_times_real @ ( inverse_inverse_real @ B4 ) @ A3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8602_divide__inverse__commute,axiom,
% 9.04/5.10      ( divide1717551699836669952omplex
% 9.04/5.10      = ( ^ [A3: complex,B4: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B4 ) @ A3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8603_divide__inverse__commute,axiom,
% 9.04/5.10      ( divide_divide_rat
% 9.04/5.10      = ( ^ [A3: rat,B4: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B4 ) @ A3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8604_divide__inverse,axiom,
% 9.04/5.10      ( divide_divide_real
% 9.04/5.10      = ( ^ [A3: real,B4: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8605_divide__inverse,axiom,
% 9.04/5.10      ( divide1717551699836669952omplex
% 9.04/5.10      = ( ^ [A3: complex,B4: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8606_divide__inverse,axiom,
% 9.04/5.10      ( divide_divide_rat
% 9.04/5.10      = ( ^ [A3: rat,B4: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8607_field__class_Ofield__divide__inverse,axiom,
% 9.04/5.10      ( divide_divide_real
% 9.04/5.10      = ( ^ [A3: real,B4: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8608_field__class_Ofield__divide__inverse,axiom,
% 9.04/5.10      ( divide1717551699836669952omplex
% 9.04/5.10      = ( ^ [A3: complex,B4: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8609_field__class_Ofield__divide__inverse,axiom,
% 9.04/5.10      ( divide_divide_rat
% 9.04/5.10      = ( ^ [A3: rat,B4: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8610_inverse__eq__divide,axiom,
% 9.04/5.10      ( inverse_inverse_real
% 9.04/5.10      = ( divide_divide_real @ one_one_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8611_inverse__eq__divide,axiom,
% 9.04/5.10      ( invers8013647133539491842omplex
% 9.04/5.10      = ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8612_inverse__eq__divide,axiom,
% 9.04/5.10      ( inverse_inverse_rat
% 9.04/5.10      = ( divide_divide_rat @ one_one_rat ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8613_power__mult__inverse__distrib,axiom,
% 9.04/5.10      ! [X: real,M: nat] :
% 9.04/5.10        ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( inverse_inverse_real @ X ) )
% 9.04/5.10        = ( times_times_real @ ( inverse_inverse_real @ X ) @ ( power_power_real @ X @ M ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8614_power__mult__inverse__distrib,axiom,
% 9.04/5.10      ! [X: complex,M: nat] :
% 9.04/5.10        ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( invers8013647133539491842omplex @ X ) )
% 9.04/5.10        = ( times_times_complex @ ( invers8013647133539491842omplex @ X ) @ ( power_power_complex @ X @ M ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8615_power__mult__inverse__distrib,axiom,
% 9.04/5.10      ! [X: rat,M: nat] :
% 9.04/5.10        ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( inverse_inverse_rat @ X ) )
% 9.04/5.10        = ( times_times_rat @ ( inverse_inverse_rat @ X ) @ ( power_power_rat @ X @ M ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8616_power__mult__power__inverse__commute,axiom,
% 9.04/5.10      ! [X: real,M: nat,N: nat] :
% 9.04/5.10        ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) )
% 9.04/5.10        = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) @ ( power_power_real @ X @ M ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8617_power__mult__power__inverse__commute,axiom,
% 9.04/5.10      ! [X: complex,M: nat,N: nat] :
% 9.04/5.10        ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) )
% 9.04/5.10        = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) @ ( power_power_complex @ X @ M ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8618_power__mult__power__inverse__commute,axiom,
% 9.04/5.10      ! [X: rat,M: nat,N: nat] :
% 9.04/5.10        ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) )
% 9.04/5.10        = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) @ ( power_power_rat @ X @ M ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8619_mult__inverse__of__nat__commute,axiom,
% 9.04/5.10      ! [Xa: nat,X: real] :
% 9.04/5.10        ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X )
% 9.04/5.10        = ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8620_mult__inverse__of__nat__commute,axiom,
% 9.04/5.10      ! [Xa: nat,X: complex] :
% 9.04/5.10        ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X )
% 9.04/5.10        = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8621_mult__inverse__of__nat__commute,axiom,
% 9.04/5.10      ! [Xa: nat,X: rat] :
% 9.04/5.10        ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X )
% 9.04/5.10        = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8622_nonzero__abs__inverse,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( A != zero_zero_real )
% 9.04/5.10       => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 9.04/5.10          = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8623_nonzero__abs__inverse,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( A != zero_zero_rat )
% 9.04/5.10       => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 9.04/5.10          = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8624_mult__inverse__of__int__commute,axiom,
% 9.04/5.10      ! [Xa: int,X: real] :
% 9.04/5.10        ( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) @ X )
% 9.04/5.10        = ( times_times_real @ X @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8625_mult__inverse__of__int__commute,axiom,
% 9.04/5.10      ! [Xa: int,X: complex] :
% 9.04/5.10        ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) @ X )
% 9.04/5.10        = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8626_mult__inverse__of__int__commute,axiom,
% 9.04/5.10      ! [Xa: int,X: rat] :
% 9.04/5.10        ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X )
% 9.04/5.10        = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8627_exp__minus,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( exp_real @ ( uminus_uminus_real @ X ) )
% 9.04/5.10        = ( inverse_inverse_real @ ( exp_real @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8628_exp__minus,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( exp_complex @ ( uminus1482373934393186551omplex @ X ) )
% 9.04/5.10        = ( invers8013647133539491842omplex @ ( exp_complex @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8629_powr__minus,axiom,
% 9.04/5.10      ! [X: real,A: real] :
% 9.04/5.10        ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
% 9.04/5.10        = ( inverse_inverse_real @ ( powr_real @ X @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8630_divide__real__def,axiom,
% 9.04/5.10      ( divide_divide_real
% 9.04/5.10      = ( ^ [X4: real,Y6: real] : ( times_times_real @ X4 @ ( inverse_inverse_real @ Y6 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8631_cot__altdef,axiom,
% 9.04/5.10      ( cot_real
% 9.04/5.10      = ( ^ [X4: real] : ( inverse_inverse_real @ ( tan_real @ X4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8632_cot__altdef,axiom,
% 9.04/5.10      ( cot_complex
% 9.04/5.10      = ( ^ [X4: complex] : ( invers8013647133539491842omplex @ ( tan_complex @ X4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8633_tan__altdef,axiom,
% 9.04/5.10      ( tan_real
% 9.04/5.10      = ( ^ [X4: real] : ( inverse_inverse_real @ ( cot_real @ X4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8634_tan__altdef,axiom,
% 9.04/5.10      ( tan_complex
% 9.04/5.10      = ( ^ [X4: complex] : ( invers8013647133539491842omplex @ ( cot_complex @ X4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8635_DeMoivre,axiom,
% 9.04/5.10      ! [A: real,N: nat] :
% 9.04/5.10        ( ( power_power_complex @ ( cis @ A ) @ N )
% 9.04/5.10        = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8636_cis__divide,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 9.04/5.10        = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8637_inverse__le__imp__le,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.10       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.10         => ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8638_inverse__le__imp__le,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.10       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.04/5.10         => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8639_le__imp__inverse__le,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ A @ B )
% 9.04/5.10       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.10         => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8640_le__imp__inverse__le,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ A @ B )
% 9.04/5.10       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.04/5.10         => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8641_inverse__le__imp__le__neg,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.10       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.04/5.10         => ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8642_inverse__le__imp__le__neg,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.10       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.04/5.10         => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8643_le__imp__inverse__le__neg,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ A @ B )
% 9.04/5.10       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.04/5.10         => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8644_le__imp__inverse__le__neg,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ A @ B )
% 9.04/5.10       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.04/5.10         => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8645_inverse__le__1__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 9.04/5.10        = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.04/5.10          | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8646_inverse__le__1__iff,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 9.04/5.10        = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.04/5.10          | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8647_one__less__inverse__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 9.04/5.10        = ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.10          & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8648_one__less__inverse__iff,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 9.04/5.10        = ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.04/5.10          & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8649_one__less__inverse,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( ord_less_real @ A @ one_one_real )
% 9.04/5.10         => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8650_one__less__inverse,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ( ord_less_rat @ A @ one_one_rat )
% 9.04/5.10         => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8651_field__class_Ofield__inverse,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( A != zero_zero_real )
% 9.04/5.10       => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 9.04/5.10          = one_one_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8652_field__class_Ofield__inverse,axiom,
% 9.04/5.10      ! [A: complex] :
% 9.04/5.10        ( ( A != zero_zero_complex )
% 9.04/5.10       => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 9.04/5.10          = one_one_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8653_field__class_Ofield__inverse,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( A != zero_zero_rat )
% 9.04/5.10       => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 9.04/5.10          = one_one_rat ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8654_division__ring__inverse__add,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( A != zero_zero_real )
% 9.04/5.10       => ( ( B != zero_zero_real )
% 9.04/5.10         => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.10            = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8655_division__ring__inverse__add,axiom,
% 9.04/5.10      ! [A: complex,B: complex] :
% 9.04/5.10        ( ( A != zero_zero_complex )
% 9.04/5.10       => ( ( B != zero_zero_complex )
% 9.04/5.10         => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 9.04/5.10            = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8656_division__ring__inverse__add,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( A != zero_zero_rat )
% 9.04/5.10       => ( ( B != zero_zero_rat )
% 9.04/5.10         => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.10            = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8657_inverse__add,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( A != zero_zero_real )
% 9.04/5.10       => ( ( B != zero_zero_real )
% 9.04/5.10         => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.10            = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8658_inverse__add,axiom,
% 9.04/5.10      ! [A: complex,B: complex] :
% 9.04/5.10        ( ( A != zero_zero_complex )
% 9.04/5.10       => ( ( B != zero_zero_complex )
% 9.04/5.10         => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 9.04/5.10            = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8659_inverse__add,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( A != zero_zero_rat )
% 9.04/5.10       => ( ( B != zero_zero_rat )
% 9.04/5.10         => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.10            = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8660_division__ring__inverse__diff,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( A != zero_zero_real )
% 9.04/5.10       => ( ( B != zero_zero_real )
% 9.04/5.10         => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.10            = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8661_division__ring__inverse__diff,axiom,
% 9.04/5.10      ! [A: complex,B: complex] :
% 9.04/5.10        ( ( A != zero_zero_complex )
% 9.04/5.10       => ( ( B != zero_zero_complex )
% 9.04/5.10         => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 9.04/5.10            = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8662_division__ring__inverse__diff,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( A != zero_zero_rat )
% 9.04/5.10       => ( ( B != zero_zero_rat )
% 9.04/5.10         => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.10            = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8663_nonzero__inverse__eq__divide,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( A != zero_zero_real )
% 9.04/5.10       => ( ( inverse_inverse_real @ A )
% 9.04/5.10          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8664_nonzero__inverse__eq__divide,axiom,
% 9.04/5.10      ! [A: complex] :
% 9.04/5.10        ( ( A != zero_zero_complex )
% 9.04/5.10       => ( ( invers8013647133539491842omplex @ A )
% 9.04/5.10          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8665_nonzero__inverse__eq__divide,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( A != zero_zero_rat )
% 9.04/5.10       => ( ( inverse_inverse_rat @ A )
% 9.04/5.10          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8666_complex__i__not__neg__numeral,axiom,
% 9.04/5.10      ! [W: num] :
% 9.04/5.10        ( imaginary_unit
% 9.04/5.10       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8667_inverse__powr,axiom,
% 9.04/5.10      ! [Y: real,A: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.10       => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
% 9.04/5.10          = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8668_Complex__eq__i,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ( complex2 @ X @ Y )
% 9.04/5.10          = imaginary_unit )
% 9.04/5.10        = ( ( X = zero_zero_real )
% 9.04/5.10          & ( Y = one_one_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8669_imaginary__unit_Ocode,axiom,
% 9.04/5.10      ( imaginary_unit
% 9.04/5.10      = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8670_inverse__le__iff,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.10        = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.04/5.10           => ( ord_less_eq_real @ B @ A ) )
% 9.04/5.10          & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 9.04/5.10           => ( ord_less_eq_real @ A @ B ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8671_inverse__le__iff,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.10        = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.04/5.10           => ( ord_less_eq_rat @ B @ A ) )
% 9.04/5.10          & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 9.04/5.10           => ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8672_inverse__less__iff,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.10        = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.04/5.10           => ( ord_less_real @ B @ A ) )
% 9.04/5.10          & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 9.04/5.10           => ( ord_less_real @ A @ B ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8673_inverse__less__iff,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.10        = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.04/5.10           => ( ord_less_rat @ B @ A ) )
% 9.04/5.10          & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 9.04/5.10           => ( ord_less_rat @ A @ B ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8674_one__le__inverse__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 9.04/5.10        = ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.10          & ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8675_one__le__inverse__iff,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 9.04/5.10        = ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.04/5.10          & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8676_inverse__less__1__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 9.04/5.10        = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.04/5.10          | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8677_inverse__less__1__iff,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 9.04/5.10        = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.04/5.10          | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8678_one__le__inverse,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.04/5.10         => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8679_one__le__inverse,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.04/5.10         => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8680_inverse__diff__inverse,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( A != zero_zero_real )
% 9.04/5.10       => ( ( B != zero_zero_real )
% 9.04/5.10         => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.04/5.10            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8681_inverse__diff__inverse,axiom,
% 9.04/5.10      ! [A: complex,B: complex] :
% 9.04/5.10        ( ( A != zero_zero_complex )
% 9.04/5.10       => ( ( B != zero_zero_complex )
% 9.04/5.10         => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 9.04/5.10            = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8682_inverse__diff__inverse,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( A != zero_zero_rat )
% 9.04/5.10       => ( ( B != zero_zero_rat )
% 9.04/5.10         => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.04/5.10            = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8683_reals__Archimedean,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.10       => ? [N2: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8684_reals__Archimedean,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.04/5.10       => ? [N2: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8685_i__complex__of__real,axiom,
% 9.04/5.10      ! [R2: real] :
% 9.04/5.10        ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
% 9.04/5.10        = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8686_complex__of__real__i,axiom,
% 9.04/5.10      ! [R2: real] :
% 9.04/5.10        ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
% 9.04/5.10        = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8687_forall__pos__mono__1,axiom,
% 9.04/5.10      ! [P: real > $o,E2: real] :
% 9.04/5.10        ( ! [D5: real,E: real] :
% 9.04/5.10            ( ( ord_less_real @ D5 @ E )
% 9.04/5.10           => ( ( P @ D5 )
% 9.04/5.10             => ( P @ E ) ) )
% 9.04/5.10       => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 9.04/5.10         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.04/5.10           => ( P @ E2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8688_forall__pos__mono,axiom,
% 9.04/5.10      ! [P: real > $o,E2: real] :
% 9.04/5.10        ( ! [D5: real,E: real] :
% 9.04/5.10            ( ( ord_less_real @ D5 @ E )
% 9.04/5.10           => ( ( P @ D5 )
% 9.04/5.10             => ( P @ E ) ) )
% 9.04/5.10       => ( ! [N2: nat] :
% 9.04/5.10              ( ( N2 != zero_zero_nat )
% 9.04/5.10             => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
% 9.04/5.10         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.04/5.10           => ( P @ E2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8689_real__arch__inverse,axiom,
% 9.04/5.10      ! [E2: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.04/5.10        = ( ? [N3: nat] :
% 9.04/5.10              ( ( N3 != zero_zero_nat )
% 9.04/5.10              & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 9.04/5.10              & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8690_even__xor__iff,axiom,
% 9.04/5.10      ! [A: code_integer,B: code_integer] :
% 9.04/5.10        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B ) )
% 9.04/5.10        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.04/5.10          = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8691_even__xor__iff,axiom,
% 9.04/5.10      ! [A: nat,B: nat] :
% 9.04/5.10        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 9.04/5.10        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.04/5.10          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8692_even__xor__iff,axiom,
% 9.04/5.10      ! [A: int,B: int] :
% 9.04/5.10        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 9.04/5.10        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.04/5.10          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8693_sqrt__divide__self__eq,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
% 9.04/5.10          = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8694_ln__inverse,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
% 9.04/5.10          = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8695_ex__inverse__of__nat__less,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.10       => ? [N2: nat] :
% 9.04/5.10            ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.04/5.10            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8696_ex__inverse__of__nat__less,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.04/5.10       => ? [N2: nat] :
% 9.04/5.10            ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.04/5.10            & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8697_power__diff__conv__inverse,axiom,
% 9.04/5.10      ! [X: real,M: nat,N: nat] :
% 9.04/5.10        ( ( X != zero_zero_real )
% 9.04/5.10       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.10         => ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M ) )
% 9.04/5.10            = ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8698_power__diff__conv__inverse,axiom,
% 9.04/5.10      ! [X: complex,M: nat,N: nat] :
% 9.04/5.10        ( ( X != zero_zero_complex )
% 9.04/5.10       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.10         => ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M ) )
% 9.04/5.10            = ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8699_power__diff__conv__inverse,axiom,
% 9.04/5.10      ! [X: rat,M: nat,N: nat] :
% 9.04/5.10        ( ( X != zero_zero_rat )
% 9.04/5.10       => ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.10         => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M ) )
% 9.04/5.10            = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8700_log__inverse,axiom,
% 9.04/5.10      ! [A: real,X: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( A != one_one_real )
% 9.04/5.10         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.10           => ( ( log @ A @ ( inverse_inverse_real @ X ) )
% 9.04/5.10              = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8701_exp__plus__inverse__exp,axiom,
% 9.04/5.10      ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8702_plus__inverse__ge__2,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8703_real__inv__sqrt__pow2,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.10          = ( inverse_inverse_real @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8704_tan__cot,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 9.04/5.10        = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8705_cmod__unit__one,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 9.04/5.10        = one_one_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8706_xor__nat__unfold,axiom,
% 9.04/5.10      ( bit_se6528837805403552850or_nat
% 9.04/5.10      = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8707_real__le__x__sinh,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8708_xor__nat__rec,axiom,
% 9.04/5.10      ( bit_se6528837805403552850or_nat
% 9.04/5.10      = ( ^ [M6: nat,N3: nat] :
% 9.04/5.10            ( plus_plus_nat
% 9.04/5.10            @ ( zero_n2687167440665602831ol_nat
% 9.04/5.10              @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
% 9.04/5.10               != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 9.04/5.10            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8709_xor__one__eq,axiom,
% 9.04/5.10      ! [A: code_integer] :
% 9.04/5.10        ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
% 9.04/5.10        = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 9.04/5.10          @ ( zero_n356916108424825756nteger
% 9.04/5.10            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8710_xor__one__eq,axiom,
% 9.04/5.10      ! [A: nat] :
% 9.04/5.10        ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
% 9.04/5.10        = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 9.04/5.10          @ ( zero_n2687167440665602831ol_nat
% 9.04/5.10            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8711_xor__one__eq,axiom,
% 9.04/5.10      ! [A: int] :
% 9.04/5.10        ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
% 9.04/5.10        = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 9.04/5.10          @ ( zero_n2684676970156552555ol_int
% 9.04/5.10            @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8712_one__xor__eq,axiom,
% 9.04/5.10      ! [A: code_integer] :
% 9.04/5.10        ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
% 9.04/5.10        = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 9.04/5.10          @ ( zero_n356916108424825756nteger
% 9.04/5.10            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8713_one__xor__eq,axiom,
% 9.04/5.10      ! [A: nat] :
% 9.04/5.10        ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
% 9.04/5.10        = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 9.04/5.10          @ ( zero_n2687167440665602831ol_nat
% 9.04/5.10            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8714_one__xor__eq,axiom,
% 9.04/5.10      ! [A: int] :
% 9.04/5.10        ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
% 9.04/5.10        = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 9.04/5.10          @ ( zero_n2684676970156552555ol_int
% 9.04/5.10            @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8715_real__le__abs__sinh,axiom,
% 9.04/5.10      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8716_tan__sec,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ( cos_real @ X )
% 9.04/5.10         != zero_zero_real )
% 9.04/5.10       => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.10          = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8717_tan__sec,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( ( cos_complex @ X )
% 9.04/5.10         != zero_zero_complex )
% 9.04/5.10       => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.10          = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8718_xor__Suc__0__eq,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.04/5.10        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.10          @ ( zero_n2687167440665602831ol_nat
% 9.04/5.10            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8719_csqrt__ii,axiom,
% 9.04/5.10      ( ( csqrt @ imaginary_unit )
% 9.04/5.10      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8720_Arg__minus__ii,axiom,
% 9.04/5.10      ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 9.04/5.10      = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8721_Arg__ii,axiom,
% 9.04/5.10      ( ( arg @ imaginary_unit )
% 9.04/5.10      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8722_Arg__correct,axiom,
% 9.04/5.10      ! [Z: complex] :
% 9.04/5.10        ( ( Z != zero_zero_complex )
% 9.04/5.10       => ( ( ( sgn_sgn_complex @ Z )
% 9.04/5.10            = ( cis @ ( arg @ Z ) ) )
% 9.04/5.10          & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 9.04/5.10          & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8723_cis__Arg__unique,axiom,
% 9.04/5.10      ! [Z: complex,X: real] :
% 9.04/5.10        ( ( ( sgn_sgn_complex @ Z )
% 9.04/5.10          = ( cis @ X ) )
% 9.04/5.10       => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 9.04/5.10         => ( ( ord_less_eq_real @ X @ pi )
% 9.04/5.10           => ( ( arg @ Z )
% 9.04/5.10              = X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8724_csqrt__eq__0,axiom,
% 9.04/5.10      ! [Z: complex] :
% 9.04/5.10        ( ( ( csqrt @ Z )
% 9.04/5.10          = zero_zero_complex )
% 9.04/5.10        = ( Z = zero_zero_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8725_csqrt__0,axiom,
% 9.04/5.10      ( ( csqrt @ zero_zero_complex )
% 9.04/5.10      = zero_zero_complex ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8726_csqrt__eq__1,axiom,
% 9.04/5.10      ! [Z: complex] :
% 9.04/5.10        ( ( ( csqrt @ Z )
% 9.04/5.10          = one_one_complex )
% 9.04/5.10        = ( Z = one_one_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8727_csqrt__1,axiom,
% 9.04/5.10      ( ( csqrt @ one_one_complex )
% 9.04/5.10      = one_one_complex ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8728_xor__nonnegative__int__iff,axiom,
% 9.04/5.10      ! [K: int,L: int] :
% 9.04/5.10        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 9.04/5.10        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.10          = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8729_xor__negative__int__iff,axiom,
% 9.04/5.10      ! [K: int,L: int] :
% 9.04/5.10        ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 9.04/5.10        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.04/5.10         != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8730_power2__csqrt,axiom,
% 9.04/5.10      ! [Z: complex] :
% 9.04/5.10        ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.10        = Z ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8731_XOR__lower,axiom,
% 9.04/5.10      ! [X: int,Y: int] :
% 9.04/5.10        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.10       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.10         => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8732_divide__complex__def,axiom,
% 9.04/5.10      ( divide1717551699836669952omplex
% 9.04/5.10      = ( ^ [X4: complex,Y6: complex] : ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ Y6 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8733_xor__nat__def,axiom,
% 9.04/5.10      ( bit_se6528837805403552850or_nat
% 9.04/5.10      = ( ^ [M6: nat,N3: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8734_Arg__zero,axiom,
% 9.04/5.10      ( ( arg @ zero_zero_complex )
% 9.04/5.10      = zero_zero_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8735_of__real__sqrt,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
% 9.04/5.10          = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8736_cis__Arg,axiom,
% 9.04/5.10      ! [Z: complex] :
% 9.04/5.10        ( ( Z != zero_zero_complex )
% 9.04/5.10       => ( ( cis @ ( arg @ Z ) )
% 9.04/5.10          = ( sgn_sgn_complex @ Z ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8737_Arg__bounded,axiom,
% 9.04/5.10      ! [Z: complex] :
% 9.04/5.10        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 9.04/5.10        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8738_XOR__upper,axiom,
% 9.04/5.10      ! [X: int,N: nat,Y: int] :
% 9.04/5.10        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.10       => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.10         => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.10           => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8739_complex__inverse,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 9.04/5.10        = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8740_xor__int__rec,axiom,
% 9.04/5.10      ( bit_se6526347334894502574or_int
% 9.04/5.10      = ( ^ [K2: int,L3: int] :
% 9.04/5.10            ( plus_plus_int
% 9.04/5.10            @ ( zero_n2684676970156552555ol_int
% 9.04/5.10              @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
% 9.04/5.10               != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
% 9.04/5.10            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8741_xor__int__unfold,axiom,
% 9.04/5.10      ( bit_se6526347334894502574or_int
% 9.04/5.10      = ( ^ [K2: int,L3: int] :
% 9.04/5.10            ( if_int
% 9.04/5.10            @ ( K2
% 9.04/5.10              = ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.10            @ ( bit_ri7919022796975470100ot_int @ L3 )
% 9.04/5.10            @ ( if_int
% 9.04/5.10              @ ( L3
% 9.04/5.10                = ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.10              @ ( bit_ri7919022796975470100ot_int @ K2 )
% 9.04/5.10              @ ( if_int @ ( K2 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = 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 @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8742_sinh__ln__real,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 9.04/5.10          = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8743_cis__multiple__2pi,axiom,
% 9.04/5.10      ! [N: real] :
% 9.04/5.10        ( ( member_real @ N @ ring_1_Ints_real )
% 9.04/5.10       => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 9.04/5.10          = one_one_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8744_cosh__ln__real,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 9.04/5.10          = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8745_sinh__real__eq__iff,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ( sinh_real @ X )
% 9.04/5.10          = ( sinh_real @ Y ) )
% 9.04/5.10        = ( X = Y ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8746_sinh__0,axiom,
% 9.04/5.10      ( ( sinh_complex @ zero_zero_complex )
% 9.04/5.10      = zero_zero_complex ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8747_sinh__0,axiom,
% 9.04/5.10      ( ( sinh_real @ zero_zero_real )
% 9.04/5.10      = zero_zero_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8748_cosh__minus,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( cosh_real @ ( uminus_uminus_real @ X ) )
% 9.04/5.10        = ( cosh_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8749_cosh__minus,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( cosh_complex @ ( uminus1482373934393186551omplex @ X ) )
% 9.04/5.10        = ( cosh_complex @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8750_sinh__minus,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( sinh_real @ ( uminus_uminus_real @ X ) )
% 9.04/5.10        = ( uminus_uminus_real @ ( sinh_real @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8751_sinh__minus,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( sinh_complex @ ( uminus1482373934393186551omplex @ X ) )
% 9.04/5.10        = ( uminus1482373934393186551omplex @ ( sinh_complex @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8752_sinh__real__zero__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ( sinh_real @ X )
% 9.04/5.10          = zero_zero_real )
% 9.04/5.10        = ( X = zero_zero_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8753_sinh__real__less__iff,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
% 9.04/5.10        = ( ord_less_real @ X @ Y ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8754_sinh__real__le__iff,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
% 9.04/5.10        = ( ord_less_eq_real @ X @ Y ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8755_cosh__real__eq__iff,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ( cosh_real @ X )
% 9.04/5.10          = ( cosh_real @ Y ) )
% 9.04/5.10        = ( ( abs_abs_real @ X )
% 9.04/5.10          = ( abs_abs_real @ Y ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8756_cosh__real__abs,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( cosh_real @ ( abs_abs_real @ X ) )
% 9.04/5.10        = ( cosh_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8757_sinh__real__abs,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( sinh_real @ ( abs_abs_real @ X ) )
% 9.04/5.10        = ( abs_abs_real @ ( sinh_real @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8758_frac__in__Ints__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( member_real @ ( archim2898591450579166408c_real @ X ) @ ring_1_Ints_real )
% 9.04/5.10        = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8759_cosh__0,axiom,
% 9.04/5.10      ( ( cosh_complex @ zero_zero_complex )
% 9.04/5.10      = one_one_complex ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8760_cosh__0,axiom,
% 9.04/5.10      ( ( cosh_real @ zero_zero_real )
% 9.04/5.10      = one_one_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8761_bit_Oconj__cancel__right,axiom,
% 9.04/5.10      ! [X: int] :
% 9.04/5.10        ( ( bit_se725231765392027082nd_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 9.04/5.10        = zero_zero_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8762_bit_Oconj__cancel__left,axiom,
% 9.04/5.10      ! [X: int] :
% 9.04/5.10        ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 9.04/5.10        = zero_zero_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8763_sinh__real__pos__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
% 9.04/5.10        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8764_sinh__real__neg__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
% 9.04/5.10        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8765_sinh__real__nonpos__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
% 9.04/5.10        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8766_sinh__real__nonneg__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
% 9.04/5.10        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8767_frac__eq__0__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ( archim2898591450579166408c_real @ X )
% 9.04/5.10          = zero_zero_real )
% 9.04/5.10        = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8768_frac__eq__0__iff,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( ( archimedean_frac_rat @ X )
% 9.04/5.10          = zero_zero_rat )
% 9.04/5.10        = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8769_bit_Ocompl__zero,axiom,
% 9.04/5.10      ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
% 9.04/5.10      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8770_bit_Ocompl__zero,axiom,
% 9.04/5.10      ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
% 9.04/5.10      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8771_bit_Ocompl__one,axiom,
% 9.04/5.10      ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.10      = zero_z3403309356797280102nteger ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8772_bit_Ocompl__one,axiom,
% 9.04/5.10      ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.10      = zero_zero_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8773_bit_Oxor__one__left,axiom,
% 9.04/5.10      ! [X: code_integer] :
% 9.04/5.10        ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 9.04/5.10        = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8774_bit_Oxor__one__left,axiom,
% 9.04/5.10      ! [X: int] :
% 9.04/5.10        ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 9.04/5.10        = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8775_bit_Oxor__one__right,axiom,
% 9.04/5.10      ! [X: code_integer] :
% 9.04/5.10        ( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.10        = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8776_bit_Oxor__one__right,axiom,
% 9.04/5.10      ! [X: int] :
% 9.04/5.10        ( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.10        = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8777_bit_Oxor__cancel__left,axiom,
% 9.04/5.10      ! [X: code_integer] :
% 9.04/5.10        ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
% 9.04/5.10        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8778_bit_Oxor__cancel__left,axiom,
% 9.04/5.10      ! [X: int] :
% 9.04/5.10        ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 9.04/5.10        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8779_bit_Oxor__cancel__right,axiom,
% 9.04/5.10      ! [X: code_integer] :
% 9.04/5.10        ( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
% 9.04/5.10        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8780_bit_Oxor__cancel__right,axiom,
% 9.04/5.10      ! [X: int] :
% 9.04/5.10        ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 9.04/5.10        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8781_not__nonnegative__int__iff,axiom,
% 9.04/5.10      ! [K: int] :
% 9.04/5.10        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 9.04/5.10        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8782_not__negative__int__iff,axiom,
% 9.04/5.10      ! [K: int] :
% 9.04/5.10        ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 9.04/5.10        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8783_floor__add2,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ( member_real @ X @ ring_1_Ints_real )
% 9.04/5.10          | ( member_real @ Y @ ring_1_Ints_real ) )
% 9.04/5.10       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.10          = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8784_floor__add2,axiom,
% 9.04/5.10      ! [X: rat,Y: rat] :
% 9.04/5.10        ( ( ( member_rat @ X @ ring_1_Ints_rat )
% 9.04/5.10          | ( member_rat @ Y @ ring_1_Ints_rat ) )
% 9.04/5.10       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 9.04/5.10          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8785_minus__not__numeral__eq,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.04/5.10        = ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8786_minus__not__numeral__eq,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.10        = ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8787_frac__gt__0__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
% 9.04/5.10        = ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8788_frac__gt__0__iff,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
% 9.04/5.10        = ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8789_even__not__iff,axiom,
% 9.04/5.10      ! [A: code_integer] :
% 9.04/5.10        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
% 9.04/5.10        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8790_even__not__iff,axiom,
% 9.04/5.10      ! [A: int] :
% 9.04/5.10        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
% 9.04/5.10        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8791_not__one__eq,axiom,
% 9.04/5.10      ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
% 9.04/5.10      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8792_not__one__eq,axiom,
% 9.04/5.10      ( ( bit_ri7919022796975470100ot_int @ one_one_int )
% 9.04/5.10      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8793_Ints__power,axiom,
% 9.04/5.10      ! [A: real,N: nat] :
% 9.04/5.10        ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10       => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8794_Ints__power,axiom,
% 9.04/5.10      ! [A: int,N: nat] :
% 9.04/5.10        ( ( member_int @ A @ ring_1_Ints_int )
% 9.04/5.10       => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8795_Ints__power,axiom,
% 9.04/5.10      ! [A: complex,N: nat] :
% 9.04/5.10        ( ( member_complex @ A @ ring_1_Ints_complex )
% 9.04/5.10       => ( member_complex @ ( power_power_complex @ A @ N ) @ ring_1_Ints_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8796_Ints__1,axiom,
% 9.04/5.10      member_rat @ one_one_rat @ ring_1_Ints_rat ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8797_Ints__1,axiom,
% 9.04/5.10      member_int @ one_one_int @ ring_1_Ints_int ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8798_Ints__1,axiom,
% 9.04/5.10      member_real @ one_one_real @ ring_1_Ints_real ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8799_Ints__1,axiom,
% 9.04/5.10      member_complex @ one_one_complex @ ring_1_Ints_complex ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8800_minus__in__Ints__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
% 9.04/5.10        = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8801_minus__in__Ints__iff,axiom,
% 9.04/5.10      ! [X: int] :
% 9.04/5.10        ( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
% 9.04/5.10        = ( member_int @ X @ ring_1_Ints_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8802_minus__in__Ints__iff,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( member_complex @ ( uminus1482373934393186551omplex @ X ) @ ring_1_Ints_complex )
% 9.04/5.10        = ( member_complex @ X @ ring_1_Ints_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8803_minus__in__Ints__iff,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( member_rat @ ( uminus_uminus_rat @ X ) @ ring_1_Ints_rat )
% 9.04/5.10        = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8804_minus__in__Ints__iff,axiom,
% 9.04/5.10      ! [X: code_integer] :
% 9.04/5.10        ( ( member_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ring_11222124179247155820nteger )
% 9.04/5.10        = ( member_Code_integer @ X @ ring_11222124179247155820nteger ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8805_Ints__minus,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10       => ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8806_Ints__minus,axiom,
% 9.04/5.10      ! [A: int] :
% 9.04/5.10        ( ( member_int @ A @ ring_1_Ints_int )
% 9.04/5.10       => ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8807_Ints__minus,axiom,
% 9.04/5.10      ! [A: complex] :
% 9.04/5.10        ( ( member_complex @ A @ ring_1_Ints_complex )
% 9.04/5.10       => ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8808_Ints__minus,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10       => ( member_rat @ ( uminus_uminus_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8809_Ints__minus,axiom,
% 9.04/5.10      ! [A: code_integer] :
% 9.04/5.10        ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 9.04/5.10       => ( member_Code_integer @ ( uminus1351360451143612070nteger @ A ) @ ring_11222124179247155820nteger ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8810_cosh__real__nonzero,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( cosh_real @ X )
% 9.04/5.10       != zero_zero_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8811_Ints__numeral,axiom,
% 9.04/5.10      ! [N: num] : ( member_complex @ ( numera6690914467698888265omplex @ N ) @ ring_1_Ints_complex ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8812_Ints__numeral,axiom,
% 9.04/5.10      ! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8813_Ints__numeral,axiom,
% 9.04/5.10      ! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8814_Ints__numeral,axiom,
% 9.04/5.10      ! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8815_Ints__0,axiom,
% 9.04/5.10      member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8816_Ints__0,axiom,
% 9.04/5.10      member_real @ zero_zero_real @ ring_1_Ints_real ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8817_Ints__0,axiom,
% 9.04/5.10      member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8818_Ints__0,axiom,
% 9.04/5.10      member_int @ zero_zero_int @ ring_1_Ints_int ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8819_Ints__cases,axiom,
% 9.04/5.10      ! [Q3: int] :
% 9.04/5.10        ( ( member_int @ Q3 @ ring_1_Ints_int )
% 9.04/5.10       => ~ ! [Z3: int] :
% 9.04/5.10              ( Q3
% 9.04/5.10             != ( ring_1_of_int_int @ Z3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8820_Ints__cases,axiom,
% 9.04/5.10      ! [Q3: real] :
% 9.04/5.10        ( ( member_real @ Q3 @ ring_1_Ints_real )
% 9.04/5.10       => ~ ! [Z3: int] :
% 9.04/5.10              ( Q3
% 9.04/5.10             != ( ring_1_of_int_real @ Z3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8821_Ints__cases,axiom,
% 9.04/5.10      ! [Q3: rat] :
% 9.04/5.10        ( ( member_rat @ Q3 @ ring_1_Ints_rat )
% 9.04/5.10       => ~ ! [Z3: int] :
% 9.04/5.10              ( Q3
% 9.04/5.10             != ( ring_1_of_int_rat @ Z3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8822_Ints__cases,axiom,
% 9.04/5.10      ! [Q3: code_integer] :
% 9.04/5.10        ( ( member_Code_integer @ Q3 @ ring_11222124179247155820nteger )
% 9.04/5.10       => ~ ! [Z3: int] :
% 9.04/5.10              ( Q3
% 9.04/5.10             != ( ring_18347121197199848620nteger @ Z3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8823_Ints__cases,axiom,
% 9.04/5.10      ! [Q3: complex] :
% 9.04/5.10        ( ( member_complex @ Q3 @ ring_1_Ints_complex )
% 9.04/5.10       => ~ ! [Z3: int] :
% 9.04/5.10              ( Q3
% 9.04/5.10             != ( ring_17405671764205052669omplex @ Z3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8824_Ints__induct,axiom,
% 9.04/5.10      ! [Q3: int,P: int > $o] :
% 9.04/5.10        ( ( member_int @ Q3 @ ring_1_Ints_int )
% 9.04/5.10       => ( ! [Z3: int] : ( P @ ( ring_1_of_int_int @ Z3 ) )
% 9.04/5.10         => ( P @ Q3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8825_Ints__induct,axiom,
% 9.04/5.10      ! [Q3: real,P: real > $o] :
% 9.04/5.10        ( ( member_real @ Q3 @ ring_1_Ints_real )
% 9.04/5.10       => ( ! [Z3: int] : ( P @ ( ring_1_of_int_real @ Z3 ) )
% 9.04/5.10         => ( P @ Q3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8826_Ints__induct,axiom,
% 9.04/5.10      ! [Q3: rat,P: rat > $o] :
% 9.04/5.10        ( ( member_rat @ Q3 @ ring_1_Ints_rat )
% 9.04/5.10       => ( ! [Z3: int] : ( P @ ( ring_1_of_int_rat @ Z3 ) )
% 9.04/5.10         => ( P @ Q3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8827_Ints__induct,axiom,
% 9.04/5.10      ! [Q3: code_integer,P: code_integer > $o] :
% 9.04/5.10        ( ( member_Code_integer @ Q3 @ ring_11222124179247155820nteger )
% 9.04/5.10       => ( ! [Z3: int] : ( P @ ( ring_18347121197199848620nteger @ Z3 ) )
% 9.04/5.10         => ( P @ Q3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8828_Ints__induct,axiom,
% 9.04/5.10      ! [Q3: complex,P: complex > $o] :
% 9.04/5.10        ( ( member_complex @ Q3 @ ring_1_Ints_complex )
% 9.04/5.10       => ( ! [Z3: int] : ( P @ ( ring_17405671764205052669omplex @ Z3 ) )
% 9.04/5.10         => ( P @ Q3 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8829_Ints__of__int,axiom,
% 9.04/5.10      ! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8830_Ints__of__int,axiom,
% 9.04/5.10      ! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8831_Ints__of__int,axiom,
% 9.04/5.10      ! [Z: int] : ( member_rat @ ( ring_1_of_int_rat @ Z ) @ ring_1_Ints_rat ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8832_Ints__of__int,axiom,
% 9.04/5.10      ! [Z: int] : ( member_Code_integer @ ( ring_18347121197199848620nteger @ Z ) @ ring_11222124179247155820nteger ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8833_Ints__of__int,axiom,
% 9.04/5.10      ! [Z: int] : ( member_complex @ ( ring_17405671764205052669omplex @ Z ) @ ring_1_Ints_complex ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8834_of__int__not__eq,axiom,
% 9.04/5.10      ! [K: int] :
% 9.04/5.10        ( ( ring_18347121197199848620nteger @ ( bit_ri7919022796975470100ot_int @ K ) )
% 9.04/5.10        = ( bit_ri7632146776885996613nteger @ ( ring_18347121197199848620nteger @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8835_of__int__not__eq,axiom,
% 9.04/5.10      ! [K: int] :
% 9.04/5.10        ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 9.04/5.10        = ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8836_sinh__less__cosh__real,axiom,
% 9.04/5.10      ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8837_sinh__le__cosh__real,axiom,
% 9.04/5.10      ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8838_Ints__abs,axiom,
% 9.04/5.10      ! [A: int] :
% 9.04/5.10        ( ( member_int @ A @ ring_1_Ints_int )
% 9.04/5.10       => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8839_Ints__abs,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10       => ( member_rat @ ( abs_abs_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8840_Ints__abs,axiom,
% 9.04/5.10      ! [A: code_integer] :
% 9.04/5.10        ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 9.04/5.10       => ( member_Code_integer @ ( abs_abs_Code_integer @ A ) @ ring_11222124179247155820nteger ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8841_Ints__abs,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10       => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8842_Ints__of__nat,axiom,
% 9.04/5.10      ! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ ring_1_Ints_complex ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8843_Ints__of__nat,axiom,
% 9.04/5.10      ! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ ring_1_Ints_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8844_Ints__of__nat,axiom,
% 9.04/5.10      ! [N: nat] : ( member_rat @ ( semiri681578069525770553at_rat @ N ) @ ring_1_Ints_rat ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8845_Ints__of__nat,axiom,
% 9.04/5.10      ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ring_1_Ints_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8846_tanh__def,axiom,
% 9.04/5.10      ( tanh_complex
% 9.04/5.10      = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ X4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8847_tanh__def,axiom,
% 9.04/5.10      ( tanh_real
% 9.04/5.10      = ( ^ [X4: real] : ( divide_divide_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8848_Ints__diff,axiom,
% 9.04/5.10      ! [A: complex,B: complex] :
% 9.04/5.10        ( ( member_complex @ A @ ring_1_Ints_complex )
% 9.04/5.10       => ( ( member_complex @ B @ ring_1_Ints_complex )
% 9.04/5.10         => ( member_complex @ ( minus_minus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8849_Ints__diff,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10       => ( ( member_real @ B @ ring_1_Ints_real )
% 9.04/5.10         => ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8850_Ints__diff,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10       => ( ( member_rat @ B @ ring_1_Ints_rat )
% 9.04/5.10         => ( member_rat @ ( minus_minus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8851_Ints__diff,axiom,
% 9.04/5.10      ! [A: int,B: int] :
% 9.04/5.10        ( ( member_int @ A @ ring_1_Ints_int )
% 9.04/5.10       => ( ( member_int @ B @ ring_1_Ints_int )
% 9.04/5.10         => ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8852_Ints__add,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10       => ( ( member_real @ B @ ring_1_Ints_real )
% 9.04/5.10         => ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8853_Ints__add,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10       => ( ( member_rat @ B @ ring_1_Ints_rat )
% 9.04/5.10         => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8854_Ints__add,axiom,
% 9.04/5.10      ! [A: int,B: int] :
% 9.04/5.10        ( ( member_int @ A @ ring_1_Ints_int )
% 9.04/5.10       => ( ( member_int @ B @ ring_1_Ints_int )
% 9.04/5.10         => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8855_Ints__add,axiom,
% 9.04/5.10      ! [A: complex,B: complex] :
% 9.04/5.10        ( ( member_complex @ A @ ring_1_Ints_complex )
% 9.04/5.10       => ( ( member_complex @ B @ ring_1_Ints_complex )
% 9.04/5.10         => ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8856_Ints__mult,axiom,
% 9.04/5.10      ! [A: complex,B: complex] :
% 9.04/5.10        ( ( member_complex @ A @ ring_1_Ints_complex )
% 9.04/5.10       => ( ( member_complex @ B @ ring_1_Ints_complex )
% 9.04/5.10         => ( member_complex @ ( times_times_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8857_Ints__mult,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10       => ( ( member_real @ B @ ring_1_Ints_real )
% 9.04/5.10         => ( member_real @ ( times_times_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8858_Ints__mult,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10       => ( ( member_rat @ B @ ring_1_Ints_rat )
% 9.04/5.10         => ( member_rat @ ( times_times_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8859_Ints__mult,axiom,
% 9.04/5.10      ! [A: int,B: int] :
% 9.04/5.10        ( ( member_int @ A @ ring_1_Ints_int )
% 9.04/5.10       => ( ( member_int @ B @ ring_1_Ints_int )
% 9.04/5.10         => ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8860_sinh__plus__cosh,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( plus_plus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
% 9.04/5.10        = ( exp_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8861_sinh__plus__cosh,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( plus_plus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
% 9.04/5.10        = ( exp_complex @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8862_cosh__plus__sinh,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( plus_plus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
% 9.04/5.10        = ( exp_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8863_cosh__plus__sinh,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( plus_plus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
% 9.04/5.10        = ( exp_complex @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8864_sinh__add,axiom,
% 9.04/5.10      ! [X: complex,Y: complex] :
% 9.04/5.10        ( ( sinh_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.04/5.10        = ( plus_plus_complex @ ( times_times_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ Y ) ) @ ( times_times_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8865_sinh__add,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( sinh_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.10        = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8866_cosh__add,axiom,
% 9.04/5.10      ! [X: complex,Y: complex] :
% 9.04/5.10        ( ( cosh_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.04/5.10        = ( plus_plus_complex @ ( times_times_complex @ ( cosh_complex @ X ) @ ( cosh_complex @ Y ) ) @ ( times_times_complex @ ( sinh_complex @ X ) @ ( sinh_complex @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8867_cosh__add,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( cosh_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.10        = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8868_sinh__diff,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( sinh_real @ ( minus_minus_real @ X @ Y ) )
% 9.04/5.10        = ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8869_cosh__diff,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( cosh_real @ ( minus_minus_real @ X @ Y ) )
% 9.04/5.10        = ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8870_of__int__not__numeral,axiom,
% 9.04/5.10      ! [K: num] :
% 9.04/5.10        ( ( ring_18347121197199848620nteger @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
% 9.04/5.10        = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8871_of__int__not__numeral,axiom,
% 9.04/5.10      ! [K: num] :
% 9.04/5.10        ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
% 9.04/5.10        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8872_arsinh__sinh__real,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( arsinh_real @ ( sinh_real @ X ) )
% 9.04/5.10        = X ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8873_sinh__minus__cosh,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( minus_minus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
% 9.04/5.10        = ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8874_sinh__minus__cosh,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( minus_minus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
% 9.04/5.10        = ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8875_cosh__minus__sinh,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( minus_minus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
% 9.04/5.10        = ( exp_real @ ( uminus_uminus_real @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8876_cosh__minus__sinh,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( minus_minus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
% 9.04/5.10        = ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8877_not__add__distrib,axiom,
% 9.04/5.10      ! [A: int,B: int] :
% 9.04/5.10        ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
% 9.04/5.10        = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8878_not__diff__distrib,axiom,
% 9.04/5.10      ! [A: int,B: int] :
% 9.04/5.10        ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
% 9.04/5.10        = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8879_cosh__real__pos,axiom,
% 9.04/5.10      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8880_cosh__real__nonpos__le__iff,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.04/5.10       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 9.04/5.10         => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 9.04/5.10            = ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8881_cosh__real__nonneg__le__iff,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.10         => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 9.04/5.10            = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8882_cosh__real__nonneg,axiom,
% 9.04/5.10      ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8883_cosh__real__ge__1,axiom,
% 9.04/5.10      ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8884_sinh__double,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.10        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X ) ) @ ( cosh_complex @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8885_sinh__double,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.10        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X ) ) @ ( cosh_real @ X ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8886_Ints__double__eq__0__iff,axiom,
% 9.04/5.10      ! [A: complex] :
% 9.04/5.10        ( ( member_complex @ A @ ring_1_Ints_complex )
% 9.04/5.10       => ( ( ( plus_plus_complex @ A @ A )
% 9.04/5.10            = zero_zero_complex )
% 9.04/5.10          = ( A = zero_zero_complex ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8887_Ints__double__eq__0__iff,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10       => ( ( ( plus_plus_real @ A @ A )
% 9.04/5.10            = zero_zero_real )
% 9.04/5.10          = ( A = zero_zero_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8888_Ints__double__eq__0__iff,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10       => ( ( ( plus_plus_rat @ A @ A )
% 9.04/5.10            = zero_zero_rat )
% 9.04/5.10          = ( A = zero_zero_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8889_Ints__double__eq__0__iff,axiom,
% 9.04/5.10      ! [A: int] :
% 9.04/5.10        ( ( member_int @ A @ ring_1_Ints_int )
% 9.04/5.10       => ( ( ( plus_plus_int @ A @ A )
% 9.04/5.10            = zero_zero_int )
% 9.04/5.10          = ( A = zero_zero_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8890_minus__eq__not__plus__1,axiom,
% 9.04/5.10      ( uminus1351360451143612070nteger
% 9.04/5.10      = ( ^ [A3: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8891_minus__eq__not__plus__1,axiom,
% 9.04/5.10      ( uminus_uminus_int
% 9.04/5.10      = ( ^ [A3: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ one_one_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8892_minus__eq__not__minus__1,axiom,
% 9.04/5.10      ( uminus1351360451143612070nteger
% 9.04/5.10      = ( ^ [A3: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A3 @ one_one_Code_integer ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8893_minus__eq__not__minus__1,axiom,
% 9.04/5.10      ( uminus_uminus_int
% 9.04/5.10      = ( ^ [A3: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A3 @ one_one_int ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8894_not__eq__complement,axiom,
% 9.04/5.10      ( bit_ri7632146776885996613nteger
% 9.04/5.10      = ( ^ [A3: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8895_not__eq__complement,axiom,
% 9.04/5.10      ( bit_ri7919022796975470100ot_int
% 9.04/5.10      = ( ^ [A3: int] : ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ one_one_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8896_not__int__def,axiom,
% 9.04/5.10      ( bit_ri7919022796975470100ot_int
% 9.04/5.10      = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8897_and__not__numerals_I1_J,axiom,
% 9.04/5.10      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.04/5.10      = zero_zero_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8898_cosh__real__nonpos__less__iff,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.04/5.10       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 9.04/5.10         => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 9.04/5.10            = ( ord_less_real @ Y @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8899_cosh__real__nonneg__less__iff,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.10         => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 9.04/5.10            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8900_cosh__real__strict__mono,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ( ord_less_real @ X @ Y )
% 9.04/5.10         => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8901_Ints__odd__nonzero,axiom,
% 9.04/5.10      ! [A: complex] :
% 9.04/5.10        ( ( member_complex @ A @ ring_1_Ints_complex )
% 9.04/5.10       => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
% 9.04/5.10         != zero_zero_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8902_Ints__odd__nonzero,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10       => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
% 9.04/5.10         != zero_zero_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8903_Ints__odd__nonzero,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10       => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
% 9.04/5.10         != zero_zero_rat ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8904_Ints__odd__nonzero,axiom,
% 9.04/5.10      ! [A: int] :
% 9.04/5.10        ( ( member_int @ A @ ring_1_Ints_int )
% 9.04/5.10       => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
% 9.04/5.10         != zero_zero_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8905_minus__numeral__inc__eq,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
% 9.04/5.10        = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8906_minus__numeral__inc__eq,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
% 9.04/5.10        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8907_cosh__square__eq,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.10        = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8908_cosh__square__eq,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.10        = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8909_hyperbolic__pythagoras,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.10        = one_one_complex ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8910_hyperbolic__pythagoras,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.10        = one_one_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8911_sinh__square__eq,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.10        = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8912_sinh__square__eq,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.10        = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8913_of__int__divide__in__Ints,axiom,
% 9.04/5.10      ! [B: int,A: int] :
% 9.04/5.10        ( ( dvd_dvd_int @ B @ A )
% 9.04/5.10       => ( member_Code_integer @ ( divide6298287555418463151nteger @ ( ring_18347121197199848620nteger @ A ) @ ( ring_18347121197199848620nteger @ B ) ) @ ring_11222124179247155820nteger ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8914_of__int__divide__in__Ints,axiom,
% 9.04/5.10      ! [B: int,A: int] :
% 9.04/5.10        ( ( dvd_dvd_int @ B @ A )
% 9.04/5.10       => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B ) ) @ ring_1_Ints_complex ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8915_of__int__divide__in__Ints,axiom,
% 9.04/5.10      ! [B: int,A: int] :
% 9.04/5.10        ( ( dvd_dvd_int @ B @ A )
% 9.04/5.10       => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) @ ring_1_Ints_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8916_of__int__divide__in__Ints,axiom,
% 9.04/5.10      ! [B: int,A: int] :
% 9.04/5.10        ( ( dvd_dvd_int @ B @ A )
% 9.04/5.10       => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) @ ring_1_Ints_rat ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8917_of__int__divide__in__Ints,axiom,
% 9.04/5.10      ! [B: int,A: int] :
% 9.04/5.10        ( ( dvd_dvd_int @ B @ A )
% 9.04/5.10       => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B ) ) @ ring_1_Ints_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8918_arcosh__cosh__real,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.10       => ( ( arcosh_real @ ( cosh_real @ X ) )
% 9.04/5.10          = X ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8919_cosh__double,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.10        = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8920_cosh__double,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 9.04/5.10        = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8921_not__int__div__2,axiom,
% 9.04/5.10      ! [K: int] :
% 9.04/5.10        ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.10        = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8922_even__not__iff__int,axiom,
% 9.04/5.10      ! [K: int] :
% 9.04/5.10        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 9.04/5.10        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8923_and__not__numerals_I2_J,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.04/5.10        = one_one_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8924_and__not__numerals_I4_J,axiom,
% 9.04/5.10      ! [M: num] :
% 9.04/5.10        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.04/5.10        = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8925_not__numeral__Bit0__eq,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
% 9.04/5.10        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8926_not__numeral__Bit0__eq,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
% 9.04/5.10        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8927_bit__minus__int__iff,axiom,
% 9.04/5.10      ! [K: int,N: nat] :
% 9.04/5.10        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 9.04/5.10        = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8928_not__numeral__BitM__eq,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bitM @ N ) ) )
% 9.04/5.10        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8929_not__numeral__BitM__eq,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bitM @ N ) ) )
% 9.04/5.10        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8930_Ints__odd__less__0,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10       => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
% 9.04/5.10          = ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8931_Ints__odd__less__0,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10       => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
% 9.04/5.10          = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8932_Ints__odd__less__0,axiom,
% 9.04/5.10      ! [A: int] :
% 9.04/5.10        ( ( member_int @ A @ ring_1_Ints_int )
% 9.04/5.10       => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
% 9.04/5.10          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8933_take__bit__not__mask__eq__0,axiom,
% 9.04/5.10      ! [M: nat,N: nat] :
% 9.04/5.10        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.10       => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) )
% 9.04/5.10          = zero_zero_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8934_Ints__nonzero__abs__ge1,axiom,
% 9.04/5.10      ! [X: code_integer] :
% 9.04/5.10        ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 9.04/5.10       => ( ( X != zero_z3403309356797280102nteger )
% 9.04/5.10         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8935_Ints__nonzero__abs__ge1,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( member_real @ X @ ring_1_Ints_real )
% 9.04/5.10       => ( ( X != zero_zero_real )
% 9.04/5.10         => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8936_Ints__nonzero__abs__ge1,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( member_rat @ X @ ring_1_Ints_rat )
% 9.04/5.10       => ( ( X != zero_zero_rat )
% 9.04/5.10         => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8937_Ints__nonzero__abs__ge1,axiom,
% 9.04/5.10      ! [X: int] :
% 9.04/5.10        ( ( member_int @ X @ ring_1_Ints_int )
% 9.04/5.10       => ( ( X != zero_zero_int )
% 9.04/5.10         => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8938_Ints__nonzero__abs__less1,axiom,
% 9.04/5.10      ! [X: code_integer] :
% 9.04/5.10        ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 9.04/5.10       => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
% 9.04/5.10         => ( X = zero_z3403309356797280102nteger ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8939_Ints__nonzero__abs__less1,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( member_real @ X @ ring_1_Ints_real )
% 9.04/5.10       => ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.10         => ( X = zero_zero_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8940_Ints__nonzero__abs__less1,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( member_rat @ X @ ring_1_Ints_rat )
% 9.04/5.10       => ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
% 9.04/5.10         => ( X = zero_zero_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8941_Ints__nonzero__abs__less1,axiom,
% 9.04/5.10      ! [X: int] :
% 9.04/5.10        ( ( member_int @ X @ ring_1_Ints_int )
% 9.04/5.10       => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
% 9.04/5.10         => ( X = zero_zero_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8942_Ints__eq__abs__less1,axiom,
% 9.04/5.10      ! [X: code_integer,Y: code_integer] :
% 9.04/5.10        ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 9.04/5.10       => ( ( member_Code_integer @ Y @ ring_11222124179247155820nteger )
% 9.04/5.10         => ( ( X = Y )
% 9.04/5.10            = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y ) ) @ one_one_Code_integer ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8943_Ints__eq__abs__less1,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( member_real @ X @ ring_1_Ints_real )
% 9.04/5.10       => ( ( member_real @ Y @ ring_1_Ints_real )
% 9.04/5.10         => ( ( X = Y )
% 9.04/5.10            = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8944_Ints__eq__abs__less1,axiom,
% 9.04/5.10      ! [X: rat,Y: rat] :
% 9.04/5.10        ( ( member_rat @ X @ ring_1_Ints_rat )
% 9.04/5.10       => ( ( member_rat @ Y @ ring_1_Ints_rat )
% 9.04/5.10         => ( ( X = Y )
% 9.04/5.10            = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y ) ) @ one_one_rat ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8945_Ints__eq__abs__less1,axiom,
% 9.04/5.10      ! [X: int,Y: int] :
% 9.04/5.10        ( ( member_int @ X @ ring_1_Ints_int )
% 9.04/5.10       => ( ( member_int @ Y @ ring_1_Ints_int )
% 9.04/5.10         => ( ( X = Y )
% 9.04/5.10            = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8946_sin__times__pi__eq__0,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
% 9.04/5.10          = zero_zero_real )
% 9.04/5.10        = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8947_and__not__numerals_I5_J,axiom,
% 9.04/5.10      ! [M: num,N: num] :
% 9.04/5.10        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.04/5.10        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8948_and__not__numerals_I7_J,axiom,
% 9.04/5.10      ! [M: num] :
% 9.04/5.10        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.04/5.10        = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8949_and__not__numerals_I3_J,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.04/5.10        = zero_zero_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8950_frac__neg,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ( member_real @ X @ ring_1_Ints_real )
% 9.04/5.10         => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 9.04/5.10            = zero_zero_real ) )
% 9.04/5.10        & ( ~ ( member_real @ X @ ring_1_Ints_real )
% 9.04/5.10         => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 9.04/5.10            = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8951_frac__neg,axiom,
% 9.04/5.10      ! [X: rat] :
% 9.04/5.10        ( ( ( member_rat @ X @ ring_1_Ints_rat )
% 9.04/5.10         => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 9.04/5.10            = zero_zero_rat ) )
% 9.04/5.10        & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
% 9.04/5.10         => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 9.04/5.10            = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8952_and__not__numerals_I9_J,axiom,
% 9.04/5.10      ! [M: num,N: num] :
% 9.04/5.10        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.04/5.10        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8953_and__not__numerals_I6_J,axiom,
% 9.04/5.10      ! [M: num,N: num] :
% 9.04/5.10        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.04/5.10        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8954_bit__not__iff__eq,axiom,
% 9.04/5.10      ! [A: int,N: nat] :
% 9.04/5.10        ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N )
% 9.04/5.10        = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.04/5.10           != zero_zero_int )
% 9.04/5.10          & ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8955_minus__exp__eq__not__mask,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.10        = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8956_minus__exp__eq__not__mask,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.10        = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8957_le__mult__floor__Ints,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10         => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8958_le__mult__floor__Ints,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10         => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8959_le__mult__floor__Ints,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10         => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8960_le__mult__floor__Ints,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10         => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8961_le__mult__floor__Ints,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10         => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8962_le__mult__floor__Ints,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10         => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8963_le__mult__floor__Ints,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10         => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8964_le__mult__floor__Ints,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10         => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8965_frac__unique__iff,axiom,
% 9.04/5.10      ! [X: real,A: real] :
% 9.04/5.10        ( ( ( archim2898591450579166408c_real @ X )
% 9.04/5.10          = A )
% 9.04/5.10        = ( ( member_real @ ( minus_minus_real @ X @ A ) @ ring_1_Ints_real )
% 9.04/5.10          & ( ord_less_eq_real @ zero_zero_real @ A )
% 9.04/5.10          & ( ord_less_real @ A @ one_one_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8966_frac__unique__iff,axiom,
% 9.04/5.10      ! [X: rat,A: rat] :
% 9.04/5.10        ( ( ( archimedean_frac_rat @ X )
% 9.04/5.10          = A )
% 9.04/5.10        = ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
% 9.04/5.10          & ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.04/5.10          & ( ord_less_rat @ A @ one_one_rat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8967_mult__ceiling__le__Ints,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10         => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8968_mult__ceiling__le__Ints,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10         => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8969_mult__ceiling__le__Ints,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10         => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8970_mult__ceiling__le__Ints,axiom,
% 9.04/5.10      ! [A: real,B: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.04/5.10       => ( ( member_real @ A @ ring_1_Ints_real )
% 9.04/5.10         => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8971_mult__ceiling__le__Ints,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10         => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8972_mult__ceiling__le__Ints,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10         => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8973_mult__ceiling__le__Ints,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10         => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8974_mult__ceiling__le__Ints,axiom,
% 9.04/5.10      ! [A: rat,B: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.04/5.10       => ( ( member_rat @ A @ ring_1_Ints_rat )
% 9.04/5.10         => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8975_tanh__add,axiom,
% 9.04/5.10      ! [X: complex,Y: complex] :
% 9.04/5.10        ( ( ( cosh_complex @ X )
% 9.04/5.10         != zero_zero_complex )
% 9.04/5.10       => ( ( ( cosh_complex @ Y )
% 9.04/5.10           != zero_zero_complex )
% 9.04/5.10         => ( ( tanh_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.04/5.10            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8976_tanh__add,axiom,
% 9.04/5.10      ! [X: real,Y: real] :
% 9.04/5.10        ( ( ( cosh_real @ X )
% 9.04/5.10         != zero_zero_real )
% 9.04/5.10       => ( ( ( cosh_real @ Y )
% 9.04/5.10           != zero_zero_real )
% 9.04/5.10         => ( ( tanh_real @ ( plus_plus_real @ X @ Y ) )
% 9.04/5.10            = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8977_sinh__zero__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ( sinh_real @ X )
% 9.04/5.10          = zero_zero_real )
% 9.04/5.10        = ( member_real @ ( exp_real @ X ) @ ( insert_real @ one_one_real @ ( insert_real @ ( uminus_uminus_real @ one_one_real ) @ bot_bot_set_real ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8978_sinh__zero__iff,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( ( sinh_complex @ X )
% 9.04/5.10          = zero_zero_complex )
% 9.04/5.10        = ( member_complex @ ( exp_complex @ X ) @ ( insert_complex @ one_one_complex @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ bot_bot_set_complex ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8979_sin__integer__2pi,axiom,
% 9.04/5.10      ! [N: real] :
% 9.04/5.10        ( ( member_real @ N @ ring_1_Ints_real )
% 9.04/5.10       => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 9.04/5.10          = zero_zero_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8980_cos__integer__2pi,axiom,
% 9.04/5.10      ! [N: real] :
% 9.04/5.10        ( ( member_real @ N @ ring_1_Ints_real )
% 9.04/5.10       => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 9.04/5.10          = one_one_real ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8981_cosh__field__def,axiom,
% 9.04/5.10      ( cosh_real
% 9.04/5.10      = ( ^ [Z4: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z4 ) @ ( exp_real @ ( uminus_uminus_real @ Z4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8982_cosh__field__def,axiom,
% 9.04/5.10      ( cosh_complex
% 9.04/5.10      = ( ^ [Z4: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z4 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8983_and__not__numerals_I8_J,axiom,
% 9.04/5.10      ! [M: num,N: num] :
% 9.04/5.10        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.04/5.10        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8984_not__int__rec,axiom,
% 9.04/5.10      ( bit_ri7919022796975470100ot_int
% 9.04/5.10      = ( ^ [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 ) ) ) ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8985_sinh__field__def,axiom,
% 9.04/5.10      ( sinh_real
% 9.04/5.10      = ( ^ [Z4: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z4 ) @ ( exp_real @ ( uminus_uminus_real @ Z4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8986_sinh__field__def,axiom,
% 9.04/5.10      ( sinh_complex
% 9.04/5.10      = ( ^ [Z4: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z4 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8987_cosh__zero__iff,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( ( cosh_real @ X )
% 9.04/5.10          = zero_zero_real )
% 9.04/5.10        = ( ( power_power_real @ ( exp_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.10          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8988_cosh__zero__iff,axiom,
% 9.04/5.10      ! [X: complex] :
% 9.04/5.10        ( ( ( cosh_complex @ X )
% 9.04/5.10          = zero_zero_complex )
% 9.04/5.10        = ( ( power_power_complex @ ( exp_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.10          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8989_gbinomial__absorption_H,axiom,
% 9.04/5.10      ! [K: nat,A: complex] :
% 9.04/5.10        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.04/5.10       => ( ( gbinomial_complex @ A @ K )
% 9.04/5.10          = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8990_gbinomial__absorption_H,axiom,
% 9.04/5.10      ! [K: nat,A: real] :
% 9.04/5.10        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.04/5.10       => ( ( gbinomial_real @ A @ K )
% 9.04/5.10          = ( 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 ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8991_gbinomial__absorption_H,axiom,
% 9.04/5.10      ! [K: nat,A: rat] :
% 9.04/5.10        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.04/5.10       => ( ( gbinomial_rat @ A @ K )
% 9.04/5.10          = ( 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 ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8992_horner__sum__of__bool__2__less,axiom,
% 9.04/5.10      ! [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 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8993_the__elem__eq,axiom,
% 9.04/5.10      ! [X: vEBT_VEBT] :
% 9.04/5.10        ( ( the_elem_VEBT_VEBT @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
% 9.04/5.10        = X ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8994_the__elem__eq,axiom,
% 9.04/5.10      ! [X: real] :
% 9.04/5.10        ( ( the_elem_real @ ( insert_real @ X @ bot_bot_set_real ) )
% 9.04/5.10        = X ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8995_the__elem__eq,axiom,
% 9.04/5.10      ! [X: $o] :
% 9.04/5.10        ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 9.04/5.10        = X ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8996_the__elem__eq,axiom,
% 9.04/5.10      ! [X: nat] :
% 9.04/5.10        ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 9.04/5.10        = X ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8997_the__elem__eq,axiom,
% 9.04/5.10      ! [X: int] :
% 9.04/5.10        ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 9.04/5.10        = X ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8998_push__bit__numeral__minus__1,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.10        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_8999_push__bit__numeral__minus__1,axiom,
% 9.04/5.10      ! [N: num] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.10        = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9000_push__bit__nonnegative__int__iff,axiom,
% 9.04/5.10      ! [N: nat,K: int] :
% 9.04/5.10        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 9.04/5.10        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9001_push__bit__negative__int__iff,axiom,
% 9.04/5.10      ! [N: nat,K: int] :
% 9.04/5.10        ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 9.04/5.10        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9002_push__bit__eq__0__iff,axiom,
% 9.04/5.10      ! [N: nat,A: int] :
% 9.04/5.10        ( ( ( bit_se545348938243370406it_int @ N @ A )
% 9.04/5.10          = zero_zero_int )
% 9.04/5.10        = ( A = zero_zero_int ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9003_push__bit__eq__0__iff,axiom,
% 9.04/5.10      ! [N: nat,A: nat] :
% 9.04/5.10        ( ( ( bit_se547839408752420682it_nat @ N @ A )
% 9.04/5.10          = zero_zero_nat )
% 9.04/5.10        = ( A = zero_zero_nat ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9004_push__bit__of__0,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
% 9.04/5.10        = zero_zero_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9005_push__bit__of__0,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
% 9.04/5.10        = zero_zero_nat ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9006_push__bit__push__bit,axiom,
% 9.04/5.10      ! [M: nat,N: nat,A: int] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 9.04/5.10        = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9007_push__bit__push__bit,axiom,
% 9.04/5.10      ! [M: nat,N: nat,A: nat] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 9.04/5.10        = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9008_concat__bit__of__zero__1,axiom,
% 9.04/5.10      ! [N: nat,L: int] :
% 9.04/5.10        ( ( bit_concat_bit @ N @ zero_zero_int @ L )
% 9.04/5.10        = ( bit_se545348938243370406it_int @ N @ L ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9009_gbinomial__0_I2_J,axiom,
% 9.04/5.10      ! [K: nat] :
% 9.04/5.10        ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
% 9.04/5.10        = zero_zero_complex ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9010_gbinomial__0_I2_J,axiom,
% 9.04/5.10      ! [K: nat] :
% 9.04/5.10        ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 9.04/5.10        = zero_zero_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9011_gbinomial__0_I2_J,axiom,
% 9.04/5.10      ! [K: nat] :
% 9.04/5.10        ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 9.04/5.10        = zero_zero_rat ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9012_gbinomial__0_I2_J,axiom,
% 9.04/5.10      ! [K: nat] :
% 9.04/5.10        ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 9.04/5.10        = zero_zero_nat ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9013_gbinomial__0_I2_J,axiom,
% 9.04/5.10      ! [K: nat] :
% 9.04/5.10        ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 9.04/5.10        = zero_zero_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9014_gbinomial__0_I1_J,axiom,
% 9.04/5.10      ! [A: complex] :
% 9.04/5.10        ( ( gbinomial_complex @ A @ zero_zero_nat )
% 9.04/5.10        = one_one_complex ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9015_gbinomial__0_I1_J,axiom,
% 9.04/5.10      ! [A: real] :
% 9.04/5.10        ( ( gbinomial_real @ A @ zero_zero_nat )
% 9.04/5.10        = one_one_real ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9016_gbinomial__0_I1_J,axiom,
% 9.04/5.10      ! [A: rat] :
% 9.04/5.10        ( ( gbinomial_rat @ A @ zero_zero_nat )
% 9.04/5.10        = one_one_rat ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9017_gbinomial__0_I1_J,axiom,
% 9.04/5.10      ! [A: nat] :
% 9.04/5.10        ( ( gbinomial_nat @ A @ zero_zero_nat )
% 9.04/5.10        = one_one_nat ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9018_gbinomial__0_I1_J,axiom,
% 9.04/5.10      ! [A: int] :
% 9.04/5.10        ( ( gbinomial_int @ A @ zero_zero_nat )
% 9.04/5.10        = one_one_int ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9019_push__bit__Suc__numeral,axiom,
% 9.04/5.10      ! [N: nat,K: num] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
% 9.04/5.10        = ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9020_push__bit__Suc__numeral,axiom,
% 9.04/5.10      ! [N: nat,K: num] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.04/5.10        = ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9021_push__bit__Suc__minus__numeral,axiom,
% 9.04/5.10      ! [N: nat,K: num] :
% 9.04/5.10        ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 9.04/5.10        = ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9022_push__bit__Suc__minus__numeral,axiom,
% 9.04/5.10      ! [N: nat,K: num] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.04/5.10        = ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9023_push__bit__numeral,axiom,
% 9.04/5.10      ! [L: num,K: num] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ K ) )
% 9.04/5.10        = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9024_push__bit__numeral,axiom,
% 9.04/5.10      ! [L: num,K: num] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ K ) )
% 9.04/5.10        = ( bit_se547839408752420682it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9025_push__bit__minus__one__eq__not__mask,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.04/5.10        = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9026_push__bit__minus__one__eq__not__mask,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.10        = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9027_push__bit__Suc,axiom,
% 9.04/5.10      ! [N: nat,A: int] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
% 9.04/5.10        = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9028_push__bit__Suc,axiom,
% 9.04/5.10      ! [N: nat,A: nat] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
% 9.04/5.10        = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9029_push__bit__of__1,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ N @ one_one_int )
% 9.04/5.10        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9030_push__bit__of__1,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
% 9.04/5.10        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9031_push__bit__of__Suc__0,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.04/5.10        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9032_even__push__bit__iff,axiom,
% 9.04/5.10      ! [N: nat,A: code_integer] :
% 9.04/5.10        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 9.04/5.10        = ( ( N != zero_zero_nat )
% 9.04/5.10          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9033_even__push__bit__iff,axiom,
% 9.04/5.10      ! [N: nat,A: int] :
% 9.04/5.10        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
% 9.04/5.10        = ( ( N != zero_zero_nat )
% 9.04/5.10          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9034_even__push__bit__iff,axiom,
% 9.04/5.10      ! [N: nat,A: nat] :
% 9.04/5.10        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 9.04/5.10        = ( ( N != zero_zero_nat )
% 9.04/5.10          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9035_push__bit__minus__numeral,axiom,
% 9.04/5.10      ! [L: num,K: num] :
% 9.04/5.10        ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 9.04/5.10        = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9036_push__bit__minus__numeral,axiom,
% 9.04/5.10      ! [L: num,K: num] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.04/5.10        = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9037_of__nat__gbinomial,axiom,
% 9.04/5.10      ! [N: nat,K: nat] :
% 9.04/5.10        ( ( semiri8010041392384452111omplex @ ( gbinomial_nat @ N @ K ) )
% 9.04/5.10        = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9038_of__nat__gbinomial,axiom,
% 9.04/5.10      ! [N: nat,K: nat] :
% 9.04/5.10        ( ( semiri5074537144036343181t_real @ ( gbinomial_nat @ N @ K ) )
% 9.04/5.10        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9039_of__nat__gbinomial,axiom,
% 9.04/5.10      ! [N: nat,K: nat] :
% 9.04/5.10        ( ( semiri681578069525770553at_rat @ ( gbinomial_nat @ N @ K ) )
% 9.04/5.10        = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9040_push__bit__of__nat,axiom,
% 9.04/5.10      ! [N: nat,M: nat] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 9.04/5.10        = ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9041_push__bit__of__nat,axiom,
% 9.04/5.10      ! [N: nat,M: nat] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 9.04/5.10        = ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9042_of__nat__push__bit,axiom,
% 9.04/5.10      ! [M: nat,N: nat] :
% 9.04/5.10        ( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 9.04/5.10        = ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9043_of__nat__push__bit,axiom,
% 9.04/5.10      ! [M: nat,N: nat] :
% 9.04/5.10        ( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 9.04/5.10        = ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9044_push__bit__of__int,axiom,
% 9.04/5.10      ! [N: nat,K: int] :
% 9.04/5.10        ( ( bit_se7788150548672797655nteger @ N @ ( ring_18347121197199848620nteger @ K ) )
% 9.04/5.10        = ( ring_18347121197199848620nteger @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9045_push__bit__of__int,axiom,
% 9.04/5.10      ! [N: nat,K: int] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ N @ ( ring_1_of_int_int @ K ) )
% 9.04/5.10        = ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9046_push__bit__add,axiom,
% 9.04/5.10      ! [N: nat,A: int,B: int] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B ) )
% 9.04/5.10        = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9047_push__bit__add,axiom,
% 9.04/5.10      ! [N: nat,A: nat,B: nat] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B ) )
% 9.04/5.10        = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9048_binomial__gbinomial,axiom,
% 9.04/5.10      ! [N: nat,K: nat] :
% 9.04/5.10        ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 9.04/5.10        = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9049_binomial__gbinomial,axiom,
% 9.04/5.10      ! [N: nat,K: nat] :
% 9.04/5.10        ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 9.04/5.10        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9050_binomial__gbinomial,axiom,
% 9.04/5.10      ! [N: nat,K: nat] :
% 9.04/5.10        ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 9.04/5.10        = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9051_push__bit__take__bit,axiom,
% 9.04/5.10      ! [M: nat,N: nat,A: int] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 9.04/5.10        = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9052_push__bit__take__bit,axiom,
% 9.04/5.10      ! [M: nat,N: nat,A: nat] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 9.04/5.10        = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9053_flip__bit__nat__def,axiom,
% 9.04/5.10      ( bit_se2161824704523386999it_nat
% 9.04/5.10      = ( ^ [M6: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9054_gbinomial__Suc__Suc,axiom,
% 9.04/5.10      ! [A: real,K: nat] :
% 9.04/5.10        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 9.04/5.10        = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9055_gbinomial__Suc__Suc,axiom,
% 9.04/5.10      ! [A: rat,K: nat] :
% 9.04/5.10        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 9.04/5.10        = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9056_gbinomial__Suc__Suc,axiom,
% 9.04/5.10      ! [A: complex,K: nat] :
% 9.04/5.10        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 9.04/5.10        = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9057_gbinomial__of__nat__symmetric,axiom,
% 9.04/5.10      ! [K: nat,N: nat] :
% 9.04/5.10        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.10       => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K )
% 9.04/5.10          = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9058_gbinomial__of__nat__symmetric,axiom,
% 9.04/5.10      ! [K: nat,N: nat] :
% 9.04/5.10        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.10       => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 9.04/5.10          = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9059_gbinomial__of__nat__symmetric,axiom,
% 9.04/5.10      ! [K: nat,N: nat] :
% 9.04/5.10        ( ( ord_less_eq_nat @ K @ N )
% 9.04/5.10       => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
% 9.04/5.10          = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9060_bit__push__bit__iff__int,axiom,
% 9.04/5.10      ! [M: nat,K: int,N: nat] :
% 9.04/5.10        ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 9.04/5.10        = ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.10          & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9061_bit__push__bit__iff__nat,axiom,
% 9.04/5.10      ! [M: nat,Q3: nat,N: nat] :
% 9.04/5.10        ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N )
% 9.04/5.10        = ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.10          & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9062_concat__bit__eq,axiom,
% 9.04/5.10      ( bit_concat_bit
% 9.04/5.10      = ( ^ [N3: nat,K2: int,L3: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L3 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9063_flip__bit__eq__xor,axiom,
% 9.04/5.10      ( bit_se2159334234014336723it_int
% 9.04/5.10      = ( ^ [N3: nat,A3: int] : ( bit_se6526347334894502574or_int @ A3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9064_flip__bit__eq__xor,axiom,
% 9.04/5.10      ( bit_se2161824704523386999it_nat
% 9.04/5.10      = ( ^ [N3: nat,A3: nat] : ( bit_se6528837805403552850or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9065_flip__bit__int__def,axiom,
% 9.04/5.10      ( bit_se2159334234014336723it_int
% 9.04/5.10      = ( ^ [N3: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9066_gbinomial__addition__formula,axiom,
% 9.04/5.10      ! [A: complex,K: nat] :
% 9.04/5.10        ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 9.04/5.10        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9067_gbinomial__addition__formula,axiom,
% 9.04/5.10      ! [A: real,K: nat] :
% 9.04/5.10        ( ( gbinomial_real @ A @ ( suc @ K ) )
% 9.04/5.10        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9068_gbinomial__addition__formula,axiom,
% 9.04/5.10      ! [A: rat,K: nat] :
% 9.04/5.10        ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 9.04/5.10        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9069_gbinomial__absorb__comp,axiom,
% 9.04/5.10      ! [A: complex,K: nat] :
% 9.04/5.10        ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
% 9.04/5.10        = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9070_gbinomial__absorb__comp,axiom,
% 9.04/5.10      ! [A: real,K: nat] :
% 9.04/5.10        ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
% 9.04/5.10        = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9071_gbinomial__absorb__comp,axiom,
% 9.04/5.10      ! [A: rat,K: nat] :
% 9.04/5.10        ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
% 9.04/5.10        = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9072_gbinomial__ge__n__over__k__pow__k,axiom,
% 9.04/5.10      ! [K: nat,A: real] :
% 9.04/5.10        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 9.04/5.10       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9073_gbinomial__ge__n__over__k__pow__k,axiom,
% 9.04/5.10      ! [K: nat,A: rat] :
% 9.04/5.10        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 9.04/5.10       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9074_gbinomial__mult__1,axiom,
% 9.04/5.10      ! [A: complex,K: nat] :
% 9.04/5.10        ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
% 9.04/5.10        = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9075_gbinomial__mult__1,axiom,
% 9.04/5.10      ! [A: real,K: nat] :
% 9.04/5.10        ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 9.04/5.10        = ( 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 ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9076_gbinomial__mult__1,axiom,
% 9.04/5.10      ! [A: rat,K: nat] :
% 9.04/5.10        ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 9.04/5.10        = ( 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 ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9077_gbinomial__mult__1_H,axiom,
% 9.04/5.10      ! [A: complex,K: nat] :
% 9.04/5.10        ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
% 9.04/5.10        = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9078_gbinomial__mult__1_H,axiom,
% 9.04/5.10      ! [A: real,K: nat] :
% 9.04/5.10        ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 9.04/5.10        = ( 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 ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9079_gbinomial__mult__1_H,axiom,
% 9.04/5.10      ! [A: rat,K: nat] :
% 9.04/5.10        ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 9.04/5.10        = ( 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 ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9080_push__bit__double,axiom,
% 9.04/5.10      ! [N: nat,A: int] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.04/5.10        = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9081_push__bit__double,axiom,
% 9.04/5.10      ! [N: nat,A: nat] :
% 9.04/5.10        ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.10        = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9082_bit__iff__and__push__bit__not__eq__0,axiom,
% 9.04/5.10      ( bit_se1146084159140164899it_int
% 9.04/5.10      = ( ^ [A3: int,N3: nat] :
% 9.04/5.10            ( ( bit_se725231765392027082nd_int @ A3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) )
% 9.04/5.10           != zero_zero_int ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9083_bit__iff__and__push__bit__not__eq__0,axiom,
% 9.04/5.10      ( bit_se1148574629649215175it_nat
% 9.04/5.10      = ( ^ [A3: nat,N3: nat] :
% 9.04/5.10            ( ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) )
% 9.04/5.10           != zero_zero_nat ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9084_push__bit__mask__eq,axiom,
% 9.04/5.10      ! [M: nat,N: nat] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N ) )
% 9.04/5.10        = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9085_unset__bit__eq__and__not,axiom,
% 9.04/5.10      ( bit_se4203085406695923979it_int
% 9.04/5.10      = ( ^ [N3: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9086_unset__bit__int__def,axiom,
% 9.04/5.10      ( bit_se4203085406695923979it_int
% 9.04/5.10      = ( ^ [N3: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9087_Suc__times__gbinomial,axiom,
% 9.04/5.10      ! [K: nat,A: complex] :
% 9.04/5.10        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 9.04/5.10        = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9088_Suc__times__gbinomial,axiom,
% 9.04/5.10      ! [K: nat,A: real] :
% 9.04/5.10        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 9.04/5.10        = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9089_Suc__times__gbinomial,axiom,
% 9.04/5.10      ! [K: nat,A: rat] :
% 9.04/5.10        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 9.04/5.10        = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9090_gbinomial__absorption,axiom,
% 9.04/5.10      ! [K: nat,A: complex] :
% 9.04/5.10        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 9.04/5.10        = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9091_gbinomial__absorption,axiom,
% 9.04/5.10      ! [K: nat,A: real] :
% 9.04/5.10        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 9.04/5.10        = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9092_gbinomial__absorption,axiom,
% 9.04/5.10      ! [K: nat,A: rat] :
% 9.04/5.10        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 9.04/5.10        = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9093_gbinomial__trinomial__revision,axiom,
% 9.04/5.10      ! [K: nat,M: nat,A: complex] :
% 9.04/5.10        ( ( ord_less_eq_nat @ K @ M )
% 9.04/5.10       => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 9.04/5.10          = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9094_gbinomial__trinomial__revision,axiom,
% 9.04/5.10      ! [K: nat,M: nat,A: real] :
% 9.04/5.10        ( ( ord_less_eq_nat @ K @ M )
% 9.04/5.10       => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 9.04/5.10          = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9095_gbinomial__trinomial__revision,axiom,
% 9.04/5.10      ! [K: nat,M: nat,A: rat] :
% 9.04/5.10        ( ( ord_less_eq_nat @ K @ M )
% 9.04/5.10       => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 9.04/5.10          = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9096_push__bit__int__def,axiom,
% 9.04/5.10      ( bit_se545348938243370406it_int
% 9.04/5.10      = ( ^ [N3: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9097_push__bit__nat__def,axiom,
% 9.04/5.10      ( bit_se547839408752420682it_nat
% 9.04/5.10      = ( ^ [N3: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9098_push__bit__eq__mult,axiom,
% 9.04/5.10      ( bit_se545348938243370406it_int
% 9.04/5.10      = ( ^ [N3: nat,A3: int] : ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9099_push__bit__eq__mult,axiom,
% 9.04/5.10      ( bit_se547839408752420682it_nat
% 9.04/5.10      = ( ^ [N3: nat,A3: nat] : ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9100_exp__dvdE,axiom,
% 9.04/5.10      ! [N: nat,A: code_integer] :
% 9.04/5.10        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A )
% 9.04/5.10       => ~ ! [B6: code_integer] :
% 9.04/5.10              ( A
% 9.04/5.10             != ( bit_se7788150548672797655nteger @ N @ B6 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9101_exp__dvdE,axiom,
% 9.04/5.10      ! [N: nat,A: int] :
% 9.04/5.10        ( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A )
% 9.04/5.10       => ~ ! [B6: int] :
% 9.04/5.10              ( A
% 9.04/5.10             != ( bit_se545348938243370406it_int @ N @ B6 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9102_exp__dvdE,axiom,
% 9.04/5.10      ! [N: nat,A: nat] :
% 9.04/5.10        ( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A )
% 9.04/5.10       => ~ ! [B6: nat] :
% 9.04/5.10              ( A
% 9.04/5.10             != ( bit_se547839408752420682it_nat @ N @ B6 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9103_the__elem__image__unique,axiom,
% 9.04/5.10      ! [A2: set_nat,F: nat > set_nat,X: nat] :
% 9.04/5.10        ( ( A2 != bot_bot_set_nat )
% 9.04/5.10       => ( ! [Y4: nat] :
% 9.04/5.10              ( ( member_nat @ Y4 @ A2 )
% 9.04/5.10             => ( ( F @ Y4 )
% 9.04/5.10                = ( F @ X ) ) )
% 9.04/5.10         => ( ( the_elem_set_nat @ ( image_nat_set_nat @ F @ A2 ) )
% 9.04/5.10            = ( F @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9104_the__elem__image__unique,axiom,
% 9.04/5.10      ! [A2: set_nat,F: nat > nat,X: nat] :
% 9.04/5.10        ( ( A2 != bot_bot_set_nat )
% 9.04/5.10       => ( ! [Y4: nat] :
% 9.04/5.10              ( ( member_nat @ Y4 @ A2 )
% 9.04/5.10             => ( ( F @ Y4 )
% 9.04/5.10                = ( F @ X ) ) )
% 9.04/5.10         => ( ( the_elem_nat @ ( image_nat_nat @ F @ A2 ) )
% 9.04/5.10            = ( F @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9105_the__elem__image__unique,axiom,
% 9.04/5.10      ! [A2: set_nat,F: nat > int,X: nat] :
% 9.04/5.10        ( ( A2 != bot_bot_set_nat )
% 9.04/5.10       => ( ! [Y4: nat] :
% 9.04/5.10              ( ( member_nat @ Y4 @ A2 )
% 9.04/5.10             => ( ( F @ Y4 )
% 9.04/5.10                = ( F @ X ) ) )
% 9.04/5.10         => ( ( the_elem_int @ ( image_nat_int @ F @ A2 ) )
% 9.04/5.10            = ( F @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9106_the__elem__image__unique,axiom,
% 9.04/5.10      ! [A2: set_int,F: int > nat,X: int] :
% 9.04/5.10        ( ( A2 != bot_bot_set_int )
% 9.04/5.10       => ( ! [Y4: int] :
% 9.04/5.10              ( ( member_int @ Y4 @ A2 )
% 9.04/5.10             => ( ( F @ Y4 )
% 9.04/5.10                = ( F @ X ) ) )
% 9.04/5.10         => ( ( the_elem_nat @ ( image_int_nat @ F @ A2 ) )
% 9.04/5.10            = ( F @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9107_the__elem__image__unique,axiom,
% 9.04/5.10      ! [A2: set_int,F: int > int,X: int] :
% 9.04/5.10        ( ( A2 != bot_bot_set_int )
% 9.04/5.10       => ( ! [Y4: int] :
% 9.04/5.10              ( ( member_int @ Y4 @ A2 )
% 9.04/5.10             => ( ( F @ Y4 )
% 9.04/5.10                = ( F @ X ) ) )
% 9.04/5.10         => ( ( the_elem_int @ ( image_int_int @ F @ A2 ) )
% 9.04/5.10            = ( F @ X ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9108_gbinomial__rec,axiom,
% 9.04/5.10      ! [A: complex,K: nat] :
% 9.04/5.10        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 9.04/5.10        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9109_gbinomial__rec,axiom,
% 9.04/5.10      ! [A: real,K: nat] :
% 9.04/5.10        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 9.04/5.10        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9110_gbinomial__rec,axiom,
% 9.04/5.10      ! [A: rat,K: nat] :
% 9.04/5.10        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 9.04/5.10        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9111_gbinomial__factors,axiom,
% 9.04/5.10      ! [A: complex,K: nat] :
% 9.04/5.10        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 9.04/5.10        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9112_gbinomial__factors,axiom,
% 9.04/5.10      ! [A: real,K: nat] :
% 9.04/5.10        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 9.04/5.10        = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9113_gbinomial__factors,axiom,
% 9.04/5.10      ! [A: rat,K: nat] :
% 9.04/5.10        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 9.04/5.10        = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9114_gbinomial__negated__upper,axiom,
% 9.04/5.10      ( gbinomial_complex
% 9.04/5.10      = ( ^ [A3: complex,K2: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A3 ) @ one_one_complex ) @ K2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9115_gbinomial__negated__upper,axiom,
% 9.04/5.10      ( gbinomial_real
% 9.04/5.10      = ( ^ [A3: 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 ) @ A3 ) @ one_one_real ) @ K2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9116_gbinomial__negated__upper,axiom,
% 9.04/5.10      ( gbinomial_rat
% 9.04/5.10      = ( ^ [A3: 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 ) @ A3 ) @ one_one_rat ) @ K2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9117_gbinomial__index__swap,axiom,
% 9.04/5.10      ! [K: nat,N: nat] :
% 9.04/5.10        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
% 9.04/5.10        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9118_gbinomial__index__swap,axiom,
% 9.04/5.10      ! [K: nat,N: nat] :
% 9.04/5.10        ( ( 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 ) )
% 9.04/5.10        = ( 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 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9119_gbinomial__index__swap,axiom,
% 9.04/5.10      ! [K: nat,N: nat] :
% 9.04/5.10        ( ( 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 ) )
% 9.04/5.10        = ( 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 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9120_push__bit__minus__one,axiom,
% 9.04/5.10      ! [N: nat] :
% 9.04/5.10        ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.10        = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9121_gbinomial__minus,axiom,
% 9.04/5.10      ! [A: complex,K: nat] :
% 9.04/5.10        ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 9.04/5.10        = ( 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 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9122_gbinomial__minus,axiom,
% 9.04/5.10      ! [A: real,K: nat] :
% 9.04/5.10        ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 9.04/5.10        = ( 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 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9123_gbinomial__minus,axiom,
% 9.04/5.10      ! [A: rat,K: nat] :
% 9.04/5.10        ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 9.04/5.10        = ( 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 ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9124_gbinomial__reduce__nat,axiom,
% 9.04/5.10      ! [K: nat,A: complex] :
% 9.04/5.10        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.04/5.10       => ( ( gbinomial_complex @ A @ K )
% 9.04/5.10          = ( 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 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9125_gbinomial__reduce__nat,axiom,
% 9.04/5.10      ! [K: nat,A: real] :
% 9.04/5.10        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.04/5.10       => ( ( gbinomial_real @ A @ K )
% 9.04/5.10          = ( 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 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9126_gbinomial__reduce__nat,axiom,
% 9.04/5.10      ! [K: nat,A: rat] :
% 9.04/5.10        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.04/5.10       => ( ( gbinomial_rat @ A @ K )
% 9.04/5.10          = ( 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 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9127_gbinomial__pochhammer,axiom,
% 9.04/5.10      ( gbinomial_rat
% 9.04/5.10      = ( ^ [A3: 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 @ A3 ) @ K2 ) ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9128_gbinomial__pochhammer,axiom,
% 9.04/5.10      ( gbinomial_real
% 9.04/5.10      = ( ^ [A3: 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 @ A3 ) @ K2 ) ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9129_gbinomial__pochhammer,axiom,
% 9.04/5.10      ( gbinomial_complex
% 9.04/5.10      = ( ^ [A3: complex,K2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A3 ) @ K2 ) ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9130_gbinomial__pochhammer_H,axiom,
% 9.04/5.10      ( gbinomial_rat
% 9.04/5.10      = ( ^ [A3: rat,K2: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9131_gbinomial__pochhammer_H,axiom,
% 9.04/5.10      ( gbinomial_real
% 9.04/5.10      = ( ^ [A3: real,K2: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9132_gbinomial__pochhammer_H,axiom,
% 9.04/5.10      ( gbinomial_complex
% 9.04/5.10      = ( ^ [A3: complex,K2: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9133_bit__horner__sum__bit__iff,axiom,
% 9.04/5.10      ! [Bs: list_o,N: nat] :
% 9.04/5.10        ( ( bit_se9216721137139052372nteger @ ( groups3417619833198082522nteger @ zero_n356916108424825756nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Bs ) @ N )
% 9.04/5.10        = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 9.04/5.10          & ( nth_o @ Bs @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9134_bit__horner__sum__bit__iff,axiom,
% 9.04/5.10      ! [Bs: list_o,N: nat] :
% 9.04/5.10        ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N )
% 9.04/5.10        = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 9.04/5.10          & ( nth_o @ Bs @ N ) ) ) ).
% 9.04/5.10  
% 9.04/5.10  thf(fact_9135_bit__horner__sum__bit__iff,axiom,
% 9.04/5.10      ! [Bs: list_o,N: nat] :
% 9.04/5.10        ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N )
% 9.04/5.11        = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 9.04/5.11          & ( nth_o @ Bs @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9136_Cauchy__iff2,axiom,
% 9.04/5.11      ( topolo4055970368930404560y_real
% 9.04/5.11      = ( ^ [X6: nat > real] :
% 9.04/5.11          ! [J2: nat] :
% 9.04/5.11          ? [M8: nat] :
% 9.04/5.11          ! [M6: nat] :
% 9.04/5.11            ( ( ord_less_eq_nat @ M8 @ M6 )
% 9.04/5.11           => ! [N3: nat] :
% 9.04/5.11                ( ( ord_less_eq_nat @ M8 @ N3 )
% 9.04/5.11               => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M6 ) @ ( X6 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J2 ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9137_is__singleton__the__elem,axiom,
% 9.04/5.11      ( is_sin24926331636114728T_VEBT
% 9.04/5.11      = ( ^ [A4: set_VEBT_VEBT] :
% 9.04/5.11            ( A4
% 9.04/5.11            = ( insert_VEBT_VEBT @ ( the_elem_VEBT_VEBT @ A4 ) @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9138_is__singleton__the__elem,axiom,
% 9.04/5.11      ( is_singleton_real
% 9.04/5.11      = ( ^ [A4: set_real] :
% 9.04/5.11            ( A4
% 9.04/5.11            = ( insert_real @ ( the_elem_real @ A4 ) @ bot_bot_set_real ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9139_is__singleton__the__elem,axiom,
% 9.04/5.11      ( is_singleton_o
% 9.04/5.11      = ( ^ [A4: set_o] :
% 9.04/5.11            ( A4
% 9.04/5.11            = ( insert_o @ ( the_elem_o @ A4 ) @ bot_bot_set_o ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9140_is__singleton__the__elem,axiom,
% 9.04/5.11      ( is_singleton_nat
% 9.04/5.11      = ( ^ [A4: set_nat] :
% 9.04/5.11            ( A4
% 9.04/5.11            = ( insert_nat @ ( the_elem_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9141_is__singleton__the__elem,axiom,
% 9.04/5.11      ( is_singleton_int
% 9.04/5.11      = ( ^ [A4: set_int] :
% 9.04/5.11            ( A4
% 9.04/5.11            = ( insert_int @ ( the_elem_int @ A4 ) @ bot_bot_set_int ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9142_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 9.04/5.11        ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 9.04/5.11          = Y )
% 9.04/5.11       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.04/5.11                ( X
% 9.04/5.11                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.04/5.11           => Y )
% 9.04/5.11         => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.04/5.11                  ( X
% 9.04/5.11                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.04/5.11             => Y )
% 9.04/5.11           => ( ! [Mi: nat,Ma2: nat] :
% 9.04/5.11                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.04/5.11                      ( X
% 9.04/5.11                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.04/5.11                 => ( Y
% 9.04/5.11                    = ( ~ ( ( Xa = Mi )
% 9.04/5.11                          | ( Xa = Ma2 ) ) ) ) )
% 9.04/5.11             => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 9.04/5.11                    ( ? [Vc: vEBT_VEBT] :
% 9.04/5.11                        ( X
% 9.04/5.11                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 9.04/5.11                   => ( Y
% 9.04/5.11                      = ( ~ ( ( Xa = Mi )
% 9.04/5.11                            | ( Xa = Ma2 )
% 9.04/5.11                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 9.04/5.11               => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 9.04/5.11                      ( ? [Vd: vEBT_VEBT] :
% 9.04/5.11                          ( X
% 9.04/5.11                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 9.04/5.11                     => ( Y
% 9.04/5.11                        = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9143_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.11        ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 9.04/5.11       => ( ! [Uu2: $o,Uv2: $o] :
% 9.04/5.11              ( X
% 9.04/5.11             != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.04/5.11         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.04/5.11                ( X
% 9.04/5.11               != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.04/5.11           => ( ! [Mi: nat,Ma2: nat] :
% 9.04/5.11                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.04/5.11                      ( X
% 9.04/5.11                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.04/5.11                 => ( ( Xa = Mi )
% 9.04/5.11                    | ( Xa = Ma2 ) ) )
% 9.04/5.11             => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 9.04/5.11                    ( ? [Vc: vEBT_VEBT] :
% 9.04/5.11                        ( X
% 9.04/5.11                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 9.04/5.11                   => ( ( Xa = Mi )
% 9.04/5.11                      | ( Xa = Ma2 )
% 9.04/5.11                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.04/5.11               => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 9.04/5.11                      ( ? [Vd: vEBT_VEBT] :
% 9.04/5.11                          ( X
% 9.04/5.11                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 9.04/5.11                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9144_image__ident,axiom,
% 9.04/5.11      ! [Y7: set_nat] :
% 9.04/5.11        ( ( image_nat_nat
% 9.04/5.11          @ ^ [X4: nat] : X4
% 9.04/5.11          @ Y7 )
% 9.04/5.11        = Y7 ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9145_image__ident,axiom,
% 9.04/5.11      ! [Y7: set_int] :
% 9.04/5.11        ( ( image_int_int
% 9.04/5.11          @ ^ [X4: int] : X4
% 9.04/5.11          @ Y7 )
% 9.04/5.11        = Y7 ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9146_singleton__conv,axiom,
% 9.04/5.11      ! [A: complex] :
% 9.04/5.11        ( ( collect_complex
% 9.04/5.11          @ ^ [X4: complex] : ( X4 = A ) )
% 9.04/5.11        = ( insert_complex @ A @ bot_bot_set_complex ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9147_singleton__conv,axiom,
% 9.04/5.11      ! [A: set_nat] :
% 9.04/5.11        ( ( collect_set_nat
% 9.04/5.11          @ ^ [X4: set_nat] : ( X4 = A ) )
% 9.04/5.11        = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9148_singleton__conv,axiom,
% 9.04/5.11      ! [A: real] :
% 9.04/5.11        ( ( collect_real
% 9.04/5.11          @ ^ [X4: real] : ( X4 = A ) )
% 9.04/5.11        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9149_singleton__conv,axiom,
% 9.04/5.11      ! [A: $o] :
% 9.04/5.11        ( ( collect_o
% 9.04/5.11          @ ^ [X4: $o] : ( X4 = A ) )
% 9.04/5.11        = ( insert_o @ A @ bot_bot_set_o ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9150_singleton__conv,axiom,
% 9.04/5.11      ! [A: nat] :
% 9.04/5.11        ( ( collect_nat
% 9.04/5.11          @ ^ [X4: nat] : ( X4 = A ) )
% 9.04/5.11        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9151_singleton__conv,axiom,
% 9.04/5.11      ! [A: int] :
% 9.04/5.11        ( ( collect_int
% 9.04/5.11          @ ^ [X4: int] : ( X4 = A ) )
% 9.04/5.11        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9152_Max__divisors__self__int,axiom,
% 9.04/5.11      ! [N: int] :
% 9.04/5.11        ( ( N != zero_zero_int )
% 9.04/5.11       => ( ( lattic8263393255366662781ax_int
% 9.04/5.11            @ ( collect_int
% 9.04/5.11              @ ^ [D3: int] : ( dvd_dvd_int @ D3 @ N ) ) )
% 9.04/5.11          = ( abs_abs_int @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9153_Max__divisors__self__nat,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( N != zero_zero_nat )
% 9.04/5.11       => ( ( lattic8265883725875713057ax_nat
% 9.04/5.11            @ ( collect_nat
% 9.04/5.11              @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ N ) ) )
% 9.04/5.11          = N ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9154_nat__less__as__int,axiom,
% 9.04/5.11      ( ord_less_nat
% 9.04/5.11      = ( ^ [A3: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9155_nat__leq__as__int,axiom,
% 9.04/5.11      ( ord_less_eq_nat
% 9.04/5.11      = ( ^ [A3: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9156_nat__plus__as__int,axiom,
% 9.04/5.11      ( plus_plus_nat
% 9.04/5.11      = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9157_nat__times__as__int,axiom,
% 9.04/5.11      ( times_times_nat
% 9.04/5.11      = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9158_nat__minus__as__int,axiom,
% 9.04/5.11      ( minus_minus_nat
% 9.04/5.11      = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9159_nat__div__as__int,axiom,
% 9.04/5.11      ( divide_divide_nat
% 9.04/5.11      = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9160_nat__mod__as__int,axiom,
% 9.04/5.11      ( modulo_modulo_nat
% 9.04/5.11      = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9161_divide__nat__def,axiom,
% 9.04/5.11      ( divide_divide_nat
% 9.04/5.11      = ( ^ [M6: nat,N3: nat] :
% 9.04/5.11            ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
% 9.04/5.11            @ ( lattic8265883725875713057ax_nat
% 9.04/5.11              @ ( collect_nat
% 9.04/5.11                @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N3 ) @ M6 ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9162_diff__nat__eq__if,axiom,
% 9.04/5.11      ! [Z6: int,Z: int] :
% 9.04/5.11        ( ( ( ord_less_int @ Z6 @ zero_zero_int )
% 9.04/5.11         => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 9.04/5.11            = ( nat2 @ Z ) ) )
% 9.04/5.11        & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
% 9.04/5.11         => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 9.04/5.11            = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9163_set__decode__def,axiom,
% 9.04/5.11      ( nat_set_decode
% 9.04/5.11      = ( ^ [X4: nat] :
% 9.04/5.11            ( collect_nat
% 9.04/5.11            @ ^ [N3: nat] :
% 9.04/5.11                ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9164_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 9.04/5.11      ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 9.04/5.11        ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
% 9.04/5.11        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.04/5.11           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9165_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 9.04/5.11      ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
% 9.04/5.11        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd2 ) @ X )
% 9.04/5.11        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.04/5.11           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9166_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 9.04/5.11      ! [Mi2: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
% 9.04/5.11        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ X )
% 9.04/5.11        = ( ( X = Mi2 )
% 9.04/5.11          | ( X = Ma )
% 9.04/5.11          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.04/5.11             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9167_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.11        ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 9.04/5.11       => ( ! [A5: $o,B6: $o] :
% 9.04/5.11              ( ( X
% 9.04/5.11                = ( vEBT_Leaf @ A5 @ B6 ) )
% 9.04/5.11             => ( ( ( Xa = zero_zero_nat )
% 9.04/5.11                 => A5 )
% 9.04/5.11                & ( ( Xa != zero_zero_nat )
% 9.04/5.11                 => ( ( ( Xa = one_one_nat )
% 9.04/5.11                     => B6 )
% 9.04/5.11                    & ( Xa = one_one_nat ) ) ) ) )
% 9.04/5.11         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.04/5.11                ( X
% 9.04/5.11               != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.04/5.11           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 9.04/5.11                  ( ? [S3: vEBT_VEBT] :
% 9.04/5.11                      ( X
% 9.04/5.11                      = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 9.04/5.11                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                     => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9168_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.11        ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 9.04/5.11       => ( ! [A5: $o,B6: $o] :
% 9.04/5.11              ( ( X
% 9.04/5.11                = ( vEBT_Leaf @ A5 @ B6 ) )
% 9.04/5.11             => ~ ( ( ( Xa = zero_zero_nat )
% 9.04/5.11                   => A5 )
% 9.04/5.11                  & ( ( Xa != zero_zero_nat )
% 9.04/5.11                   => ( ( ( Xa = one_one_nat )
% 9.04/5.11                       => B6 )
% 9.04/5.11                      & ( Xa = one_one_nat ) ) ) ) )
% 9.04/5.11         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 9.04/5.11                ( ? [S3: vEBT_VEBT] :
% 9.04/5.11                    ( X
% 9.04/5.11                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 9.04/5.11               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                     => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9169_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 9.04/5.11        ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 9.04/5.11          = Y )
% 9.04/5.11       => ( ! [A5: $o,B6: $o] :
% 9.04/5.11              ( ( X
% 9.04/5.11                = ( vEBT_Leaf @ A5 @ B6 ) )
% 9.04/5.11             => ( Y
% 9.04/5.11                = ( ~ ( ( ( Xa = zero_zero_nat )
% 9.04/5.11                       => A5 )
% 9.04/5.11                      & ( ( Xa != zero_zero_nat )
% 9.04/5.11                       => ( ( ( Xa = one_one_nat )
% 9.04/5.11                           => B6 )
% 9.04/5.11                          & ( Xa = one_one_nat ) ) ) ) ) ) )
% 9.04/5.11         => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.04/5.11                  ( X
% 9.04/5.11                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.04/5.11             => Y )
% 9.04/5.11           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 9.04/5.11                  ( ? [S3: vEBT_VEBT] :
% 9.04/5.11                      ( X
% 9.04/5.11                      = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 9.04/5.11                 => ( Y
% 9.04/5.11                    = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9170_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.11        ( ( vEBT_VEBT_membermima @ X @ Xa )
% 9.04/5.11       => ( ! [Mi: nat,Ma2: nat] :
% 9.04/5.11              ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.04/5.11                  ( X
% 9.04/5.11                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.04/5.11             => ~ ( ( Xa = Mi )
% 9.04/5.11                  | ( Xa = Ma2 ) ) )
% 9.04/5.11         => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 9.04/5.11                ( ? [Vc: vEBT_VEBT] :
% 9.04/5.11                    ( X
% 9.04/5.11                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 9.04/5.11               => ~ ( ( Xa = Mi )
% 9.04/5.11                    | ( Xa = Ma2 )
% 9.04/5.11                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.04/5.11           => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 9.04/5.11                  ( ? [Vd: vEBT_VEBT] :
% 9.04/5.11                      ( X
% 9.04/5.11                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 9.04/5.11                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9171_monoseq__arctan__series,axiom,
% 9.04/5.11      ! [X: real] :
% 9.04/5.11        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.11       => ( topolo6980174941875973593q_real
% 9.04/5.11          @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9172_ln__series,axiom,
% 9.04/5.11      ! [X: real] :
% 9.04/5.11        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.11       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.11         => ( ( ln_ln_real @ X )
% 9.04/5.11            = ( suminf_real
% 9.04/5.11              @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9173_arctan__series,axiom,
% 9.04/5.11      ! [X: real] :
% 9.04/5.11        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.11       => ( ( arctan @ X )
% 9.04/5.11          = ( suminf_real
% 9.04/5.11            @ ^ [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 @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9174_monoseq__realpow,axiom,
% 9.04/5.11      ! [X: real] :
% 9.04/5.11        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.11       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.11         => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9175_pi__series,axiom,
% 9.04/5.11      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.04/5.11      = ( suminf_real
% 9.04/5.11        @ ^ [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 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9176_summable__arctan__series,axiom,
% 9.04/5.11      ! [X: real] :
% 9.04/5.11        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.11       => ( summable_real
% 9.04/5.11          @ ^ [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 @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9177_vebt__buildup_Oelims,axiom,
% 9.04/5.11      ! [X: nat,Y: vEBT_VEBT] :
% 9.04/5.11        ( ( ( vEBT_vebt_buildup @ X )
% 9.04/5.11          = Y )
% 9.04/5.11       => ( ( ( X = zero_zero_nat )
% 9.04/5.11           => ( Y
% 9.04/5.11             != ( vEBT_Leaf @ $false @ $false ) ) )
% 9.04/5.11         => ( ( ( X
% 9.04/5.11                = ( suc @ zero_zero_nat ) )
% 9.04/5.11             => ( Y
% 9.04/5.11               != ( vEBT_Leaf @ $false @ $false ) ) )
% 9.04/5.11           => ~ ! [Va: nat] :
% 9.04/5.11                  ( ( X
% 9.04/5.11                    = ( suc @ ( suc @ Va ) ) )
% 9.04/5.11                 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.04/5.11                       => ( Y
% 9.04/5.11                          = ( 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 ) ) ) ) ) ) )
% 9.04/5.11                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.04/5.11                       => ( Y
% 9.04/5.11                          = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9178_summable__rabs__comparison__test,axiom,
% 9.04/5.11      ! [F: nat > real,G: nat > real] :
% 9.04/5.11        ( ? [N6: nat] :
% 9.04/5.11          ! [N2: nat] :
% 9.04/5.11            ( ( ord_less_eq_nat @ N6 @ N2 )
% 9.04/5.11           => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 9.04/5.11       => ( ( summable_real @ G )
% 9.04/5.11         => ( summable_real
% 9.04/5.11            @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9179_summable__rabs,axiom,
% 9.04/5.11      ! [F: nat > real] :
% 9.04/5.11        ( ( summable_real
% 9.04/5.11          @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) )
% 9.04/5.11       => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 9.04/5.11          @ ( suminf_real
% 9.04/5.11            @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9180_summable__power__series,axiom,
% 9.04/5.11      ! [F: nat > real,Z: real] :
% 9.04/5.11        ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
% 9.04/5.11       => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 9.04/5.11         => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 9.04/5.11           => ( ( ord_less_real @ Z @ one_one_real )
% 9.04/5.11             => ( summable_real
% 9.04/5.11                @ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9181_vebt__buildup_Osimps_I3_J,axiom,
% 9.04/5.11      ! [Va2: nat] :
% 9.04/5.11        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.04/5.11         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 9.04/5.11            = ( 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 ) ) ) ) ) ) )
% 9.04/5.11        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.04/5.11         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 9.04/5.11            = ( 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 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9182_sin__paired,axiom,
% 9.04/5.11      ! [X: real] :
% 9.04/5.11        ( sums_real
% 9.04/5.11        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 9.04/5.11        @ ( sin_real @ X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9183_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.11        ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 9.04/5.11       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 9.04/5.11         => ( ! [Uu2: $o,Uv2: $o] :
% 9.04/5.11                ( ( X
% 9.04/5.11                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.04/5.11               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
% 9.04/5.11           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.04/5.11                  ( ( X
% 9.04/5.11                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.04/5.11                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
% 9.04/5.11             => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.04/5.11                    ( ( X
% 9.04/5.11                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.04/5.11                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
% 9.04/5.11                     => ( ( Xa = Mi )
% 9.04/5.11                        | ( Xa = Ma2 ) ) ) )
% 9.04/5.11               => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 9.04/5.11                      ( ( X
% 9.04/5.11                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 9.04/5.11                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) @ Xa ) )
% 9.04/5.11                       => ( ( Xa = Mi )
% 9.04/5.11                          | ( Xa = Ma2 )
% 9.04/5.11                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 9.04/5.11                 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 9.04/5.11                        ( ( X
% 9.04/5.11                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 9.04/5.11                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa ) )
% 9.04/5.11                         => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9184_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 9.04/5.11        ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 9.04/5.11          = Y )
% 9.04/5.11       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 9.04/5.11         => ( ! [Uu2: $o,Uv2: $o] :
% 9.04/5.11                ( ( X
% 9.04/5.11                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.04/5.11               => ( ~ Y
% 9.04/5.11                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 9.04/5.11           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.04/5.11                  ( ( X
% 9.04/5.11                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.04/5.11                 => ( ~ Y
% 9.04/5.11                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
% 9.04/5.11             => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.04/5.11                    ( ( X
% 9.04/5.11                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.04/5.11                   => ( ( Y
% 9.04/5.11                        = ( ( Xa = Mi )
% 9.04/5.11                          | ( Xa = Ma2 ) ) )
% 9.04/5.11                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) ) ) )
% 9.04/5.11               => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 9.04/5.11                      ( ( X
% 9.04/5.11                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 9.04/5.11                     => ( ( Y
% 9.04/5.11                          = ( ( Xa = Mi )
% 9.04/5.11                            | ( Xa = Ma2 )
% 9.04/5.11                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.04/5.11                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) @ Xa ) ) ) )
% 9.04/5.11                 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 9.04/5.11                        ( ( X
% 9.04/5.11                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 9.04/5.11                       => ( ( Y
% 9.04/5.11                            = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 9.04/5.11                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9185_power__half__series,axiom,
% 9.04/5.11      ( sums_real
% 9.04/5.11      @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
% 9.04/5.11      @ one_one_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9186_sums__if_H,axiom,
% 9.04/5.11      ! [G: nat > real,X: real] :
% 9.04/5.11        ( ( sums_real @ G @ X )
% 9.04/5.11       => ( sums_real
% 9.04/5.11          @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.11          @ X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9187_sums__if,axiom,
% 9.04/5.11      ! [G: nat > real,X: real,F: nat > real,Y: real] :
% 9.04/5.11        ( ( sums_real @ G @ X )
% 9.04/5.11       => ( ( sums_real @ F @ Y )
% 9.04/5.11         => ( sums_real
% 9.04/5.11            @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.11            @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9188_cos__paired,axiom,
% 9.04/5.11      ! [X: real] :
% 9.04/5.11        ( sums_real
% 9.04/5.11        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 9.04/5.11        @ ( cos_real @ X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9189_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.11        ( ( vEBT_VEBT_membermima @ X @ Xa )
% 9.04/5.11       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 9.04/5.11         => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.04/5.11                ( ( X
% 9.04/5.11                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.04/5.11               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
% 9.04/5.11                 => ~ ( ( Xa = Mi )
% 9.04/5.11                      | ( Xa = Ma2 ) ) ) )
% 9.04/5.11           => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 9.04/5.11                  ( ( X
% 9.04/5.11                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) )
% 9.04/5.11                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc ) @ Xa ) )
% 9.04/5.11                   => ~ ( ( Xa = Mi )
% 9.04/5.11                        | ( Xa = Ma2 )
% 9.04/5.11                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 9.04/5.11             => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 9.04/5.11                    ( ( X
% 9.04/5.11                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
% 9.04/5.11                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa ) )
% 9.04/5.11                     => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9190_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 9.04/5.11        ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 9.04/5.11          = Y )
% 9.04/5.11       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 9.04/5.11         => ( ! [A5: $o,B6: $o] :
% 9.04/5.11                ( ( X
% 9.04/5.11                  = ( vEBT_Leaf @ A5 @ B6 ) )
% 9.04/5.11               => ( ( Y
% 9.04/5.11                    = ( ( ( Xa = zero_zero_nat )
% 9.04/5.11                       => A5 )
% 9.04/5.11                      & ( ( Xa != zero_zero_nat )
% 9.04/5.11                       => ( ( ( Xa = one_one_nat )
% 9.04/5.11                           => B6 )
% 9.04/5.11                          & ( Xa = one_one_nat ) ) ) ) )
% 9.04/5.11                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa ) ) ) )
% 9.04/5.11           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.04/5.11                  ( ( X
% 9.04/5.11                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.04/5.11                 => ( ~ Y
% 9.04/5.11                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
% 9.04/5.11             => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.04/5.11                    ( ( X
% 9.04/5.11                      = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 9.04/5.11                   => ( ( Y
% 9.04/5.11                        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 9.04/5.11                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9191_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.11        ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 9.04/5.11       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 9.04/5.11         => ( ! [A5: $o,B6: $o] :
% 9.04/5.11                ( ( X
% 9.04/5.11                  = ( vEBT_Leaf @ A5 @ B6 ) )
% 9.04/5.11               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa ) )
% 9.04/5.11                 => ~ ( ( ( Xa = zero_zero_nat )
% 9.04/5.11                       => A5 )
% 9.04/5.11                      & ( ( Xa != zero_zero_nat )
% 9.04/5.11                       => ( ( ( Xa = one_one_nat )
% 9.04/5.11                           => B6 )
% 9.04/5.11                          & ( Xa = one_one_nat ) ) ) ) ) )
% 9.04/5.11           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.04/5.11                  ( ( X
% 9.04/5.11                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 9.04/5.11                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 9.04/5.11                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9192_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 9.04/5.11      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.11        ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 9.04/5.11       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 9.04/5.11         => ( ! [A5: $o,B6: $o] :
% 9.04/5.11                ( ( X
% 9.04/5.11                  = ( vEBT_Leaf @ A5 @ B6 ) )
% 9.04/5.11               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa ) )
% 9.04/5.11                 => ( ( ( Xa = zero_zero_nat )
% 9.04/5.11                     => A5 )
% 9.04/5.11                    & ( ( Xa != zero_zero_nat )
% 9.04/5.11                     => ( ( ( Xa = one_one_nat )
% 9.04/5.11                         => B6 )
% 9.04/5.11                        & ( Xa = one_one_nat ) ) ) ) ) )
% 9.04/5.11           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.04/5.11                  ( ( X
% 9.04/5.11                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.04/5.11                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
% 9.04/5.11             => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.04/5.11                    ( ( X
% 9.04/5.11                      = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 9.04/5.11                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 9.04/5.11                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.04/5.11                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.11                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9193_diffs__sin__coeff,axiom,
% 9.04/5.11      ( ( diffs_real @ sin_coeff )
% 9.04/5.11      = cos_coeff ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9194_diffs__cos__coeff,axiom,
% 9.04/5.11      ( ( diffs_real @ cos_coeff )
% 9.04/5.11      = ( ^ [N3: nat] : ( uminus_uminus_real @ ( sin_coeff @ N3 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9195_Arg__def,axiom,
% 9.04/5.11      ( arg
% 9.04/5.11      = ( ^ [Z4: complex] :
% 9.04/5.11            ( if_real @ ( Z4 = zero_zero_complex ) @ zero_zero_real
% 9.04/5.11            @ ( fChoice_real
% 9.04/5.11              @ ^ [A3: real] :
% 9.04/5.11                  ( ( ( sgn_sgn_complex @ Z4 )
% 9.04/5.11                    = ( cis @ A3 ) )
% 9.04/5.11                  & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
% 9.04/5.11                  & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9196_set__vebt__def,axiom,
% 9.04/5.11      ( vEBT_set_vebt
% 9.04/5.11      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9197_atMost__0,axiom,
% 9.04/5.11      ( ( set_ord_atMost_nat @ zero_zero_nat )
% 9.04/5.11      = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9198_sum__choose__upper,axiom,
% 9.04/5.11      ! [M: nat,N: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [K2: nat] : ( binomial @ K2 @ M )
% 9.04/5.11          @ ( set_ord_atMost_nat @ N ) )
% 9.04/5.11        = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9199_sum__choose__lower,axiom,
% 9.04/5.11      ! [R2: nat,N: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K2 ) @ K2 )
% 9.04/5.11          @ ( set_ord_atMost_nat @ N ) )
% 9.04/5.11        = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9200_choose__rising__sum_I1_J,axiom,
% 9.04/5.11      ! [N: nat,M: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N @ J2 ) @ N )
% 9.04/5.11          @ ( set_ord_atMost_nat @ M ) )
% 9.04/5.11        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9201_choose__rising__sum_I2_J,axiom,
% 9.04/5.11      ! [N: nat,M: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N @ J2 ) @ N )
% 9.04/5.11          @ ( set_ord_atMost_nat @ M ) )
% 9.04/5.11        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9202_sum__choose__diagonal,axiom,
% 9.04/5.11      ! [M: nat,N: nat] :
% 9.04/5.11        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.11       => ( ( groups3542108847815614940at_nat
% 9.04/5.11            @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
% 9.04/5.11            @ ( set_ord_atMost_nat @ M ) )
% 9.04/5.11          = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9203_vandermonde,axiom,
% 9.04/5.11      ! [M: nat,N: nat,R2: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K2 ) ) )
% 9.04/5.11          @ ( set_ord_atMost_nat @ R2 ) )
% 9.04/5.11        = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9204_atMost__Suc,axiom,
% 9.04/5.11      ! [K: nat] :
% 9.04/5.11        ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 9.04/5.11        = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9205_choose__row__sum,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 9.04/5.11        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9206_binomial,axiom,
% 9.04/5.11      ! [A: nat,B: nat,N: nat] :
% 9.04/5.11        ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 9.04/5.11        = ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 9.04/5.11          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9207_polynomial__product__nat,axiom,
% 9.04/5.11      ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
% 9.04/5.11        ( ! [I2: nat] :
% 9.04/5.11            ( ( ord_less_nat @ M @ I2 )
% 9.04/5.11           => ( ( A @ I2 )
% 9.04/5.11              = zero_zero_nat ) )
% 9.04/5.11       => ( ! [J3: nat] :
% 9.04/5.11              ( ( ord_less_nat @ N @ J3 )
% 9.04/5.11             => ( ( B @ J3 )
% 9.04/5.11                = zero_zero_nat ) )
% 9.04/5.11         => ( ( times_times_nat
% 9.04/5.11              @ ( groups3542108847815614940at_nat
% 9.04/5.11                @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
% 9.04/5.11                @ ( set_ord_atMost_nat @ M ) )
% 9.04/5.11              @ ( groups3542108847815614940at_nat
% 9.04/5.11                @ ^ [J2: nat] : ( times_times_nat @ ( B @ J2 ) @ ( power_power_nat @ X @ J2 ) )
% 9.04/5.11                @ ( set_ord_atMost_nat @ N ) ) )
% 9.04/5.11            = ( groups3542108847815614940at_nat
% 9.04/5.11              @ ^ [R5: nat] :
% 9.04/5.11                  ( times_times_nat
% 9.04/5.11                  @ ( groups3542108847815614940at_nat
% 9.04/5.11                    @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 9.04/5.11                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.04/5.11                  @ ( power_power_nat @ X @ R5 ) )
% 9.04/5.11              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9208_choose__square__sum,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.11          @ ( set_ord_atMost_nat @ N ) )
% 9.04/5.11        = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9209_atMost__nat__numeral,axiom,
% 9.04/5.11      ! [K: num] :
% 9.04/5.11        ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 9.04/5.11        = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9210_binomial__r__part__sum,axiom,
% 9.04/5.11      ! [M: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 9.04/5.11        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9211_choose__linear__sum,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N @ I3 ) )
% 9.04/5.11          @ ( set_ord_atMost_nat @ N ) )
% 9.04/5.11        = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9212_atMost__Suc__eq__insert__0,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 9.04/5.11        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9213_mask__eq__sum__exp__nat,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 9.04/5.11        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.11          @ ( collect_nat
% 9.04/5.11            @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9214_sum__nth__roots,axiom,
% 9.04/5.11      ! [N: nat,C: complex] :
% 9.04/5.11        ( ( ord_less_nat @ one_one_nat @ N )
% 9.04/5.11       => ( ( groups7754918857620584856omplex
% 9.04/5.11            @ ^ [X4: complex] : X4
% 9.04/5.11            @ ( collect_complex
% 9.04/5.11              @ ^ [Z4: complex] :
% 9.04/5.11                  ( ( power_power_complex @ Z4 @ N )
% 9.04/5.11                  = C ) ) )
% 9.04/5.11          = zero_zero_complex ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9215_sum__roots__unity,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( ord_less_nat @ one_one_nat @ N )
% 9.04/5.11       => ( ( groups7754918857620584856omplex
% 9.04/5.11            @ ^ [X4: complex] : X4
% 9.04/5.11            @ ( collect_complex
% 9.04/5.11              @ ^ [Z4: complex] :
% 9.04/5.11                  ( ( power_power_complex @ Z4 @ N )
% 9.04/5.11                  = one_one_complex ) ) )
% 9.04/5.11          = zero_zero_complex ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9216_Maclaurin__minus__cos__expansion,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ X @ zero_zero_real )
% 9.04/5.11         => ? [T3: real] :
% 9.04/5.11              ( ( ord_less_real @ X @ T3 )
% 9.04/5.11              & ( ord_less_real @ T3 @ zero_zero_real )
% 9.04/5.11              & ( ( cos_real @ X )
% 9.04/5.11                = ( plus_plus_real
% 9.04/5.11                  @ ( groups6591440286371151544t_real
% 9.04/5.11                    @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.11                    @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9217_Maclaurin__cos__expansion2,axiom,
% 9.04/5.11      ! [X: real,N: nat] :
% 9.04/5.11        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.11       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11         => ? [T3: real] :
% 9.04/5.11              ( ( ord_less_real @ zero_zero_real @ T3 )
% 9.04/5.11              & ( ord_less_real @ T3 @ X )
% 9.04/5.11              & ( ( cos_real @ X )
% 9.04/5.11                = ( plus_plus_real
% 9.04/5.11                  @ ( groups6591440286371151544t_real
% 9.04/5.11                    @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.11                    @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9218_Maclaurin__sin__expansion3,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.11         => ? [T3: real] :
% 9.04/5.11              ( ( ord_less_real @ zero_zero_real @ T3 )
% 9.04/5.11              & ( ord_less_real @ T3 @ X )
% 9.04/5.11              & ( ( sin_real @ X )
% 9.04/5.11                = ( plus_plus_real
% 9.04/5.11                  @ ( groups6591440286371151544t_real
% 9.04/5.11                    @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.11                    @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11                  @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9219_Maclaurin__sin__expansion4,axiom,
% 9.04/5.11      ! [X: real,N: nat] :
% 9.04/5.11        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.11       => ? [T3: real] :
% 9.04/5.11            ( ( ord_less_real @ zero_zero_real @ T3 )
% 9.04/5.11            & ( ord_less_eq_real @ T3 @ X )
% 9.04/5.11            & ( ( sin_real @ X )
% 9.04/5.11              = ( plus_plus_real
% 9.04/5.11                @ ( groups6591440286371151544t_real
% 9.04/5.11                  @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.11                  @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9220_lessThan__0,axiom,
% 9.04/5.11      ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 9.04/5.11      = bot_bot_set_nat ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9221_sumr__cos__zero__one,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( groups6591440286371151544t_real
% 9.04/5.11          @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
% 9.04/5.11          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.04/5.11        = one_one_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9222_lessThan__Suc__atMost,axiom,
% 9.04/5.11      ! [K: nat] :
% 9.04/5.11        ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 9.04/5.11        = ( set_ord_atMost_nat @ K ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9223_lessThan__Suc,axiom,
% 9.04/5.11      ! [K: nat] :
% 9.04/5.11        ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 9.04/5.11        = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9224_lessThan__empty__iff,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( ( set_ord_lessThan_nat @ N )
% 9.04/5.11          = bot_bot_set_nat )
% 9.04/5.11        = ( N = zero_zero_nat ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9225_lessThan__nat__numeral,axiom,
% 9.04/5.11      ! [K: num] :
% 9.04/5.11        ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 9.04/5.11        = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9226_lessThan__Suc__eq__insert__0,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 9.04/5.11        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9227_Maclaurin__lemma,axiom,
% 9.04/5.11      ! [H: real,F: real > real,J: nat > real,N: nat] :
% 9.04/5.11        ( ( ord_less_real @ zero_zero_real @ H )
% 9.04/5.11       => ? [B3: real] :
% 9.04/5.11            ( ( F @ H )
% 9.04/5.11            = ( plus_plus_real
% 9.04/5.11              @ ( groups6591440286371151544t_real
% 9.04/5.11                @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
% 9.04/5.11                @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11              @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ H @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9228_sum__split__even__odd,axiom,
% 9.04/5.11      ! [F: nat > real,G: nat > real,N: nat] :
% 9.04/5.11        ( ( groups6591440286371151544t_real
% 9.04/5.11          @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
% 9.04/5.11          @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.11        = ( plus_plus_real
% 9.04/5.11          @ ( groups6591440286371151544t_real
% 9.04/5.11            @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
% 9.04/5.11            @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11          @ ( groups6591440286371151544t_real
% 9.04/5.11            @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
% 9.04/5.11            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9229_Maclaurin__exp__le,axiom,
% 9.04/5.11      ! [X: real,N: nat] :
% 9.04/5.11      ? [T3: real] :
% 9.04/5.11        ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 9.04/5.11        & ( ( exp_real @ X )
% 9.04/5.11          = ( plus_plus_real
% 9.04/5.11            @ ( groups6591440286371151544t_real
% 9.04/5.11              @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 9.04/5.11              @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11            @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9230_Maclaurin__sin__bound,axiom,
% 9.04/5.11      ! [X: real,N: nat] :
% 9.04/5.11        ( ord_less_eq_real
% 9.04/5.11        @ ( abs_abs_real
% 9.04/5.11          @ ( minus_minus_real @ ( sin_real @ X )
% 9.04/5.11            @ ( groups6591440286371151544t_real
% 9.04/5.11              @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.11              @ ( set_ord_lessThan_nat @ N ) ) ) )
% 9.04/5.11        @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9231_sum__pos__lt__pair,axiom,
% 9.04/5.11      ! [F: nat > real,K: nat] :
% 9.04/5.11        ( ( summable_real @ F )
% 9.04/5.11       => ( ! [D5: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) @ one_one_nat ) ) ) ) )
% 9.04/5.11         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9232_Maclaurin__exp__lt,axiom,
% 9.04/5.11      ! [X: real,N: nat] :
% 9.04/5.11        ( ( X != zero_zero_real )
% 9.04/5.11       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11         => ? [T3: real] :
% 9.04/5.11              ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 9.04/5.11              & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 9.04/5.11              & ( ( exp_real @ X )
% 9.04/5.11                = ( plus_plus_real
% 9.04/5.11                  @ ( groups6591440286371151544t_real
% 9.04/5.11                    @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 9.04/5.11                    @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11                  @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9233_Maclaurin__sin__expansion,axiom,
% 9.04/5.11      ! [X: real,N: nat] :
% 9.04/5.11      ? [T3: real] :
% 9.04/5.11        ( ( sin_real @ X )
% 9.04/5.11        = ( plus_plus_real
% 9.04/5.11          @ ( groups6591440286371151544t_real
% 9.04/5.11            @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.11            @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9234_Maclaurin__sin__expansion2,axiom,
% 9.04/5.11      ! [X: real,N: nat] :
% 9.04/5.11      ? [T3: real] :
% 9.04/5.11        ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 9.04/5.11        & ( ( sin_real @ X )
% 9.04/5.11          = ( plus_plus_real
% 9.04/5.11            @ ( groups6591440286371151544t_real
% 9.04/5.11              @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.11              @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11            @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9235_Maclaurin__cos__expansion,axiom,
% 9.04/5.11      ! [X: real,N: nat] :
% 9.04/5.11      ? [T3: real] :
% 9.04/5.11        ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 9.04/5.11        & ( ( cos_real @ X )
% 9.04/5.11          = ( plus_plus_real
% 9.04/5.11            @ ( groups6591440286371151544t_real
% 9.04/5.11              @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.11              @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11            @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9236_bij__betw__roots__unity,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( bij_betw_nat_complex
% 9.04/5.11          @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.04/5.11          @ ( set_ord_lessThan_nat @ N )
% 9.04/5.11          @ ( collect_complex
% 9.04/5.11            @ ^ [Z4: complex] :
% 9.04/5.11                ( ( power_power_complex @ Z4 @ N )
% 9.04/5.11                = one_one_complex ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9237_image__Suc__atLeastAtMost,axiom,
% 9.04/5.11      ! [I: nat,J: nat] :
% 9.04/5.11        ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 9.04/5.11        = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9238_all__nat__less,axiom,
% 9.04/5.11      ! [N: nat,P: nat > $o] :
% 9.04/5.11        ( ( ! [M6: nat] :
% 9.04/5.11              ( ( ord_less_eq_nat @ M6 @ N )
% 9.04/5.11             => ( P @ M6 ) ) )
% 9.04/5.11        = ( ! [X4: nat] :
% 9.04/5.11              ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.04/5.11             => ( P @ X4 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9239_ex__nat__less,axiom,
% 9.04/5.11      ! [N: nat,P: nat > $o] :
% 9.04/5.11        ( ( ? [M6: nat] :
% 9.04/5.11              ( ( ord_less_eq_nat @ M6 @ N )
% 9.04/5.11              & ( P @ M6 ) ) )
% 9.04/5.11        = ( ? [X4: nat] :
% 9.04/5.11              ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.04/5.11              & ( P @ X4 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9240_atMost__atLeast0,axiom,
% 9.04/5.11      ( set_ord_atMost_nat
% 9.04/5.11      = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9241_atLeast0__atMost__Suc,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.04/5.11        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9242_atLeastAtMost__insertL,axiom,
% 9.04/5.11      ! [M: nat,N: nat] :
% 9.04/5.11        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.11       => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.04/5.11          = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9243_atLeastAtMostSuc__conv,axiom,
% 9.04/5.11      ! [M: nat,N: nat] :
% 9.04/5.11        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.04/5.11       => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 9.04/5.11          = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9244_Icc__eq__insert__lb__nat,axiom,
% 9.04/5.11      ! [M: nat,N: nat] :
% 9.04/5.11        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.11       => ( ( set_or1269000886237332187st_nat @ M @ N )
% 9.04/5.11          = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9245_image__Suc__lessThan,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.11        = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9246_image__Suc__atMost,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 9.04/5.11        = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9247_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.04/5.11        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9248_atLeast1__atMost__eq__remove0,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.04/5.11        = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9249_gauss__sum__nat,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [X4: nat] : X4
% 9.04/5.11          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.04/5.11        = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9250_arith__series__nat,axiom,
% 9.04/5.11      ! [A: nat,D: nat,N: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
% 9.04/5.11          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.04/5.11        = ( 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 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9251_Sum__Icc__nat,axiom,
% 9.04/5.11      ! [M: nat,N: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [X4: nat] : X4
% 9.04/5.11          @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.04/5.11        = ( 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 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9252_vebt__buildup_Opelims,axiom,
% 9.04/5.11      ! [X: nat,Y: vEBT_VEBT] :
% 9.04/5.11        ( ( ( vEBT_vebt_buildup @ X )
% 9.04/5.11          = Y )
% 9.04/5.11       => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 9.04/5.11         => ( ( ( X = zero_zero_nat )
% 9.04/5.11             => ( ( Y
% 9.04/5.11                  = ( vEBT_Leaf @ $false @ $false ) )
% 9.04/5.11               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 9.04/5.11           => ( ( ( X
% 9.04/5.11                  = ( suc @ zero_zero_nat ) )
% 9.04/5.11               => ( ( Y
% 9.04/5.11                    = ( vEBT_Leaf @ $false @ $false ) )
% 9.04/5.11                 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 9.04/5.11             => ~ ! [Va: nat] :
% 9.04/5.11                    ( ( X
% 9.04/5.11                      = ( suc @ ( suc @ Va ) ) )
% 9.04/5.11                   => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.04/5.11                         => ( Y
% 9.04/5.11                            = ( 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 ) ) ) ) ) ) )
% 9.04/5.11                        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.04/5.11                         => ( Y
% 9.04/5.11                            = ( 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 ) ) ) ) ) ) ) ) )
% 9.04/5.11                     => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9253_arctan__def,axiom,
% 9.04/5.11      ( arctan
% 9.04/5.11      = ( ^ [Y6: real] :
% 9.04/5.11            ( the_real
% 9.04/5.11            @ ^ [X4: real] :
% 9.04/5.11                ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 9.04/5.11                & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.11                & ( ( tan_real @ X4 )
% 9.04/5.11                  = Y6 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9254_bij__betw__Suc,axiom,
% 9.04/5.11      ! [M2: set_nat,N4: set_nat] :
% 9.04/5.11        ( ( bij_betw_nat_nat @ suc @ M2 @ N4 )
% 9.04/5.11        = ( ( image_nat_nat @ suc @ M2 )
% 9.04/5.11          = N4 ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9255_image__int__atLeastAtMost,axiom,
% 9.04/5.11      ! [A: nat,B: nat] :
% 9.04/5.11        ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 9.04/5.11        = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9256_ln__real__def,axiom,
% 9.04/5.11      ( ln_ln_real
% 9.04/5.11      = ( ^ [X4: real] :
% 9.04/5.11            ( the_real
% 9.04/5.11            @ ^ [U2: real] :
% 9.04/5.11                ( ( exp_real @ U2 )
% 9.04/5.11                = X4 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9257_bset_I1_J,axiom,
% 9.04/5.11      ! [D2: int,B2: set_int,P: int > $o,Q: int > $o] :
% 9.04/5.11        ( ! [X3: int] :
% 9.04/5.11            ( ! [Xa2: int] :
% 9.04/5.11                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb2: int] :
% 9.04/5.11                    ( ( member_int @ Xb2 @ B2 )
% 9.04/5.11                   => ( X3
% 9.04/5.11                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 9.04/5.11           => ( ( P @ X3 )
% 9.04/5.11             => ( P @ ( minus_minus_int @ X3 @ D2 ) ) ) )
% 9.04/5.11       => ( ! [X3: int] :
% 9.04/5.11              ( ! [Xa2: int] :
% 9.04/5.11                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb2: int] :
% 9.04/5.11                      ( ( member_int @ Xb2 @ B2 )
% 9.04/5.11                     => ( X3
% 9.04/5.11                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 9.04/5.11             => ( ( Q @ X3 )
% 9.04/5.11               => ( Q @ ( minus_minus_int @ X3 @ D2 ) ) ) )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ B2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( ( P @ X2 )
% 9.04/5.11                  & ( Q @ X2 ) )
% 9.04/5.11               => ( ( P @ ( minus_minus_int @ X2 @ D2 ) )
% 9.04/5.11                  & ( Q @ ( minus_minus_int @ X2 @ D2 ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9258_bset_I2_J,axiom,
% 9.04/5.11      ! [D2: int,B2: set_int,P: int > $o,Q: int > $o] :
% 9.04/5.11        ( ! [X3: int] :
% 9.04/5.11            ( ! [Xa2: int] :
% 9.04/5.11                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb2: int] :
% 9.04/5.11                    ( ( member_int @ Xb2 @ B2 )
% 9.04/5.11                   => ( X3
% 9.04/5.11                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 9.04/5.11           => ( ( P @ X3 )
% 9.04/5.11             => ( P @ ( minus_minus_int @ X3 @ D2 ) ) ) )
% 9.04/5.11       => ( ! [X3: int] :
% 9.04/5.11              ( ! [Xa2: int] :
% 9.04/5.11                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb2: int] :
% 9.04/5.11                      ( ( member_int @ Xb2 @ B2 )
% 9.04/5.11                     => ( X3
% 9.04/5.11                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 9.04/5.11             => ( ( Q @ X3 )
% 9.04/5.11               => ( Q @ ( minus_minus_int @ X3 @ D2 ) ) ) )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ B2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( ( P @ X2 )
% 9.04/5.11                  | ( Q @ X2 ) )
% 9.04/5.11               => ( ( P @ ( minus_minus_int @ X2 @ D2 ) )
% 9.04/5.11                  | ( Q @ ( minus_minus_int @ X2 @ D2 ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9259_aset_I1_J,axiom,
% 9.04/5.11      ! [D2: int,A2: set_int,P: int > $o,Q: int > $o] :
% 9.04/5.11        ( ! [X3: int] :
% 9.04/5.11            ( ! [Xa2: int] :
% 9.04/5.11                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb2: int] :
% 9.04/5.11                    ( ( member_int @ Xb2 @ A2 )
% 9.04/5.11                   => ( X3
% 9.04/5.11                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 9.04/5.11           => ( ( P @ X3 )
% 9.04/5.11             => ( P @ ( plus_plus_int @ X3 @ D2 ) ) ) )
% 9.04/5.11       => ( ! [X3: int] :
% 9.04/5.11              ( ! [Xa2: int] :
% 9.04/5.11                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb2: int] :
% 9.04/5.11                      ( ( member_int @ Xb2 @ A2 )
% 9.04/5.11                     => ( X3
% 9.04/5.11                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 9.04/5.11             => ( ( Q @ X3 )
% 9.04/5.11               => ( Q @ ( plus_plus_int @ X3 @ D2 ) ) ) )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ A2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( ( P @ X2 )
% 9.04/5.11                  & ( Q @ X2 ) )
% 9.04/5.11               => ( ( P @ ( plus_plus_int @ X2 @ D2 ) )
% 9.04/5.11                  & ( Q @ ( plus_plus_int @ X2 @ D2 ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9260_aset_I2_J,axiom,
% 9.04/5.11      ! [D2: int,A2: set_int,P: int > $o,Q: int > $o] :
% 9.04/5.11        ( ! [X3: int] :
% 9.04/5.11            ( ! [Xa2: int] :
% 9.04/5.11                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb2: int] :
% 9.04/5.11                    ( ( member_int @ Xb2 @ A2 )
% 9.04/5.11                   => ( X3
% 9.04/5.11                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 9.04/5.11           => ( ( P @ X3 )
% 9.04/5.11             => ( P @ ( plus_plus_int @ X3 @ D2 ) ) ) )
% 9.04/5.11       => ( ! [X3: int] :
% 9.04/5.11              ( ! [Xa2: int] :
% 9.04/5.11                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb2: int] :
% 9.04/5.11                      ( ( member_int @ Xb2 @ A2 )
% 9.04/5.11                     => ( X3
% 9.04/5.11                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 9.04/5.11             => ( ( Q @ X3 )
% 9.04/5.11               => ( Q @ ( plus_plus_int @ X3 @ D2 ) ) ) )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ A2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( ( P @ X2 )
% 9.04/5.11                  | ( Q @ X2 ) )
% 9.04/5.11               => ( ( P @ ( plus_plus_int @ X2 @ D2 ) )
% 9.04/5.11                  | ( Q @ ( plus_plus_int @ X2 @ D2 ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9261_ln__neg__is__const,axiom,
% 9.04/5.11      ! [X: real] :
% 9.04/5.11        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.04/5.11       => ( ( ln_ln_real @ X )
% 9.04/5.11          = ( the_real
% 9.04/5.11            @ ^ [X4: real] : $false ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9262_bset_I9_J,axiom,
% 9.04/5.11      ! [D: int,D2: int,B2: set_int,T: int] :
% 9.04/5.11        ( ( dvd_dvd_int @ D @ D2 )
% 9.04/5.11       => ! [X2: int] :
% 9.04/5.11            ( ! [Xa3: int] :
% 9.04/5.11                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb3: int] :
% 9.04/5.11                    ( ( member_int @ Xb3 @ B2 )
% 9.04/5.11                   => ( X2
% 9.04/5.11                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11           => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 9.04/5.11             => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9263_bset_I10_J,axiom,
% 9.04/5.11      ! [D: int,D2: int,B2: set_int,T: int] :
% 9.04/5.11        ( ( dvd_dvd_int @ D @ D2 )
% 9.04/5.11       => ! [X2: int] :
% 9.04/5.11            ( ! [Xa3: int] :
% 9.04/5.11                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb3: int] :
% 9.04/5.11                    ( ( member_int @ Xb3 @ B2 )
% 9.04/5.11                   => ( X2
% 9.04/5.11                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11           => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 9.04/5.11             => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9264_aset_I9_J,axiom,
% 9.04/5.11      ! [D: int,D2: int,A2: set_int,T: int] :
% 9.04/5.11        ( ( dvd_dvd_int @ D @ D2 )
% 9.04/5.11       => ! [X2: int] :
% 9.04/5.11            ( ! [Xa3: int] :
% 9.04/5.11                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb3: int] :
% 9.04/5.11                    ( ( member_int @ Xb3 @ A2 )
% 9.04/5.11                   => ( X2
% 9.04/5.11                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11           => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 9.04/5.11             => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9265_aset_I10_J,axiom,
% 9.04/5.11      ! [D: int,D2: int,A2: set_int,T: int] :
% 9.04/5.11        ( ( dvd_dvd_int @ D @ D2 )
% 9.04/5.11       => ! [X2: int] :
% 9.04/5.11            ( ! [Xa3: int] :
% 9.04/5.11                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb3: int] :
% 9.04/5.11                    ( ( member_int @ Xb3 @ A2 )
% 9.04/5.11                   => ( X2
% 9.04/5.11                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11           => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 9.04/5.11             => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9266_atLeastAtMostPlus1__int__conv,axiom,
% 9.04/5.11      ! [M: int,N: int] :
% 9.04/5.11        ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 9.04/5.11       => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 9.04/5.11          = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9267_periodic__finite__ex,axiom,
% 9.04/5.11      ! [D: int,P: int > $o] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D )
% 9.04/5.11       => ( ! [X3: int,K3: int] :
% 9.04/5.11              ( ( P @ X3 )
% 9.04/5.11              = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 9.04/5.11         => ( ( ? [X6: int] : ( P @ X6 ) )
% 9.04/5.11            = ( ? [X4: int] :
% 9.04/5.11                  ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 9.04/5.11                  & ( P @ X4 ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9268_aset_I7_J,axiom,
% 9.04/5.11      ! [D2: int,A2: set_int,T: int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ! [X2: int] :
% 9.04/5.11            ( ! [Xa3: int] :
% 9.04/5.11                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb3: int] :
% 9.04/5.11                    ( ( member_int @ Xb3 @ A2 )
% 9.04/5.11                   => ( X2
% 9.04/5.11                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11           => ( ( ord_less_int @ T @ X2 )
% 9.04/5.11             => ( ord_less_int @ T @ ( plus_plus_int @ X2 @ D2 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9269_aset_I5_J,axiom,
% 9.04/5.11      ! [D2: int,T: int,A2: set_int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ( ( member_int @ T @ A2 )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ A2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( ord_less_int @ X2 @ T )
% 9.04/5.11               => ( ord_less_int @ ( plus_plus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9270_aset_I4_J,axiom,
% 9.04/5.11      ! [D2: int,T: int,A2: set_int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ( ( member_int @ T @ A2 )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ A2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( X2 != T )
% 9.04/5.11               => ( ( plus_plus_int @ X2 @ D2 )
% 9.04/5.11                 != T ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9271_aset_I3_J,axiom,
% 9.04/5.11      ! [D2: int,T: int,A2: set_int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ A2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( X2 = T )
% 9.04/5.11               => ( ( plus_plus_int @ X2 @ D2 )
% 9.04/5.11                  = T ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9272_bset_I7_J,axiom,
% 9.04/5.11      ! [D2: int,T: int,B2: set_int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ( ( member_int @ T @ B2 )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ B2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( ord_less_int @ T @ X2 )
% 9.04/5.11               => ( ord_less_int @ T @ ( minus_minus_int @ X2 @ D2 ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9273_bset_I5_J,axiom,
% 9.04/5.11      ! [D2: int,B2: set_int,T: int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ! [X2: int] :
% 9.04/5.11            ( ! [Xa3: int] :
% 9.04/5.11                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb3: int] :
% 9.04/5.11                    ( ( member_int @ Xb3 @ B2 )
% 9.04/5.11                   => ( X2
% 9.04/5.11                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11           => ( ( ord_less_int @ X2 @ T )
% 9.04/5.11             => ( ord_less_int @ ( minus_minus_int @ X2 @ D2 ) @ T ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9274_bset_I4_J,axiom,
% 9.04/5.11      ! [D2: int,T: int,B2: set_int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ( ( member_int @ T @ B2 )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ B2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( X2 != T )
% 9.04/5.11               => ( ( minus_minus_int @ X2 @ D2 )
% 9.04/5.11                 != T ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9275_bset_I3_J,axiom,
% 9.04/5.11      ! [D2: int,T: int,B2: set_int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ B2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( X2 = T )
% 9.04/5.11               => ( ( minus_minus_int @ X2 @ D2 )
% 9.04/5.11                  = T ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9276_fact__eq__fact__times,axiom,
% 9.04/5.11      ! [N: nat,M: nat] :
% 9.04/5.11        ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.11       => ( ( semiri1408675320244567234ct_nat @ M )
% 9.04/5.11          = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 9.04/5.11            @ ( groups708209901874060359at_nat
% 9.04/5.11              @ ^ [X4: nat] : X4
% 9.04/5.11              @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9277_simp__from__to,axiom,
% 9.04/5.11      ( set_or1266510415728281911st_int
% 9.04/5.11      = ( ^ [I3: int,J2: int] : ( if_set_int @ ( ord_less_int @ J2 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9278_bset_I6_J,axiom,
% 9.04/5.11      ! [D2: int,B2: set_int,T: int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ! [X2: int] :
% 9.04/5.11            ( ! [Xa3: int] :
% 9.04/5.11                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb3: int] :
% 9.04/5.11                    ( ( member_int @ Xb3 @ B2 )
% 9.04/5.11                   => ( X2
% 9.04/5.11                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11           => ( ( ord_less_eq_int @ X2 @ T )
% 9.04/5.11             => ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D2 ) @ T ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9279_bset_I8_J,axiom,
% 9.04/5.11      ! [D2: int,T: int,B2: set_int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ B2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( ord_less_eq_int @ T @ X2 )
% 9.04/5.11               => ( ord_less_eq_int @ T @ ( minus_minus_int @ X2 @ D2 ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9280_aset_I6_J,axiom,
% 9.04/5.11      ! [D2: int,T: int,A2: set_int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 9.04/5.11         => ! [X2: int] :
% 9.04/5.11              ( ! [Xa3: int] :
% 9.04/5.11                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                 => ! [Xb3: int] :
% 9.04/5.11                      ( ( member_int @ Xb3 @ A2 )
% 9.04/5.11                     => ( X2
% 9.04/5.11                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11             => ( ( ord_less_eq_int @ X2 @ T )
% 9.04/5.11               => ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D2 ) @ T ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9281_aset_I8_J,axiom,
% 9.04/5.11      ! [D2: int,A2: set_int,T: int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ! [X2: int] :
% 9.04/5.11            ( ! [Xa3: int] :
% 9.04/5.11                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11               => ! [Xb3: int] :
% 9.04/5.11                    ( ( member_int @ Xb3 @ A2 )
% 9.04/5.11                   => ( X2
% 9.04/5.11                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.04/5.11           => ( ( ord_less_eq_int @ T @ X2 )
% 9.04/5.11             => ( ord_less_eq_int @ T @ ( plus_plus_int @ X2 @ D2 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9282_cpmi,axiom,
% 9.04/5.11      ! [D2: int,P: int > $o,P6: int > $o,B2: set_int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ( ? [Z5: int] :
% 9.04/5.11            ! [X3: int] :
% 9.04/5.11              ( ( ord_less_int @ X3 @ Z5 )
% 9.04/5.11             => ( ( P @ X3 )
% 9.04/5.11                = ( P6 @ X3 ) ) )
% 9.04/5.11         => ( ! [X3: int] :
% 9.04/5.11                ( ! [Xa2: int] :
% 9.04/5.11                    ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                   => ! [Xb2: int] :
% 9.04/5.11                        ( ( member_int @ Xb2 @ B2 )
% 9.04/5.11                       => ( X3
% 9.04/5.11                         != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 9.04/5.11               => ( ( P @ X3 )
% 9.04/5.11                 => ( P @ ( minus_minus_int @ X3 @ D2 ) ) ) )
% 9.04/5.11           => ( ! [X3: int,K3: int] :
% 9.04/5.11                  ( ( P6 @ X3 )
% 9.04/5.11                  = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D2 ) ) ) )
% 9.04/5.11             => ( ( ? [X6: int] : ( P @ X6 ) )
% 9.04/5.11                = ( ? [X4: int] :
% 9.04/5.11                      ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                      & ( P6 @ X4 ) )
% 9.04/5.11                  | ? [X4: int] :
% 9.04/5.11                      ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                      & ? [Y6: int] :
% 9.04/5.11                          ( ( member_int @ Y6 @ B2 )
% 9.04/5.11                          & ( P @ ( plus_plus_int @ Y6 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9283_cppi,axiom,
% 9.04/5.11      ! [D2: int,P: int > $o,P6: int > $o,A2: set_int] :
% 9.04/5.11        ( ( ord_less_int @ zero_zero_int @ D2 )
% 9.04/5.11       => ( ? [Z5: int] :
% 9.04/5.11            ! [X3: int] :
% 9.04/5.11              ( ( ord_less_int @ Z5 @ X3 )
% 9.04/5.11             => ( ( P @ X3 )
% 9.04/5.11                = ( P6 @ X3 ) ) )
% 9.04/5.11         => ( ! [X3: int] :
% 9.04/5.11                ( ! [Xa2: int] :
% 9.04/5.11                    ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                   => ! [Xb2: int] :
% 9.04/5.11                        ( ( member_int @ Xb2 @ A2 )
% 9.04/5.11                       => ( X3
% 9.04/5.11                         != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 9.04/5.11               => ( ( P @ X3 )
% 9.04/5.11                 => ( P @ ( plus_plus_int @ X3 @ D2 ) ) ) )
% 9.04/5.11           => ( ! [X3: int,K3: int] :
% 9.04/5.11                  ( ( P6 @ X3 )
% 9.04/5.11                  = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D2 ) ) ) )
% 9.04/5.11             => ( ( ? [X6: int] : ( P @ X6 ) )
% 9.04/5.11                = ( ? [X4: int] :
% 9.04/5.11                      ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                      & ( P6 @ X4 ) )
% 9.04/5.11                  | ? [X4: int] :
% 9.04/5.11                      ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 9.04/5.11                      & ? [Y6: int] :
% 9.04/5.11                          ( ( member_int @ Y6 @ A2 )
% 9.04/5.11                          & ( P @ ( minus_minus_int @ Y6 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9284_arccos__def,axiom,
% 9.04/5.11      ( arccos
% 9.04/5.11      = ( ^ [Y6: real] :
% 9.04/5.11            ( the_real
% 9.04/5.11            @ ^ [X4: real] :
% 9.04/5.11                ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 9.04/5.11                & ( ord_less_eq_real @ X4 @ pi )
% 9.04/5.11                & ( ( cos_real @ X4 )
% 9.04/5.11                  = Y6 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9285_fact__div__fact,axiom,
% 9.04/5.11      ! [N: nat,M: nat] :
% 9.04/5.11        ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.11       => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 9.04/5.11          = ( groups708209901874060359at_nat
% 9.04/5.11            @ ^ [X4: nat] : X4
% 9.04/5.11            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9286_divmod__step__nat__def,axiom,
% 9.04/5.11      ( unique5026877609467782581ep_nat
% 9.04/5.11      = ( ^ [L3: num] :
% 9.04/5.11            ( produc2626176000494625587at_nat
% 9.04/5.11            @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ 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 @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9287_divmod__step__int__def,axiom,
% 9.04/5.11      ( unique5024387138958732305ep_int
% 9.04/5.11      = ( ^ [L3: num] :
% 9.04/5.11            ( produc4245557441103728435nt_int
% 9.04/5.11            @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ 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 @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9288_pi__half,axiom,
% 9.04/5.11      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.11      = ( the_real
% 9.04/5.11        @ ^ [X4: real] :
% 9.04/5.11            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 9.04/5.11            & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.11            & ( ( cos_real @ X4 )
% 9.04/5.11              = zero_zero_real ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9289_pi__def,axiom,
% 9.04/5.11      ( pi
% 9.04/5.11      = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.04/5.11        @ ( the_real
% 9.04/5.11          @ ^ [X4: real] :
% 9.04/5.11              ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 9.04/5.11              & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.04/5.11              & ( ( cos_real @ X4 )
% 9.04/5.11                = zero_zero_real ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9290_Sum__Icc__int,axiom,
% 9.04/5.11      ! [M: int,N: int] :
% 9.04/5.11        ( ( ord_less_eq_int @ M @ N )
% 9.04/5.11       => ( ( groups4538972089207619220nt_int
% 9.04/5.11            @ ^ [X4: int] : X4
% 9.04/5.11            @ ( set_or1266510415728281911st_int @ M @ N ) )
% 9.04/5.11          = ( 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 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9291_arcsin__def,axiom,
% 9.04/5.11      ( arcsin
% 9.04/5.11      = ( ^ [Y6: real] :
% 9.04/5.11            ( the_real
% 9.04/5.11            @ ^ [X4: real] :
% 9.04/5.11                ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 9.04/5.11                & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.11                & ( ( sin_real @ X4 )
% 9.04/5.11                  = Y6 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9292_bij__betw__nth__root__unity,axiom,
% 9.04/5.11      ! [C: complex,N: nat] :
% 9.04/5.11        ( ( C != zero_zero_complex )
% 9.04/5.11       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11         => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 9.04/5.11            @ ( collect_complex
% 9.04/5.11              @ ^ [Z4: complex] :
% 9.04/5.11                  ( ( power_power_complex @ Z4 @ N )
% 9.04/5.11                  = one_one_complex ) )
% 9.04/5.11            @ ( collect_complex
% 9.04/5.11              @ ^ [Z4: complex] :
% 9.04/5.11                  ( ( power_power_complex @ Z4 @ N )
% 9.04/5.11                  = C ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9293_divmod__nat__if,axiom,
% 9.04/5.11      ( divmod_nat
% 9.04/5.11      = ( ^ [M6: nat,N3: nat] :
% 9.04/5.11            ( if_Pro6206227464963214023at_nat
% 9.04/5.11            @ ( ( N3 = zero_zero_nat )
% 9.04/5.11              | ( ord_less_nat @ M6 @ N3 ) )
% 9.04/5.11            @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
% 9.04/5.11            @ ( produc2626176000494625587at_nat
% 9.04/5.11              @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
% 9.04/5.11              @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9294_set__encode__def,axiom,
% 9.04/5.11      ( nat_set_encode
% 9.04/5.11      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9295_real__root__zero,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( root @ N @ zero_zero_real )
% 9.04/5.11        = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9296_real__root__Suc__0,axiom,
% 9.04/5.11      ! [X: real] :
% 9.04/5.11        ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 9.04/5.11        = X ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9297_root__0,axiom,
% 9.04/5.11      ! [X: real] :
% 9.04/5.11        ( ( root @ zero_zero_nat @ X )
% 9.04/5.11        = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9298_real__root__eq__iff,axiom,
% 9.04/5.11      ! [N: nat,X: real,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ( root @ N @ X )
% 9.04/5.11            = ( root @ N @ Y ) )
% 9.04/5.11          = ( X = Y ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9299_set__encode__empty,axiom,
% 9.04/5.11      ( ( nat_set_encode @ bot_bot_set_nat )
% 9.04/5.11      = zero_zero_nat ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9300_real__root__eq__0__iff,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ( root @ N @ X )
% 9.04/5.11            = zero_zero_real )
% 9.04/5.11          = ( X = zero_zero_real ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9301_real__root__less__iff,axiom,
% 9.04/5.11      ! [N: nat,X: real,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 9.04/5.11          = ( ord_less_real @ X @ Y ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9302_real__root__le__iff,axiom,
% 9.04/5.11      ! [N: nat,X: real,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 9.04/5.11          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9303_real__root__one,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( root @ N @ one_one_real )
% 9.04/5.11          = one_one_real ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9304_real__root__eq__1__iff,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ( root @ N @ X )
% 9.04/5.11            = one_one_real )
% 9.04/5.11          = ( X = one_one_real ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9305_real__root__lt__0__iff,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
% 9.04/5.11          = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9306_real__root__gt__0__iff,axiom,
% 9.04/5.11      ! [N: nat,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
% 9.04/5.11          = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9307_real__root__le__0__iff,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
% 9.04/5.11          = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9308_real__root__ge__0__iff,axiom,
% 9.04/5.11      ! [N: nat,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
% 9.04/5.11          = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9309_real__root__gt__1__iff,axiom,
% 9.04/5.11      ! [N: nat,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
% 9.04/5.11          = ( ord_less_real @ one_one_real @ Y ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9310_real__root__lt__1__iff,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
% 9.04/5.11          = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9311_real__root__ge__1__iff,axiom,
% 9.04/5.11      ! [N: nat,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
% 9.04/5.11          = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9312_real__root__le__1__iff,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
% 9.04/5.11          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9313_real__root__pow__pos2,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.11         => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 9.04/5.11            = X ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9314_real__root__divide,axiom,
% 9.04/5.11      ! [N: nat,X: real,Y: real] :
% 9.04/5.11        ( ( root @ N @ ( divide_divide_real @ X @ Y ) )
% 9.04/5.11        = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9315_real__root__pos__pos__le,axiom,
% 9.04/5.11      ! [X: real,N: nat] :
% 9.04/5.11        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.11       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9316_prod__int__eq,axiom,
% 9.04/5.11      ! [I: nat,J: nat] :
% 9.04/5.11        ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 9.04/5.11        = ( groups1705073143266064639nt_int
% 9.04/5.11          @ ^ [X4: int] : X4
% 9.04/5.11          @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9317_real__root__less__mono,axiom,
% 9.04/5.11      ! [N: nat,X: real,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ X @ Y )
% 9.04/5.11         => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9318_real__root__le__mono,axiom,
% 9.04/5.11      ! [N: nat,X: real,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_real @ X @ Y )
% 9.04/5.11         => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9319_real__root__power,axiom,
% 9.04/5.11      ! [N: nat,X: real,K: nat] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( root @ N @ ( power_power_real @ X @ K ) )
% 9.04/5.11          = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9320_real__root__abs,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( root @ N @ ( abs_abs_real @ X ) )
% 9.04/5.11          = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9321_sgn__root,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( sgn_sgn_real @ ( root @ N @ X ) )
% 9.04/5.11          = ( sgn_sgn_real @ X ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9322_prod__int__plus__eq,axiom,
% 9.04/5.11      ! [I: nat,J: nat] :
% 9.04/5.11        ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 9.04/5.11        = ( groups1705073143266064639nt_int
% 9.04/5.11          @ ^ [X4: int] : X4
% 9.04/5.11          @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9323_real__root__gt__zero,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.11         => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9324_real__root__strict__decreasing,axiom,
% 9.04/5.11      ! [N: nat,N4: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_nat @ N @ N4 )
% 9.04/5.11         => ( ( ord_less_real @ one_one_real @ X )
% 9.04/5.11           => ( ord_less_real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9325_sqrt__def,axiom,
% 9.04/5.11      ( sqrt
% 9.04/5.11      = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9326_root__abs__power,axiom,
% 9.04/5.11      ! [N: nat,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
% 9.04/5.11          = ( abs_abs_real @ Y ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9327_real__root__pos__pos,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.11         => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9328_real__root__strict__increasing,axiom,
% 9.04/5.11      ! [N: nat,N4: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_nat @ N @ N4 )
% 9.04/5.11         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.11           => ( ( ord_less_real @ X @ one_one_real )
% 9.04/5.11             => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9329_real__root__decreasing,axiom,
% 9.04/5.11      ! [N: nat,N4: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_nat @ N @ N4 )
% 9.04/5.11         => ( ( ord_less_eq_real @ one_one_real @ X )
% 9.04/5.11           => ( ord_less_eq_real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9330_real__root__pow__pos,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.11         => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 9.04/5.11            = X ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9331_real__root__power__cancel,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.11         => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 9.04/5.11            = X ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9332_real__root__pos__unique,axiom,
% 9.04/5.11      ! [N: nat,Y: real,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.04/5.11         => ( ( ( power_power_real @ Y @ N )
% 9.04/5.11              = X )
% 9.04/5.11           => ( ( root @ N @ X )
% 9.04/5.11              = Y ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9333_odd__real__root__power__cancel,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.11       => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 9.04/5.11          = X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9334_odd__real__root__unique,axiom,
% 9.04/5.11      ! [N: nat,Y: real,X: real] :
% 9.04/5.11        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.11       => ( ( ( power_power_real @ Y @ N )
% 9.04/5.11            = X )
% 9.04/5.11         => ( ( root @ N @ X )
% 9.04/5.11            = Y ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9335_odd__real__root__pow,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.11       => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 9.04/5.11          = X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9336_real__root__increasing,axiom,
% 9.04/5.11      ! [N: nat,N4: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_eq_nat @ N @ N4 )
% 9.04/5.11         => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.11           => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.11             => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9337_root__sgn__power,axiom,
% 9.04/5.11      ! [N: nat,Y: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
% 9.04/5.11          = Y ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9338_sgn__power__root,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
% 9.04/5.11          = X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9339_log__root,axiom,
% 9.04/5.11      ! [N: nat,A: real,B: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.04/5.11         => ( ( log @ B @ ( root @ N @ A ) )
% 9.04/5.11            = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9340_log__base__root,axiom,
% 9.04/5.11      ! [N: nat,B: real,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.04/5.11         => ( ( log @ ( root @ N @ B ) @ X )
% 9.04/5.11            = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9341_ln__root,axiom,
% 9.04/5.11      ! [N: nat,B: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.04/5.11         => ( ( ln_ln_real @ ( root @ N @ B ) )
% 9.04/5.11            = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9342_split__root,axiom,
% 9.04/5.11      ! [P: real > $o,N: nat,X: real] :
% 9.04/5.11        ( ( P @ ( root @ N @ X ) )
% 9.04/5.11        = ( ( ( N = zero_zero_nat )
% 9.04/5.11           => ( P @ zero_zero_real ) )
% 9.04/5.11          & ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11           => ! [Y6: real] :
% 9.04/5.11                ( ( ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 9.04/5.11                  = X )
% 9.04/5.11               => ( P @ Y6 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9343_divmod__nat__def,axiom,
% 9.04/5.11      ( divmod_nat
% 9.04/5.11      = ( ^ [M6: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N3 ) @ ( modulo_modulo_nat @ M6 @ N3 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9344_floor__real__def,axiom,
% 9.04/5.11      ( archim6058952711729229775r_real
% 9.04/5.11      = ( ^ [X4: real] :
% 9.04/5.11            ( the_int
% 9.04/5.11            @ ^ [Z4: int] :
% 9.04/5.11                ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X4 )
% 9.04/5.11                & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9345_root__powr__inverse,axiom,
% 9.04/5.11      ! [N: nat,X: real] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.11         => ( ( root @ N @ X )
% 9.04/5.11            = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9346_set__encode__insert,axiom,
% 9.04/5.11      ! [A2: set_nat,N: nat] :
% 9.04/5.11        ( ( finite_finite_nat @ A2 )
% 9.04/5.11       => ( ~ ( member_nat @ N @ A2 )
% 9.04/5.11         => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
% 9.04/5.11            = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9347_floor__rat__def,axiom,
% 9.04/5.11      ( archim3151403230148437115or_rat
% 9.04/5.11      = ( ^ [X4: rat] :
% 9.04/5.11            ( the_int
% 9.04/5.11            @ ^ [Z4: int] :
% 9.04/5.11                ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X4 )
% 9.04/5.11                & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9348_VEBT_Osize_I3_J,axiom,
% 9.04/5.11      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 9.04/5.11        ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 9.04/5.11        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9349_finite__M__bounded__by__nat,axiom,
% 9.04/5.11      ! [P: nat > $o,I: nat] :
% 9.04/5.11        ( finite_finite_nat
% 9.04/5.11        @ ( collect_nat
% 9.04/5.11          @ ^ [K2: nat] :
% 9.04/5.11              ( ( P @ K2 )
% 9.04/5.11              & ( ord_less_nat @ K2 @ I ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9350_finite__less__ub,axiom,
% 9.04/5.11      ! [F: nat > nat,U: nat] :
% 9.04/5.11        ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
% 9.04/5.11       => ( finite_finite_nat
% 9.04/5.11          @ ( collect_nat
% 9.04/5.11            @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9351_less__eq__rat__def,axiom,
% 9.04/5.11      ( ord_less_eq_rat
% 9.04/5.11      = ( ^ [X4: rat,Y6: rat] :
% 9.04/5.11            ( ( ord_less_rat @ X4 @ Y6 )
% 9.04/5.11            | ( X4 = Y6 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9352_finite__nat__set__iff__bounded__le,axiom,
% 9.04/5.11      ( finite_finite_nat
% 9.04/5.11      = ( ^ [N7: set_nat] :
% 9.04/5.11          ? [M6: nat] :
% 9.04/5.11          ! [X4: nat] :
% 9.04/5.11            ( ( member_nat @ X4 @ N7 )
% 9.04/5.11           => ( ord_less_eq_nat @ X4 @ M6 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9353_finite__nat__set__iff__bounded,axiom,
% 9.04/5.11      ( finite_finite_nat
% 9.04/5.11      = ( ^ [N7: set_nat] :
% 9.04/5.11          ? [M6: nat] :
% 9.04/5.11          ! [X4: nat] :
% 9.04/5.11            ( ( member_nat @ X4 @ N7 )
% 9.04/5.11           => ( ord_less_nat @ X4 @ M6 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9354_bounded__nat__set__is__finite,axiom,
% 9.04/5.11      ! [N4: set_nat,N: nat] :
% 9.04/5.11        ( ! [X3: nat] :
% 9.04/5.11            ( ( member_nat @ X3 @ N4 )
% 9.04/5.11           => ( ord_less_nat @ X3 @ N ) )
% 9.04/5.11       => ( finite_finite_nat @ N4 ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9355_sgn__rat__def,axiom,
% 9.04/5.11      ( sgn_sgn_rat
% 9.04/5.11      = ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9356_abs__rat__def,axiom,
% 9.04/5.11      ( abs_abs_rat
% 9.04/5.11      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9357_obtain__pos__sum,axiom,
% 9.04/5.11      ! [R2: rat] :
% 9.04/5.11        ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 9.04/5.11       => ~ ! [S3: rat] :
% 9.04/5.11              ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 9.04/5.11             => ! [T3: rat] :
% 9.04/5.11                  ( ( ord_less_rat @ zero_zero_rat @ T3 )
% 9.04/5.11                 => ( R2
% 9.04/5.11                   != ( plus_plus_rat @ S3 @ T3 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9358_set__encode__inf,axiom,
% 9.04/5.11      ! [A2: set_nat] :
% 9.04/5.11        ( ~ ( finite_finite_nat @ A2 )
% 9.04/5.11       => ( ( nat_set_encode @ A2 )
% 9.04/5.11          = zero_zero_nat ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9359_finite__divisors__nat,axiom,
% 9.04/5.11      ! [M: nat] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.11       => ( finite_finite_nat
% 9.04/5.11          @ ( collect_nat
% 9.04/5.11            @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9360_subset__eq__atLeast0__atMost__finite,axiom,
% 9.04/5.11      ! [N4: set_nat,N: nat] :
% 9.04/5.11        ( ( ord_less_eq_set_nat @ N4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.04/5.11       => ( finite_finite_nat @ N4 ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9361_even__set__encode__iff,axiom,
% 9.04/5.11      ! [A2: set_nat] :
% 9.04/5.11        ( ( finite_finite_nat @ A2 )
% 9.04/5.11       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 9.04/5.11          = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9362_finite__Collect__le__nat,axiom,
% 9.04/5.11      ! [K: nat] :
% 9.04/5.11        ( finite_finite_nat
% 9.04/5.11        @ ( collect_nat
% 9.04/5.11          @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9363_finite__Collect__less__nat,axiom,
% 9.04/5.11      ! [K: nat] :
% 9.04/5.11        ( finite_finite_nat
% 9.04/5.11        @ ( collect_nat
% 9.04/5.11          @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9364_finite__interval__int1,axiom,
% 9.04/5.11      ! [A: int,B: int] :
% 9.04/5.11        ( finite_finite_int
% 9.04/5.11        @ ( collect_int
% 9.04/5.11          @ ^ [I3: int] :
% 9.04/5.11              ( ( ord_less_eq_int @ A @ I3 )
% 9.04/5.11              & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9365_finite__interval__int4,axiom,
% 9.04/5.11      ! [A: int,B: int] :
% 9.04/5.11        ( finite_finite_int
% 9.04/5.11        @ ( collect_int
% 9.04/5.11          @ ^ [I3: int] :
% 9.04/5.11              ( ( ord_less_int @ A @ I3 )
% 9.04/5.11              & ( ord_less_int @ I3 @ B ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9366_finite__interval__int3,axiom,
% 9.04/5.11      ! [A: int,B: int] :
% 9.04/5.11        ( finite_finite_int
% 9.04/5.11        @ ( collect_int
% 9.04/5.11          @ ^ [I3: int] :
% 9.04/5.11              ( ( ord_less_int @ A @ I3 )
% 9.04/5.11              & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9367_finite__interval__int2,axiom,
% 9.04/5.11      ! [A: int,B: int] :
% 9.04/5.11        ( finite_finite_int
% 9.04/5.11        @ ( collect_int
% 9.04/5.11          @ ^ [I3: int] :
% 9.04/5.11              ( ( ord_less_eq_int @ A @ I3 )
% 9.04/5.11              & ( ord_less_int @ I3 @ B ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9368_finite__nth__roots,axiom,
% 9.04/5.11      ! [N: nat,C: complex] :
% 9.04/5.11        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.11       => ( finite3207457112153483333omplex
% 9.04/5.11          @ ( collect_complex
% 9.04/5.11            @ ^ [Z4: complex] :
% 9.04/5.11                ( ( power_power_complex @ Z4 @ N )
% 9.04/5.11                = C ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9369_finite__divisors__int,axiom,
% 9.04/5.11      ! [I: int] :
% 9.04/5.11        ( ( I != zero_zero_int )
% 9.04/5.11       => ( finite_finite_int
% 9.04/5.11          @ ( collect_int
% 9.04/5.11            @ ^ [D3: int] : ( dvd_dvd_int @ D3 @ I ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9370_VEBT_Osize__gen_I1_J,axiom,
% 9.04/5.11      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 9.04/5.11        ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 9.04/5.11        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9371_finite__nat__iff__bounded__le,axiom,
% 9.04/5.11      ( finite_finite_nat
% 9.04/5.11      = ( ^ [S4: set_nat] :
% 9.04/5.11          ? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9372_finite__int__iff__bounded__le,axiom,
% 9.04/5.11      ( finite_finite_int
% 9.04/5.11      = ( ^ [S4: set_int] :
% 9.04/5.11          ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9373_finite__int__iff__bounded,axiom,
% 9.04/5.11      ( finite_finite_int
% 9.04/5.11      = ( ^ [S4: set_int] :
% 9.04/5.11          ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9374_VEBT_Osize__gen_I2_J,axiom,
% 9.04/5.11      ! [X21: $o,X222: $o] :
% 9.04/5.11        ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 9.04/5.11        = zero_zero_nat ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9375_infinite__int__iff__unbounded__le,axiom,
% 9.04/5.11      ! [S2: set_int] :
% 9.04/5.11        ( ( ~ ( finite_finite_int @ S2 ) )
% 9.04/5.11        = ( ! [M6: int] :
% 9.04/5.11            ? [N3: int] :
% 9.04/5.11              ( ( ord_less_eq_int @ M6 @ ( abs_abs_int @ N3 ) )
% 9.04/5.11              & ( member_int @ N3 @ S2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9376_infinite__int__iff__unbounded,axiom,
% 9.04/5.11      ! [S2: set_int] :
% 9.04/5.11        ( ( ~ ( finite_finite_int @ S2 ) )
% 9.04/5.11        = ( ! [M6: int] :
% 9.04/5.11            ? [N3: int] :
% 9.04/5.11              ( ( ord_less_int @ M6 @ ( abs_abs_int @ N3 ) )
% 9.04/5.11              & ( member_int @ N3 @ S2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9377_unbounded__k__infinite,axiom,
% 9.04/5.11      ! [K: nat,S2: set_nat] :
% 9.04/5.11        ( ! [M5: nat] :
% 9.04/5.11            ( ( ord_less_nat @ K @ M5 )
% 9.04/5.11           => ? [N8: nat] :
% 9.04/5.11                ( ( ord_less_nat @ M5 @ N8 )
% 9.04/5.11                & ( member_nat @ N8 @ S2 ) ) )
% 9.04/5.11       => ~ ( finite_finite_nat @ S2 ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9378_infinite__nat__iff__unbounded,axiom,
% 9.04/5.11      ! [S2: set_nat] :
% 9.04/5.11        ( ( ~ ( finite_finite_nat @ S2 ) )
% 9.04/5.11        = ( ! [M6: nat] :
% 9.04/5.11            ? [N3: nat] :
% 9.04/5.11              ( ( ord_less_nat @ M6 @ N3 )
% 9.04/5.11              & ( member_nat @ N3 @ S2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9379_infinite__nat__iff__unbounded__le,axiom,
% 9.04/5.11      ! [S2: set_nat] :
% 9.04/5.11        ( ( ~ ( finite_finite_nat @ S2 ) )
% 9.04/5.11        = ( ! [M6: nat] :
% 9.04/5.11            ? [N3: nat] :
% 9.04/5.11              ( ( ord_less_eq_nat @ M6 @ N3 )
% 9.04/5.11              & ( member_nat @ N3 @ S2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9380_finite__nat__bounded,axiom,
% 9.04/5.11      ! [S2: set_nat] :
% 9.04/5.11        ( ( finite_finite_nat @ S2 )
% 9.04/5.11       => ? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9381_finite__nat__iff__bounded,axiom,
% 9.04/5.11      ( finite_finite_nat
% 9.04/5.11      = ( ^ [S4: set_nat] :
% 9.04/5.11          ? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9382_int__ge__less__than2__def,axiom,
% 9.04/5.11      ( int_ge_less_than2
% 9.04/5.11      = ( ^ [D3: int] :
% 9.04/5.11            ( collec213857154873943460nt_int
% 9.04/5.11            @ ( produc4947309494688390418_int_o
% 9.04/5.11              @ ^ [Z7: int,Z4: int] :
% 9.04/5.11                  ( ( ord_less_eq_int @ D3 @ Z4 )
% 9.04/5.11                  & ( ord_less_int @ Z7 @ Z4 ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9383_int__ge__less__than__def,axiom,
% 9.04/5.11      ( int_ge_less_than
% 9.04/5.11      = ( ^ [D3: int] :
% 9.04/5.11            ( collec213857154873943460nt_int
% 9.04/5.11            @ ( produc4947309494688390418_int_o
% 9.04/5.11              @ ^ [Z7: int,Z4: int] :
% 9.04/5.11                  ( ( ord_less_eq_int @ D3 @ Z7 )
% 9.04/5.11                  & ( ord_less_int @ Z7 @ Z4 ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9384_rat__inverse__code,axiom,
% 9.04/5.11      ! [P5: rat] :
% 9.04/5.11        ( ( quotient_of @ ( inverse_inverse_rat @ P5 ) )
% 9.04/5.11        = ( produc4245557441103728435nt_int
% 9.04/5.11          @ ^ [A3: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B4 ) @ ( abs_abs_int @ A3 ) ) )
% 9.04/5.11          @ ( quotient_of @ P5 ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9385_normalize__negative,axiom,
% 9.04/5.11      ! [Q3: int,P5: int] :
% 9.04/5.11        ( ( ord_less_int @ Q3 @ zero_zero_int )
% 9.04/5.11       => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q3 ) )
% 9.04/5.11          = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P5 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9386_quotient__of__number_I3_J,axiom,
% 9.04/5.11      ! [K: num] :
% 9.04/5.11        ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 9.04/5.11        = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9387_normalize__denom__zero,axiom,
% 9.04/5.11      ! [P5: int] :
% 9.04/5.11        ( ( normalize @ ( product_Pair_int_int @ P5 @ zero_zero_int ) )
% 9.04/5.11        = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9388_rat__one__code,axiom,
% 9.04/5.11      ( ( quotient_of @ one_one_rat )
% 9.04/5.11      = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9389_rat__zero__code,axiom,
% 9.04/5.11      ( ( quotient_of @ zero_zero_rat )
% 9.04/5.11      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9390_quotient__of__number_I5_J,axiom,
% 9.04/5.11      ! [K: num] :
% 9.04/5.11        ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 9.04/5.11        = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9391_quotient__of__number_I4_J,axiom,
% 9.04/5.11      ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.04/5.11      = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9392_divide__rat__def,axiom,
% 9.04/5.11      ( divide_divide_rat
% 9.04/5.11      = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9393_rat__divide__code,axiom,
% 9.04/5.11      ! [P5: rat,Q3: rat] :
% 9.04/5.11        ( ( quotient_of @ ( divide_divide_rat @ P5 @ Q3 ) )
% 9.04/5.11        = ( produc4245557441103728435nt_int
% 9.04/5.11          @ ^ [A3: int,C3: int] :
% 9.04/5.11              ( produc4245557441103728435nt_int
% 9.04/5.11              @ ^ [B4: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ C3 @ B4 ) ) )
% 9.04/5.11              @ ( quotient_of @ Q3 ) )
% 9.04/5.11          @ ( quotient_of @ P5 ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9394_quotient__of__div,axiom,
% 9.04/5.11      ! [R2: rat,N: int,D: int] :
% 9.04/5.11        ( ( ( quotient_of @ R2 )
% 9.04/5.11          = ( product_Pair_int_int @ N @ D ) )
% 9.04/5.11       => ( R2
% 9.04/5.11          = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9395_quotient__of__denom__pos,axiom,
% 9.04/5.11      ! [R2: rat,P5: int,Q3: int] :
% 9.04/5.11        ( ( ( quotient_of @ R2 )
% 9.04/5.11          = ( product_Pair_int_int @ P5 @ Q3 ) )
% 9.04/5.11       => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9396_rat__plus__code,axiom,
% 9.04/5.11      ! [P5: rat,Q3: rat] :
% 9.04/5.11        ( ( quotient_of @ ( plus_plus_rat @ P5 @ Q3 ) )
% 9.04/5.11        = ( produc4245557441103728435nt_int
% 9.04/5.11          @ ^ [A3: int,C3: int] :
% 9.04/5.11              ( produc4245557441103728435nt_int
% 9.04/5.11              @ ^ [B4: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D3 ) ) )
% 9.04/5.11              @ ( quotient_of @ Q3 ) )
% 9.04/5.11          @ ( quotient_of @ P5 ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9397_normalize__denom__pos,axiom,
% 9.04/5.11      ! [R2: product_prod_int_int,P5: int,Q3: int] :
% 9.04/5.11        ( ( ( normalize @ R2 )
% 9.04/5.11          = ( product_Pair_int_int @ P5 @ Q3 ) )
% 9.04/5.11       => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9398_normalize__crossproduct,axiom,
% 9.04/5.11      ! [Q3: int,S: int,P5: int,R2: int] :
% 9.04/5.11        ( ( Q3 != zero_zero_int )
% 9.04/5.11       => ( ( S != zero_zero_int )
% 9.04/5.11         => ( ( ( normalize @ ( product_Pair_int_int @ P5 @ Q3 ) )
% 9.04/5.11              = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
% 9.04/5.11           => ( ( times_times_int @ P5 @ S )
% 9.04/5.11              = ( times_times_int @ R2 @ Q3 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9399_rat__less__code,axiom,
% 9.04/5.11      ( ord_less_rat
% 9.04/5.11      = ( ^ [P4: rat,Q4: rat] :
% 9.04/5.11            ( produc4947309494688390418_int_o
% 9.04/5.11            @ ^ [A3: int,C3: int] :
% 9.04/5.11                ( produc4947309494688390418_int_o
% 9.04/5.11                @ ^ [B4: int,D3: int] : ( ord_less_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ C3 @ B4 ) )
% 9.04/5.11                @ ( quotient_of @ Q4 ) )
% 9.04/5.11            @ ( quotient_of @ P4 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9400_rat__less__eq__code,axiom,
% 9.04/5.11      ( ord_less_eq_rat
% 9.04/5.11      = ( ^ [P4: rat,Q4: rat] :
% 9.04/5.11            ( produc4947309494688390418_int_o
% 9.04/5.11            @ ^ [A3: int,C3: int] :
% 9.04/5.11                ( produc4947309494688390418_int_o
% 9.04/5.11                @ ^ [B4: int,D3: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ C3 @ B4 ) )
% 9.04/5.11                @ ( quotient_of @ Q4 ) )
% 9.04/5.11            @ ( quotient_of @ P4 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9401_quotient__of__int,axiom,
% 9.04/5.11      ! [A: int] :
% 9.04/5.11        ( ( quotient_of @ ( of_int @ A ) )
% 9.04/5.11        = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9402_Sum__Ico__nat,axiom,
% 9.04/5.11      ! [M: nat,N: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat
% 9.04/5.11          @ ^ [X4: nat] : X4
% 9.04/5.11          @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 9.04/5.11        = ( 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 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9403_sum__power2,axiom,
% 9.04/5.11      ! [K: nat] :
% 9.04/5.11        ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 9.04/5.11        = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9404_image__Suc__atLeastLessThan,axiom,
% 9.04/5.11      ! [I: nat,J: nat] :
% 9.04/5.11        ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 9.04/5.11        = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9405_atLeastLessThan__singleton,axiom,
% 9.04/5.11      ! [M: nat] :
% 9.04/5.11        ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 9.04/5.11        = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9406_all__nat__less__eq,axiom,
% 9.04/5.11      ! [N: nat,P: nat > $o] :
% 9.04/5.11        ( ( ! [M6: nat] :
% 9.04/5.11              ( ( ord_less_nat @ M6 @ N )
% 9.04/5.11             => ( P @ M6 ) ) )
% 9.04/5.11        = ( ! [X4: nat] :
% 9.04/5.11              ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.04/5.11             => ( P @ X4 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9407_ex__nat__less__eq,axiom,
% 9.04/5.11      ! [N: nat,P: nat > $o] :
% 9.04/5.11        ( ( ? [M6: nat] :
% 9.04/5.11              ( ( ord_less_nat @ M6 @ N )
% 9.04/5.11              & ( P @ M6 ) ) )
% 9.04/5.11        = ( ? [X4: nat] :
% 9.04/5.11              ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.04/5.11              & ( P @ X4 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9408_atLeastLessThanSuc__atLeastAtMost,axiom,
% 9.04/5.11      ! [L: nat,U: nat] :
% 9.04/5.11        ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 9.04/5.11        = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9409_lessThan__atLeast0,axiom,
% 9.04/5.11      ( set_ord_lessThan_nat
% 9.04/5.11      = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9410_atLeastLessThan0,axiom,
% 9.04/5.11      ! [M: nat] :
% 9.04/5.11        ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 9.04/5.11        = bot_bot_set_nat ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9411_atLeast0__lessThan__Suc,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.04/5.11        = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9412_subset__eq__atLeast0__lessThan__finite,axiom,
% 9.04/5.11      ! [N4: set_nat,N: nat] :
% 9.04/5.11        ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.04/5.11       => ( finite_finite_nat @ N4 ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9413_atLeastLessThanSuc,axiom,
% 9.04/5.11      ! [M: nat,N: nat] :
% 9.04/5.11        ( ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.11         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 9.04/5.11            = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 9.04/5.11        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.04/5.11         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 9.04/5.11            = bot_bot_set_nat ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9414_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.04/5.11        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9415_prod__Suc__fact,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.04/5.11        = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9416_prod__Suc__Suc__fact,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.04/5.11        = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9417_atLeastLessThan__nat__numeral,axiom,
% 9.04/5.11      ! [M: nat,K: num] :
% 9.04/5.11        ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 9.04/5.11         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 9.04/5.11            = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 9.04/5.11        & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 9.04/5.11         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 9.04/5.11            = bot_bot_set_nat ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9418_Rat_Oof__int__def,axiom,
% 9.04/5.11      of_int = ring_1_of_int_rat ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9419_image__minus__const__atLeastLessThan__nat,axiom,
% 9.04/5.11      ! [C: nat,Y: nat,X: nat] :
% 9.04/5.11        ( ( ( ord_less_nat @ C @ Y )
% 9.04/5.11         => ( ( image_nat_nat
% 9.04/5.11              @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 9.04/5.11              @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 9.04/5.11            = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
% 9.04/5.11        & ( ~ ( ord_less_nat @ C @ Y )
% 9.04/5.11         => ( ( ( ord_less_nat @ X @ Y )
% 9.04/5.11             => ( ( image_nat_nat
% 9.04/5.11                  @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 9.04/5.11                  @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 9.04/5.11                = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 9.04/5.11            & ( ~ ( ord_less_nat @ X @ Y )
% 9.04/5.11             => ( ( image_nat_nat
% 9.04/5.11                  @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 9.04/5.11                  @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 9.04/5.11                = bot_bot_set_nat ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9420_atLeast1__lessThan__eq__remove0,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.04/5.11        = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9421_Chebyshev__sum__upper__nat,axiom,
% 9.04/5.11      ! [N: nat,A: nat > nat,B: nat > nat] :
% 9.04/5.11        ( ! [I2: nat,J3: nat] :
% 9.04/5.11            ( ( ord_less_eq_nat @ I2 @ J3 )
% 9.04/5.11           => ( ( ord_less_nat @ J3 @ N )
% 9.04/5.11             => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J3 ) ) ) )
% 9.04/5.11       => ( ! [I2: nat,J3: nat] :
% 9.04/5.11              ( ( ord_less_eq_nat @ I2 @ J3 )
% 9.04/5.11             => ( ( ord_less_nat @ J3 @ N )
% 9.04/5.11               => ( ord_less_eq_nat @ ( B @ J3 ) @ ( B @ I2 ) ) ) )
% 9.04/5.11         => ( ord_less_eq_nat
% 9.04/5.11            @ ( times_times_nat @ N
% 9.04/5.11              @ ( groups3542108847815614940at_nat
% 9.04/5.11                @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
% 9.04/5.11                @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 9.04/5.11            @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9422_finite__atLeastZeroLessThan__int,axiom,
% 9.04/5.11      ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9423_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 9.04/5.11      ! [L: int,U: int] :
% 9.04/5.11        ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
% 9.04/5.11        = ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9424_image__int__atLeastLessThan,axiom,
% 9.04/5.11      ! [A: nat,B: nat] :
% 9.04/5.11        ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
% 9.04/5.11        = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9425_image__add__int__atLeastLessThan,axiom,
% 9.04/5.11      ! [L: int,U: int] :
% 9.04/5.11        ( ( image_int_int
% 9.04/5.11          @ ^ [X4: int] : ( plus_plus_int @ X4 @ L )
% 9.04/5.11          @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
% 9.04/5.11        = ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9426_image__atLeastZeroLessThan__int,axiom,
% 9.04/5.11      ! [U: int] :
% 9.04/5.11        ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.04/5.11       => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 9.04/5.11          = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9427_Frct__code__post_I5_J,axiom,
% 9.04/5.11      ! [K: num] :
% 9.04/5.11        ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 9.04/5.11        = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9428_int__of__nat__def,axiom,
% 9.04/5.11      code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9429_Frct__code__post_I2_J,axiom,
% 9.04/5.11      ! [A: int] :
% 9.04/5.11        ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 9.04/5.11        = zero_zero_rat ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9430_Frct__code__post_I1_J,axiom,
% 9.04/5.11      ! [A: int] :
% 9.04/5.11        ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 9.04/5.11        = zero_zero_rat ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9431_Frct__code__post_I3_J,axiom,
% 9.04/5.11      ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 9.04/5.11      = one_one_rat ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9432_Frct__code__post_I4_J,axiom,
% 9.04/5.11      ! [K: num] :
% 9.04/5.11        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 9.04/5.11        = ( numeral_numeral_rat @ K ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9433_Frct__code__post_I6_J,axiom,
% 9.04/5.11      ! [K: num,L: num] :
% 9.04/5.11        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 9.04/5.11        = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9434_Code__Target__Int_Opositive__def,axiom,
% 9.04/5.11      code_Target_positive = numeral_numeral_int ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9435_divmod__step__integer__def,axiom,
% 9.04/5.11      ( unique4921790084139445826nteger
% 9.04/5.11      = ( ^ [L3: num] :
% 9.04/5.11            ( produc6916734918728496179nteger
% 9.04/5.11            @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9436_times__integer__code_I2_J,axiom,
% 9.04/5.11      ! [L: code_integer] :
% 9.04/5.11        ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 9.04/5.11        = zero_z3403309356797280102nteger ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9437_times__integer__code_I1_J,axiom,
% 9.04/5.11      ! [K: code_integer] :
% 9.04/5.11        ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 9.04/5.11        = zero_z3403309356797280102nteger ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9438_divmod__integer_H__def,axiom,
% 9.04/5.11      ( unique3479559517661332726nteger
% 9.04/5.11      = ( ^ [M6: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9439_less__eq__integer__code_I1_J,axiom,
% 9.04/5.11      ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9440_sgn__integer__code,axiom,
% 9.04/5.11      ( sgn_sgn_Code_integer
% 9.04/5.11      = ( ^ [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 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9441_minus__integer__code_I2_J,axiom,
% 9.04/5.11      ! [L: code_integer] :
% 9.04/5.11        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
% 9.04/5.11        = ( uminus1351360451143612070nteger @ L ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9442_minus__integer__code_I1_J,axiom,
% 9.04/5.11      ! [K: code_integer] :
% 9.04/5.11        ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 9.04/5.11        = K ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9443_plus__integer__code_I2_J,axiom,
% 9.04/5.11      ! [L: code_integer] :
% 9.04/5.11        ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
% 9.04/5.11        = L ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9444_plus__integer__code_I1_J,axiom,
% 9.04/5.11      ! [K: code_integer] :
% 9.04/5.11        ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
% 9.04/5.11        = K ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9445_zero__natural_Orsp,axiom,
% 9.04/5.11      zero_zero_nat = zero_zero_nat ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9446_zero__integer_Orsp,axiom,
% 9.04/5.11      zero_zero_int = zero_zero_int ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9447_one__integer_Orsp,axiom,
% 9.04/5.11      one_one_int = one_one_int ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9448_one__natural_Orsp,axiom,
% 9.04/5.11      one_one_nat = one_one_nat ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9449_integer__of__int__code,axiom,
% 9.04/5.11      ( code_integer_of_int
% 9.04/5.11      = ( ^ [K2: int] :
% 9.04/5.11            ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
% 9.04/5.11            @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 9.04/5.11              @ ( if_Code_integer
% 9.04/5.11                @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.04/5.11                  = zero_zero_int )
% 9.04/5.11                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.04/5.11                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9450_Code__Numeral_Opositive__def,axiom,
% 9.04/5.11      code_positive = numera6620942414471956472nteger ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9451_integer__of__int__eq__of__int,axiom,
% 9.04/5.11      code_integer_of_int = ring_18347121197199848620nteger ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9452_divide__integer_Oabs__eq,axiom,
% 9.04/5.11      ! [Xa: int,X: int] :
% 9.04/5.11        ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 9.04/5.11        = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9453_abs__integer__code,axiom,
% 9.04/5.11      ( abs_abs_Code_integer
% 9.04/5.11      = ( ^ [K2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K2 ) @ K2 ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9454_less__integer__code_I1_J,axiom,
% 9.04/5.11      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9455_zero__integer__def,axiom,
% 9.04/5.11      ( zero_z3403309356797280102nteger
% 9.04/5.11      = ( code_integer_of_int @ zero_zero_int ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9456_uminus__integer__code_I1_J,axiom,
% 9.04/5.11      ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 9.04/5.11      = zero_z3403309356797280102nteger ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9457_less__integer_Oabs__eq,axiom,
% 9.04/5.11      ! [Xa: int,X: int] :
% 9.04/5.11        ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 9.04/5.11        = ( ord_less_int @ Xa @ X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9458_plus__integer_Oabs__eq,axiom,
% 9.04/5.11      ! [Xa: int,X: int] :
% 9.04/5.11        ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 9.04/5.11        = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9459_one__integer__def,axiom,
% 9.04/5.11      ( one_one_Code_integer
% 9.04/5.11      = ( code_integer_of_int @ one_one_int ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9460_less__eq__integer_Oabs__eq,axiom,
% 9.04/5.11      ! [Xa: int,X: int] :
% 9.04/5.11        ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 9.04/5.11        = ( ord_less_eq_int @ Xa @ X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9461_integer__of__num_I3_J,axiom,
% 9.04/5.11      ! [N: num] :
% 9.04/5.11        ( ( code_integer_of_num @ ( bit1 @ N ) )
% 9.04/5.11        = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9462_bit__cut__integer__def,axiom,
% 9.04/5.11      ( code_bit_cut_integer
% 9.04/5.11      = ( ^ [K2: code_integer] :
% 9.04/5.11            ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.04/5.11            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9463_integer__of__num__def,axiom,
% 9.04/5.11      code_integer_of_num = numera6620942414471956472nteger ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9464_integer__of__num__triv_I1_J,axiom,
% 9.04/5.11      ( ( code_integer_of_num @ one )
% 9.04/5.11      = one_one_Code_integer ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9465_integer__of__num_I2_J,axiom,
% 9.04/5.11      ! [N: num] :
% 9.04/5.11        ( ( code_integer_of_num @ ( bit0 @ N ) )
% 9.04/5.11        = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9466_integer__of__num__triv_I2_J,axiom,
% 9.04/5.11      ( ( code_integer_of_num @ ( bit0 @ one ) )
% 9.04/5.11      = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9467_bit__cut__integer__code,axiom,
% 9.04/5.11      ( code_bit_cut_integer
% 9.04/5.11      = ( ^ [K2: code_integer] :
% 9.04/5.11            ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 9.04/5.11            @ ( produc9125791028180074456eger_o
% 9.04/5.11              @ ^ [R5: code_integer,S5: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S5 ) ) @ ( S5 = one_one_Code_integer ) )
% 9.04/5.11              @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9468_divmod__integer__def,axiom,
% 9.04/5.11      ( code_divmod_integer
% 9.04/5.11      = ( ^ [K2: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K2 @ L3 ) @ ( modulo364778990260209775nteger @ K2 @ L3 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9469_csqrt_Osimps_I1_J,axiom,
% 9.04/5.11      ! [Z: complex] :
% 9.04/5.11        ( ( re @ ( csqrt @ Z ) )
% 9.04/5.11        = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9470_complex__Re__of__nat,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( re @ ( semiri8010041392384452111omplex @ N ) )
% 9.04/5.11        = ( semiri5074537144036343181t_real @ N ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9471_complex__Re__of__int,axiom,
% 9.04/5.11      ! [Z: int] :
% 9.04/5.11        ( ( re @ ( ring_17405671764205052669omplex @ Z ) )
% 9.04/5.11        = ( ring_1_of_int_real @ Z ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9472_complex__Re__numeral,axiom,
% 9.04/5.11      ! [V: num] :
% 9.04/5.11        ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 9.04/5.11        = ( numeral_numeral_real @ V ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9473_Re__divide__of__nat,axiom,
% 9.04/5.11      ! [Z: complex,N: nat] :
% 9.04/5.11        ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 9.04/5.11        = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9474_Re__divide__of__real,axiom,
% 9.04/5.11      ! [Z: complex,R2: real] :
% 9.04/5.11        ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 9.04/5.11        = ( divide_divide_real @ ( re @ Z ) @ R2 ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9475_Re__sgn,axiom,
% 9.04/5.11      ! [Z: complex] :
% 9.04/5.11        ( ( re @ ( sgn_sgn_complex @ Z ) )
% 9.04/5.11        = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9476_Re__divide__numeral,axiom,
% 9.04/5.11      ! [Z: complex,W: num] :
% 9.04/5.11        ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 9.04/5.11        = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9477_cos__Arg__i__mult__zero,axiom,
% 9.04/5.11      ! [Y: complex] :
% 9.04/5.11        ( ( Y != zero_zero_complex )
% 9.04/5.11       => ( ( ( re @ Y )
% 9.04/5.11            = zero_zero_real )
% 9.04/5.11         => ( ( cos_real @ ( arg @ Y ) )
% 9.04/5.11            = zero_zero_real ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9478_imaginary__unit_Osimps_I1_J,axiom,
% 9.04/5.11      ( ( re @ imaginary_unit )
% 9.04/5.11      = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9479_complex__Re__le__cmod,axiom,
% 9.04/5.11      ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9480_zero__complex_Osimps_I1_J,axiom,
% 9.04/5.11      ( ( re @ zero_zero_complex )
% 9.04/5.11      = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9481_one__complex_Osimps_I1_J,axiom,
% 9.04/5.11      ( ( re @ one_one_complex )
% 9.04/5.11      = one_one_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9482_abs__Re__le__cmod,axiom,
% 9.04/5.11      ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9483_Re__csqrt,axiom,
% 9.04/5.11      ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9484_divmod__abs__code_I6_J,axiom,
% 9.04/5.11      ! [J: code_integer] :
% 9.04/5.11        ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
% 9.04/5.11        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9485_cmod__plus__Re__le__0__iff,axiom,
% 9.04/5.11      ! [Z: complex] :
% 9.04/5.11        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 9.04/5.11        = ( ( re @ Z )
% 9.04/5.11          = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9486_cos__n__Re__cis__pow__n,axiom,
% 9.04/5.11      ! [N: nat,A: real] :
% 9.04/5.11        ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 9.04/5.11        = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9487_divmod__abs__code_I5_J,axiom,
% 9.04/5.11      ! [J: code_integer] :
% 9.04/5.11        ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
% 9.04/5.11        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9488_divmod__abs__def,axiom,
% 9.04/5.11      ( code_divmod_abs
% 9.04/5.11      = ( ^ [K2: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L3 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L3 ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9489_divmod__integer__code,axiom,
% 9.04/5.11      ( code_divmod_integer
% 9.04/5.11      = ( ^ [K2: code_integer,L3: code_integer] :
% 9.04/5.11            ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 9.04/5.11            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L3 )
% 9.04/5.11              @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L3 )
% 9.04/5.11                @ ( produc6916734918728496179nteger
% 9.04/5.11                  @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L3 @ S5 ) ) )
% 9.04/5.11                  @ ( code_divmod_abs @ K2 @ L3 ) ) )
% 9.04/5.11              @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 9.04/5.11                @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 9.04/5.11                  @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L3 )
% 9.04/5.11                    @ ( produc6916734918728496179nteger
% 9.04/5.11                      @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L3 ) @ S5 ) ) )
% 9.04/5.11                      @ ( code_divmod_abs @ K2 @ L3 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9490_csqrt_Ocode,axiom,
% 9.04/5.11      ( csqrt
% 9.04/5.11      = ( ^ [Z4: complex] :
% 9.04/5.11            ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.11            @ ( times_times_real
% 9.04/5.11              @ ( if_real
% 9.04/5.11                @ ( ( im @ Z4 )
% 9.04/5.11                  = zero_zero_real )
% 9.04/5.11                @ one_one_real
% 9.04/5.11                @ ( sgn_sgn_real @ ( im @ Z4 ) ) )
% 9.04/5.11              @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9491_csqrt_Osimps_I2_J,axiom,
% 9.04/5.11      ! [Z: complex] :
% 9.04/5.11        ( ( im @ ( csqrt @ Z ) )
% 9.04/5.11        = ( times_times_real
% 9.04/5.11          @ ( if_real
% 9.04/5.11            @ ( ( im @ Z )
% 9.04/5.11              = zero_zero_real )
% 9.04/5.11            @ one_one_real
% 9.04/5.11            @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 9.04/5.11          @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9492_complex__Im__fact,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( im @ ( semiri5044797733671781792omplex @ N ) )
% 9.04/5.11        = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9493_Im__complex__of__real,axiom,
% 9.04/5.11      ! [Z: real] :
% 9.04/5.11        ( ( im @ ( real_V4546457046886955230omplex @ Z ) )
% 9.04/5.11        = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9494_Im__power__real,axiom,
% 9.04/5.11      ! [X: complex,N: nat] :
% 9.04/5.11        ( ( ( im @ X )
% 9.04/5.11          = zero_zero_real )
% 9.04/5.11       => ( ( im @ ( power_power_complex @ X @ N ) )
% 9.04/5.11          = zero_zero_real ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9495_complex__Im__numeral,axiom,
% 9.04/5.11      ! [V: num] :
% 9.04/5.11        ( ( im @ ( numera6690914467698888265omplex @ V ) )
% 9.04/5.11        = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9496_complex__Im__of__nat,axiom,
% 9.04/5.11      ! [N: nat] :
% 9.04/5.11        ( ( im @ ( semiri8010041392384452111omplex @ N ) )
% 9.04/5.11        = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9497_complex__Im__of__int,axiom,
% 9.04/5.11      ! [Z: int] :
% 9.04/5.11        ( ( im @ ( ring_17405671764205052669omplex @ Z ) )
% 9.04/5.11        = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9498_Im__divide__of__real,axiom,
% 9.04/5.11      ! [Z: complex,R2: real] :
% 9.04/5.11        ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 9.04/5.11        = ( divide_divide_real @ ( im @ Z ) @ R2 ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9499_Im__sgn,axiom,
% 9.04/5.11      ! [Z: complex] :
% 9.04/5.11        ( ( im @ ( sgn_sgn_complex @ Z ) )
% 9.04/5.11        = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9500_Re__power__real,axiom,
% 9.04/5.11      ! [X: complex,N: nat] :
% 9.04/5.11        ( ( ( im @ X )
% 9.04/5.11          = zero_zero_real )
% 9.04/5.11       => ( ( re @ ( power_power_complex @ X @ N ) )
% 9.04/5.11          = ( power_power_real @ ( re @ X ) @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9501_Im__divide__numeral,axiom,
% 9.04/5.11      ! [Z: complex,W: num] :
% 9.04/5.11        ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 9.04/5.11        = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9502_Im__divide__of__nat,axiom,
% 9.04/5.11      ! [Z: complex,N: nat] :
% 9.04/5.11        ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 9.04/5.11        = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9503_csqrt__of__real__nonneg,axiom,
% 9.04/5.11      ! [X: complex] :
% 9.04/5.11        ( ( ( im @ X )
% 9.04/5.11          = zero_zero_real )
% 9.04/5.11       => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
% 9.04/5.11         => ( ( csqrt @ X )
% 9.04/5.11            = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9504_csqrt__minus,axiom,
% 9.04/5.11      ! [X: complex] :
% 9.04/5.11        ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
% 9.04/5.11          | ( ( ( im @ X )
% 9.04/5.11              = zero_zero_real )
% 9.04/5.11            & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
% 9.04/5.11       => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
% 9.04/5.11          = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9505_csqrt__of__real__nonpos,axiom,
% 9.04/5.11      ! [X: complex] :
% 9.04/5.11        ( ( ( im @ X )
% 9.04/5.11          = zero_zero_real )
% 9.04/5.11       => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
% 9.04/5.11         => ( ( csqrt @ X )
% 9.04/5.11            = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9506_imaginary__unit_Osimps_I2_J,axiom,
% 9.04/5.11      ( ( im @ imaginary_unit )
% 9.04/5.11      = one_one_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9507_zero__complex_Osimps_I2_J,axiom,
% 9.04/5.11      ( ( im @ zero_zero_complex )
% 9.04/5.11      = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9508_one__complex_Osimps_I2_J,axiom,
% 9.04/5.11      ( ( im @ one_one_complex )
% 9.04/5.11      = zero_zero_real ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9509_complex__is__Int__iff,axiom,
% 9.04/5.11      ! [Z: complex] :
% 9.04/5.11        ( ( member_complex @ Z @ ring_1_Ints_complex )
% 9.04/5.11        = ( ( ( im @ Z )
% 9.04/5.11            = zero_zero_real )
% 9.04/5.11          & ? [I3: int] :
% 9.04/5.11              ( ( re @ Z )
% 9.04/5.11              = ( ring_1_of_int_real @ I3 ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9510_abs__Im__le__cmod,axiom,
% 9.04/5.11      ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9511_cmod__eq__Re,axiom,
% 9.04/5.11      ! [Z: complex] :
% 9.04/5.11        ( ( ( im @ Z )
% 9.04/5.11          = zero_zero_real )
% 9.04/5.11       => ( ( real_V1022390504157884413omplex @ Z )
% 9.04/5.11          = ( abs_abs_real @ ( re @ Z ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9512_cmod__eq__Im,axiom,
% 9.04/5.11      ! [Z: complex] :
% 9.04/5.11        ( ( ( re @ Z )
% 9.04/5.11          = zero_zero_real )
% 9.04/5.11       => ( ( real_V1022390504157884413omplex @ Z )
% 9.04/5.11          = ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9513_Im__eq__0,axiom,
% 9.04/5.11      ! [Z: complex] :
% 9.04/5.11        ( ( ( abs_abs_real @ ( re @ Z ) )
% 9.04/5.11          = ( real_V1022390504157884413omplex @ Z ) )
% 9.04/5.11       => ( ( im @ Z )
% 9.04/5.11          = zero_zero_real ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9514_cmod__Im__le__iff,axiom,
% 9.04/5.11      ! [X: complex,Y: complex] :
% 9.04/5.11        ( ( ( re @ X )
% 9.04/5.11          = ( re @ Y ) )
% 9.04/5.11       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
% 9.04/5.11          = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9515_cmod__Re__le__iff,axiom,
% 9.04/5.11      ! [X: complex,Y: complex] :
% 9.04/5.11        ( ( ( im @ X )
% 9.04/5.11          = ( im @ Y ) )
% 9.04/5.11       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
% 9.04/5.11          = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9516_csqrt__principal,axiom,
% 9.04/5.11      ! [Z: complex] :
% 9.04/5.11        ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 9.04/5.11        | ( ( ( re @ ( csqrt @ Z ) )
% 9.04/5.11            = zero_zero_real )
% 9.04/5.11          & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9517_cmod__le,axiom,
% 9.04/5.11      ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 9.04/5.11  
% 9.04/5.11  thf(fact_9518_sin__n__Im__cis__pow__n,axiom,
% 9.04/5.11      ! [N: nat,A: real] :
% 9.04/5.12        ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 9.04/5.12        = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9519_cmod__power2,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.12        = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9520_Im__power2,axiom,
% 9.04/5.12      ! [X: complex] :
% 9.04/5.12        ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.12        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9521_Re__power2,axiom,
% 9.04/5.12      ! [X: complex] :
% 9.04/5.12        ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.12        = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9522_complex__eq__0,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( Z = zero_zero_complex )
% 9.04/5.12        = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.12          = zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9523_norm__complex__def,axiom,
% 9.04/5.12      ( real_V1022390504157884413omplex
% 9.04/5.12      = ( ^ [Z4: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9524_inverse__complex_Osimps_I1_J,axiom,
% 9.04/5.12      ! [X: complex] :
% 9.04/5.12        ( ( re @ ( invers8013647133539491842omplex @ X ) )
% 9.04/5.12        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9525_complex__neq__0,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( Z != zero_zero_complex )
% 9.04/5.12        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9526_Re__divide,axiom,
% 9.04/5.12      ! [X: complex,Y: complex] :
% 9.04/5.12        ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
% 9.04/5.12        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9527_csqrt__square,axiom,
% 9.04/5.12      ! [B: complex] :
% 9.04/5.12        ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 9.04/5.12          | ( ( ( re @ B )
% 9.04/5.12              = zero_zero_real )
% 9.04/5.12            & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 9.04/5.12       => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.12          = B ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9528_csqrt__unique,axiom,
% 9.04/5.12      ! [W: complex,Z: complex] :
% 9.04/5.12        ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.04/5.12          = Z )
% 9.04/5.12       => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 9.04/5.12            | ( ( ( re @ W )
% 9.04/5.12                = zero_zero_real )
% 9.04/5.12              & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 9.04/5.12         => ( ( csqrt @ Z )
% 9.04/5.12            = W ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9529_inverse__complex_Osimps_I2_J,axiom,
% 9.04/5.12      ! [X: complex] :
% 9.04/5.12        ( ( im @ ( invers8013647133539491842omplex @ X ) )
% 9.04/5.12        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9530_Im__divide,axiom,
% 9.04/5.12      ! [X: complex,Y: complex] :
% 9.04/5.12        ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
% 9.04/5.12        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9531_complex__abs__le__norm,axiom,
% 9.04/5.12      ! [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 ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9532_complex__unit__circle,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( Z != zero_zero_complex )
% 9.04/5.12       => ( ( 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 ) ) ) )
% 9.04/5.12          = one_one_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9533_inverse__complex_Ocode,axiom,
% 9.04/5.12      ( invers8013647133539491842omplex
% 9.04/5.12      = ( ^ [X4: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X4 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X4 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9534_Complex__divide,axiom,
% 9.04/5.12      ( divide1717551699836669952omplex
% 9.04/5.12      = ( ^ [X4: complex,Y6: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9535_Im__Reals__divide,axiom,
% 9.04/5.12      ! [R2: complex,Z: complex] :
% 9.04/5.12        ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 9.04/5.12       => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 9.04/5.12          = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9536_Re__Reals__divide,axiom,
% 9.04/5.12      ! [R2: complex,Z: complex] :
% 9.04/5.12        ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 9.04/5.12       => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 9.04/5.12          = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9537_complex__mult__cnj,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 9.04/5.12        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9538_complex__cnj__divide,axiom,
% 9.04/5.12      ! [X: complex,Y: complex] :
% 9.04/5.12        ( ( cnj @ ( divide1717551699836669952omplex @ X @ Y ) )
% 9.04/5.12        = ( divide1717551699836669952omplex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9539_complex__cnj__zero,axiom,
% 9.04/5.12      ( ( cnj @ zero_zero_complex )
% 9.04/5.12      = zero_zero_complex ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9540_complex__cnj__zero__iff,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( ( cnj @ Z )
% 9.04/5.12          = zero_zero_complex )
% 9.04/5.12        = ( Z = zero_zero_complex ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9541_complex__cnj__one,axiom,
% 9.04/5.12      ( ( cnj @ one_one_complex )
% 9.04/5.12      = one_one_complex ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9542_complex__cnj__one__iff,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( ( cnj @ Z )
% 9.04/5.12          = one_one_complex )
% 9.04/5.12        = ( Z = one_one_complex ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9543_complex__cnj__power,axiom,
% 9.04/5.12      ! [X: complex,N: nat] :
% 9.04/5.12        ( ( cnj @ ( power_power_complex @ X @ N ) )
% 9.04/5.12        = ( power_power_complex @ ( cnj @ X ) @ N ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9544_complex__cnj__numeral,axiom,
% 9.04/5.12      ! [W: num] :
% 9.04/5.12        ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
% 9.04/5.12        = ( numera6690914467698888265omplex @ W ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9545_complex__cnj__of__int,axiom,
% 9.04/5.12      ! [Z: int] :
% 9.04/5.12        ( ( cnj @ ( ring_17405671764205052669omplex @ Z ) )
% 9.04/5.12        = ( ring_17405671764205052669omplex @ Z ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9546_complex__cnj__neg__numeral,axiom,
% 9.04/5.12      ! [W: num] :
% 9.04/5.12        ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.04/5.12        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9547_complex__In__mult__cnj__zero,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 9.04/5.12        = zero_zero_real ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9548_Re__divide__Reals,axiom,
% 9.04/5.12      ! [R2: complex,Z: complex] :
% 9.04/5.12        ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 9.04/5.12       => ( ( re @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 9.04/5.12          = ( divide_divide_real @ ( re @ Z ) @ ( re @ R2 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9549_real__eq__imaginary__iff,axiom,
% 9.04/5.12      ! [Y: complex,X: complex] :
% 9.04/5.12        ( ( member_complex @ Y @ real_V2521375963428798218omplex )
% 9.04/5.12       => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 9.04/5.12         => ( ( X
% 9.04/5.12              = ( times_times_complex @ imaginary_unit @ Y ) )
% 9.04/5.12            = ( ( X = zero_zero_complex )
% 9.04/5.12              & ( Y = zero_zero_complex ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9550_imaginary__eq__real__iff,axiom,
% 9.04/5.12      ! [Y: complex,X: complex] :
% 9.04/5.12        ( ( member_complex @ Y @ real_V2521375963428798218omplex )
% 9.04/5.12       => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 9.04/5.12         => ( ( ( times_times_complex @ imaginary_unit @ Y )
% 9.04/5.12              = X )
% 9.04/5.12            = ( ( X = zero_zero_complex )
% 9.04/5.12              & ( Y = zero_zero_complex ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9551_Im__divide__Reals,axiom,
% 9.04/5.12      ! [R2: complex,Z: complex] :
% 9.04/5.12        ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 9.04/5.12       => ( ( im @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 9.04/5.12          = ( divide_divide_real @ ( im @ Z ) @ ( re @ R2 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9552_complex__is__Real__iff,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( member_complex @ Z @ real_V2521375963428798218omplex )
% 9.04/5.12        = ( ( im @ Z )
% 9.04/5.12          = zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9553_Complex__in__Reals,axiom,
% 9.04/5.12      ! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9554_Re__complex__div__eq__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.04/5.12          = zero_zero_real )
% 9.04/5.12        = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 9.04/5.12          = zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9555_Im__complex__div__eq__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.04/5.12          = zero_zero_real )
% 9.04/5.12        = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 9.04/5.12          = zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9556_Re__complex__div__gt__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 9.04/5.12        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9557_Re__complex__div__lt__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 9.04/5.12        = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9558_Re__complex__div__le__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 9.04/5.12        = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9559_Re__complex__div__ge__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 9.04/5.12        = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9560_Im__complex__div__gt__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 9.04/5.12        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9561_Im__complex__div__lt__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 9.04/5.12        = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9562_Im__complex__div__le__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 9.04/5.12        = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9563_Im__complex__div__ge__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 9.04/5.12        = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9564_complex__mod__mult__cnj,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 9.04/5.12        = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9565_complex__div__gt__0,axiom,
% 9.04/5.12      ! [A: complex,B: complex] :
% 9.04/5.12        ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 9.04/5.12          = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 9.04/5.12        & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 9.04/5.12          = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9566_complex__norm__square,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.04/5.12        = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9567_complex__add__cnj,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 9.04/5.12        = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9568_complex__diff__cnj,axiom,
% 9.04/5.12      ! [Z: complex] :
% 9.04/5.12        ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 9.04/5.12        = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9569_complex__div__cnj,axiom,
% 9.04/5.12      ( divide1717551699836669952omplex
% 9.04/5.12      = ( ^ [A3: complex,B4: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B4 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9570_cnj__add__mult__eq__Re,axiom,
% 9.04/5.12      ! [Z: complex,W: complex] :
% 9.04/5.12        ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 9.04/5.12        = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9571_card__Collect__less__nat,axiom,
% 9.04/5.12      ! [N: nat] :
% 9.04/5.12        ( ( finite_card_nat
% 9.04/5.12          @ ( collect_nat
% 9.04/5.12            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
% 9.04/5.12        = N ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9572_card__atMost,axiom,
% 9.04/5.12      ! [U: nat] :
% 9.04/5.12        ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 9.04/5.12        = ( suc @ U ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9573_card__Collect__le__nat,axiom,
% 9.04/5.12      ! [N: nat] :
% 9.04/5.12        ( ( finite_card_nat
% 9.04/5.12          @ ( collect_nat
% 9.04/5.12            @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N ) ) )
% 9.04/5.12        = ( suc @ N ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9574_card__atLeastAtMost,axiom,
% 9.04/5.12      ! [L: nat,U: nat] :
% 9.04/5.12        ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 9.04/5.12        = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9575_card__atLeastAtMost__int,axiom,
% 9.04/5.12      ! [L: int,U: int] :
% 9.04/5.12        ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
% 9.04/5.12        = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9576_nat_Odisc__eq__case_I1_J,axiom,
% 9.04/5.12      ! [Nat: nat] :
% 9.04/5.12        ( ( Nat = zero_zero_nat )
% 9.04/5.12        = ( case_nat_o @ $true
% 9.04/5.12          @ ^ [Uu3: nat] : $false
% 9.04/5.12          @ Nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9577_nat_Odisc__eq__case_I2_J,axiom,
% 9.04/5.12      ! [Nat: nat] :
% 9.04/5.12        ( ( Nat != zero_zero_nat )
% 9.04/5.12        = ( case_nat_o @ $false
% 9.04/5.12          @ ^ [Uu3: nat] : $true
% 9.04/5.12          @ Nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9578_card__less,axiom,
% 9.04/5.12      ! [M2: set_nat,I: nat] :
% 9.04/5.12        ( ( member_nat @ zero_zero_nat @ M2 )
% 9.04/5.12       => ( ( finite_card_nat
% 9.04/5.12            @ ( collect_nat
% 9.04/5.12              @ ^ [K2: nat] :
% 9.04/5.12                  ( ( member_nat @ K2 @ M2 )
% 9.04/5.12                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
% 9.04/5.12         != zero_zero_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9579_card__less__Suc,axiom,
% 9.04/5.12      ! [M2: set_nat,I: nat] :
% 9.04/5.12        ( ( member_nat @ zero_zero_nat @ M2 )
% 9.04/5.12       => ( ( suc
% 9.04/5.12            @ ( finite_card_nat
% 9.04/5.12              @ ( collect_nat
% 9.04/5.12                @ ^ [K2: nat] :
% 9.04/5.12                    ( ( member_nat @ ( suc @ K2 ) @ M2 )
% 9.04/5.12                    & ( ord_less_nat @ K2 @ I ) ) ) ) )
% 9.04/5.12          = ( finite_card_nat
% 9.04/5.12            @ ( collect_nat
% 9.04/5.12              @ ^ [K2: nat] :
% 9.04/5.12                  ( ( member_nat @ K2 @ M2 )
% 9.04/5.12                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9580_card__less__Suc2,axiom,
% 9.04/5.12      ! [M2: set_nat,I: nat] :
% 9.04/5.12        ( ~ ( member_nat @ zero_zero_nat @ M2 )
% 9.04/5.12       => ( ( finite_card_nat
% 9.04/5.12            @ ( collect_nat
% 9.04/5.12              @ ^ [K2: nat] :
% 9.04/5.12                  ( ( member_nat @ ( suc @ K2 ) @ M2 )
% 9.04/5.12                  & ( ord_less_nat @ K2 @ I ) ) ) )
% 9.04/5.12          = ( finite_card_nat
% 9.04/5.12            @ ( collect_nat
% 9.04/5.12              @ ^ [K2: nat] :
% 9.04/5.12                  ( ( member_nat @ K2 @ M2 )
% 9.04/5.12                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9581_card__atLeastZeroLessThan__int,axiom,
% 9.04/5.12      ! [U: int] :
% 9.04/5.12        ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
% 9.04/5.12        = ( nat2 @ U ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9582_subset__card__intvl__is__intvl,axiom,
% 9.04/5.12      ! [A2: set_nat,K: nat] :
% 9.04/5.12        ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 9.04/5.12       => ( A2
% 9.04/5.12          = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9583_less__eq__nat_Osimps_I2_J,axiom,
% 9.04/5.12      ! [M: nat,N: nat] :
% 9.04/5.12        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.04/5.12        = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9584_card__le__Suc__Max,axiom,
% 9.04/5.12      ! [S2: set_nat] :
% 9.04/5.12        ( ( finite_finite_nat @ S2 )
% 9.04/5.12       => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9585_subset__eq__atLeast0__lessThan__card,axiom,
% 9.04/5.12      ! [N4: set_nat,N: nat] :
% 9.04/5.12        ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.04/5.12       => ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9586_card__sum__le__nat__sum,axiom,
% 9.04/5.12      ! [S2: set_nat] :
% 9.04/5.12        ( ord_less_eq_nat
% 9.04/5.12        @ ( groups3542108847815614940at_nat
% 9.04/5.12          @ ^ [X4: nat] : X4
% 9.04/5.12          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 9.04/5.12        @ ( groups3542108847815614940at_nat
% 9.04/5.12          @ ^ [X4: nat] : X4
% 9.04/5.12          @ S2 ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9587_card__nth__roots,axiom,
% 9.04/5.12      ! [C: complex,N: nat] :
% 9.04/5.12        ( ( C != zero_zero_complex )
% 9.04/5.12       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12         => ( ( finite_card_complex
% 9.04/5.12              @ ( collect_complex
% 9.04/5.12                @ ^ [Z4: complex] :
% 9.04/5.12                    ( ( power_power_complex @ Z4 @ N )
% 9.04/5.12                    = C ) ) )
% 9.04/5.12            = N ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9588_card__roots__unity__eq,axiom,
% 9.04/5.12      ! [N: nat] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( finite_card_complex
% 9.04/5.12            @ ( collect_complex
% 9.04/5.12              @ ^ [Z4: complex] :
% 9.04/5.12                  ( ( power_power_complex @ Z4 @ N )
% 9.04/5.12                  = one_one_complex ) ) )
% 9.04/5.12          = N ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9589_diff__Suc,axiom,
% 9.04/5.12      ! [M: nat,N: nat] :
% 9.04/5.12        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 9.04/5.12        = ( case_nat_nat @ zero_zero_nat
% 9.04/5.12          @ ^ [K2: nat] : K2
% 9.04/5.12          @ ( minus_minus_nat @ M @ N ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9590_pred__def,axiom,
% 9.04/5.12      ( pred
% 9.04/5.12      = ( case_nat_nat @ zero_zero_nat
% 9.04/5.12        @ ^ [X24: nat] : X24 ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9591_or__int__rec,axiom,
% 9.04/5.12      ( bit_se1409905431419307370or_int
% 9.04/5.12      = ( ^ [K2: int,L3: int] :
% 9.04/5.12            ( plus_plus_int
% 9.04/5.12            @ ( zero_n2684676970156552555ol_int
% 9.04/5.12              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 9.04/5.12                | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 9.04/5.12            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9592_or__nonnegative__int__iff,axiom,
% 9.04/5.12      ! [K: int,L: int] :
% 9.04/5.12        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 9.04/5.12        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.04/5.12          & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9593_or__negative__int__iff,axiom,
% 9.04/5.12      ! [K: int,L: int] :
% 9.04/5.12        ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 9.04/5.12        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.04/5.12          | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9594_or__minus__numerals_I6_J,axiom,
% 9.04/5.12      ! [N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 9.04/5.12        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9595_or__minus__numerals_I2_J,axiom,
% 9.04/5.12      ! [N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9596_or__minus__minus__numerals,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.12        = ( 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 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9597_and__minus__minus__numerals,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.12        = ( 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 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9598_or__greater__eq,axiom,
% 9.04/5.12      ! [L: int,K: int] :
% 9.04/5.12        ( ( ord_less_eq_int @ zero_zero_int @ L )
% 9.04/5.12       => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9599_OR__lower,axiom,
% 9.04/5.12      ! [X: int,Y: int] :
% 9.04/5.12        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.12         => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9600_plus__and__or,axiom,
% 9.04/5.12      ! [X: int,Y: int] :
% 9.04/5.12        ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
% 9.04/5.12        = ( plus_plus_int @ X @ Y ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9601_or__not__numerals_I1_J,axiom,
% 9.04/5.12      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.04/5.12      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9602_set__bit__int__def,axiom,
% 9.04/5.12      ( bit_se7879613467334960850it_int
% 9.04/5.12      = ( ^ [N3: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9603_or__not__numerals_I2_J,axiom,
% 9.04/5.12      ! [N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.04/5.12        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9604_or__not__numerals_I4_J,axiom,
% 9.04/5.12      ! [M: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.04/5.12        = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9605_or__not__numerals_I3_J,axiom,
% 9.04/5.12      ! [N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.04/5.12        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9606_or__not__numerals_I7_J,axiom,
% 9.04/5.12      ! [M: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.04/5.12        = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9607_binomial__def,axiom,
% 9.04/5.12      ( binomial
% 9.04/5.12      = ( ^ [N3: nat,K2: nat] :
% 9.04/5.12            ( finite_card_set_nat
% 9.04/5.12            @ ( collect_set_nat
% 9.04/5.12              @ ^ [K7: set_nat] :
% 9.04/5.12                  ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
% 9.04/5.12                  & ( ( finite_card_nat @ K7 )
% 9.04/5.12                    = K2 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9608_or__not__numerals_I6_J,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.04/5.12        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9609_OR__upper,axiom,
% 9.04/5.12      ! [X: int,N: nat,Y: int] :
% 9.04/5.12        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.12       => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.12         => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.04/5.12           => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9610_or__not__numerals_I5_J,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.04/5.12        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9611_or__not__numerals_I8_J,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.04/5.12        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9612_or__not__numerals_I9_J,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.04/5.12        = ( 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 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9613_or__minus__numerals_I1_J,axiom,
% 9.04/5.12      ! [N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9614_or__minus__numerals_I5_J,axiom,
% 9.04/5.12      ! [N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 9.04/5.12        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9615_or__nat__numerals_I4_J,axiom,
% 9.04/5.12      ! [X: num] :
% 9.04/5.12        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.04/5.12        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9616_or__nat__numerals_I2_J,axiom,
% 9.04/5.12      ! [Y: num] :
% 9.04/5.12        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.04/5.12        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9617_or__nat__numerals_I3_J,axiom,
% 9.04/5.12      ! [X: num] :
% 9.04/5.12        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.04/5.12        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9618_or__nat__numerals_I1_J,axiom,
% 9.04/5.12      ! [Y: num] :
% 9.04/5.12        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.04/5.12        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9619_or__minus__numerals_I8_J,axiom,
% 9.04/5.12      ! [N: num,M: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9620_or__minus__numerals_I4_J,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9621_or__minus__numerals_I3_J,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9622_or__minus__numerals_I7_J,axiom,
% 9.04/5.12      ! [N: num,M: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9623_or__not__num__neg_Osimps_I1_J,axiom,
% 9.04/5.12      ( ( bit_or_not_num_neg @ one @ one )
% 9.04/5.12      = one ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9624_set__bit__nat__def,axiom,
% 9.04/5.12      ( bit_se7882103937844011126it_nat
% 9.04/5.12      = ( ^ [M6: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9625_or__not__num__neg_Osimps_I4_J,axiom,
% 9.04/5.12      ! [N: num] :
% 9.04/5.12        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 9.04/5.12        = ( bit0 @ one ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9626_or__not__num__neg_Osimps_I6_J,axiom,
% 9.04/5.12      ! [N: num,M: num] :
% 9.04/5.12        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 9.04/5.12        = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9627_or__not__num__neg_Osimps_I7_J,axiom,
% 9.04/5.12      ! [N: num] :
% 9.04/5.12        ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 9.04/5.12        = one ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9628_or__not__num__neg_Osimps_I3_J,axiom,
% 9.04/5.12      ! [M: num] :
% 9.04/5.12        ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 9.04/5.12        = ( bit1 @ M ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9629_or__not__num__neg_Osimps_I5_J,axiom,
% 9.04/5.12      ! [N: num,M: num] :
% 9.04/5.12        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 9.04/5.12        = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9630_or__nat__def,axiom,
% 9.04/5.12      ( bit_se1412395901928357646or_nat
% 9.04/5.12      = ( ^ [M6: nat,N3: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9631_or__not__num__neg_Osimps_I2_J,axiom,
% 9.04/5.12      ! [M: num] :
% 9.04/5.12        ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 9.04/5.12        = ( bit1 @ M ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9632_or__not__num__neg_Osimps_I8_J,axiom,
% 9.04/5.12      ! [N: num,M: num] :
% 9.04/5.12        ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 9.04/5.12        = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9633_or__not__num__neg_Oelims,axiom,
% 9.04/5.12      ! [X: num,Xa: num,Y: num] :
% 9.04/5.12        ( ( ( bit_or_not_num_neg @ X @ Xa )
% 9.04/5.12          = Y )
% 9.04/5.12       => ( ( ( X = one )
% 9.04/5.12           => ( ( Xa = one )
% 9.04/5.12             => ( Y != one ) ) )
% 9.04/5.12         => ( ( ( X = one )
% 9.04/5.12             => ! [M5: num] :
% 9.04/5.12                  ( ( Xa
% 9.04/5.12                    = ( bit0 @ M5 ) )
% 9.04/5.12                 => ( Y
% 9.04/5.12                   != ( bit1 @ M5 ) ) ) )
% 9.04/5.12           => ( ( ( X = one )
% 9.04/5.12               => ! [M5: num] :
% 9.04/5.12                    ( ( Xa
% 9.04/5.12                      = ( bit1 @ M5 ) )
% 9.04/5.12                   => ( Y
% 9.04/5.12                     != ( bit1 @ M5 ) ) ) )
% 9.04/5.12             => ( ( ? [N2: num] :
% 9.04/5.12                      ( X
% 9.04/5.12                      = ( bit0 @ N2 ) )
% 9.04/5.12                 => ( ( Xa = one )
% 9.04/5.12                   => ( Y
% 9.04/5.12                     != ( bit0 @ one ) ) ) )
% 9.04/5.12               => ( ! [N2: num] :
% 9.04/5.12                      ( ( X
% 9.04/5.12                        = ( bit0 @ N2 ) )
% 9.04/5.12                     => ! [M5: num] :
% 9.04/5.12                          ( ( Xa
% 9.04/5.12                            = ( bit0 @ M5 ) )
% 9.04/5.12                         => ( Y
% 9.04/5.12                           != ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) ) ) )
% 9.04/5.12                 => ( ! [N2: num] :
% 9.04/5.12                        ( ( X
% 9.04/5.12                          = ( bit0 @ N2 ) )
% 9.04/5.12                       => ! [M5: num] :
% 9.04/5.12                            ( ( Xa
% 9.04/5.12                              = ( bit1 @ M5 ) )
% 9.04/5.12                           => ( Y
% 9.04/5.12                             != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M5 ) ) ) ) )
% 9.04/5.12                   => ( ( ? [N2: num] :
% 9.04/5.12                            ( X
% 9.04/5.12                            = ( bit1 @ N2 ) )
% 9.04/5.12                       => ( ( Xa = one )
% 9.04/5.12                         => ( Y != one ) ) )
% 9.04/5.12                     => ( ! [N2: num] :
% 9.04/5.12                            ( ( X
% 9.04/5.12                              = ( bit1 @ N2 ) )
% 9.04/5.12                           => ! [M5: num] :
% 9.04/5.12                                ( ( Xa
% 9.04/5.12                                  = ( bit0 @ M5 ) )
% 9.04/5.12                               => ( Y
% 9.04/5.12                                 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) ) ) )
% 9.04/5.12                       => ~ ! [N2: num] :
% 9.04/5.12                              ( ( X
% 9.04/5.12                                = ( bit1 @ N2 ) )
% 9.04/5.12                             => ! [M5: num] :
% 9.04/5.12                                  ( ( Xa
% 9.04/5.12                                    = ( bit1 @ M5 ) )
% 9.04/5.12                                 => ( Y
% 9.04/5.12                                   != ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9634_numeral__or__not__num__eq,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9635_int__numeral__not__or__num__neg,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9636_int__numeral__or__not__num__neg,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9637_Suc__0__or__eq,axiom,
% 9.04/5.12      ! [N: nat] :
% 9.04/5.12        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.04/5.12        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9638_or__Suc__0__eq,axiom,
% 9.04/5.12      ! [N: nat] :
% 9.04/5.12        ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.04/5.12        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9639_or__nat__rec,axiom,
% 9.04/5.12      ( bit_se1412395901928357646or_nat
% 9.04/5.12      = ( ^ [M6: nat,N3: nat] :
% 9.04/5.12            ( plus_plus_nat
% 9.04/5.12            @ ( zero_n2687167440665602831ol_nat
% 9.04/5.12              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 9.04/5.12                | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 9.04/5.12            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9640_or__not__num__neg_Opelims,axiom,
% 9.04/5.12      ! [X: num,Xa: num,Y: num] :
% 9.04/5.12        ( ( ( bit_or_not_num_neg @ X @ Xa )
% 9.04/5.12          = Y )
% 9.04/5.12       => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 9.04/5.12         => ( ( ( X = one )
% 9.04/5.12             => ( ( Xa = one )
% 9.04/5.12               => ( ( Y = one )
% 9.04/5.12                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.04/5.12           => ( ( ( X = one )
% 9.04/5.12               => ! [M5: num] :
% 9.04/5.12                    ( ( Xa
% 9.04/5.12                      = ( bit0 @ M5 ) )
% 9.04/5.12                   => ( ( Y
% 9.04/5.12                        = ( bit1 @ M5 ) )
% 9.04/5.12                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M5 ) ) ) ) ) )
% 9.04/5.12             => ( ( ( X = one )
% 9.04/5.12                 => ! [M5: num] :
% 9.04/5.12                      ( ( Xa
% 9.04/5.12                        = ( bit1 @ M5 ) )
% 9.04/5.12                     => ( ( Y
% 9.04/5.12                          = ( bit1 @ M5 ) )
% 9.04/5.12                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M5 ) ) ) ) ) )
% 9.04/5.12               => ( ! [N2: num] :
% 9.04/5.12                      ( ( X
% 9.04/5.12                        = ( bit0 @ N2 ) )
% 9.04/5.12                     => ( ( Xa = one )
% 9.04/5.12                       => ( ( Y
% 9.04/5.12                            = ( bit0 @ one ) )
% 9.04/5.12                         => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ one ) ) ) ) )
% 9.04/5.12                 => ( ! [N2: num] :
% 9.04/5.12                        ( ( X
% 9.04/5.12                          = ( bit0 @ N2 ) )
% 9.04/5.12                       => ! [M5: num] :
% 9.04/5.12                            ( ( Xa
% 9.04/5.12                              = ( bit0 @ M5 ) )
% 9.04/5.12                           => ( ( Y
% 9.04/5.12                                = ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) )
% 9.04/5.12                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit0 @ M5 ) ) ) ) ) )
% 9.04/5.12                   => ( ! [N2: num] :
% 9.04/5.12                          ( ( X
% 9.04/5.12                            = ( bit0 @ N2 ) )
% 9.04/5.12                         => ! [M5: num] :
% 9.04/5.12                              ( ( Xa
% 9.04/5.12                                = ( bit1 @ M5 ) )
% 9.04/5.12                             => ( ( Y
% 9.04/5.12                                  = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M5 ) ) )
% 9.04/5.12                               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit1 @ M5 ) ) ) ) ) )
% 9.04/5.12                     => ( ! [N2: num] :
% 9.04/5.12                            ( ( X
% 9.04/5.12                              = ( bit1 @ N2 ) )
% 9.04/5.12                           => ( ( Xa = one )
% 9.04/5.12                             => ( ( Y = one )
% 9.04/5.12                               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ one ) ) ) ) )
% 9.04/5.12                       => ( ! [N2: num] :
% 9.04/5.12                              ( ( X
% 9.04/5.12                                = ( bit1 @ N2 ) )
% 9.04/5.12                             => ! [M5: num] :
% 9.04/5.12                                  ( ( Xa
% 9.04/5.12                                    = ( bit0 @ M5 ) )
% 9.04/5.12                                 => ( ( Y
% 9.04/5.12                                      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) )
% 9.04/5.12                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit0 @ M5 ) ) ) ) ) )
% 9.04/5.12                         => ~ ! [N2: num] :
% 9.04/5.12                                ( ( X
% 9.04/5.12                                  = ( bit1 @ N2 ) )
% 9.04/5.12                               => ! [M5: num] :
% 9.04/5.12                                    ( ( Xa
% 9.04/5.12                                      = ( bit1 @ M5 ) )
% 9.04/5.12                                   => ( ( Y
% 9.04/5.12                                        = ( bitM @ ( bit_or_not_num_neg @ N2 @ M5 ) ) )
% 9.04/5.12                                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit1 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9641_or__int__unfold,axiom,
% 9.04/5.12      ( bit_se1409905431419307370or_int
% 9.04/5.12      = ( ^ [K2: int,L3: int] :
% 9.04/5.12            ( if_int
% 9.04/5.12            @ ( ( K2
% 9.04/5.12                = ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.12              | ( L3
% 9.04/5.12                = ( uminus_uminus_int @ one_one_int ) ) )
% 9.04/5.12            @ ( uminus_uminus_int @ one_one_int )
% 9.04/5.12            @ ( if_int @ ( K2 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9642_max__enat__simps_I2_J,axiom,
% 9.04/5.12      ! [Q3: extended_enat] :
% 9.04/5.12        ( ( ord_ma741700101516333627d_enat @ Q3 @ zero_z5237406670263579293d_enat )
% 9.04/5.12        = Q3 ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9643_max__enat__simps_I3_J,axiom,
% 9.04/5.12      ! [Q3: extended_enat] :
% 9.04/5.12        ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q3 )
% 9.04/5.12        = Q3 ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9644_max__Suc__Suc,axiom,
% 9.04/5.12      ! [M: nat,N: nat] :
% 9.04/5.12        ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.04/5.12        = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9645_max__nat_Oeq__neutr__iff,axiom,
% 9.04/5.12      ! [A: nat,B: nat] :
% 9.04/5.12        ( ( ( ord_max_nat @ A @ B )
% 9.04/5.12          = zero_zero_nat )
% 9.04/5.12        = ( ( A = zero_zero_nat )
% 9.04/5.12          & ( B = zero_zero_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9646_max__nat_Oleft__neutral,axiom,
% 9.04/5.12      ! [A: nat] :
% 9.04/5.12        ( ( ord_max_nat @ zero_zero_nat @ A )
% 9.04/5.12        = A ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9647_max__nat_Oneutr__eq__iff,axiom,
% 9.04/5.12      ! [A: nat,B: nat] :
% 9.04/5.12        ( ( zero_zero_nat
% 9.04/5.12          = ( ord_max_nat @ A @ B ) )
% 9.04/5.12        = ( ( A = zero_zero_nat )
% 9.04/5.12          & ( B = zero_zero_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9648_max__nat_Oright__neutral,axiom,
% 9.04/5.12      ! [A: nat] :
% 9.04/5.12        ( ( ord_max_nat @ A @ zero_zero_nat )
% 9.04/5.12        = A ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9649_max__0L,axiom,
% 9.04/5.12      ! [N: nat] :
% 9.04/5.12        ( ( ord_max_nat @ zero_zero_nat @ N )
% 9.04/5.12        = N ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9650_max__0R,axiom,
% 9.04/5.12      ! [N: nat] :
% 9.04/5.12        ( ( ord_max_nat @ N @ zero_zero_nat )
% 9.04/5.12        = N ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9651_max__Suc__numeral,axiom,
% 9.04/5.12      ! [N: nat,K: num] :
% 9.04/5.12        ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.04/5.12        = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9652_max__numeral__Suc,axiom,
% 9.04/5.12      ! [K: num,N: nat] :
% 9.04/5.12        ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.04/5.12        = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9653_nat__add__max__right,axiom,
% 9.04/5.12      ! [M: nat,N: nat,Q3: nat] :
% 9.04/5.12        ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 9.04/5.12        = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9654_nat__add__max__left,axiom,
% 9.04/5.12      ! [M: nat,N: nat,Q3: nat] :
% 9.04/5.12        ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 9.04/5.12        = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9655_nat__mult__max__right,axiom,
% 9.04/5.12      ! [M: nat,N: nat,Q3: nat] :
% 9.04/5.12        ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 9.04/5.12        = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9656_nat__mult__max__left,axiom,
% 9.04/5.12      ! [M: nat,N: nat,Q3: nat] :
% 9.04/5.12        ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 9.04/5.12        = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9657_nat__minus__add__max,axiom,
% 9.04/5.12      ! [N: nat,M: nat] :
% 9.04/5.12        ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 9.04/5.12        = ( ord_max_nat @ N @ M ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9658_max__Suc2,axiom,
% 9.04/5.12      ! [M: nat,N: nat] :
% 9.04/5.12        ( ( ord_max_nat @ M @ ( suc @ N ) )
% 9.04/5.12        = ( case_nat_nat @ ( suc @ N )
% 9.04/5.12          @ ^ [M7: nat] : ( suc @ ( ord_max_nat @ M7 @ N ) )
% 9.04/5.12          @ M ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9659_max__Suc1,axiom,
% 9.04/5.12      ! [N: nat,M: nat] :
% 9.04/5.12        ( ( ord_max_nat @ ( suc @ N ) @ M )
% 9.04/5.12        = ( case_nat_nat @ ( suc @ N )
% 9.04/5.12          @ ^ [M7: nat] : ( suc @ ( ord_max_nat @ N @ M7 ) )
% 9.04/5.12          @ M ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9660_or__nat__unfold,axiom,
% 9.04/5.12      ( bit_se1412395901928357646or_nat
% 9.04/5.12      = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9661_prod__decode__aux_Osimps,axiom,
% 9.04/5.12      ( nat_prod_decode_aux
% 9.04/5.12      = ( ^ [K2: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K2 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K2 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M6 @ ( suc @ K2 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9662_prod__decode__aux_Oelims,axiom,
% 9.04/5.12      ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
% 9.04/5.12        ( ( ( nat_prod_decode_aux @ X @ Xa )
% 9.04/5.12          = Y )
% 9.04/5.12       => ( ( ( ord_less_eq_nat @ Xa @ X )
% 9.04/5.12           => ( Y
% 9.04/5.12              = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 9.04/5.12          & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 9.04/5.12           => ( Y
% 9.04/5.12              = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9663_bezw__0,axiom,
% 9.04/5.12      ! [X: nat] :
% 9.04/5.12        ( ( bezw @ X @ zero_zero_nat )
% 9.04/5.12        = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9664_prod__decode__aux_Opelims,axiom,
% 9.04/5.12      ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
% 9.04/5.12        ( ( ( nat_prod_decode_aux @ X @ Xa )
% 9.04/5.12          = Y )
% 9.04/5.12       => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 9.04/5.12         => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
% 9.04/5.12                 => ( Y
% 9.04/5.12                    = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 9.04/5.12                & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 9.04/5.12                 => ( Y
% 9.04/5.12                    = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
% 9.04/5.12             => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9665_Suc__0__div__numeral,axiom,
% 9.04/5.12      ! [K: num] :
% 9.04/5.12        ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 9.04/5.12        = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9666_drop__bit__numeral__minus__bit1,axiom,
% 9.04/5.12      ! [L: num,K: num] :
% 9.04/5.12        ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.04/5.12        = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9667_drop__bit__nonnegative__int__iff,axiom,
% 9.04/5.12      ! [N: nat,K: int] :
% 9.04/5.12        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 9.04/5.12        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9668_drop__bit__negative__int__iff,axiom,
% 9.04/5.12      ! [N: nat,K: int] :
% 9.04/5.12        ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 9.04/5.12        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9669_drop__bit__minus__one,axiom,
% 9.04/5.12      ! [N: nat] :
% 9.04/5.12        ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 9.04/5.12        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9670_fst__divmod__nat,axiom,
% 9.04/5.12      ! [M: nat,N: nat] :
% 9.04/5.12        ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 9.04/5.12        = ( divide_divide_nat @ M @ N ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9671_drop__bit__Suc__minus__bit0,axiom,
% 9.04/5.12      ! [N: nat,K: num] :
% 9.04/5.12        ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.04/5.12        = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9672_drop__bit__numeral__minus__bit0,axiom,
% 9.04/5.12      ! [L: num,K: num] :
% 9.04/5.12        ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.04/5.12        = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9673_drop__bit__Suc__minus__bit1,axiom,
% 9.04/5.12      ! [N: nat,K: num] :
% 9.04/5.12        ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.04/5.12        = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9674_drop__bit__int__def,axiom,
% 9.04/5.12      ( bit_se8568078237143864401it_int
% 9.04/5.12      = ( ^ [N3: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9675_Suc__0__mod__numeral,axiom,
% 9.04/5.12      ! [K: num] :
% 9.04/5.12        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 9.04/5.12        = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9676_finite__enumerate,axiom,
% 9.04/5.12      ! [S2: set_nat] :
% 9.04/5.12        ( ( finite_finite_nat @ S2 )
% 9.04/5.12       => ? [R3: nat > nat] :
% 9.04/5.12            ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
% 9.04/5.12            & ! [N8: nat] :
% 9.04/5.12                ( ( ord_less_nat @ N8 @ ( finite_card_nat @ S2 ) )
% 9.04/5.12               => ( member_nat @ ( R3 @ N8 ) @ S2 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9677_fst__divmod__integer,axiom,
% 9.04/5.12      ! [K: code_integer,L: code_integer] :
% 9.04/5.12        ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L ) )
% 9.04/5.12        = ( divide6298287555418463151nteger @ K @ L ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9678_fst__divmod__abs,axiom,
% 9.04/5.12      ! [K: code_integer,L: code_integer] :
% 9.04/5.12        ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L ) )
% 9.04/5.12        = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9679_drop__bit__of__Suc__0,axiom,
% 9.04/5.12      ! [N: nat] :
% 9.04/5.12        ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.04/5.12        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9680_drop__bit__nat__def,axiom,
% 9.04/5.12      ( bit_se8570568707652914677it_nat
% 9.04/5.12      = ( ^ [N3: nat,M6: nat] : ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9681_rat__sgn__code,axiom,
% 9.04/5.12      ! [P5: rat] :
% 9.04/5.12        ( ( quotient_of @ ( sgn_sgn_rat @ P5 ) )
% 9.04/5.12        = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P5 ) ) ) @ one_one_int ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9682_Sup__nat__empty,axiom,
% 9.04/5.12      ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 9.04/5.12      = zero_zero_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9683_Inf__real__def,axiom,
% 9.04/5.12      ( comple4887499456419720421f_real
% 9.04/5.12      = ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9684_Sup__nat__def,axiom,
% 9.04/5.12      ( complete_Sup_Sup_nat
% 9.04/5.12      = ( ^ [X6: set_nat] : ( if_nat @ ( X6 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X6 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9685_quotient__of__denom__pos_H,axiom,
% 9.04/5.12      ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9686_bezw__non__0,axiom,
% 9.04/5.12      ! [Y: nat,X: nat] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ Y )
% 9.04/5.12       => ( ( bezw @ X @ Y )
% 9.04/5.12          = ( 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 ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9687_bezw_Osimps,axiom,
% 9.04/5.12      ( bezw
% 9.04/5.12      = ( ^ [X4: nat,Y6: nat] : ( if_Pro3027730157355071871nt_int @ ( Y6 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X4 @ Y6 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X4 @ Y6 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X4 @ Y6 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y6 ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9688_bezw_Oelims,axiom,
% 9.04/5.12      ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 9.04/5.12        ( ( ( bezw @ X @ Xa )
% 9.04/5.12          = Y )
% 9.04/5.12       => ( ( ( Xa = zero_zero_nat )
% 9.04/5.12           => ( Y
% 9.04/5.12              = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 9.04/5.12          & ( ( Xa != zero_zero_nat )
% 9.04/5.12           => ( Y
% 9.04/5.12              = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9689_bezw_Opelims,axiom,
% 9.04/5.12      ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 9.04/5.12        ( ( ( bezw @ X @ Xa )
% 9.04/5.12          = Y )
% 9.04/5.12       => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 9.04/5.12         => ~ ( ( ( ( Xa = zero_zero_nat )
% 9.04/5.12                 => ( Y
% 9.04/5.12                    = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 9.04/5.12                & ( ( Xa != zero_zero_nat )
% 9.04/5.12                 => ( Y
% 9.04/5.12                    = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
% 9.04/5.12             => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9690_minus__one__mod__numeral,axiom,
% 9.04/5.12      ! [N: num] :
% 9.04/5.12        ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.12        = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9691_numeral__mod__minus__numeral,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9692_minus__numeral__mod__numeral,axiom,
% 9.04/5.12      ! [M: num,N: num] :
% 9.04/5.12        ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.12        = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9693_one__mod__minus__numeral,axiom,
% 9.04/5.12      ! [N: num] :
% 9.04/5.12        ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.12        = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9694_Divides_Oadjust__mod__def,axiom,
% 9.04/5.12      ( adjust_mod
% 9.04/5.12      = ( ^ [L3: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L3 @ R5 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9695_normalize__def,axiom,
% 9.04/5.12      ( normalize
% 9.04/5.12      = ( ^ [P4: product_prod_int_int] :
% 9.04/5.12            ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P4 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) )
% 9.04/5.12            @ ( if_Pro3027730157355071871nt_int
% 9.04/5.12              @ ( ( product_snd_int_int @ P4 )
% 9.04/5.12                = zero_zero_int )
% 9.04/5.12              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.04/5.12              @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9696_gcd__1__int,axiom,
% 9.04/5.12      ! [M: int] :
% 9.04/5.12        ( ( gcd_gcd_int @ M @ one_one_int )
% 9.04/5.12        = one_one_int ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9697_gcd__pos__int,axiom,
% 9.04/5.12      ! [M: int,N: int] :
% 9.04/5.12        ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 9.04/5.12        = ( ( M != zero_zero_int )
% 9.04/5.12          | ( N != zero_zero_int ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9698_gcd__neg__numeral__2__int,axiom,
% 9.04/5.12      ! [X: int,N: num] :
% 9.04/5.12        ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.12        = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9699_gcd__neg__numeral__1__int,axiom,
% 9.04/5.12      ! [N: num,X: int] :
% 9.04/5.12        ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
% 9.04/5.12        = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9700_gcd__0__left__int,axiom,
% 9.04/5.12      ! [X: int] :
% 9.04/5.12        ( ( gcd_gcd_int @ zero_zero_int @ X )
% 9.04/5.12        = ( abs_abs_int @ X ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9701_gcd__0__int,axiom,
% 9.04/5.12      ! [X: int] :
% 9.04/5.12        ( ( gcd_gcd_int @ X @ zero_zero_int )
% 9.04/5.12        = ( abs_abs_int @ X ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9702_bezout__int,axiom,
% 9.04/5.12      ! [X: int,Y: int] :
% 9.04/5.12      ? [U3: int,V3: int] :
% 9.04/5.12        ( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V3 @ Y ) )
% 9.04/5.12        = ( gcd_gcd_int @ X @ Y ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9703_gcd__ge__0__int,axiom,
% 9.04/5.12      ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9704_gcd__le1__int,axiom,
% 9.04/5.12      ! [A: int,B: int] :
% 9.04/5.12        ( ( ord_less_int @ zero_zero_int @ A )
% 9.04/5.12       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9705_gcd__le2__int,axiom,
% 9.04/5.12      ! [B: int,A: int] :
% 9.04/5.12        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.12       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9706_gcd__cases__int,axiom,
% 9.04/5.12      ! [X: int,Y: int,P: int > $o] :
% 9.04/5.12        ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.12         => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.12           => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
% 9.04/5.12       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.04/5.12           => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 9.04/5.12             => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
% 9.04/5.12         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 9.04/5.12             => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.04/5.12               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
% 9.04/5.12           => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 9.04/5.12               => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 9.04/5.12                 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
% 9.04/5.12             => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9707_gcd__unique__int,axiom,
% 9.04/5.12      ! [D: int,A: int,B: int] :
% 9.04/5.12        ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 9.04/5.12          & ( dvd_dvd_int @ D @ A )
% 9.04/5.12          & ( dvd_dvd_int @ D @ B )
% 9.04/5.12          & ! [E3: int] :
% 9.04/5.12              ( ( ( dvd_dvd_int @ E3 @ A )
% 9.04/5.12                & ( dvd_dvd_int @ E3 @ B ) )
% 9.04/5.12             => ( dvd_dvd_int @ E3 @ D ) ) )
% 9.04/5.12        = ( D
% 9.04/5.12          = ( gcd_gcd_int @ A @ B ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9708_gcd__non__0__int,axiom,
% 9.04/5.12      ! [Y: int,X: int] :
% 9.04/5.12        ( ( ord_less_int @ zero_zero_int @ Y )
% 9.04/5.12       => ( ( gcd_gcd_int @ X @ Y )
% 9.04/5.12          = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9709_gcd__code__int,axiom,
% 9.04/5.12      ( gcd_gcd_int
% 9.04/5.12      = ( ^ [K2: int,L3: int] : ( abs_abs_int @ ( if_int @ ( L3 = zero_zero_int ) @ K2 @ ( gcd_gcd_int @ L3 @ ( modulo_modulo_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9710_gcd__is__Max__divisors__int,axiom,
% 9.04/5.12      ! [N: int,M: int] :
% 9.04/5.12        ( ( N != zero_zero_int )
% 9.04/5.12       => ( ( gcd_gcd_int @ M @ N )
% 9.04/5.12          = ( lattic8263393255366662781ax_int
% 9.04/5.12            @ ( collect_int
% 9.04/5.12              @ ^ [D3: int] :
% 9.04/5.12                  ( ( dvd_dvd_int @ D3 @ M )
% 9.04/5.12                  & ( dvd_dvd_int @ D3 @ N ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9711_suminf__eq__SUP__real,axiom,
% 9.04/5.12      ! [X8: nat > real] :
% 9.04/5.12        ( ( summable_real @ X8 )
% 9.04/5.12       => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I2 ) )
% 9.04/5.12         => ( ( suminf_real @ X8 )
% 9.04/5.12            = ( comple1385675409528146559p_real
% 9.04/5.12              @ ( image_nat_real
% 9.04/5.12                @ ^ [I3: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I3 ) )
% 9.04/5.12                @ top_top_set_nat ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9712_gcd__0__left__nat,axiom,
% 9.04/5.12      ! [X: nat] :
% 9.04/5.12        ( ( gcd_gcd_nat @ zero_zero_nat @ X )
% 9.04/5.12        = X ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9713_gcd__0__nat,axiom,
% 9.04/5.12      ! [X: nat] :
% 9.04/5.12        ( ( gcd_gcd_nat @ X @ zero_zero_nat )
% 9.04/5.12        = X ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9714_gcd__nat_Oright__neutral,axiom,
% 9.04/5.12      ! [A: nat] :
% 9.04/5.12        ( ( gcd_gcd_nat @ A @ zero_zero_nat )
% 9.04/5.12        = A ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9715_gcd__nat_Oneutr__eq__iff,axiom,
% 9.04/5.12      ! [A: nat,B: nat] :
% 9.04/5.12        ( ( zero_zero_nat
% 9.04/5.12          = ( gcd_gcd_nat @ A @ B ) )
% 9.04/5.12        = ( ( A = zero_zero_nat )
% 9.04/5.12          & ( B = zero_zero_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9716_gcd__nat_Oleft__neutral,axiom,
% 9.04/5.12      ! [A: nat] :
% 9.04/5.12        ( ( gcd_gcd_nat @ zero_zero_nat @ A )
% 9.04/5.12        = A ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9717_gcd__nat_Oeq__neutr__iff,axiom,
% 9.04/5.12      ! [A: nat,B: nat] :
% 9.04/5.12        ( ( ( gcd_gcd_nat @ A @ B )
% 9.04/5.12          = zero_zero_nat )
% 9.04/5.12        = ( ( A = zero_zero_nat )
% 9.04/5.12          & ( B = zero_zero_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9718_gcd__1__nat,axiom,
% 9.04/5.12      ! [M: nat] :
% 9.04/5.12        ( ( gcd_gcd_nat @ M @ one_one_nat )
% 9.04/5.12        = one_one_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9719_gcd__Suc__0,axiom,
% 9.04/5.12      ! [M: nat] :
% 9.04/5.12        ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.04/5.12        = ( suc @ zero_zero_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9720_gcd__pos__nat,axiom,
% 9.04/5.12      ! [M: nat,N: nat] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 9.04/5.12        = ( ( M != zero_zero_nat )
% 9.04/5.12          | ( N != zero_zero_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9721_gcd__int__int__eq,axiom,
% 9.04/5.12      ! [M: nat,N: nat] :
% 9.04/5.12        ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.04/5.12        = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9722_gcd__nat__abs__left__eq,axiom,
% 9.04/5.12      ! [K: int,N: nat] :
% 9.04/5.12        ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 9.04/5.12        = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9723_gcd__nat__abs__right__eq,axiom,
% 9.04/5.12      ! [N: nat,K: int] :
% 9.04/5.12        ( ( gcd_gcd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 9.04/5.12        = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9724_gcd__non__0__nat,axiom,
% 9.04/5.12      ! [Y: nat,X: nat] :
% 9.04/5.12        ( ( Y != zero_zero_nat )
% 9.04/5.12       => ( ( gcd_gcd_nat @ X @ Y )
% 9.04/5.12          = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9725_gcd__nat_Osimps,axiom,
% 9.04/5.12      ( gcd_gcd_nat
% 9.04/5.12      = ( ^ [X4: nat,Y6: nat] : ( if_nat @ ( Y6 = zero_zero_nat ) @ X4 @ ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X4 @ Y6 ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9726_gcd__nat_Oelims,axiom,
% 9.04/5.12      ! [X: nat,Xa: nat,Y: nat] :
% 9.04/5.12        ( ( ( gcd_gcd_nat @ X @ Xa )
% 9.04/5.12          = Y )
% 9.04/5.12       => ( ( ( Xa = zero_zero_nat )
% 9.04/5.12           => ( Y = X ) )
% 9.04/5.12          & ( ( Xa != zero_zero_nat )
% 9.04/5.12           => ( Y
% 9.04/5.12              = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9727_gcd__le1__nat,axiom,
% 9.04/5.12      ! [A: nat,B: nat] :
% 9.04/5.12        ( ( A != zero_zero_nat )
% 9.04/5.12       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9728_gcd__le2__nat,axiom,
% 9.04/5.12      ! [B: nat,A: nat] :
% 9.04/5.12        ( ( B != zero_zero_nat )
% 9.04/5.12       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9729_gcd__diff1__nat,axiom,
% 9.04/5.12      ! [N: nat,M: nat] :
% 9.04/5.12        ( ( ord_less_eq_nat @ N @ M )
% 9.04/5.12       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 9.04/5.12          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9730_gcd__diff2__nat,axiom,
% 9.04/5.12      ! [M: nat,N: nat] :
% 9.04/5.12        ( ( ord_less_eq_nat @ M @ N )
% 9.04/5.12       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 9.04/5.12          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9731_bezout__nat,axiom,
% 9.04/5.12      ! [A: nat,B: nat] :
% 9.04/5.12        ( ( A != zero_zero_nat )
% 9.04/5.12       => ? [X3: nat,Y4: nat] :
% 9.04/5.12            ( ( times_times_nat @ A @ X3 )
% 9.04/5.12            = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9732_bezout__gcd__nat_H,axiom,
% 9.04/5.12      ! [B: nat,A: nat] :
% 9.04/5.12      ? [X3: nat,Y4: nat] :
% 9.04/5.12        ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X3 ) )
% 9.04/5.12          & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
% 9.04/5.12            = ( gcd_gcd_nat @ A @ B ) ) )
% 9.04/5.12        | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X3 ) )
% 9.04/5.12          & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
% 9.04/5.12            = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9733_gcd__code__integer,axiom,
% 9.04/5.12      ( gcd_gcd_Code_integer
% 9.04/5.12      = ( ^ [K2: code_integer,L3: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L3 = zero_z3403309356797280102nteger ) @ K2 @ ( gcd_gcd_Code_integer @ L3 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L3 ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9734_gcd__int__def,axiom,
% 9.04/5.12      ( gcd_gcd_int
% 9.04/5.12      = ( ^ [X4: int,Y6: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X4 ) ) @ ( nat2 @ ( abs_abs_int @ Y6 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9735_gcd__is__Max__divisors__nat,axiom,
% 9.04/5.12      ! [N: nat,M: nat] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( gcd_gcd_nat @ M @ N )
% 9.04/5.12          = ( lattic8265883725875713057ax_nat
% 9.04/5.12            @ ( collect_nat
% 9.04/5.12              @ ^ [D3: nat] :
% 9.04/5.12                  ( ( dvd_dvd_nat @ D3 @ M )
% 9.04/5.12                  & ( dvd_dvd_nat @ D3 @ N ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9736_UNIV__nat__eq,axiom,
% 9.04/5.12      ( top_top_set_nat
% 9.04/5.12      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9737_range__mod,axiom,
% 9.04/5.12      ! [N: nat] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( image_nat_nat
% 9.04/5.12            @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N )
% 9.04/5.12            @ top_top_set_nat )
% 9.04/5.12          = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9738_bezw__aux,axiom,
% 9.04/5.12      ! [X: nat,Y: nat] :
% 9.04/5.12        ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
% 9.04/5.12        = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9739_gcd__nat_Opelims,axiom,
% 9.04/5.12      ! [X: nat,Xa: nat,Y: nat] :
% 9.04/5.12        ( ( ( gcd_gcd_nat @ X @ Xa )
% 9.04/5.12          = Y )
% 9.04/5.12       => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 9.04/5.12         => ~ ( ( ( ( Xa = zero_zero_nat )
% 9.04/5.12                 => ( Y = X ) )
% 9.04/5.12                & ( ( Xa != zero_zero_nat )
% 9.04/5.12                 => ( Y
% 9.04/5.12                    = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
% 9.04/5.12             => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9740_card__UNIV__unit,axiom,
% 9.04/5.12      ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 9.04/5.12      = one_one_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9741_card__UNIV__bool,axiom,
% 9.04/5.12      ( ( finite_card_o @ top_top_set_o )
% 9.04/5.12      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9742_range__mult,axiom,
% 9.04/5.12      ! [A: real] :
% 9.04/5.12        ( ( ( A = zero_zero_real )
% 9.04/5.12         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 9.04/5.12            = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 9.04/5.12        & ( ( A != zero_zero_real )
% 9.04/5.12         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 9.04/5.12            = top_top_set_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9743_infinite__UNIV__int,axiom,
% 9.04/5.12      ~ ( finite_finite_int @ top_top_set_int ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9744_int__in__range__abs,axiom,
% 9.04/5.12      ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9745_nat__descend__induct,axiom,
% 9.04/5.12      ! [N: nat,P: nat > $o,M: nat] :
% 9.04/5.12        ( ! [K3: nat] :
% 9.04/5.12            ( ( ord_less_nat @ N @ K3 )
% 9.04/5.12           => ( P @ K3 ) )
% 9.04/5.12       => ( ! [K3: nat] :
% 9.04/5.12              ( ( ord_less_eq_nat @ K3 @ N )
% 9.04/5.12             => ( ! [I4: nat] :
% 9.04/5.12                    ( ( ord_less_nat @ K3 @ I4 )
% 9.04/5.12                   => ( P @ I4 ) )
% 9.04/5.12               => ( P @ K3 ) ) )
% 9.04/5.12         => ( P @ M ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9746_root__def,axiom,
% 9.04/5.12      ( root
% 9.04/5.12      = ( ^ [N3: nat,X4: real] :
% 9.04/5.12            ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
% 9.04/5.12            @ ( the_in5290026491893676941l_real @ top_top_set_real
% 9.04/5.12              @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N3 ) )
% 9.04/5.12              @ X4 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9747_card__UNIV__char,axiom,
% 9.04/5.12      ( ( finite_card_char @ top_top_set_char )
% 9.04/5.12      = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9748_UNIV__bool,axiom,
% 9.04/5.12      ( top_top_set_o
% 9.04/5.12      = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9749_UNIV__char__of__nat,axiom,
% 9.04/5.12      ( top_top_set_char
% 9.04/5.12      = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9750_char_Osize_I2_J,axiom,
% 9.04/5.12      ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 9.04/5.12        ( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 9.04/5.12        = zero_zero_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9751_nat__of__char__less__256,axiom,
% 9.04/5.12      ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9752_range__nat__of__char,axiom,
% 9.04/5.12      ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 9.04/5.12      = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9753_integer__of__char__code,axiom,
% 9.04/5.12      ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 9.04/5.12        ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 9.04/5.12        = ( 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 ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9754_char_Osize__gen,axiom,
% 9.04/5.12      ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 9.04/5.12        ( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 9.04/5.12        = zero_zero_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9755_String_Ochar__of__ascii__of,axiom,
% 9.04/5.12      ! [C: char] :
% 9.04/5.12        ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 9.04/5.12        = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9756_DERIV__real__root__generic,axiom,
% 9.04/5.12      ! [N: nat,X: real,D2: real] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( X != zero_zero_real )
% 9.04/5.12         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.12             => ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.12               => ( D2
% 9.04/5.12                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 9.04/5.12           => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.12               => ( ( ord_less_real @ X @ zero_zero_real )
% 9.04/5.12                 => ( D2
% 9.04/5.12                    = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 9.04/5.12             => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.12                 => ( D2
% 9.04/5.12                    = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 9.04/5.12               => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9757_DERIV__even__real__root,axiom,
% 9.04/5.12      ! [N: nat,X: real] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.12         => ( ( ord_less_real @ X @ zero_zero_real )
% 9.04/5.12           => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9758_DERIV__arctan__series,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.12       => ( has_fi5821293074295781190e_real
% 9.04/5.12          @ ^ [X9: real] :
% 9.04/5.12              ( suminf_real
% 9.04/5.12              @ ^ [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 @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 9.04/5.12          @ ( suminf_real
% 9.04/5.12            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.12          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9759_has__real__derivative__pos__inc__right,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real,S2: set_real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.04/5.12         => ? [D5: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.04/5.12              & ! [H2: real] :
% 9.04/5.12                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.04/5.12                 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S2 )
% 9.04/5.12                   => ( ( ord_less_real @ H2 @ D5 )
% 9.04/5.12                     => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9760_has__real__derivative__neg__dec__right,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real,S2: set_real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 9.04/5.12       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.04/5.12         => ? [D5: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.04/5.12              & ! [H2: real] :
% 9.04/5.12                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.04/5.12                 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S2 )
% 9.04/5.12                   => ( ( ord_less_real @ H2 @ D5 )
% 9.04/5.12                     => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9761_has__real__derivative__neg__dec__left,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real,S2: set_real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 9.04/5.12       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.04/5.12         => ? [D5: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.04/5.12              & ! [H2: real] :
% 9.04/5.12                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.04/5.12                 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S2 )
% 9.04/5.12                   => ( ( ord_less_real @ H2 @ D5 )
% 9.04/5.12                     => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9762_has__real__derivative__pos__inc__left,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real,S2: set_real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.04/5.12         => ? [D5: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.04/5.12              & ! [H2: real] :
% 9.04/5.12                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.04/5.12                 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S2 )
% 9.04/5.12                   => ( ( ord_less_real @ H2 @ D5 )
% 9.04/5.12                     => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9763_DERIV__const__ratio__const2,axiom,
% 9.04/5.12      ! [A: real,B: real,F: real > real,K: real] :
% 9.04/5.12        ( ( A != B )
% 9.04/5.12       => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12         => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 9.04/5.12            = K ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9764_DERIV__pos__inc__left,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.04/5.12         => ? [D5: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.04/5.12              & ! [H2: real] :
% 9.04/5.12                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.04/5.12                 => ( ( ord_less_real @ H2 @ D5 )
% 9.04/5.12                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9765_DERIV__neg__dec__left,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.04/5.12         => ? [D5: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.04/5.12              & ! [H2: real] :
% 9.04/5.12                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.04/5.12                 => ( ( ord_less_real @ H2 @ D5 )
% 9.04/5.12                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9766_DERIV__isconst__all,axiom,
% 9.04/5.12      ! [F: real > real,X: real,Y: real] :
% 9.04/5.12        ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12       => ( ( F @ X )
% 9.04/5.12          = ( F @ Y ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9767_DERIV__pos__inc__right,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.04/5.12         => ? [D5: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.04/5.12              & ! [H2: real] :
% 9.04/5.12                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.04/5.12                 => ( ( ord_less_real @ H2 @ D5 )
% 9.04/5.12                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9768_DERIV__neg__dec__right,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.04/5.12         => ? [D5: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.04/5.12              & ! [H2: real] :
% 9.04/5.12                  ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.04/5.12                 => ( ( ord_less_real @ H2 @ D5 )
% 9.04/5.12                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9769_DERIV__neg__imp__decreasing,axiom,
% 9.04/5.12      ! [A: real,B: real,F: real > real] :
% 9.04/5.12        ( ( ord_less_real @ A @ B )
% 9.04/5.12       => ( ! [X3: real] :
% 9.04/5.12              ( ( ord_less_eq_real @ A @ X3 )
% 9.04/5.12             => ( ( ord_less_eq_real @ X3 @ B )
% 9.04/5.12               => ? [Y5: real] :
% 9.04/5.12                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12                    & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 9.04/5.12         => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9770_DERIV__pos__imp__increasing,axiom,
% 9.04/5.12      ! [A: real,B: real,F: real > real] :
% 9.04/5.12        ( ( ord_less_real @ A @ B )
% 9.04/5.12       => ( ! [X3: real] :
% 9.04/5.12              ( ( ord_less_eq_real @ A @ X3 )
% 9.04/5.12             => ( ( ord_less_eq_real @ X3 @ B )
% 9.04/5.12               => ? [Y5: real] :
% 9.04/5.12                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12                    & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 9.04/5.12         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9771_DERIV__nonneg__imp__nondecreasing,axiom,
% 9.04/5.12      ! [A: real,B: real,F: real > real] :
% 9.04/5.12        ( ( ord_less_eq_real @ A @ B )
% 9.04/5.12       => ( ! [X3: real] :
% 9.04/5.12              ( ( ord_less_eq_real @ A @ X3 )
% 9.04/5.12             => ( ( ord_less_eq_real @ X3 @ B )
% 9.04/5.12               => ? [Y5: real] :
% 9.04/5.12                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12                    & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 9.04/5.12         => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9772_DERIV__nonpos__imp__nonincreasing,axiom,
% 9.04/5.12      ! [A: real,B: real,F: real > real] :
% 9.04/5.12        ( ( ord_less_eq_real @ A @ B )
% 9.04/5.12       => ( ! [X3: real] :
% 9.04/5.12              ( ( ord_less_eq_real @ A @ X3 )
% 9.04/5.12             => ( ( ord_less_eq_real @ X3 @ B )
% 9.04/5.12               => ? [Y5: real] :
% 9.04/5.12                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12                    & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 9.04/5.12         => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9773_deriv__nonneg__imp__mono,axiom,
% 9.04/5.12      ! [A: real,B: real,G: real > real,G2: real > real] :
% 9.04/5.12        ( ! [X3: real] :
% 9.04/5.12            ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 9.04/5.12           => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 9.04/5.12       => ( ! [X3: real] :
% 9.04/5.12              ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 9.04/5.12             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
% 9.04/5.12         => ( ( ord_less_eq_real @ A @ B )
% 9.04/5.12           => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9774_MVT2,axiom,
% 9.04/5.12      ! [A: real,B: real,F: real > real,F3: real > real] :
% 9.04/5.12        ( ( ord_less_real @ A @ B )
% 9.04/5.12       => ( ! [X3: real] :
% 9.04/5.12              ( ( ord_less_eq_real @ A @ X3 )
% 9.04/5.12             => ( ( ord_less_eq_real @ X3 @ B )
% 9.04/5.12               => ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 9.04/5.12         => ? [Z3: real] :
% 9.04/5.12              ( ( ord_less_real @ A @ Z3 )
% 9.04/5.12              & ( ord_less_real @ Z3 @ B )
% 9.04/5.12              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 9.04/5.12                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F3 @ Z3 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9775_DERIV__local__const,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real,D: real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.04/5.12         => ( ! [Y4: real] :
% 9.04/5.12                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 9.04/5.12               => ( ( F @ X )
% 9.04/5.12                  = ( F @ Y4 ) ) )
% 9.04/5.12           => ( L = zero_zero_real ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9776_DERIV__ln,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.12       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9777_DERIV__const__average,axiom,
% 9.04/5.12      ! [A: real,B: real,V: real > real,K: real] :
% 9.04/5.12        ( ( A != B )
% 9.04/5.12       => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12         => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.04/5.12            = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9778_DERIV__local__min,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real,D: real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.04/5.12         => ( ! [Y4: real] :
% 9.04/5.12                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 9.04/5.12               => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
% 9.04/5.12           => ( L = zero_zero_real ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9779_DERIV__local__max,axiom,
% 9.04/5.12      ! [F: real > real,L: real,X: real,D: real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.04/5.12         => ( ! [Y4: real] :
% 9.04/5.12                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 9.04/5.12               => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
% 9.04/5.12           => ( L = zero_zero_real ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9780_DERIV__ln__divide,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.12       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9781_DERIV__pow,axiom,
% 9.04/5.12      ! [N: nat,X: real,S: set_real] :
% 9.04/5.12        ( has_fi5821293074295781190e_real
% 9.04/5.12        @ ^ [X4: real] : ( power_power_real @ X4 @ N )
% 9.04/5.12        @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 9.04/5.12        @ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9782_DERIV__fun__pow,axiom,
% 9.04/5.12      ! [G: real > real,M: real,X: real,N: nat] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( has_fi5821293074295781190e_real
% 9.04/5.12          @ ^ [X4: real] : ( power_power_real @ ( G @ X4 ) @ N )
% 9.04/5.12          @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 9.04/5.12          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9783_has__real__derivative__powr,axiom,
% 9.04/5.12      ! [Z: real,R2: real] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ Z )
% 9.04/5.12       => ( has_fi5821293074295781190e_real
% 9.04/5.12          @ ^ [Z4: real] : ( powr_real @ Z4 @ R2 )
% 9.04/5.12          @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 9.04/5.12          @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9784_DERIV__log,axiom,
% 9.04/5.12      ! [X: real,B: real] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.12       => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9785_DERIV__fun__powr,axiom,
% 9.04/5.12      ! [G: real > real,M: real,X: real,R2: real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 9.04/5.12         => ( has_fi5821293074295781190e_real
% 9.04/5.12            @ ^ [X4: real] : ( powr_real @ ( G @ X4 ) @ R2 )
% 9.04/5.12            @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 9.04/5.12            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9786_DERIV__powr,axiom,
% 9.04/5.12      ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 9.04/5.12         => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12           => ( has_fi5821293074295781190e_real
% 9.04/5.12              @ ^ [X4: real] : ( powr_real @ ( G @ X4 ) @ ( F @ X4 ) )
% 9.04/5.12              @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 9.04/5.12              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9787_DERIV__real__sqrt,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.12       => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9788_DERIV__arctan,axiom,
% 9.04/5.12      ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9789_arsinh__real__has__field__derivative,axiom,
% 9.04/5.12      ! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9790_DERIV__real__sqrt__generic,axiom,
% 9.04/5.12      ! [X: real,D2: real] :
% 9.04/5.12        ( ( X != zero_zero_real )
% 9.04/5.12       => ( ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.12           => ( D2
% 9.04/5.12              = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.04/5.12         => ( ( ( ord_less_real @ X @ zero_zero_real )
% 9.04/5.12             => ( D2
% 9.04/5.12                = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.04/5.12           => ( has_fi5821293074295781190e_real @ sqrt @ D2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9791_arcosh__real__has__field__derivative,axiom,
% 9.04/5.12      ! [X: real,A2: set_real] :
% 9.04/5.12        ( ( ord_less_real @ one_one_real @ X )
% 9.04/5.12       => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9792_artanh__real__has__field__derivative,axiom,
% 9.04/5.12      ! [X: real,A2: set_real] :
% 9.04/5.12        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.12       => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9793_DERIV__real__root,axiom,
% 9.04/5.12      ! [N: nat,X: real] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.04/5.12         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9794_DERIV__arccos,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.12       => ( ( ord_less_real @ X @ one_one_real )
% 9.04/5.12         => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9795_DERIV__arcsin,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.12       => ( ( ord_less_real @ X @ one_one_real )
% 9.04/5.12         => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9796_Maclaurin__all__le,axiom,
% 9.04/5.12      ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 9.04/5.12        ( ( ( Diff @ zero_zero_nat )
% 9.04/5.12          = F )
% 9.04/5.12       => ( ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12         => ? [T3: real] :
% 9.04/5.12              ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 9.04/5.12              & ( ( F @ X )
% 9.04/5.12                = ( plus_plus_real
% 9.04/5.12                  @ ( groups6591440286371151544t_real
% 9.04/5.12                    @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.12                    @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.12                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9797_Maclaurin__all__le__objl,axiom,
% 9.04/5.12      ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 9.04/5.12        ( ( ( ( Diff @ zero_zero_nat )
% 9.04/5.12            = F )
% 9.04/5.12          & ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 9.04/5.12       => ? [T3: real] :
% 9.04/5.12            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 9.04/5.12            & ( ( F @ X )
% 9.04/5.12              = ( plus_plus_real
% 9.04/5.12                @ ( groups6591440286371151544t_real
% 9.04/5.12                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.12                  @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.12                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9798_DERIV__odd__real__root,axiom,
% 9.04/5.12      ! [N: nat,X: real] :
% 9.04/5.12        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.12       => ( ( X != zero_zero_real )
% 9.04/5.12         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9799_Maclaurin__minus,axiom,
% 9.04/5.12      ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 9.04/5.12        ( ( ord_less_real @ H @ zero_zero_real )
% 9.04/5.12       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12         => ( ( ( Diff @ zero_zero_nat )
% 9.04/5.12              = F )
% 9.04/5.12           => ( ! [M5: nat,T3: real] :
% 9.04/5.12                  ( ( ( ord_less_nat @ M5 @ N )
% 9.04/5.12                    & ( ord_less_eq_real @ H @ T3 )
% 9.04/5.12                    & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
% 9.04/5.12                 => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 9.04/5.12             => ? [T3: real] :
% 9.04/5.12                  ( ( ord_less_real @ H @ T3 )
% 9.04/5.12                  & ( ord_less_real @ T3 @ zero_zero_real )
% 9.04/5.12                  & ( ( F @ H )
% 9.04/5.12                    = ( plus_plus_real
% 9.04/5.12                      @ ( groups6591440286371151544t_real
% 9.04/5.12                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
% 9.04/5.12                        @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.12                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9800_Maclaurin2,axiom,
% 9.04/5.12      ! [H: real,Diff: nat > real > real,F: real > real,N: nat] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ H )
% 9.04/5.12       => ( ( ( Diff @ zero_zero_nat )
% 9.04/5.12            = F )
% 9.04/5.12         => ( ! [M5: nat,T3: real] :
% 9.04/5.12                ( ( ( ord_less_nat @ M5 @ N )
% 9.04/5.12                  & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 9.04/5.12                  & ( ord_less_eq_real @ T3 @ H ) )
% 9.04/5.12               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 9.04/5.12           => ? [T3: real] :
% 9.04/5.12                ( ( ord_less_real @ zero_zero_real @ T3 )
% 9.04/5.12                & ( ord_less_eq_real @ T3 @ H )
% 9.04/5.12                & ( ( F @ H )
% 9.04/5.12                  = ( plus_plus_real
% 9.04/5.12                    @ ( groups6591440286371151544t_real
% 9.04/5.12                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
% 9.04/5.12                      @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.12                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9801_Maclaurin,axiom,
% 9.04/5.12      ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ H )
% 9.04/5.12       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12         => ( ( ( Diff @ zero_zero_nat )
% 9.04/5.12              = F )
% 9.04/5.12           => ( ! [M5: nat,T3: real] :
% 9.04/5.12                  ( ( ( ord_less_nat @ M5 @ N )
% 9.04/5.12                    & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 9.04/5.12                    & ( ord_less_eq_real @ T3 @ H ) )
% 9.04/5.12                 => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 9.04/5.12             => ? [T3: real] :
% 9.04/5.12                  ( ( ord_less_real @ zero_zero_real @ T3 )
% 9.04/5.12                  & ( ord_less_real @ T3 @ H )
% 9.04/5.12                  & ( ( F @ H )
% 9.04/5.12                    = ( plus_plus_real
% 9.04/5.12                      @ ( groups6591440286371151544t_real
% 9.04/5.12                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
% 9.04/5.12                        @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.12                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9802_Maclaurin__all__lt,axiom,
% 9.04/5.12      ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 9.04/5.12        ( ( ( Diff @ zero_zero_nat )
% 9.04/5.12          = F )
% 9.04/5.12       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12         => ( ( X != zero_zero_real )
% 9.04/5.12           => ( ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12             => ? [T3: real] :
% 9.04/5.12                  ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 9.04/5.12                  & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 9.04/5.12                  & ( ( F @ X )
% 9.04/5.12                    = ( plus_plus_real
% 9.04/5.12                      @ ( groups6591440286371151544t_real
% 9.04/5.12                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.12                        @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.12                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9803_Maclaurin__bi__le,axiom,
% 9.04/5.12      ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 9.04/5.12        ( ( ( Diff @ zero_zero_nat )
% 9.04/5.12          = F )
% 9.04/5.12       => ( ! [M5: nat,T3: real] :
% 9.04/5.12              ( ( ( ord_less_nat @ M5 @ N )
% 9.04/5.12                & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
% 9.04/5.12             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 9.04/5.12         => ? [T3: real] :
% 9.04/5.12              ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 9.04/5.12              & ( ( F @ X )
% 9.04/5.12                = ( plus_plus_real
% 9.04/5.12                  @ ( groups6591440286371151544t_real
% 9.04/5.12                    @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 9.04/5.12                    @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.12                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9804_Taylor__down,axiom,
% 9.04/5.12      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( ( Diff @ zero_zero_nat )
% 9.04/5.12            = F )
% 9.04/5.12         => ( ! [M5: nat,T3: real] :
% 9.04/5.12                ( ( ( ord_less_nat @ M5 @ N )
% 9.04/5.12                  & ( ord_less_eq_real @ A @ T3 )
% 9.04/5.12                  & ( ord_less_eq_real @ T3 @ B ) )
% 9.04/5.12               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 9.04/5.12           => ( ( ord_less_real @ A @ C )
% 9.04/5.12             => ( ( ord_less_eq_real @ C @ B )
% 9.04/5.12               => ? [T3: real] :
% 9.04/5.12                    ( ( ord_less_real @ A @ T3 )
% 9.04/5.12                    & ( ord_less_real @ T3 @ C )
% 9.04/5.12                    & ( ( F @ A )
% 9.04/5.12                      = ( plus_plus_real
% 9.04/5.12                        @ ( groups6591440286371151544t_real
% 9.04/5.12                          @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
% 9.04/5.12                          @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.12                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9805_Taylor__up,axiom,
% 9.04/5.12      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( ( Diff @ zero_zero_nat )
% 9.04/5.12            = F )
% 9.04/5.12         => ( ! [M5: nat,T3: real] :
% 9.04/5.12                ( ( ( ord_less_nat @ M5 @ N )
% 9.04/5.12                  & ( ord_less_eq_real @ A @ T3 )
% 9.04/5.12                  & ( ord_less_eq_real @ T3 @ B ) )
% 9.04/5.12               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 9.04/5.12           => ( ( ord_less_eq_real @ A @ C )
% 9.04/5.12             => ( ( ord_less_real @ C @ B )
% 9.04/5.12               => ? [T3: real] :
% 9.04/5.12                    ( ( ord_less_real @ C @ T3 )
% 9.04/5.12                    & ( ord_less_real @ T3 @ B )
% 9.04/5.12                    & ( ( F @ B )
% 9.04/5.12                      = ( plus_plus_real
% 9.04/5.12                        @ ( groups6591440286371151544t_real
% 9.04/5.12                          @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M6 ) )
% 9.04/5.12                          @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.12                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9806_Taylor,axiom,
% 9.04/5.12      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( ( Diff @ zero_zero_nat )
% 9.04/5.12            = F )
% 9.04/5.12         => ( ! [M5: nat,T3: real] :
% 9.04/5.12                ( ( ( ord_less_nat @ M5 @ N )
% 9.04/5.12                  & ( ord_less_eq_real @ A @ T3 )
% 9.04/5.12                  & ( ord_less_eq_real @ T3 @ B ) )
% 9.04/5.12               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 9.04/5.12           => ( ( ord_less_eq_real @ A @ C )
% 9.04/5.12             => ( ( ord_less_eq_real @ C @ B )
% 9.04/5.12               => ( ( ord_less_eq_real @ A @ X )
% 9.04/5.12                 => ( ( ord_less_eq_real @ X @ B )
% 9.04/5.12                   => ( ( X != C )
% 9.04/5.12                     => ? [T3: real] :
% 9.04/5.12                          ( ( ( ord_less_real @ X @ C )
% 9.04/5.12                           => ( ( ord_less_real @ X @ T3 )
% 9.04/5.12                              & ( ord_less_real @ T3 @ C ) ) )
% 9.04/5.12                          & ( ~ ( ord_less_real @ X @ C )
% 9.04/5.12                           => ( ( ord_less_real @ C @ T3 )
% 9.04/5.12                              & ( ord_less_real @ T3 @ X ) ) )
% 9.04/5.12                          & ( ( F @ X )
% 9.04/5.12                            = ( plus_plus_real
% 9.04/5.12                              @ ( groups6591440286371151544t_real
% 9.04/5.12                                @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M6 ) )
% 9.04/5.12                                @ ( set_ord_lessThan_nat @ N ) )
% 9.04/5.12                              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9807_Maclaurin__lemma2,axiom,
% 9.04/5.12      ! [N: nat,H: real,Diff: nat > real > real,K: nat,B2: real] :
% 9.04/5.12        ( ! [M5: nat,T3: real] :
% 9.04/5.12            ( ( ( ord_less_nat @ M5 @ N )
% 9.04/5.12              & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 9.04/5.12              & ( ord_less_eq_real @ T3 @ H ) )
% 9.04/5.12           => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 9.04/5.12       => ( ( N
% 9.04/5.12            = ( suc @ K ) )
% 9.04/5.12         => ! [M3: nat,T4: real] :
% 9.04/5.12              ( ( ( ord_less_nat @ M3 @ N )
% 9.04/5.12                & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 9.04/5.12                & ( ord_less_eq_real @ T4 @ H ) )
% 9.04/5.12             => ( has_fi5821293074295781190e_real
% 9.04/5.12                @ ^ [U2: real] :
% 9.04/5.12                    ( minus_minus_real @ ( Diff @ M3 @ U2 )
% 9.04/5.12                    @ ( plus_plus_real
% 9.04/5.12                      @ ( groups6591440286371151544t_real
% 9.04/5.12                        @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M3 @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ U2 @ P4 ) )
% 9.04/5.12                        @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M3 ) ) )
% 9.04/5.12                      @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M3 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M3 ) ) ) ) ) )
% 9.04/5.12                @ ( minus_minus_real @ ( Diff @ ( suc @ M3 ) @ T4 )
% 9.04/5.12                  @ ( plus_plus_real
% 9.04/5.12                    @ ( groups6591440286371151544t_real
% 9.04/5.12                      @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M3 ) @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ T4 @ P4 ) )
% 9.04/5.12                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) )
% 9.04/5.12                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
% 9.04/5.12                @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9808_DERIV__power__series_H,axiom,
% 9.04/5.12      ! [R: real,F: nat > real,X0: real] :
% 9.04/5.12        ( ! [X3: real] :
% 9.04/5.12            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 9.04/5.12           => ( summable_real
% 9.04/5.12              @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X3 @ N3 ) ) ) )
% 9.04/5.12       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 9.04/5.12         => ( ( ord_less_real @ zero_zero_real @ R )
% 9.04/5.12           => ( has_fi5821293074295781190e_real
% 9.04/5.12              @ ^ [X4: real] :
% 9.04/5.12                  ( suminf_real
% 9.04/5.12                  @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X4 @ ( suc @ N3 ) ) ) )
% 9.04/5.12              @ ( suminf_real
% 9.04/5.12                @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
% 9.04/5.12              @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9809_tanh__real__bounds,axiom,
% 9.04/5.12      ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9810_DERIV__isconst3,axiom,
% 9.04/5.12      ! [A: real,B: real,X: real,Y: real,F: real > real] :
% 9.04/5.12        ( ( ord_less_real @ A @ B )
% 9.04/5.12       => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.04/5.12         => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.04/5.12           => ( ! [X3: real] :
% 9.04/5.12                  ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.04/5.12                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 9.04/5.12             => ( ( F @ X )
% 9.04/5.12                = ( F @ Y ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9811_DERIV__series_H,axiom,
% 9.04/5.12      ! [F: real > nat > real,F3: real > nat > real,X0: real,A: real,B: real,L4: nat > real] :
% 9.04/5.12        ( ! [N2: nat] :
% 9.04/5.12            ( has_fi5821293074295781190e_real
% 9.04/5.12            @ ^ [X4: real] : ( F @ X4 @ N2 )
% 9.04/5.12            @ ( F3 @ X0 @ N2 )
% 9.04/5.12            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 9.04/5.12       => ( ! [X3: real] :
% 9.04/5.12              ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.04/5.12             => ( summable_real @ ( F @ X3 ) ) )
% 9.04/5.12         => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.04/5.12           => ( ( summable_real @ ( F3 @ X0 ) )
% 9.04/5.12             => ( ( summable_real @ L4 )
% 9.04/5.12               => ( ! [N2: nat,X3: real,Y4: real] :
% 9.04/5.12                      ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.04/5.12                     => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.04/5.12                       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N2 ) @ ( F @ Y4 @ N2 ) ) ) @ ( times_times_real @ ( L4 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) ) ) ) )
% 9.04/5.12                 => ( has_fi5821293074295781190e_real
% 9.04/5.12                    @ ^ [X4: real] : ( suminf_real @ ( F @ X4 ) )
% 9.04/5.12                    @ ( suminf_real @ ( F3 @ X0 ) )
% 9.04/5.12                    @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9812_divmod__integer__eq__cases,axiom,
% 9.04/5.12      ( code_divmod_integer
% 9.04/5.12      = ( ^ [K2: code_integer,L3: code_integer] :
% 9.04/5.12            ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 9.04/5.12            @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 9.04/5.12              @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L3
% 9.04/5.12                @ ( if_Pro6119634080678213985nteger
% 9.04/5.12                  @ ( ( sgn_sgn_Code_integer @ K2 )
% 9.04/5.12                    = ( sgn_sgn_Code_integer @ L3 ) )
% 9.04/5.12                  @ ( code_divmod_abs @ K2 @ L3 )
% 9.04/5.12                  @ ( produc6916734918728496179nteger
% 9.04/5.12                    @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L3 ) @ S5 ) ) )
% 9.04/5.12                    @ ( code_divmod_abs @ K2 @ L3 ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9813_card__greaterThanLessThan,axiom,
% 9.04/5.12      ! [L: nat,U: nat] :
% 9.04/5.12        ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 9.04/5.12        = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9814_card__greaterThanLessThan__int,axiom,
% 9.04/5.12      ! [L: int,U: int] :
% 9.04/5.12        ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
% 9.04/5.12        = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9815_atLeastSucLessThan__greaterThanLessThan,axiom,
% 9.04/5.12      ! [L: nat,U: nat] :
% 9.04/5.12        ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 9.04/5.12        = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9816_isCont__Lb__Ub,axiom,
% 9.04/5.12      ! [A: real,B: real,F: real > real] :
% 9.04/5.12        ( ( ord_less_eq_real @ A @ B )
% 9.04/5.12       => ( ! [X3: real] :
% 9.04/5.12              ( ( ( ord_less_eq_real @ A @ X3 )
% 9.04/5.12                & ( ord_less_eq_real @ X3 @ B ) )
% 9.04/5.12             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 9.04/5.12         => ? [L5: real,M9: real] :
% 9.04/5.12              ( ! [X2: real] :
% 9.04/5.12                  ( ( ( ord_less_eq_real @ A @ X2 )
% 9.04/5.12                    & ( ord_less_eq_real @ X2 @ B ) )
% 9.04/5.12                 => ( ( ord_less_eq_real @ L5 @ ( F @ X2 ) )
% 9.04/5.12                    & ( ord_less_eq_real @ ( F @ X2 ) @ M9 ) ) )
% 9.04/5.12              & ! [Y5: real] :
% 9.04/5.12                  ( ( ( ord_less_eq_real @ L5 @ Y5 )
% 9.04/5.12                    & ( ord_less_eq_real @ Y5 @ M9 ) )
% 9.04/5.12                 => ? [X3: real] :
% 9.04/5.12                      ( ( ord_less_eq_real @ A @ X3 )
% 9.04/5.12                      & ( ord_less_eq_real @ X3 @ B )
% 9.04/5.12                      & ( ( F @ X3 )
% 9.04/5.12                        = Y5 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9817_LIM__fun__gt__zero,axiom,
% 9.04/5.12      ! [F: real > real,L: real,C: real] :
% 9.04/5.12        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.04/5.12         => ? [R3: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.04/5.12              & ! [X2: real] :
% 9.04/5.12                  ( ( ( X2 != C )
% 9.04/5.12                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 9.04/5.12                 => ( ord_less_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9818_LIM__fun__not__zero,axiom,
% 9.04/5.12      ! [F: real > real,L: real,C: real] :
% 9.04/5.12        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.04/5.12       => ( ( L != zero_zero_real )
% 9.04/5.12         => ? [R3: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.04/5.12              & ! [X2: real] :
% 9.04/5.12                  ( ( ( X2 != C )
% 9.04/5.12                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 9.04/5.12                 => ( ( F @ X2 )
% 9.04/5.12                   != zero_zero_real ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9819_LIM__fun__less__zero,axiom,
% 9.04/5.12      ! [F: real > real,L: real,C: real] :
% 9.04/5.12        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.04/5.12       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.04/5.12         => ? [R3: real] :
% 9.04/5.12              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.04/5.12              & ! [X2: real] :
% 9.04/5.12                  ( ( ( X2 != C )
% 9.04/5.12                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 9.04/5.12                 => ( ord_less_real @ ( F @ X2 ) @ zero_zero_real ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9820_isCont__arctan,axiom,
% 9.04/5.12      ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arctan ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9821_isCont__arsinh,axiom,
% 9.04/5.12      ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arsinh_real ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9822_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 9.04/5.12      ! [L: int,U: int] :
% 9.04/5.12        ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 9.04/5.12        = ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9823_isCont__inverse__function2,axiom,
% 9.04/5.12      ! [A: real,X: real,B: real,G: real > real,F: real > real] :
% 9.04/5.12        ( ( ord_less_real @ A @ X )
% 9.04/5.12       => ( ( ord_less_real @ X @ B )
% 9.04/5.12         => ( ! [Z3: real] :
% 9.04/5.12                ( ( ord_less_eq_real @ A @ Z3 )
% 9.04/5.12               => ( ( ord_less_eq_real @ Z3 @ B )
% 9.04/5.12                 => ( ( G @ ( F @ Z3 ) )
% 9.04/5.12                    = Z3 ) ) )
% 9.04/5.12           => ( ! [Z3: real] :
% 9.04/5.12                  ( ( ord_less_eq_real @ A @ Z3 )
% 9.04/5.12                 => ( ( ord_less_eq_real @ Z3 @ B )
% 9.04/5.12                   => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
% 9.04/5.12             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9824_isCont__ln,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( X != zero_zero_real )
% 9.04/5.12       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ln_ln_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9825_isCont__arcosh,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ one_one_real @ X )
% 9.04/5.12       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9826_LIM__cos__div__sin,axiom,
% 9.04/5.12      ( filterlim_real_real
% 9.04/5.12      @ ^ [X4: real] : ( divide_divide_real @ ( cos_real @ X4 ) @ ( sin_real @ X4 ) )
% 9.04/5.12      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12      @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9827_DERIV__inverse__function,axiom,
% 9.04/5.12      ! [F: real > real,D2: real,G: real > real,X: real,A: real,B: real] :
% 9.04/5.12        ( ( has_fi5821293074295781190e_real @ F @ D2 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
% 9.04/5.12       => ( ( D2 != zero_zero_real )
% 9.04/5.12         => ( ( ord_less_real @ A @ X )
% 9.04/5.12           => ( ( ord_less_real @ X @ B )
% 9.04/5.12             => ( ! [Y4: real] :
% 9.04/5.12                    ( ( ord_less_real @ A @ Y4 )
% 9.04/5.12                   => ( ( ord_less_real @ Y4 @ B )
% 9.04/5.12                     => ( ( F @ ( G @ Y4 ) )
% 9.04/5.12                        = Y4 ) ) )
% 9.04/5.12               => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
% 9.04/5.12                 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D2 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9828_isCont__arccos,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.12       => ( ( ord_less_real @ X @ one_one_real )
% 9.04/5.12         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9829_isCont__arcsin,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.12       => ( ( ord_less_real @ X @ one_one_real )
% 9.04/5.12         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9830_LIM__less__bound,axiom,
% 9.04/5.12      ! [B: real,X: real,F: real > real] :
% 9.04/5.12        ( ( ord_less_real @ B @ X )
% 9.04/5.12       => ( ! [X3: real] :
% 9.04/5.12              ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ X ) )
% 9.04/5.12             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 9.04/5.12         => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
% 9.04/5.12           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9831_isCont__artanh,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.04/5.12       => ( ( ord_less_real @ X @ one_one_real )
% 9.04/5.12         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9832_isCont__inverse__function,axiom,
% 9.04/5.12      ! [D: real,X: real,G: real > real,F: real > real] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ D )
% 9.04/5.12       => ( ! [Z3: real] :
% 9.04/5.12              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
% 9.04/5.12             => ( ( G @ ( F @ Z3 ) )
% 9.04/5.12                = Z3 ) )
% 9.04/5.12         => ( ! [Z3: real] :
% 9.04/5.12                ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
% 9.04/5.12               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
% 9.04/5.12           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9833_GMVT_H,axiom,
% 9.04/5.12      ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F3: real > real] :
% 9.04/5.12        ( ( ord_less_real @ A @ B )
% 9.04/5.12       => ( ! [Z3: real] :
% 9.04/5.12              ( ( ord_less_eq_real @ A @ Z3 )
% 9.04/5.12             => ( ( ord_less_eq_real @ Z3 @ B )
% 9.04/5.12               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
% 9.04/5.12         => ( ! [Z3: real] :
% 9.04/5.12                ( ( ord_less_eq_real @ A @ Z3 )
% 9.04/5.12               => ( ( ord_less_eq_real @ Z3 @ B )
% 9.04/5.12                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
% 9.04/5.12           => ( ! [Z3: real] :
% 9.04/5.12                  ( ( ord_less_real @ A @ Z3 )
% 9.04/5.12                 => ( ( ord_less_real @ Z3 @ B )
% 9.04/5.12                   => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
% 9.04/5.12             => ( ! [Z3: real] :
% 9.04/5.12                    ( ( ord_less_real @ A @ Z3 )
% 9.04/5.12                   => ( ( ord_less_real @ Z3 @ B )
% 9.04/5.12                     => ( has_fi5821293074295781190e_real @ F @ ( F3 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
% 9.04/5.12               => ? [C4: real] :
% 9.04/5.12                    ( ( ord_less_real @ A @ C4 )
% 9.04/5.12                    & ( ord_less_real @ C4 @ B )
% 9.04/5.12                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C4 ) )
% 9.04/5.12                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F3 @ C4 ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9834_summable__Leibniz_I3_J,axiom,
% 9.04/5.12      ! [A: nat > real] :
% 9.04/5.12        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12       => ( ( topolo6980174941875973593q_real @ A )
% 9.04/5.12         => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 9.04/5.12           => ! [N8: nat] :
% 9.04/5.12                ( member_real
% 9.04/5.12                @ ( suminf_real
% 9.04/5.12                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 9.04/5.12                @ ( set_or1222579329274155063t_real
% 9.04/5.12                  @ ( groups6591440286371151544t_real
% 9.04/5.12                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) )
% 9.04/5.12                  @ ( groups6591440286371151544t_real
% 9.04/5.12                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9835_summable__Leibniz_I2_J,axiom,
% 9.04/5.12      ! [A: nat > real] :
% 9.04/5.12        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12       => ( ( topolo6980174941875973593q_real @ A )
% 9.04/5.12         => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 9.04/5.12           => ! [N8: nat] :
% 9.04/5.12                ( member_real
% 9.04/5.12                @ ( suminf_real
% 9.04/5.12                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 9.04/5.12                @ ( set_or1222579329274155063t_real
% 9.04/5.12                  @ ( groups6591440286371151544t_real
% 9.04/5.12                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) )
% 9.04/5.12                  @ ( groups6591440286371151544t_real
% 9.04/5.12                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9836_summable__Leibniz_H_I4_J,axiom,
% 9.04/5.12      ! [A: nat > real,N: nat] :
% 9.04/5.12        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 9.04/5.12         => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 9.04/5.12           => ( ord_less_eq_real
% 9.04/5.12              @ ( suminf_real
% 9.04/5.12                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 9.04/5.12              @ ( groups6591440286371151544t_real
% 9.04/5.12                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9837_filterlim__Suc,axiom,
% 9.04/5.12      filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9838_card_Ocomp__fun__commute__on,axiom,
% 9.04/5.12      ( ( comp_nat_nat_nat @ suc @ suc )
% 9.04/5.12      = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9839_mult__nat__right__at__top,axiom,
% 9.04/5.12      ! [C: nat] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.04/5.12       => ( filterlim_nat_nat
% 9.04/5.12          @ ^ [X4: nat] : ( times_times_nat @ X4 @ C )
% 9.04/5.12          @ at_top_nat
% 9.04/5.12          @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9840_mult__nat__left__at__top,axiom,
% 9.04/5.12      ! [C: nat] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.04/5.12       => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9841_monoseq__convergent,axiom,
% 9.04/5.12      ! [X8: nat > real,B2: real] :
% 9.04/5.12        ( ( topolo6980174941875973593q_real @ X8 )
% 9.04/5.12       => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I2 ) ) @ B2 )
% 9.04/5.12         => ~ ! [L5: real] :
% 9.04/5.12                ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9842_LIMSEQ__root,axiom,
% 9.04/5.12      ( filterlim_nat_real
% 9.04/5.12      @ ^ [N3: nat] : ( root @ N3 @ ( semiri5074537144036343181t_real @ N3 ) )
% 9.04/5.12      @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.04/5.12      @ at_top_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9843_nested__sequence__unique,axiom,
% 9.04/5.12      ! [F: nat > real,G: nat > real] :
% 9.04/5.12        ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.04/5.12       => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
% 9.04/5.12         => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.04/5.12           => ( ( filterlim_nat_real
% 9.04/5.12                @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.04/5.12                @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12                @ at_top_nat )
% 9.04/5.12             => ? [L2: real] :
% 9.04/5.12                  ( ! [N8: nat] : ( ord_less_eq_real @ ( F @ N8 ) @ L2 )
% 9.04/5.12                  & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat )
% 9.04/5.12                  & ! [N8: nat] : ( ord_less_eq_real @ L2 @ ( G @ N8 ) )
% 9.04/5.12                  & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9844_LIMSEQ__inverse__zero,axiom,
% 9.04/5.12      ! [X8: nat > real] :
% 9.04/5.12        ( ! [R3: real] :
% 9.04/5.12          ? [N6: nat] :
% 9.04/5.12          ! [N2: nat] :
% 9.04/5.12            ( ( ord_less_eq_nat @ N6 @ N2 )
% 9.04/5.12           => ( ord_less_real @ R3 @ ( X8 @ N2 ) ) )
% 9.04/5.12       => ( filterlim_nat_real
% 9.04/5.12          @ ^ [N3: nat] : ( inverse_inverse_real @ ( X8 @ N3 ) )
% 9.04/5.12          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12          @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9845_lim__inverse__n_H,axiom,
% 9.04/5.12      ( filterlim_nat_real
% 9.04/5.12      @ ^ [N3: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
% 9.04/5.12      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12      @ at_top_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9846_LIMSEQ__root__const,axiom,
% 9.04/5.12      ! [C: real] :
% 9.04/5.12        ( ( ord_less_real @ zero_zero_real @ C )
% 9.04/5.12       => ( filterlim_nat_real
% 9.04/5.12          @ ^ [N3: nat] : ( root @ N3 @ C )
% 9.04/5.12          @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.04/5.12          @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9847_LIMSEQ__inverse__real__of__nat,axiom,
% 9.04/5.12      ( filterlim_nat_real
% 9.04/5.12      @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
% 9.04/5.12      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12      @ at_top_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9848_LIMSEQ__inverse__real__of__nat__add,axiom,
% 9.04/5.12      ! [R2: real] :
% 9.04/5.12        ( filterlim_nat_real
% 9.04/5.12        @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 9.04/5.12        @ ( topolo2815343760600316023s_real @ R2 )
% 9.04/5.12        @ at_top_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9849_increasing__LIMSEQ,axiom,
% 9.04/5.12      ! [F: nat > real,L: real] :
% 9.04/5.12        ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.04/5.12       => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
% 9.04/5.12         => ( ! [E: real] :
% 9.04/5.12                ( ( ord_less_real @ zero_zero_real @ E )
% 9.04/5.12               => ? [N8: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N8 ) @ E ) ) )
% 9.04/5.12           => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9850_continuous__floor,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ~ ( member_real @ X @ ring_1_Ints_real )
% 9.04/5.12       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( comp_int_real_real @ ring_1_of_int_real @ archim6058952711729229775r_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9851_LIMSEQ__realpow__zero,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.12       => ( ( ord_less_real @ X @ one_one_real )
% 9.04/5.12         => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9852_LIMSEQ__divide__realpow__zero,axiom,
% 9.04/5.12      ! [X: real,A: real] :
% 9.04/5.12        ( ( ord_less_real @ one_one_real @ X )
% 9.04/5.12       => ( filterlim_nat_real
% 9.04/5.12          @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N3 ) )
% 9.04/5.12          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12          @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9853_LIMSEQ__abs__realpow__zero2,axiom,
% 9.04/5.12      ! [C: real] :
% 9.04/5.12        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 9.04/5.12       => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9854_LIMSEQ__abs__realpow__zero,axiom,
% 9.04/5.12      ! [C: real] :
% 9.04/5.12        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 9.04/5.12       => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9855_LIMSEQ__inverse__realpow__zero,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_real @ one_one_real @ X )
% 9.04/5.12       => ( filterlim_nat_real
% 9.04/5.12          @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N3 ) )
% 9.04/5.12          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12          @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9856_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 9.04/5.12      ! [R2: real] :
% 9.04/5.12        ( filterlim_nat_real
% 9.04/5.12        @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
% 9.04/5.12        @ ( topolo2815343760600316023s_real @ R2 )
% 9.04/5.12        @ at_top_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9857_tendsto__exp__limit__sequentially,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( filterlim_nat_real
% 9.04/5.12        @ ^ [N3: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 )
% 9.04/5.12        @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 9.04/5.12        @ at_top_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9858_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 9.04/5.12      ! [R2: real] :
% 9.04/5.12        ( filterlim_nat_real
% 9.04/5.12        @ ^ [N3: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
% 9.04/5.12        @ ( topolo2815343760600316023s_real @ R2 )
% 9.04/5.12        @ at_top_nat ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9859_summable__Leibniz_I1_J,axiom,
% 9.04/5.12      ! [A: nat > real] :
% 9.04/5.12        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12       => ( ( topolo6980174941875973593q_real @ A )
% 9.04/5.12         => ( summable_real
% 9.04/5.12            @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9860_summable,axiom,
% 9.04/5.12      ! [A: nat > real] :
% 9.04/5.12        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 9.04/5.12         => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 9.04/5.12           => ( summable_real
% 9.04/5.12              @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9861_cos__diff__limit__1,axiom,
% 9.04/5.12      ! [Theta: nat > real,Theta2: real] :
% 9.04/5.12        ( ( filterlim_nat_real
% 9.04/5.12          @ ^ [J2: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J2 ) @ Theta2 ) )
% 9.04/5.12          @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.04/5.12          @ at_top_nat )
% 9.04/5.12       => ~ ! [K3: nat > int] :
% 9.04/5.12              ~ ( filterlim_nat_real
% 9.04/5.12                @ ^ [J2: nat] : ( minus_minus_real @ ( Theta @ J2 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.04/5.12                @ ( topolo2815343760600316023s_real @ Theta2 )
% 9.04/5.12                @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9862_cos__limit__1,axiom,
% 9.04/5.12      ! [Theta: nat > real] :
% 9.04/5.12        ( ( filterlim_nat_real
% 9.04/5.12          @ ^ [J2: nat] : ( cos_real @ ( Theta @ J2 ) )
% 9.04/5.12          @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.04/5.12          @ at_top_nat )
% 9.04/5.12       => ? [K3: nat > int] :
% 9.04/5.12            ( filterlim_nat_real
% 9.04/5.12            @ ^ [J2: nat] : ( minus_minus_real @ ( Theta @ J2 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.04/5.12            @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12            @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9863_summable__Leibniz_I4_J,axiom,
% 9.04/5.12      ! [A: nat > real] :
% 9.04/5.12        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12       => ( ( topolo6980174941875973593q_real @ A )
% 9.04/5.12         => ( filterlim_nat_real
% 9.04/5.12            @ ^ [N3: nat] :
% 9.04/5.12                ( groups6591440286371151544t_real
% 9.04/5.12                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 9.04/5.12            @ ( topolo2815343760600316023s_real
% 9.04/5.12              @ ( suminf_real
% 9.04/5.12                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 9.04/5.12            @ at_top_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9864_zeroseq__arctan__series,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.04/5.12       => ( filterlim_nat_real
% 9.04/5.12          @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 9.04/5.12          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12          @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9865_summable__Leibniz_H_I3_J,axiom,
% 9.04/5.12      ! [A: nat > real] :
% 9.04/5.12        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 9.04/5.12         => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 9.04/5.12           => ( filterlim_nat_real
% 9.04/5.12              @ ^ [N3: nat] :
% 9.04/5.12                  ( groups6591440286371151544t_real
% 9.04/5.12                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 9.04/5.12              @ ( topolo2815343760600316023s_real
% 9.04/5.12                @ ( suminf_real
% 9.04/5.12                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 9.04/5.12              @ at_top_nat ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9866_summable__Leibniz_H_I2_J,axiom,
% 9.04/5.12      ! [A: nat > real,N: nat] :
% 9.04/5.12        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 9.04/5.12         => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 9.04/5.12           => ( ord_less_eq_real
% 9.04/5.12              @ ( groups6591440286371151544t_real
% 9.04/5.12                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.04/5.12              @ ( suminf_real
% 9.04/5.12                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9867_sums__alternating__upper__lower,axiom,
% 9.04/5.12      ! [A: nat > real] :
% 9.04/5.12        ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 9.04/5.12       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 9.04/5.12         => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12           => ? [L2: real] :
% 9.04/5.12                ( ! [N8: nat] :
% 9.04/5.12                    ( ord_less_eq_real
% 9.04/5.12                    @ ( groups6591440286371151544t_real
% 9.04/5.12                      @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) )
% 9.04/5.12                    @ L2 )
% 9.04/5.12                & ( filterlim_nat_real
% 9.04/5.12                  @ ^ [N3: nat] :
% 9.04/5.12                      ( groups6591440286371151544t_real
% 9.04/5.12                      @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 9.04/5.12                  @ ( topolo2815343760600316023s_real @ L2 )
% 9.04/5.12                  @ at_top_nat )
% 9.04/5.12                & ! [N8: nat] :
% 9.04/5.12                    ( ord_less_eq_real @ L2
% 9.04/5.12                    @ ( groups6591440286371151544t_real
% 9.04/5.12                      @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) ) )
% 9.04/5.12                & ( filterlim_nat_real
% 9.04/5.12                  @ ^ [N3: nat] :
% 9.04/5.12                      ( groups6591440286371151544t_real
% 9.04/5.12                      @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 9.04/5.12                  @ ( topolo2815343760600316023s_real @ L2 )
% 9.04/5.12                  @ at_top_nat ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9868_summable__Leibniz_I5_J,axiom,
% 9.04/5.12      ! [A: nat > real] :
% 9.04/5.12        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12       => ( ( topolo6980174941875973593q_real @ A )
% 9.04/5.12         => ( filterlim_nat_real
% 9.04/5.12            @ ^ [N3: nat] :
% 9.04/5.12                ( groups6591440286371151544t_real
% 9.04/5.12                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 9.04/5.12            @ ( topolo2815343760600316023s_real
% 9.04/5.12              @ ( suminf_real
% 9.04/5.12                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 9.04/5.12            @ at_top_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9869_summable__Leibniz_H_I5_J,axiom,
% 9.04/5.12      ! [A: nat > real] :
% 9.04/5.12        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.04/5.12       => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 9.04/5.12         => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 9.04/5.12           => ( filterlim_nat_real
% 9.04/5.12              @ ^ [N3: nat] :
% 9.04/5.12                  ( groups6591440286371151544t_real
% 9.04/5.12                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 9.04/5.12                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 9.04/5.12              @ ( topolo2815343760600316023s_real
% 9.04/5.12                @ ( suminf_real
% 9.04/5.12                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 9.04/5.12              @ at_top_nat ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9870_Code__Numeral_Onegative__def,axiom,
% 9.04/5.12      ( code_negative
% 9.04/5.12      = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9871_Code__Target__Int_Onegative__def,axiom,
% 9.04/5.12      ( code_Target_negative
% 9.04/5.12      = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9872_tendsto__exp__limit__at__right,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( filterlim_real_real
% 9.04/5.12        @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y6 ) ) @ ( divide_divide_real @ one_one_real @ Y6 ) )
% 9.04/5.12        @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 9.04/5.12        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9873_tendsto__arcosh__at__left__1,axiom,
% 9.04/5.12      filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9874_filterlim__tan__at__right,axiom,
% 9.04/5.12      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 ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9875_sinh__real__at__bot,axiom,
% 9.04/5.12      filterlim_real_real @ sinh_real @ at_bot_real @ at_bot_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9876_arsinh__real__at__bot,axiom,
% 9.04/5.12      filterlim_real_real @ arsinh_real @ at_bot_real @ at_bot_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9877_greaterThan__0,axiom,
% 9.04/5.12      ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 9.04/5.12      = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9878_exp__at__bot,axiom,
% 9.04/5.12      filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9879_greaterThan__Suc,axiom,
% 9.04/5.12      ! [K: nat] :
% 9.04/5.12        ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 9.04/5.12        = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9880_filterlim__inverse__at__bot__neg,axiom,
% 9.04/5.12      filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9881_tanh__real__at__bot,axiom,
% 9.04/5.12      filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9882_ln__at__0,axiom,
% 9.04/5.12      filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9883_artanh__real__at__right__1,axiom,
% 9.04/5.12      filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9884_DERIV__pos__imp__increasing__at__bot,axiom,
% 9.04/5.12      ! [B: real,F: real > real,Flim: real] :
% 9.04/5.12        ( ! [X3: real] :
% 9.04/5.12            ( ( ord_less_eq_real @ X3 @ B )
% 9.04/5.12           => ? [Y5: real] :
% 9.04/5.12                ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12                & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 9.04/5.12       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 9.04/5.12         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9885_filterlim__pow__at__bot__odd,axiom,
% 9.04/5.12      ! [N: nat,F: real > real,F4: filter_real] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 9.04/5.12         => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.12           => ( filterlim_real_real
% 9.04/5.12              @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
% 9.04/5.12              @ at_bot_real
% 9.04/5.12              @ F4 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9886_tendsto__arctan__at__bot,axiom,
% 9.04/5.12      filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9887_filterlim__pow__at__bot__even,axiom,
% 9.04/5.12      ! [N: nat,F: real > real,F4: filter_real] :
% 9.04/5.12        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.12       => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 9.04/5.12         => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.04/5.12           => ( filterlim_real_real
% 9.04/5.12              @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
% 9.04/5.12              @ at_top_real
% 9.04/5.12              @ F4 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9888_at__bot__le__at__infinity,axiom,
% 9.04/5.12      ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9889_cosh__real__at__top,axiom,
% 9.04/5.12      filterlim_real_real @ cosh_real @ at_top_real @ at_top_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9890_sinh__real__at__top,axiom,
% 9.04/5.12      filterlim_real_real @ sinh_real @ at_top_real @ at_top_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9891_at__top__le__at__infinity,axiom,
% 9.04/5.12      ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9892_arcosh__real__at__top,axiom,
% 9.04/5.12      filterlim_real_real @ arcosh_real @ at_top_real @ at_top_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9893_arsinh__real__at__top,axiom,
% 9.04/5.12      filterlim_real_real @ arsinh_real @ at_top_real @ at_top_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9894_ln__at__top,axiom,
% 9.04/5.12      filterlim_real_real @ ln_ln_real @ at_top_real @ at_top_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9895_exp__at__top,axiom,
% 9.04/5.12      filterlim_real_real @ exp_real @ at_top_real @ at_top_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9896_cosh__real__at__bot,axiom,
% 9.04/5.12      filterlim_real_real @ cosh_real @ at_top_real @ at_bot_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9897_filterlim__real__sequentially,axiom,
% 9.04/5.12      filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9898_filterlim__real__at__infinity__sequentially,axiom,
% 9.04/5.12      filterlim_nat_real @ semiri5074537144036343181t_real @ at_infinity_real @ at_top_nat ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9899_tanh__real__at__top,axiom,
% 9.04/5.12      filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9900_artanh__real__at__left__1,axiom,
% 9.04/5.12      filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9901_ln__x__over__x__tendsto__0,axiom,
% 9.04/5.12      ( filterlim_real_real
% 9.04/5.12      @ ^ [X4: real] : ( divide_divide_real @ ( ln_ln_real @ X4 ) @ X4 )
% 9.04/5.12      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12      @ at_top_real ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9902_filterlim__inverse__at__right__top,axiom,
% 9.04/5.12      filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9903_filterlim__inverse__at__top__right,axiom,
% 9.04/5.12      filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9904_tendsto__power__div__exp__0,axiom,
% 9.04/5.12      ! [K: nat] :
% 9.04/5.12        ( filterlim_real_real
% 9.04/5.12        @ ^ [X4: real] : ( divide_divide_real @ ( power_power_real @ X4 @ K ) @ ( exp_real @ X4 ) )
% 9.04/5.12        @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.04/5.12        @ at_top_real ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9905_tendsto__exp__limit__at__top,axiom,
% 9.04/5.12      ! [X: real] :
% 9.04/5.12        ( filterlim_real_real
% 9.04/5.12        @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y6 ) ) @ Y6 )
% 9.04/5.12        @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 9.04/5.12        @ at_top_real ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9906_filterlim__tan__at__left,axiom,
% 9.04/5.12      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 ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9907_DERIV__neg__imp__decreasing__at__top,axiom,
% 9.04/5.12      ! [B: real,F: real > real,Flim: real] :
% 9.04/5.12        ( ! [X3: real] :
% 9.04/5.12            ( ( ord_less_eq_real @ B @ X3 )
% 9.04/5.12           => ? [Y5: real] :
% 9.04/5.12                ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.12                & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 9.04/5.12       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 9.04/5.12         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9908_tendsto__arctan__at__top,axiom,
% 9.04/5.12      filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9909_lhopital__left__at__top,axiom,
% 9.04/5.12      ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 9.04/5.12        ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.12       => ( ( eventually_real
% 9.04/5.12            @ ^ [X4: real] :
% 9.04/5.12                ( ( G2 @ X4 )
% 9.04/5.12               != zero_zero_real )
% 9.04/5.12            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.12         => ( ( eventually_real
% 9.04/5.12              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.12              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.12           => ( ( eventually_real
% 9.04/5.12                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.12                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.12             => ( ( filterlim_real_real
% 9.04/5.12                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.12                  @ ( topolo2815343760600316023s_real @ Y )
% 9.04/5.12                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.12               => ( filterlim_real_real
% 9.04/5.12                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.12                  @ ( topolo2815343760600316023s_real @ Y )
% 9.04/5.12                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9910_eventually__sequentially__Suc,axiom,
% 9.04/5.12      ! [P: nat > $o] :
% 9.04/5.12        ( ( eventually_nat
% 9.04/5.12          @ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
% 9.04/5.12          @ at_top_nat )
% 9.04/5.12        = ( eventually_nat @ P @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9911_eventually__sequentially__seg,axiom,
% 9.04/5.12      ! [P: nat > $o,K: nat] :
% 9.04/5.12        ( ( eventually_nat
% 9.04/5.12          @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K ) )
% 9.04/5.12          @ at_top_nat )
% 9.04/5.12        = ( eventually_nat @ P @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9912_eventually__sequentially,axiom,
% 9.04/5.12      ! [P: nat > $o] :
% 9.04/5.12        ( ( eventually_nat @ P @ at_top_nat )
% 9.04/5.12        = ( ? [N7: nat] :
% 9.04/5.12            ! [N3: nat] :
% 9.04/5.12              ( ( ord_less_eq_nat @ N7 @ N3 )
% 9.04/5.12             => ( P @ N3 ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9913_eventually__sequentiallyI,axiom,
% 9.04/5.12      ! [C: nat,P: nat > $o] :
% 9.04/5.12        ( ! [X3: nat] :
% 9.04/5.12            ( ( ord_less_eq_nat @ C @ X3 )
% 9.04/5.12           => ( P @ X3 ) )
% 9.04/5.12       => ( eventually_nat @ P @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9914_sequentially__offset,axiom,
% 9.04/5.12      ! [P: nat > $o,K: nat] :
% 9.04/5.12        ( ( eventually_nat @ P @ at_top_nat )
% 9.04/5.12       => ( eventually_nat
% 9.04/5.12          @ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
% 9.04/5.12          @ at_top_nat ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9915_le__sequentially,axiom,
% 9.04/5.12      ! [F4: filter_nat] :
% 9.04/5.12        ( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
% 9.04/5.12        = ( ! [N7: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N7 ) @ F4 ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9916_eventually__at__right__to__0,axiom,
% 9.04/5.12      ! [P: real > $o,A: real] :
% 9.04/5.12        ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.12        = ( eventually_real
% 9.04/5.12          @ ^ [X4: real] : ( P @ ( plus_plus_real @ X4 @ A ) )
% 9.04/5.12          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9917_eventually__at__right__real,axiom,
% 9.04/5.12      ! [A: real,B: real] :
% 9.04/5.12        ( ( ord_less_real @ A @ B )
% 9.04/5.12       => ( eventually_real
% 9.04/5.12          @ ^ [X4: real] : ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.04/5.12          @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9918_eventually__at__left__real,axiom,
% 9.04/5.12      ! [B: real,A: real] :
% 9.04/5.12        ( ( ord_less_real @ B @ A )
% 9.04/5.12       => ( eventually_real
% 9.04/5.12          @ ^ [X4: real] : ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ A ) )
% 9.04/5.12          @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9919_eventually__at__top__to__right,axiom,
% 9.04/5.12      ! [P: real > $o] :
% 9.04/5.12        ( ( eventually_real @ P @ at_top_real )
% 9.04/5.12        = ( eventually_real
% 9.04/5.12          @ ^ [X4: real] : ( P @ ( inverse_inverse_real @ X4 ) )
% 9.04/5.12          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9920_eventually__at__right__to__top,axiom,
% 9.04/5.12      ! [P: real > $o] :
% 9.04/5.12        ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.12        = ( eventually_real
% 9.04/5.12          @ ^ [X4: real] : ( P @ ( inverse_inverse_real @ X4 ) )
% 9.04/5.12          @ at_top_real ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9921_lhopital__at__top__at__top,axiom,
% 9.04/5.12      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 9.04/5.12        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 9.04/5.12       => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 9.04/5.12         => ( ( eventually_real
% 9.04/5.12              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.12              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 9.04/5.12           => ( ( eventually_real
% 9.04/5.12                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.12                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 9.04/5.12             => ( ( filterlim_real_real
% 9.04/5.12                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.12                  @ at_top_real
% 9.04/5.12                  @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 9.04/5.12               => ( filterlim_real_real
% 9.04/5.12                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.12                  @ at_top_real
% 9.04/5.12                  @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9922_lhopital,axiom,
% 9.04/5.12      ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 9.04/5.12        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12       => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12         => ( ( eventually_real
% 9.04/5.12              @ ^ [X4: real] :
% 9.04/5.12                  ( ( G @ X4 )
% 9.04/5.12                 != zero_zero_real )
% 9.04/5.12              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12           => ( ( eventually_real
% 9.04/5.12                @ ^ [X4: real] :
% 9.04/5.12                    ( ( G2 @ X4 )
% 9.04/5.12                   != zero_zero_real )
% 9.04/5.12                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12             => ( ( eventually_real
% 9.04/5.12                  @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.12                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12               => ( ( eventually_real
% 9.04/5.12                    @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.12                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12                 => ( ( filterlim_real_real
% 9.04/5.12                      @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.12                      @ F4
% 9.04/5.12                      @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.12                   => ( filterlim_real_real
% 9.04/5.12                      @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.12                      @ F4
% 9.04/5.12                      @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9923_lhopital__right__at__top__at__top,axiom,
% 9.04/5.12      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 9.04/5.12        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.12       => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.12         => ( ( eventually_real
% 9.04/5.12              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.12              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.12           => ( ( eventually_real
% 9.04/5.12                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.12                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.12             => ( ( filterlim_real_real
% 9.04/5.12                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.12                  @ at_top_real
% 9.04/5.12                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.12               => ( filterlim_real_real
% 9.04/5.12                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.12                  @ at_top_real
% 9.04/5.12                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 9.04/5.12  
% 9.04/5.12  thf(fact_9924_lhopital__at__top__at__bot,axiom,
% 9.04/5.12      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 9.04/5.12        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 9.04/5.12       => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 9.04/5.12         => ( ( eventually_real
% 9.04/5.12              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 9.04/5.13             => ( ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                  @ at_bot_real
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 9.04/5.13               => ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.13                  @ at_bot_real
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9925_lhopital__left__at__top__at__top,axiom,
% 9.04/5.13      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 9.04/5.13        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 9.04/5.13       => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 9.04/5.13         => ( ( eventually_real
% 9.04/5.13              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 9.04/5.13             => ( ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                  @ at_top_real
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 9.04/5.13               => ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.13                  @ at_top_real
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9926_lhospital__at__top__at__top,axiom,
% 9.04/5.13      ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 9.04/5.13        ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 9.04/5.13       => ( ( eventually_real
% 9.04/5.13            @ ^ [X4: real] :
% 9.04/5.13                ( ( G2 @ X4 )
% 9.04/5.13               != zero_zero_real )
% 9.04/5.13            @ at_top_real )
% 9.04/5.13         => ( ( eventually_real
% 9.04/5.13              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13              @ at_top_real )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                @ at_top_real )
% 9.04/5.13             => ( ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                  @ ( topolo2815343760600316023s_real @ X )
% 9.04/5.13                  @ at_top_real )
% 9.04/5.13               => ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.13                  @ ( topolo2815343760600316023s_real @ X )
% 9.04/5.13                  @ at_top_real ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9927_lhopital__at__top,axiom,
% 9.04/5.13      ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 9.04/5.13        ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.13       => ( ( eventually_real
% 9.04/5.13            @ ^ [X4: real] :
% 9.04/5.13                ( ( G2 @ X4 )
% 9.04/5.13               != zero_zero_real )
% 9.04/5.13            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.13         => ( ( eventually_real
% 9.04/5.13              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.13             => ( ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                  @ ( topolo2815343760600316023s_real @ Y )
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.04/5.13               => ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.13                  @ ( topolo2815343760600316023s_real @ Y )
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9928_lhopital__right__0,axiom,
% 9.04/5.13      ! [F0: real > real,G0: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 9.04/5.13        ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13       => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13         => ( ( eventually_real
% 9.04/5.13              @ ^ [X4: real] :
% 9.04/5.13                  ( ( G0 @ X4 )
% 9.04/5.13                 != zero_zero_real )
% 9.04/5.13              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] :
% 9.04/5.13                    ( ( G2 @ X4 )
% 9.04/5.13                   != zero_zero_real )
% 9.04/5.13                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13             => ( ( eventually_real
% 9.04/5.13                  @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13               => ( ( eventually_real
% 9.04/5.13                    @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13                 => ( ( filterlim_real_real
% 9.04/5.13                      @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                      @ F4
% 9.04/5.13                      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13                   => ( filterlim_real_real
% 9.04/5.13                      @ ^ [X4: real] : ( divide_divide_real @ ( F0 @ X4 ) @ ( G0 @ X4 ) )
% 9.04/5.13                      @ F4
% 9.04/5.13                      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9929_lhopital__right,axiom,
% 9.04/5.13      ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 9.04/5.13        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13       => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13         => ( ( eventually_real
% 9.04/5.13              @ ^ [X4: real] :
% 9.04/5.13                  ( ( G @ X4 )
% 9.04/5.13                 != zero_zero_real )
% 9.04/5.13              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] :
% 9.04/5.13                    ( ( G2 @ X4 )
% 9.04/5.13                   != zero_zero_real )
% 9.04/5.13                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13             => ( ( eventually_real
% 9.04/5.13                  @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13               => ( ( eventually_real
% 9.04/5.13                    @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13                 => ( ( filterlim_real_real
% 9.04/5.13                      @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                      @ F4
% 9.04/5.13                      @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13                   => ( filterlim_real_real
% 9.04/5.13                      @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.13                      @ F4
% 9.04/5.13                      @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9930_lhopital__left,axiom,
% 9.04/5.13      ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 9.04/5.13        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.13       => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.13         => ( ( eventually_real
% 9.04/5.13              @ ^ [X4: real] :
% 9.04/5.13                  ( ( G @ X4 )
% 9.04/5.13                 != zero_zero_real )
% 9.04/5.13              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] :
% 9.04/5.13                    ( ( G2 @ X4 )
% 9.04/5.13                   != zero_zero_real )
% 9.04/5.13                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.13             => ( ( eventually_real
% 9.04/5.13                  @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.13               => ( ( eventually_real
% 9.04/5.13                    @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.13                 => ( ( filterlim_real_real
% 9.04/5.13                      @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                      @ F4
% 9.04/5.13                      @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 9.04/5.13                   => ( filterlim_real_real
% 9.04/5.13                      @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.13                      @ F4
% 9.04/5.13                      @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9931_lhopital__right__at__top__at__bot,axiom,
% 9.04/5.13      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 9.04/5.13        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.13       => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.13         => ( ( eventually_real
% 9.04/5.13              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.13             => ( ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                  @ at_bot_real
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 9.04/5.13               => ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.13                  @ at_bot_real
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9932_lhopital__left__at__top__at__bot,axiom,
% 9.04/5.13      ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 9.04/5.13        ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 9.04/5.13       => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 9.04/5.13         => ( ( eventually_real
% 9.04/5.13              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 9.04/5.13             => ( ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                  @ at_bot_real
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 9.04/5.13               => ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.13                  @ at_bot_real
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9933_lhopital__right__at__top,axiom,
% 9.04/5.13      ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 9.04/5.13        ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13       => ( ( eventually_real
% 9.04/5.13            @ ^ [X4: real] :
% 9.04/5.13                ( ( G2 @ X4 )
% 9.04/5.13               != zero_zero_real )
% 9.04/5.13            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13         => ( ( eventually_real
% 9.04/5.13              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13             => ( ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                  @ ( topolo2815343760600316023s_real @ Y )
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 9.04/5.13               => ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.13                  @ ( topolo2815343760600316023s_real @ Y )
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9934_lhopital__right__0__at__top,axiom,
% 9.04/5.13      ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 9.04/5.13        ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13       => ( ( eventually_real
% 9.04/5.13            @ ^ [X4: real] :
% 9.04/5.13                ( ( G2 @ X4 )
% 9.04/5.13               != zero_zero_real )
% 9.04/5.13            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13         => ( ( eventually_real
% 9.04/5.13              @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13           => ( ( eventually_real
% 9.04/5.13                @ ^ [X4: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.04/5.13                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13             => ( ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 9.04/5.13                  @ ( topolo2815343760600316023s_real @ X )
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 9.04/5.13               => ( filterlim_real_real
% 9.04/5.13                  @ ^ [X4: real] : ( divide_divide_real @ ( F @ X4 ) @ ( G @ X4 ) )
% 9.04/5.13                  @ ( topolo2815343760600316023s_real @ X )
% 9.04/5.13                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9935_filterlim__int__sequentially,axiom,
% 9.04/5.13      filterlim_nat_int @ semiri1314217659103216013at_int @ at_top_int @ at_top_nat ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9936_filterlim__real__of__int__at__top,axiom,
% 9.04/5.13      filterlim_int_real @ ring_1_of_int_real @ at_top_real @ at_top_int ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9937_Bseq__eq__bounded,axiom,
% 9.04/5.13      ! [F: nat > real,A: real,B: real] :
% 9.04/5.13        ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
% 9.04/5.13       => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9938_Bseq__realpow,axiom,
% 9.04/5.13      ! [X: real] :
% 9.04/5.13        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.13       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.04/5.13         => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9939_atLeastSucAtMost__greaterThanAtMost,axiom,
% 9.04/5.13      ! [L: nat,U: nat] :
% 9.04/5.13        ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 9.04/5.13        = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9940_GreatestI__nat,axiom,
% 9.04/5.13      ! [P: nat > $o,K: nat,B: nat] :
% 9.04/5.13        ( ( P @ K )
% 9.04/5.13       => ( ! [Y4: nat] :
% 9.04/5.13              ( ( P @ Y4 )
% 9.04/5.13             => ( ord_less_eq_nat @ Y4 @ B ) )
% 9.04/5.13         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9941_Greatest__le__nat,axiom,
% 9.04/5.13      ! [P: nat > $o,K: nat,B: nat] :
% 9.04/5.13        ( ( P @ K )
% 9.04/5.13       => ( ! [Y4: nat] :
% 9.04/5.13              ( ( P @ Y4 )
% 9.04/5.13             => ( ord_less_eq_nat @ Y4 @ B ) )
% 9.04/5.13         => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9942_GreatestI__ex__nat,axiom,
% 9.04/5.13      ! [P: nat > $o,B: nat] :
% 9.04/5.13        ( ? [X_1: nat] : ( P @ X_1 )
% 9.04/5.13       => ( ! [Y4: nat] :
% 9.04/5.13              ( ( P @ Y4 )
% 9.04/5.13             => ( ord_less_eq_nat @ Y4 @ B ) )
% 9.04/5.13         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9943_atLeast__0,axiom,
% 9.04/5.13      ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 9.04/5.13      = top_top_set_nat ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9944_atLeast__Suc__greaterThan,axiom,
% 9.04/5.13      ! [K: nat] :
% 9.04/5.13        ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 9.04/5.13        = ( set_or1210151606488870762an_nat @ K ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9945_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 9.04/5.13      ! [L: int,U: int] :
% 9.04/5.13        ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 9.04/5.13        = ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9946_decseq__bounded,axiom,
% 9.04/5.13      ! [X8: nat > real,B2: real] :
% 9.04/5.13        ( ( order_9091379641038594480t_real @ X8 )
% 9.04/5.13       => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X8 @ I2 ) )
% 9.04/5.13         => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9947_decseq__convergent,axiom,
% 9.04/5.13      ! [X8: nat > real,B2: real] :
% 9.04/5.13        ( ( order_9091379641038594480t_real @ X8 )
% 9.04/5.13       => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X8 @ I2 ) )
% 9.04/5.13         => ~ ! [L5: real] :
% 9.04/5.13                ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat )
% 9.04/5.13               => ~ ! [I4: nat] : ( ord_less_eq_real @ L5 @ ( X8 @ I4 ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9948_atLeast__Suc,axiom,
% 9.04/5.13      ! [K: nat] :
% 9.04/5.13        ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 9.04/5.13        = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9949_GMVT,axiom,
% 9.04/5.13      ! [A: real,B: real,F: real > real,G: real > real] :
% 9.04/5.13        ( ( ord_less_real @ A @ B )
% 9.04/5.13       => ( ! [X3: real] :
% 9.04/5.13              ( ( ( ord_less_eq_real @ A @ X3 )
% 9.04/5.13                & ( ord_less_eq_real @ X3 @ B ) )
% 9.04/5.13             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 9.04/5.13         => ( ! [X3: real] :
% 9.04/5.13                ( ( ( ord_less_real @ A @ X3 )
% 9.04/5.13                  & ( ord_less_real @ X3 @ B ) )
% 9.04/5.13               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 9.04/5.13           => ( ! [X3: real] :
% 9.04/5.13                  ( ( ( ord_less_eq_real @ A @ X3 )
% 9.04/5.13                    & ( ord_less_eq_real @ X3 @ B ) )
% 9.04/5.13                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
% 9.04/5.13             => ( ! [X3: real] :
% 9.04/5.13                    ( ( ( ord_less_real @ A @ X3 )
% 9.04/5.13                      & ( ord_less_real @ X3 @ B ) )
% 9.04/5.13                   => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 9.04/5.13               => ? [G_c: real,F_c: real,C4: real] :
% 9.04/5.13                    ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C4 @ top_top_set_real ) )
% 9.04/5.13                    & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C4 @ top_top_set_real ) )
% 9.04/5.13                    & ( ord_less_real @ A @ C4 )
% 9.04/5.13                    & ( ord_less_real @ C4 @ B )
% 9.04/5.13                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 9.04/5.13                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9950_MVT,axiom,
% 9.04/5.13      ! [A: real,B: real,F: real > real] :
% 9.04/5.13        ( ( ord_less_real @ A @ B )
% 9.04/5.13       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.04/5.13         => ( ! [X3: real] :
% 9.04/5.13                ( ( ord_less_real @ A @ X3 )
% 9.04/5.13               => ( ( ord_less_real @ X3 @ B )
% 9.04/5.13                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 9.04/5.13           => ? [L2: real,Z3: real] :
% 9.04/5.13                ( ( ord_less_real @ A @ Z3 )
% 9.04/5.13                & ( ord_less_real @ Z3 @ B )
% 9.04/5.13                & ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
% 9.04/5.13                & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 9.04/5.13                  = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L2 ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9951_continuous__on__arsinh,axiom,
% 9.04/5.13      ! [A2: set_real] : ( topolo5044208981011980120l_real @ A2 @ arsinh_real ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9952_continuous__on__arsinh_H,axiom,
% 9.04/5.13      ! [A2: set_real,F: real > real] :
% 9.04/5.13        ( ( topolo5044208981011980120l_real @ A2 @ F )
% 9.04/5.13       => ( topolo5044208981011980120l_real @ A2
% 9.04/5.13          @ ^ [X4: real] : ( arsinh_real @ ( F @ X4 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9953_continuous__on__cos__real,axiom,
% 9.04/5.13      ! [A: real,B: real] : ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ cos_real ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9954_continuous__on__sin__real,axiom,
% 9.04/5.13      ! [A: real,B: real] : ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ sin_real ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9955_continuous__on__arcosh_H,axiom,
% 9.04/5.13      ! [A2: set_real,F: real > real] :
% 9.04/5.13        ( ( topolo5044208981011980120l_real @ A2 @ F )
% 9.04/5.13       => ( ! [X3: real] :
% 9.04/5.13              ( ( member_real @ X3 @ A2 )
% 9.04/5.13             => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 9.04/5.13         => ( topolo5044208981011980120l_real @ A2
% 9.04/5.13            @ ^ [X4: real] : ( arcosh_real @ ( F @ X4 ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9956_continuous__image__closed__interval,axiom,
% 9.04/5.13      ! [A: real,B: real,F: real > real] :
% 9.04/5.13        ( ( ord_less_eq_real @ A @ B )
% 9.04/5.13       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.04/5.13         => ? [C4: real,D5: real] :
% 9.04/5.13              ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
% 9.04/5.13                = ( set_or1222579329274155063t_real @ C4 @ D5 ) )
% 9.04/5.13              & ( ord_less_eq_real @ C4 @ D5 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9957_continuous__on__arcosh,axiom,
% 9.04/5.13      ! [A2: set_real] :
% 9.04/5.13        ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
% 9.04/5.13       => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9958_continuous__on__arccos_H,axiom,
% 9.04/5.13      topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9959_continuous__on__arcsin_H,axiom,
% 9.04/5.13      topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9960_continuous__on__artanh_H,axiom,
% 9.04/5.13      ! [A2: set_real,F: real > real] :
% 9.04/5.13        ( ( topolo5044208981011980120l_real @ A2 @ F )
% 9.04/5.13       => ( ! [X3: real] :
% 9.04/5.13              ( ( member_real @ X3 @ A2 )
% 9.04/5.13             => ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 9.04/5.13         => ( topolo5044208981011980120l_real @ A2
% 9.04/5.13            @ ^ [X4: real] : ( artanh_real @ ( F @ X4 ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9961_Rolle__deriv,axiom,
% 9.04/5.13      ! [A: real,B: real,F: real > real,F3: real > real > real] :
% 9.04/5.13        ( ( ord_less_real @ A @ B )
% 9.04/5.13       => ( ( ( F @ A )
% 9.04/5.13            = ( F @ B ) )
% 9.04/5.13         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.04/5.13           => ( ! [X3: real] :
% 9.04/5.13                  ( ( ord_less_real @ A @ X3 )
% 9.04/5.13                 => ( ( ord_less_real @ X3 @ B )
% 9.04/5.13                   => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 9.04/5.13             => ? [Z3: real] :
% 9.04/5.13                  ( ( ord_less_real @ A @ Z3 )
% 9.04/5.13                  & ( ord_less_real @ Z3 @ B )
% 9.04/5.13                  & ( ( F3 @ Z3 )
% 9.04/5.13                    = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9962_mvt,axiom,
% 9.04/5.13      ! [A: real,B: real,F: real > real,F3: real > real > real] :
% 9.04/5.13        ( ( ord_less_real @ A @ B )
% 9.04/5.13       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.04/5.13         => ( ! [X3: real] :
% 9.04/5.13                ( ( ord_less_real @ A @ X3 )
% 9.04/5.13               => ( ( ord_less_real @ X3 @ B )
% 9.04/5.13                 => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 9.04/5.13           => ~ ! [Xi: real] :
% 9.04/5.13                  ( ( ord_less_real @ A @ Xi )
% 9.04/5.13                 => ( ( ord_less_real @ Xi @ B )
% 9.04/5.13                   => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 9.04/5.13                     != ( F3 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9963_DERIV__isconst__end,axiom,
% 9.04/5.13      ! [A: real,B: real,F: real > real] :
% 9.04/5.13        ( ( ord_less_real @ A @ B )
% 9.04/5.13       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.04/5.13         => ( ! [X3: real] :
% 9.04/5.13                ( ( ord_less_real @ A @ X3 )
% 9.04/5.13               => ( ( ord_less_real @ X3 @ B )
% 9.04/5.13                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 9.04/5.13           => ( ( F @ B )
% 9.04/5.13              = ( F @ A ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9964_DERIV__neg__imp__decreasing__open,axiom,
% 9.04/5.13      ! [A: real,B: real,F: real > real] :
% 9.04/5.13        ( ( ord_less_real @ A @ B )
% 9.04/5.13       => ( ! [X3: real] :
% 9.04/5.13              ( ( ord_less_real @ A @ X3 )
% 9.04/5.13             => ( ( ord_less_real @ X3 @ B )
% 9.04/5.13               => ? [Y5: real] :
% 9.04/5.13                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.13                    & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 9.04/5.13         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.04/5.13           => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9965_DERIV__pos__imp__increasing__open,axiom,
% 9.04/5.13      ! [A: real,B: real,F: real > real] :
% 9.04/5.13        ( ( ord_less_real @ A @ B )
% 9.04/5.13       => ( ! [X3: real] :
% 9.04/5.13              ( ( ord_less_real @ A @ X3 )
% 9.04/5.13             => ( ( ord_less_real @ X3 @ B )
% 9.04/5.13               => ? [Y5: real] :
% 9.04/5.13                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.13                    & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 9.04/5.13         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.04/5.13           => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9966_continuous__on__artanh,axiom,
% 9.04/5.13      ! [A2: set_real] :
% 9.04/5.13        ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 9.04/5.13       => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9967_DERIV__isconst2,axiom,
% 9.04/5.13      ! [A: real,B: real,F: real > real,X: real] :
% 9.04/5.13        ( ( ord_less_real @ A @ B )
% 9.04/5.13       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.04/5.13         => ( ! [X3: real] :
% 9.04/5.13                ( ( ord_less_real @ A @ X3 )
% 9.04/5.13               => ( ( ord_less_real @ X3 @ B )
% 9.04/5.13                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 9.04/5.13           => ( ( ord_less_eq_real @ A @ X )
% 9.04/5.13             => ( ( ord_less_eq_real @ X @ B )
% 9.04/5.13               => ( ( F @ X )
% 9.04/5.13                  = ( F @ A ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9968_Rolle,axiom,
% 9.04/5.13      ! [A: real,B: real,F: real > real] :
% 9.04/5.13        ( ( ord_less_real @ A @ B )
% 9.04/5.13       => ( ( ( F @ A )
% 9.04/5.13            = ( F @ B ) )
% 9.04/5.13         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.04/5.13           => ( ! [X3: real] :
% 9.04/5.13                  ( ( ord_less_real @ A @ X3 )
% 9.04/5.13                 => ( ( ord_less_real @ X3 @ B )
% 9.04/5.13                   => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 9.04/5.13             => ? [Z3: real] :
% 9.04/5.13                  ( ( ord_less_real @ A @ Z3 )
% 9.04/5.13                  & ( ord_less_real @ Z3 @ B )
% 9.04/5.13                  & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9969_mono__Suc,axiom,
% 9.04/5.13      order_mono_nat_nat @ suc ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9970_mono__times__nat,axiom,
% 9.04/5.13      ! [N: nat] :
% 9.04/5.13        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.13       => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9971_incseq__bounded,axiom,
% 9.04/5.13      ! [X8: nat > real,B2: real] :
% 9.04/5.13        ( ( order_mono_nat_real @ X8 )
% 9.04/5.13       => ( ! [I2: nat] : ( ord_less_eq_real @ ( X8 @ I2 ) @ B2 )
% 9.04/5.13         => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9972_incseq__convergent,axiom,
% 9.04/5.13      ! [X8: nat > real,B2: real] :
% 9.04/5.13        ( ( order_mono_nat_real @ X8 )
% 9.04/5.13       => ( ! [I2: nat] : ( ord_less_eq_real @ ( X8 @ I2 ) @ B2 )
% 9.04/5.13         => ~ ! [L5: real] :
% 9.04/5.13                ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat )
% 9.04/5.13               => ~ ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ L5 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9973_mono__ge2__power__minus__self,axiom,
% 9.04/5.13      ! [K: nat] :
% 9.04/5.13        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.04/5.13       => ( order_mono_nat_nat
% 9.04/5.13          @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9974_xor__minus__numerals_I2_J,axiom,
% 9.04/5.13      ! [K: int,N: num] :
% 9.04/5.13        ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13        = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9975_xor__minus__numerals_I1_J,axiom,
% 9.04/5.13      ! [N: num,K: int] :
% 9.04/5.13        ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 9.04/5.13        = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9976_sub__BitM__One__eq,axiom,
% 9.04/5.13      ! [N: num] :
% 9.04/5.13        ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 9.04/5.13        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9977_tendsto__at__topI__sequentially__real,axiom,
% 9.04/5.13      ! [F: real > real,Y: real] :
% 9.04/5.13        ( ( order_mono_real_real @ F )
% 9.04/5.13       => ( ( filterlim_nat_real
% 9.04/5.13            @ ^ [N3: nat] : ( F @ ( semiri5074537144036343181t_real @ N3 ) )
% 9.04/5.13            @ ( topolo2815343760600316023s_real @ Y )
% 9.04/5.13            @ at_top_nat )
% 9.04/5.13         => ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Y ) @ at_top_real ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9978_nonneg__incseq__Bseq__subseq__iff,axiom,
% 9.04/5.13      ! [F: nat > real,G: nat > nat] :
% 9.04/5.13        ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 9.04/5.13       => ( ( order_mono_nat_real @ F )
% 9.04/5.13         => ( ( order_5726023648592871131at_nat @ G )
% 9.04/5.13           => ( ( bfun_nat_real
% 9.04/5.13                @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
% 9.04/5.13                @ at_top_nat )
% 9.04/5.13              = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9979_range__abs__Nats,axiom,
% 9.04/5.13      ( ( image_int_int @ abs_abs_int @ top_top_set_int )
% 9.04/5.13      = semiring_1_Nats_int ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9980_strict__mono__imp__increasing,axiom,
% 9.04/5.13      ! [F: nat > nat,N: nat] :
% 9.04/5.13        ( ( order_5726023648592871131at_nat @ F )
% 9.04/5.13       => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9981_tanh__real__strict__mono,axiom,
% 9.04/5.13      order_7092887310737990675l_real @ tanh_real ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9982_sinh__real__strict__mono,axiom,
% 9.04/5.13      order_7092887310737990675l_real @ sinh_real ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9983_complex__is__Nat__iff,axiom,
% 9.04/5.13      ! [Z: complex] :
% 9.04/5.13        ( ( member_complex @ Z @ semiri3842193898606819883omplex )
% 9.04/5.13        = ( ( ( im @ Z )
% 9.04/5.13            = zero_zero_real )
% 9.04/5.13          & ? [I3: nat] :
% 9.04/5.13              ( ( re @ Z )
% 9.04/5.13              = ( semiri5074537144036343181t_real @ I3 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9984_pos__deriv__imp__strict__mono,axiom,
% 9.04/5.13      ! [F: real > real,F3: real > real] :
% 9.04/5.13        ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.04/5.13       => ( ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( F3 @ X3 ) )
% 9.04/5.13         => ( order_7092887310737990675l_real @ F ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9985_nat__of__integer__non__positive,axiom,
% 9.04/5.13      ! [K: code_integer] :
% 9.04/5.13        ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 9.04/5.13       => ( ( code_nat_of_integer @ K )
% 9.04/5.13          = zero_zero_nat ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9986_Suc__funpow,axiom,
% 9.04/5.13      ! [N: nat] :
% 9.04/5.13        ( ( compow_nat_nat @ N @ suc )
% 9.04/5.13        = ( plus_plus_nat @ N ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9987_of__nat__of__integer,axiom,
% 9.04/5.13      ! [K: code_integer] :
% 9.04/5.13        ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
% 9.04/5.13        = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9988_nat__of__integer__code__post_I1_J,axiom,
% 9.04/5.13      ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 9.04/5.13      = zero_zero_nat ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9989_nat__of__integer__code__post_I3_J,axiom,
% 9.04/5.13      ! [K: num] :
% 9.04/5.13        ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 9.04/5.13        = ( numeral_numeral_nat @ K ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9990_nat__of__integer__code__post_I2_J,axiom,
% 9.04/5.13      ( ( code_nat_of_integer @ one_one_Code_integer )
% 9.04/5.13      = one_one_nat ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9991_nat__of__integer__code,axiom,
% 9.04/5.13      ( code_nat_of_integer
% 9.04/5.13      = ( ^ [K2: code_integer] :
% 9.04/5.13            ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 9.04/5.13            @ ( produc1555791787009142072er_nat
% 9.04/5.13              @ ^ [L3: code_integer,J2: code_integer] : ( if_nat @ ( J2 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ one_one_nat ) )
% 9.04/5.13              @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9992_inj__sgn__power,axiom,
% 9.04/5.13      ! [N: nat] :
% 9.04/5.13        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.13       => ( inj_on_real_real
% 9.04/5.13          @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 9.04/5.13          @ top_top_set_real ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9993_log__inj,axiom,
% 9.04/5.13      ! [B: real] :
% 9.04/5.13        ( ( ord_less_real @ one_one_real @ B )
% 9.04/5.13       => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9994_inj__on__diff__nat,axiom,
% 9.04/5.13      ! [N4: set_nat,K: nat] :
% 9.04/5.13        ( ! [N2: nat] :
% 9.04/5.13            ( ( member_nat @ N2 @ N4 )
% 9.04/5.13           => ( ord_less_eq_nat @ K @ N2 ) )
% 9.04/5.13       => ( inj_on_nat_nat
% 9.04/5.13          @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
% 9.04/5.13          @ N4 ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9995_inj__Suc,axiom,
% 9.04/5.13      ! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9996_summable__reindex,axiom,
% 9.04/5.13      ! [F: nat > real,G: nat > nat] :
% 9.04/5.13        ( ( summable_real @ F )
% 9.04/5.13       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 9.04/5.13         => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 9.04/5.13           => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9997_suminf__reindex__mono,axiom,
% 9.04/5.13      ! [F: nat > real,G: nat > nat] :
% 9.04/5.13        ( ( summable_real @ F )
% 9.04/5.13       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 9.04/5.13         => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 9.04/5.13           => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9998_inj__on__char__of__nat,axiom,
% 9.04/5.13      inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_9999_suminf__reindex,axiom,
% 9.04/5.13      ! [F: nat > real,G: nat > nat] :
% 9.04/5.13        ( ( summable_real @ F )
% 9.04/5.13       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 9.04/5.13         => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 9.04/5.13           => ( ! [X3: nat] :
% 9.04/5.13                  ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 9.04/5.13                 => ( ( F @ X3 )
% 9.04/5.13                    = zero_zero_real ) )
% 9.04/5.13             => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 9.04/5.13                = ( suminf_real @ F ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10000_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 9.04/5.13      ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 9.04/5.13      @ ^ [X4: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X4 )
% 9.04/5.13      @ ^ [X4: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X4 ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10001_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 9.04/5.13      ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 9.04/5.13      @ ^ [M6: nat,N3: nat] :
% 9.04/5.13          ( ( dvd_dvd_nat @ M6 @ N3 )
% 9.04/5.13          & ( M6 != N3 ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10002_int__of__integer__code,axiom,
% 9.04/5.13      ( code_int_of_integer
% 9.04/5.13      = ( ^ [K2: code_integer] :
% 9.04/5.13            ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
% 9.04/5.13            @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 9.04/5.13              @ ( produc1553301316500091796er_int
% 9.04/5.13                @ ^ [L3: code_integer,J2: code_integer] : ( if_int @ ( J2 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ one_one_int ) )
% 9.04/5.13                @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10003_of__int__integer__of,axiom,
% 9.04/5.13      ! [K: code_integer] :
% 9.04/5.13        ( ( ring_18347121197199848620nteger @ ( code_int_of_integer @ K ) )
% 9.04/5.13        = K ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10004_int__of__integer__of__int,axiom,
% 9.04/5.13      ! [K: int] :
% 9.04/5.13        ( ( code_int_of_integer @ ( ring_18347121197199848620nteger @ K ) )
% 9.04/5.13        = K ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10005_zero__integer_Orep__eq,axiom,
% 9.04/5.13      ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
% 9.04/5.13      = zero_zero_int ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10006_int__of__integer__numeral,axiom,
% 9.04/5.13      ! [K: num] :
% 9.04/5.13        ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 9.04/5.13        = ( numeral_numeral_int @ K ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10007_plus__integer_Orep__eq,axiom,
% 9.04/5.13      ! [X: code_integer,Xa: code_integer] :
% 9.04/5.13        ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa ) )
% 9.04/5.13        = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10008_one__integer_Orep__eq,axiom,
% 9.04/5.13      ( ( code_int_of_integer @ one_one_Code_integer )
% 9.04/5.13      = one_one_int ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10009_divide__integer_Orep__eq,axiom,
% 9.04/5.13      ! [X: code_integer,Xa: code_integer] :
% 9.04/5.13        ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
% 9.04/5.13        = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10010_int__of__integer__of__nat,axiom,
% 9.04/5.13      ! [N: nat] :
% 9.04/5.13        ( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
% 9.04/5.13        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10011_less__eq__integer_Orep__eq,axiom,
% 9.04/5.13      ( ord_le3102999989581377725nteger
% 9.04/5.13      = ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10012_integer__less__eq__iff,axiom,
% 9.04/5.13      ( ord_le3102999989581377725nteger
% 9.04/5.13      = ( ^ [K2: code_integer,L3: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10013_integer__less__iff,axiom,
% 9.04/5.13      ( ord_le6747313008572928689nteger
% 9.04/5.13      = ( ^ [K2: code_integer,L3: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10014_less__integer_Orep__eq,axiom,
% 9.04/5.13      ( ord_le6747313008572928689nteger
% 9.04/5.13      = ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10015_times__int_Oabs__eq,axiom,
% 9.04/5.13      ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 9.04/5.13        ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 9.04/5.13        = ( abs_Integ
% 9.04/5.13          @ ( produc27273713700761075at_nat
% 9.04/5.13            @ ^ [X4: nat,Y6: nat] :
% 9.04/5.13                ( produc2626176000494625587at_nat
% 9.04/5.13                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y6 @ U2 ) ) ) )
% 9.04/5.13            @ Xa
% 9.04/5.13            @ X ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10016_nat_Oabs__eq,axiom,
% 9.04/5.13      ! [X: product_prod_nat_nat] :
% 9.04/5.13        ( ( nat2 @ ( abs_Integ @ X ) )
% 9.04/5.13        = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10017_int_Oabs__induct,axiom,
% 9.04/5.13      ! [P: int > $o,X: int] :
% 9.04/5.13        ( ! [Y4: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y4 ) )
% 9.04/5.13       => ( P @ X ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10018_eq__Abs__Integ,axiom,
% 9.04/5.13      ! [Z: int] :
% 9.04/5.13        ~ ! [X3: nat,Y4: nat] :
% 9.04/5.13            ( Z
% 9.04/5.13           != ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y4 ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10019_zero__int__def,axiom,
% 9.04/5.13      ( zero_zero_int
% 9.04/5.13      = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10020_int__def,axiom,
% 9.04/5.13      ( semiri1314217659103216013at_int
% 9.04/5.13      = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10021_uminus__int_Oabs__eq,axiom,
% 9.04/5.13      ! [X: product_prod_nat_nat] :
% 9.04/5.13        ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
% 9.04/5.13        = ( abs_Integ
% 9.04/5.13          @ ( produc2626176000494625587at_nat
% 9.04/5.13            @ ^ [X4: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X4 )
% 9.04/5.13            @ X ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10022_one__int__def,axiom,
% 9.04/5.13      ( one_one_int
% 9.04/5.13      = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10023_less__int_Oabs__eq,axiom,
% 9.04/5.13      ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 9.04/5.13        ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 9.04/5.13        = ( produc8739625826339149834_nat_o
% 9.04/5.13          @ ^ [X4: nat,Y6: nat] :
% 9.04/5.13              ( produc6081775807080527818_nat_o
% 9.04/5.13              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y6 ) ) )
% 9.04/5.13          @ Xa
% 9.04/5.13          @ X ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10024_less__eq__int_Oabs__eq,axiom,
% 9.04/5.13      ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 9.04/5.13        ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 9.04/5.13        = ( produc8739625826339149834_nat_o
% 9.04/5.13          @ ^ [X4: nat,Y6: nat] :
% 9.04/5.13              ( produc6081775807080527818_nat_o
% 9.04/5.13              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y6 ) ) )
% 9.04/5.13          @ Xa
% 9.04/5.13          @ X ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10025_plus__int_Oabs__eq,axiom,
% 9.04/5.13      ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 9.04/5.13        ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 9.04/5.13        = ( abs_Integ
% 9.04/5.13          @ ( produc27273713700761075at_nat
% 9.04/5.13            @ ^ [X4: nat,Y6: nat] :
% 9.04/5.13                ( produc2626176000494625587at_nat
% 9.04/5.13                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) )
% 9.04/5.13            @ Xa
% 9.04/5.13            @ X ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10026_minus__int_Oabs__eq,axiom,
% 9.04/5.13      ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 9.04/5.13        ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 9.04/5.13        = ( abs_Integ
% 9.04/5.13          @ ( produc27273713700761075at_nat
% 9.04/5.13            @ ^ [X4: nat,Y6: nat] :
% 9.04/5.13                ( produc2626176000494625587at_nat
% 9.04/5.13                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y6 @ U2 ) ) )
% 9.04/5.13            @ Xa
% 9.04/5.13            @ X ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10027_Gcd__remove0__nat,axiom,
% 9.04/5.13      ! [M2: set_nat] :
% 9.04/5.13        ( ( finite_finite_nat @ M2 )
% 9.04/5.13       => ( ( gcd_Gcd_nat @ M2 )
% 9.04/5.13          = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10028_Gcd__nat__eq__one,axiom,
% 9.04/5.13      ! [N4: set_nat] :
% 9.04/5.13        ( ( member_nat @ one_one_nat @ N4 )
% 9.04/5.13       => ( ( gcd_Gcd_nat @ N4 )
% 9.04/5.13          = one_one_nat ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10029_Gcd__eq__Max,axiom,
% 9.04/5.13      ! [M2: set_nat] :
% 9.04/5.13        ( ( finite_finite_nat @ M2 )
% 9.04/5.13       => ( ( M2 != bot_bot_set_nat )
% 9.04/5.13         => ( ~ ( member_nat @ zero_zero_nat @ M2 )
% 9.04/5.13           => ( ( gcd_Gcd_nat @ M2 )
% 9.04/5.13              = ( lattic8265883725875713057ax_nat
% 9.04/5.13                @ ( comple7806235888213564991et_nat
% 9.04/5.13                  @ ( image_nat_set_nat
% 9.04/5.13                    @ ^ [M6: nat] :
% 9.04/5.13                        ( collect_nat
% 9.04/5.13                        @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M6 ) )
% 9.04/5.13                    @ M2 ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10030_less__eq__int_Orep__eq,axiom,
% 9.04/5.13      ( ord_less_eq_int
% 9.04/5.13      = ( ^ [X4: int,Xa4: int] :
% 9.04/5.13            ( produc8739625826339149834_nat_o
% 9.04/5.13            @ ^ [Y6: nat,Z4: nat] :
% 9.04/5.13                ( produc6081775807080527818_nat_o
% 9.04/5.13                @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U2 @ Z4 ) ) )
% 9.04/5.13            @ ( rep_Integ @ X4 )
% 9.04/5.13            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10031_less__int_Orep__eq,axiom,
% 9.04/5.13      ( ord_less_int
% 9.04/5.13      = ( ^ [X4: int,Xa4: int] :
% 9.04/5.13            ( produc8739625826339149834_nat_o
% 9.04/5.13            @ ^ [Y6: nat,Z4: nat] :
% 9.04/5.13                ( produc6081775807080527818_nat_o
% 9.04/5.13                @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U2 @ Z4 ) ) )
% 9.04/5.13            @ ( rep_Integ @ X4 )
% 9.04/5.13            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10032_Gcd__int__eq,axiom,
% 9.04/5.13      ! [N4: set_nat] :
% 9.04/5.13        ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N4 ) )
% 9.04/5.13        = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N4 ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10033_Gcd__int__greater__eq__0,axiom,
% 9.04/5.13      ! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K4 ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10034_nat_Orep__eq,axiom,
% 9.04/5.13      ( nat2
% 9.04/5.13      = ( ^ [X4: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X4 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10035_Gcd__int__def,axiom,
% 9.04/5.13      ( gcd_Gcd_int
% 9.04/5.13      = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10036_uminus__int__def,axiom,
% 9.04/5.13      ( uminus_uminus_int
% 9.04/5.13      = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 9.04/5.13        @ ( produc2626176000494625587at_nat
% 9.04/5.13          @ ^ [X4: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X4 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10037_times__int__def,axiom,
% 9.04/5.13      ( times_times_int
% 9.04/5.13      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.04/5.13        @ ( produc27273713700761075at_nat
% 9.04/5.13          @ ^ [X4: nat,Y6: nat] :
% 9.04/5.13              ( produc2626176000494625587at_nat
% 9.04/5.13              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y6 @ U2 ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10038_minus__int__def,axiom,
% 9.04/5.13      ( minus_minus_int
% 9.04/5.13      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.04/5.13        @ ( produc27273713700761075at_nat
% 9.04/5.13          @ ^ [X4: nat,Y6: nat] :
% 9.04/5.13              ( produc2626176000494625587at_nat
% 9.04/5.13              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y6 @ U2 ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10039_plus__int__def,axiom,
% 9.04/5.13      ( plus_plus_int
% 9.04/5.13      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.04/5.13        @ ( produc27273713700761075at_nat
% 9.04/5.13          @ ^ [X4: nat,Y6: nat] :
% 9.04/5.13              ( produc2626176000494625587at_nat
% 9.04/5.13              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10040_measure__function__int,axiom,
% 9.04/5.13      fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10041_prod__encode__def,axiom,
% 9.04/5.13      ( nat_prod_encode
% 9.04/5.13      = ( produc6842872674320459806at_nat
% 9.04/5.13        @ ^ [M6: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N3 ) ) @ M6 ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10042_le__prod__encode__2,axiom,
% 9.04/5.13      ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10043_le__prod__encode__1,axiom,
% 9.04/5.13      ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10044_prod__encode__prod__decode__aux,axiom,
% 9.04/5.13      ! [K: nat,M: nat] :
% 9.04/5.13        ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 9.04/5.13        = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10045_powr__real__of__int_H,axiom,
% 9.04/5.13      ! [X: real,N: int] :
% 9.04/5.13        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.04/5.13       => ( ( ( X != zero_zero_real )
% 9.04/5.13            | ( ord_less_int @ zero_zero_int @ N ) )
% 9.04/5.13         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 9.04/5.13            = ( power_int_real @ X @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10046_uniformity__complex__def,axiom,
% 9.04/5.13      ( topolo896644834953643431omplex
% 9.04/5.13      = ( comple8358262395181532106omplex
% 9.04/5.13        @ ( image_5971271580939081552omplex
% 9.04/5.13          @ ^ [E3: real] :
% 9.04/5.13              ( princi3496590319149328850omplex
% 9.04/5.13              @ ( collec8663557070575231912omplex
% 9.04/5.13                @ ( produc6771430404735790350plex_o
% 9.04/5.13                  @ ^ [X4: complex,Y6: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X4 @ Y6 ) @ E3 ) ) ) )
% 9.04/5.13          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10047_uniformity__real__def,axiom,
% 9.04/5.13      ( topolo1511823702728130853y_real
% 9.04/5.13      = ( comple2936214249959783750l_real
% 9.04/5.13        @ ( image_2178119161166701260l_real
% 9.04/5.13          @ ^ [E3: real] :
% 9.04/5.13              ( princi6114159922880469582l_real
% 9.04/5.13              @ ( collec3799799289383736868l_real
% 9.04/5.13                @ ( produc5414030515140494994real_o
% 9.04/5.13                  @ ^ [X4: real,Y6: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X4 @ Y6 ) @ E3 ) ) ) )
% 9.04/5.13          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10048_num__of__nat_Osimps_I2_J,axiom,
% 9.04/5.13      ! [N: nat] :
% 9.04/5.13        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.13         => ( ( num_of_nat @ ( suc @ N ) )
% 9.04/5.13            = ( inc @ ( num_of_nat @ N ) ) ) )
% 9.04/5.13        & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.13         => ( ( num_of_nat @ ( suc @ N ) )
% 9.04/5.13            = one ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10049_num__of__nat__numeral__eq,axiom,
% 9.04/5.13      ! [Q3: num] :
% 9.04/5.13        ( ( num_of_nat @ ( numeral_numeral_nat @ Q3 ) )
% 9.04/5.13        = Q3 ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10050_num__of__nat_Osimps_I1_J,axiom,
% 9.04/5.13      ( ( num_of_nat @ zero_zero_nat )
% 9.04/5.13      = one ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10051_numeral__num__of__nat,axiom,
% 9.04/5.13      ! [N: nat] :
% 9.04/5.13        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.13       => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 9.04/5.13          = N ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10052_num__of__nat__One,axiom,
% 9.04/5.13      ! [N: nat] :
% 9.04/5.13        ( ( ord_less_eq_nat @ N @ one_one_nat )
% 9.04/5.13       => ( ( num_of_nat @ N )
% 9.04/5.13          = one ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10053_num__of__nat__double,axiom,
% 9.04/5.13      ! [N: nat] :
% 9.04/5.13        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.13       => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 9.04/5.13          = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10054_num__of__nat__plus__distrib,axiom,
% 9.04/5.13      ! [M: nat,N: nat] :
% 9.04/5.13        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.04/5.13       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.04/5.13         => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 9.04/5.13            = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10055_upto__aux__rec,axiom,
% 9.04/5.13      ( upto_aux
% 9.04/5.13      = ( ^ [I3: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10056_upto_Opsimps,axiom,
% 9.04/5.13      ! [I: int,J: int] :
% 9.04/5.13        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 9.04/5.13       => ( ( ( ord_less_eq_int @ I @ J )
% 9.04/5.13           => ( ( upto @ I @ J )
% 9.04/5.13              = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 9.04/5.13          & ( ~ ( ord_less_eq_int @ I @ J )
% 9.04/5.13           => ( ( upto @ I @ J )
% 9.04/5.13              = nil_int ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10057_upto_Opelims,axiom,
% 9.04/5.13      ! [X: int,Xa: int,Y: list_int] :
% 9.04/5.13        ( ( ( upto @ X @ Xa )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 9.04/5.13         => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
% 9.04/5.13                 => ( Y
% 9.04/5.13                    = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 9.04/5.13                & ( ~ ( ord_less_eq_int @ X @ Xa )
% 9.04/5.13                 => ( Y = nil_int ) ) )
% 9.04/5.13             => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10058_upto__empty,axiom,
% 9.04/5.13      ! [J: int,I: int] :
% 9.04/5.13        ( ( ord_less_int @ J @ I )
% 9.04/5.13       => ( ( upto @ I @ J )
% 9.04/5.13          = nil_int ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10059_upto__Nil2,axiom,
% 9.04/5.13      ! [I: int,J: int] :
% 9.04/5.13        ( ( nil_int
% 9.04/5.13          = ( upto @ I @ J ) )
% 9.04/5.13        = ( ord_less_int @ J @ I ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10060_upto__Nil,axiom,
% 9.04/5.13      ! [I: int,J: int] :
% 9.04/5.13        ( ( ( upto @ I @ J )
% 9.04/5.13          = nil_int )
% 9.04/5.13        = ( ord_less_int @ J @ I ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10061_upto__single,axiom,
% 9.04/5.13      ! [I: int] :
% 9.04/5.13        ( ( upto @ I @ I )
% 9.04/5.13        = ( cons_int @ I @ nil_int ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10062_nth__upto,axiom,
% 9.04/5.13      ! [I: int,K: nat,J: int] :
% 9.04/5.13        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 9.04/5.13       => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 9.04/5.13          = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10063_length__upto,axiom,
% 9.04/5.13      ! [I: int,J: int] :
% 9.04/5.13        ( ( size_size_list_int @ ( upto @ I @ J ) )
% 9.04/5.13        = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10064_upto__rec__numeral_I1_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.13         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.13            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 9.04/5.13        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.13         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.13            = nil_int ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10065_upto__rec__numeral_I4_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13            = ( 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 ) ) ) ) ) )
% 9.04/5.13        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13            = nil_int ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10066_upto__rec__numeral_I3_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.13         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.13            = ( 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 ) ) ) ) )
% 9.04/5.13        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.13         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.13            = nil_int ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10067_upto__rec__numeral_I2_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 9.04/5.13        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13            = nil_int ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10068_upto__aux__def,axiom,
% 9.04/5.13      ( upto_aux
% 9.04/5.13      = ( ^ [I3: int,J2: int] : ( append_int @ ( upto @ I3 @ J2 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10069_upto__code,axiom,
% 9.04/5.13      ( upto
% 9.04/5.13      = ( ^ [I3: int,J2: int] : ( upto_aux @ I3 @ J2 @ nil_int ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10070_atLeastAtMost__upto,axiom,
% 9.04/5.13      ( set_or1266510415728281911st_int
% 9.04/5.13      = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ I3 @ J2 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10071_distinct__upto,axiom,
% 9.04/5.13      ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10072_upto__split2,axiom,
% 9.04/5.13      ! [I: int,J: int,K: int] :
% 9.04/5.13        ( ( ord_less_eq_int @ I @ J )
% 9.04/5.13       => ( ( ord_less_eq_int @ J @ K )
% 9.04/5.13         => ( ( upto @ I @ K )
% 9.04/5.13            = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10073_upto__split1,axiom,
% 9.04/5.13      ! [I: int,J: int,K: int] :
% 9.04/5.13        ( ( ord_less_eq_int @ I @ J )
% 9.04/5.13       => ( ( ord_less_eq_int @ J @ K )
% 9.04/5.13         => ( ( upto @ I @ K )
% 9.04/5.13            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10074_atLeastLessThan__upto,axiom,
% 9.04/5.13      ( set_or4662586982721622107an_int
% 9.04/5.13      = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10075_greaterThanAtMost__upto,axiom,
% 9.04/5.13      ( set_or6656581121297822940st_int
% 9.04/5.13      = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10076_upto__rec1,axiom,
% 9.04/5.13      ! [I: int,J: int] :
% 9.04/5.13        ( ( ord_less_eq_int @ I @ J )
% 9.04/5.13       => ( ( upto @ I @ J )
% 9.04/5.13          = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10077_upto_Oelims,axiom,
% 9.04/5.13      ! [X: int,Xa: int,Y: list_int] :
% 9.04/5.13        ( ( ( upto @ X @ Xa )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( ( ord_less_eq_int @ X @ Xa )
% 9.04/5.13           => ( Y
% 9.04/5.13              = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 9.04/5.13          & ( ~ ( ord_less_eq_int @ X @ Xa )
% 9.04/5.13           => ( Y = nil_int ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10078_upto_Osimps,axiom,
% 9.04/5.13      ( upto
% 9.04/5.13      = ( ^ [I3: int,J2: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J2 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) @ nil_int ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10079_upto__rec2,axiom,
% 9.04/5.13      ! [I: int,J: int] :
% 9.04/5.13        ( ( ord_less_eq_int @ I @ J )
% 9.04/5.13       => ( ( upto @ I @ J )
% 9.04/5.13          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10080_greaterThanLessThan__upto,axiom,
% 9.04/5.13      ( set_or5832277885323065728an_int
% 9.04/5.13      = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10081_upto__split3,axiom,
% 9.04/5.13      ! [I: int,J: int,K: int] :
% 9.04/5.13        ( ( ord_less_eq_int @ I @ J )
% 9.04/5.13       => ( ( ord_less_eq_int @ J @ K )
% 9.04/5.13         => ( ( upto @ I @ K )
% 9.04/5.13            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10082_sup__enat__def,axiom,
% 9.04/5.13      sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10083_sup__nat__def,axiom,
% 9.04/5.13      sup_sup_nat = ord_max_nat ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10084_sup__int__def,axiom,
% 9.04/5.13      sup_sup_int = ord_max_int ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10085_atLeastLessThan__add__Un,axiom,
% 9.04/5.13      ! [I: nat,J: nat,K: nat] :
% 9.04/5.13        ( ( ord_less_eq_nat @ I @ J )
% 9.04/5.13       => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 9.04/5.13          = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10086_list__encode_Oelims,axiom,
% 9.04/5.13      ! [X: list_nat,Y: nat] :
% 9.04/5.13        ( ( ( nat_list_encode @ X )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( ( X = nil_nat )
% 9.04/5.13           => ( Y != zero_zero_nat ) )
% 9.04/5.13         => ~ ! [X3: nat,Xs2: list_nat] :
% 9.04/5.13                ( ( X
% 9.04/5.13                  = ( cons_nat @ X3 @ Xs2 ) )
% 9.04/5.13               => ( Y
% 9.04/5.13                 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10087_list__encode_Osimps_I1_J,axiom,
% 9.04/5.13      ( ( nat_list_encode @ nil_nat )
% 9.04/5.13      = zero_zero_nat ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10088_list__encode_Osimps_I2_J,axiom,
% 9.04/5.13      ! [X: nat,Xs: list_nat] :
% 9.04/5.13        ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
% 9.04/5.13        = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10089_list__encode_Opelims,axiom,
% 9.04/5.13      ! [X: list_nat,Y: nat] :
% 9.04/5.13        ( ( ( nat_list_encode @ X )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( accp_list_nat @ nat_list_encode_rel @ X )
% 9.04/5.13         => ( ( ( X = nil_nat )
% 9.04/5.13             => ( ( Y = zero_zero_nat )
% 9.04/5.13               => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 9.04/5.13           => ~ ! [X3: nat,Xs2: list_nat] :
% 9.04/5.13                  ( ( X
% 9.04/5.13                    = ( cons_nat @ X3 @ Xs2 ) )
% 9.04/5.13                 => ( ( Y
% 9.04/5.13                      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) )
% 9.04/5.13                   => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10090_pred__nat__def,axiom,
% 9.04/5.13      ( pred_nat
% 9.04/5.13      = ( collec3392354462482085612at_nat
% 9.04/5.13        @ ( produc6081775807080527818_nat_o
% 9.04/5.13          @ ^ [M6: nat,N3: nat] :
% 9.04/5.13              ( N3
% 9.04/5.13              = ( suc @ M6 ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10091_sorted__list__of__set__lessThan__Suc,axiom,
% 9.04/5.13      ! [K: nat] :
% 9.04/5.13        ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 9.04/5.13        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10092_sorted__list__of__set__atMost__Suc,axiom,
% 9.04/5.13      ! [K: nat] :
% 9.04/5.13        ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 9.04/5.13        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10093_sorted__list__of__set__greaterThanAtMost,axiom,
% 9.04/5.13      ! [I: nat,J: nat] :
% 9.04/5.13        ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 9.04/5.13       => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 9.04/5.13          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10094_sorted__list__of__set__greaterThanLessThan,axiom,
% 9.04/5.13      ! [I: nat,J: nat] :
% 9.04/5.13        ( ( ord_less_nat @ ( suc @ I ) @ J )
% 9.04/5.13       => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 9.04/5.13          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10095_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 9.04/5.13      ! [N: nat,J: nat,I: nat] :
% 9.04/5.13        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 9.04/5.13       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 9.04/5.13          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10096_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 9.04/5.13      ! [N: nat,J: nat,I: nat] :
% 9.04/5.13        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 9.04/5.13       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 9.04/5.13          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10097_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 9.04/5.13      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.13        ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 9.04/5.13       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.04/5.13                ( X
% 9.04/5.13                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.04/5.13           => ( Xa = one_one_nat ) )
% 9.04/5.13         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.04/5.13                ( ( X
% 9.04/5.13                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.13               => ( ( Deg2 = Xa )
% 9.04/5.13                  & ! [X3: vEBT_VEBT] :
% 9.04/5.13                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                     => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.04/5.13                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                  & ( case_o184042715313410164at_nat
% 9.04/5.13                    @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 9.04/5.13                      & ! [X4: vEBT_VEBT] :
% 9.04/5.13                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                         => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                    @ ( produc6081775807080527818_nat_o
% 9.04/5.13                      @ ^ [Mi3: nat,Ma3: nat] :
% 9.04/5.13                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.04/5.13                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                          & ! [I3: nat] :
% 9.04/5.13                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                             => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 9.04/5.13                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 9.04/5.13                          & ( ( Mi3 = Ma3 )
% 9.04/5.13                           => ! [X4: vEBT_VEBT] :
% 9.04/5.13                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                          & ( ( Mi3 != Ma3 )
% 9.04/5.13                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.04/5.13                              & ! [X4: nat] :
% 9.04/5.13                                  ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 9.04/5.13                                   => ( ( ord_less_nat @ Mi3 @ X4 )
% 9.04/5.13                                      & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 9.04/5.13                    @ Mima ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10098_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 9.04/5.13      ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 9.04/5.13        ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
% 9.04/5.13        = ( ( Deg = Deg4 )
% 9.04/5.13          & ! [X4: vEBT_VEBT] :
% 9.04/5.13              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.04/5.13             => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13          & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13          & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.04/5.13            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13          & ( case_o184042715313410164at_nat
% 9.04/5.13            @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
% 9.04/5.13              & ! [X4: vEBT_VEBT] :
% 9.04/5.13                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.04/5.13                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13            @ ( produc6081775807080527818_nat_o
% 9.04/5.13              @ ^ [Mi3: nat,Ma3: nat] :
% 9.04/5.13                  ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.04/5.13                  & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.04/5.13                  & ! [I3: nat] :
% 9.04/5.13                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
% 9.04/5.13                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 9.04/5.13                  & ( ( Mi3 = Ma3 )
% 9.04/5.13                   => ! [X4: vEBT_VEBT] :
% 9.04/5.13                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.04/5.13                       => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                  & ( ( Mi3 != Ma3 )
% 9.04/5.13                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 9.04/5.13                      & ! [X4: nat] :
% 9.04/5.13                          ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.04/5.13                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
% 9.04/5.13                           => ( ( ord_less_nat @ Mi3 @ X4 )
% 9.04/5.13                              & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 9.04/5.13            @ Mima2 ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10099_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 9.04/5.13      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 9.04/5.13        ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.04/5.13                ( X
% 9.04/5.13                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.04/5.13           => ( Y
% 9.04/5.13              = ( Xa != one_one_nat ) ) )
% 9.04/5.13         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.04/5.13                ( ( X
% 9.04/5.13                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.13               => ( Y
% 9.04/5.13                  = ( ~ ( ( Deg2 = Xa )
% 9.04/5.13                        & ! [X4: vEBT_VEBT] :
% 9.04/5.13                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                           => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.04/5.13                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                        & ( case_o184042715313410164at_nat
% 9.04/5.13                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 9.04/5.13                            & ! [X4: vEBT_VEBT] :
% 9.04/5.13                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                          @ ( produc6081775807080527818_nat_o
% 9.04/5.13                            @ ^ [Mi3: nat,Ma3: nat] :
% 9.04/5.13                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.04/5.13                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                                & ! [I3: nat] :
% 9.04/5.13                                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 9.04/5.13                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 9.04/5.13                                & ( ( Mi3 = Ma3 )
% 9.04/5.13                                 => ! [X4: vEBT_VEBT] :
% 9.04/5.13                                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                                & ( ( Mi3 != Ma3 )
% 9.04/5.13                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.04/5.13                                    & ! [X4: nat] :
% 9.04/5.13                                        ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 9.04/5.13                                         => ( ( ord_less_nat @ Mi3 @ X4 )
% 9.04/5.13                                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 9.04/5.13                          @ Mima ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10100_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 9.04/5.13      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.13        ( ( vEBT_VEBT_valid @ X @ Xa )
% 9.04/5.13       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.04/5.13                ( X
% 9.04/5.13                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.04/5.13           => ( Xa != one_one_nat ) )
% 9.04/5.13         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.04/5.13                ( ( X
% 9.04/5.13                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.13               => ~ ( ( Deg2 = Xa )
% 9.04/5.13                    & ! [X2: vEBT_VEBT] :
% 9.04/5.13                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                       => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.04/5.13                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                    & ( case_o184042715313410164at_nat
% 9.04/5.13                      @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 9.04/5.13                        & ! [X4: vEBT_VEBT] :
% 9.04/5.13                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                           => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                      @ ( produc6081775807080527818_nat_o
% 9.04/5.13                        @ ^ [Mi3: nat,Ma3: nat] :
% 9.04/5.13                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.04/5.13                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                            & ! [I3: nat] :
% 9.04/5.13                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                               => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 9.04/5.13                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 9.04/5.13                            & ( ( Mi3 = Ma3 )
% 9.04/5.13                             => ! [X4: vEBT_VEBT] :
% 9.04/5.13                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                            & ( ( Mi3 != Ma3 )
% 9.04/5.13                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.04/5.13                                & ! [X4: nat] :
% 9.04/5.13                                    ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 9.04/5.13                                     => ( ( ord_less_nat @ Mi3 @ X4 )
% 9.04/5.13                                        & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 9.04/5.13                      @ Mima ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10101_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 9.04/5.13      ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 9.04/5.13        ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 9.04/5.13         => ( ! [Uu2: $o,Uv2: $o] :
% 9.04/5.13                ( ( X
% 9.04/5.13                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.04/5.13               => ( ( Y
% 9.04/5.13                    = ( Xa = one_one_nat ) )
% 9.04/5.13                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 9.04/5.13           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.04/5.13                  ( ( X
% 9.04/5.13                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.13                 => ( ( Y
% 9.04/5.13                      = ( ( Deg2 = Xa )
% 9.04/5.13                        & ! [X4: vEBT_VEBT] :
% 9.04/5.13                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                           => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.04/5.13                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                        & ( case_o184042715313410164at_nat
% 9.04/5.13                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 9.04/5.13                            & ! [X4: vEBT_VEBT] :
% 9.04/5.13                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                          @ ( produc6081775807080527818_nat_o
% 9.04/5.13                            @ ^ [Mi3: nat,Ma3: nat] :
% 9.04/5.13                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.04/5.13                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                                & ! [I3: nat] :
% 9.04/5.13                                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 9.04/5.13                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 9.04/5.13                                & ( ( Mi3 = Ma3 )
% 9.04/5.13                                 => ! [X4: vEBT_VEBT] :
% 9.04/5.13                                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                                & ( ( Mi3 != Ma3 )
% 9.04/5.13                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.04/5.13                                    & ! [X4: nat] :
% 9.04/5.13                                        ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 9.04/5.13                                         => ( ( ord_less_nat @ Mi3 @ X4 )
% 9.04/5.13                                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 9.04/5.13                          @ Mima ) ) )
% 9.04/5.13                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10102_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 9.04/5.13      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.13        ( ( vEBT_VEBT_valid @ X @ Xa )
% 9.04/5.13       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 9.04/5.13         => ( ! [Uu2: $o,Uv2: $o] :
% 9.04/5.13                ( ( X
% 9.04/5.13                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.04/5.13               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 9.04/5.13                 => ( Xa != one_one_nat ) ) )
% 9.04/5.13           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.04/5.13                  ( ( X
% 9.04/5.13                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.13                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 9.04/5.13                   => ~ ( ( Deg2 = Xa )
% 9.04/5.13                        & ! [X2: vEBT_VEBT] :
% 9.04/5.13                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                           => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.04/5.13                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                        & ( case_o184042715313410164at_nat
% 9.04/5.13                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 9.04/5.13                            & ! [X4: vEBT_VEBT] :
% 9.04/5.13                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                          @ ( produc6081775807080527818_nat_o
% 9.04/5.13                            @ ^ [Mi3: nat,Ma3: nat] :
% 9.04/5.13                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.04/5.13                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                                & ! [I3: nat] :
% 9.04/5.13                                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 9.04/5.13                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 9.04/5.13                                & ( ( Mi3 = Ma3 )
% 9.04/5.13                                 => ! [X4: vEBT_VEBT] :
% 9.04/5.13                                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                                & ( ( Mi3 != Ma3 )
% 9.04/5.13                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.04/5.13                                    & ! [X4: nat] :
% 9.04/5.13                                        ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 9.04/5.13                                         => ( ( ord_less_nat @ Mi3 @ X4 )
% 9.04/5.13                                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 9.04/5.13                          @ Mima ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10103_Sup__int__def,axiom,
% 9.04/5.13      ( complete_Sup_Sup_int
% 9.04/5.13      = ( ^ [X6: set_int] :
% 9.04/5.13            ( the_int
% 9.04/5.13            @ ^ [X4: int] :
% 9.04/5.13                ( ( member_int @ X4 @ X6 )
% 9.04/5.13                & ! [Y6: int] :
% 9.04/5.13                    ( ( member_int @ Y6 @ X6 )
% 9.04/5.13                   => ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10104_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 9.04/5.13      ! [X: vEBT_VEBT,Xa: nat] :
% 9.04/5.13        ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 9.04/5.13       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 9.04/5.13         => ( ! [Uu2: $o,Uv2: $o] :
% 9.04/5.13                ( ( X
% 9.04/5.13                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.04/5.13               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 9.04/5.13                 => ( Xa = one_one_nat ) ) )
% 9.04/5.13           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.04/5.13                  ( ( X
% 9.04/5.13                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.04/5.13                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 9.04/5.13                   => ( ( Deg2 = Xa )
% 9.04/5.13                      & ! [X3: vEBT_VEBT] :
% 9.04/5.13                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                         => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.04/5.13                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.04/5.13                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                      & ( case_o184042715313410164at_nat
% 9.04/5.13                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 9.04/5.13                          & ! [X4: vEBT_VEBT] :
% 9.04/5.13                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                        @ ( produc6081775807080527818_nat_o
% 9.04/5.13                          @ ^ [Mi3: nat,Ma3: nat] :
% 9.04/5.13                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.04/5.13                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                              & ! [I3: nat] :
% 9.04/5.13                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.04/5.13                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 9.04/5.13                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 9.04/5.13                              & ( ( Mi3 = Ma3 )
% 9.04/5.13                               => ! [X4: vEBT_VEBT] :
% 9.04/5.13                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.04/5.13                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
% 9.04/5.13                              & ( ( Mi3 != Ma3 )
% 9.04/5.13                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.04/5.13                                  & ! [X4: nat] :
% 9.04/5.13                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.04/5.13                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
% 9.04/5.13                                       => ( ( ord_less_nat @ Mi3 @ X4 )
% 9.04/5.13                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
% 9.04/5.13                        @ Mima ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10105_take__bit__numeral__minus__numeral__int,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13        = ( case_option_int_num @ zero_zero_int
% 9.04/5.13          @ ^ [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 ) ) )
% 9.04/5.13          @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10106_take__bit__num__simps_I1_J,axiom,
% 9.04/5.13      ! [M: num] :
% 9.04/5.13        ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 9.04/5.13        = none_num ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10107_take__bit__num__simps_I2_J,axiom,
% 9.04/5.13      ! [N: nat] :
% 9.04/5.13        ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 9.04/5.13        = ( some_num @ one ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10108_take__bit__num__simps_I5_J,axiom,
% 9.04/5.13      ! [R2: num] :
% 9.04/5.13        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
% 9.04/5.13        = ( some_num @ one ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10109_take__bit__num__simps_I3_J,axiom,
% 9.04/5.13      ! [N: nat,M: num] :
% 9.04/5.13        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 9.04/5.13        = ( case_o6005452278849405969um_num @ none_num
% 9.04/5.13          @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 9.04/5.13          @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10110_take__bit__num__simps_I4_J,axiom,
% 9.04/5.13      ! [N: nat,M: num] :
% 9.04/5.13        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 9.04/5.13        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10111_take__bit__num__simps_I6_J,axiom,
% 9.04/5.13      ! [R2: num,M: num] :
% 9.04/5.13        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 9.04/5.13        = ( case_o6005452278849405969um_num @ none_num
% 9.04/5.13          @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 9.04/5.13          @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10112_take__bit__num__simps_I7_J,axiom,
% 9.04/5.13      ! [R2: num,M: num] :
% 9.04/5.13        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
% 9.04/5.13        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10113_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 9.04/5.13      ! [N: nat,M: num] :
% 9.04/5.13        ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 9.04/5.13        = ( case_nat_option_num @ none_num
% 9.04/5.13          @ ^ [N3: nat] :
% 9.04/5.13              ( case_o6005452278849405969um_num @ none_num
% 9.04/5.13              @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 9.04/5.13              @ ( bit_take_bit_num @ N3 @ M ) )
% 9.04/5.13          @ N ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10114_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 9.04/5.13      ! [N: nat,M: num] :
% 9.04/5.13        ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 9.04/5.13        = ( case_nat_option_num @ none_num
% 9.04/5.13          @ ^ [N3: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N3 @ M ) ) )
% 9.04/5.13          @ N ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10115_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 9.04/5.13      ! [N: nat] :
% 9.04/5.13        ( ( bit_take_bit_num @ N @ one )
% 9.04/5.13        = ( case_nat_option_num @ none_num
% 9.04/5.13          @ ^ [N3: nat] : ( some_num @ one )
% 9.04/5.13          @ N ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10116_take__bit__num__def,axiom,
% 9.04/5.13      ( bit_take_bit_num
% 9.04/5.13      = ( ^ [N3: nat,M6: num] :
% 9.04/5.13            ( if_option_num
% 9.04/5.13            @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M6 ) )
% 9.04/5.13              = zero_zero_nat )
% 9.04/5.13            @ none_num
% 9.04/5.13            @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10117_and__minus__numerals_I7_J,axiom,
% 9.04/5.13      ! [N: num,M: num] :
% 9.04/5.13        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 9.04/5.13        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10118_and__minus__numerals_I3_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.04/5.13        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10119_and__minus__numerals_I8_J,axiom,
% 9.04/5.13      ! [N: num,M: num] :
% 9.04/5.13        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 9.04/5.13        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10120_and__minus__numerals_I4_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.04/5.13        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10121_and__not__num_Osimps_I2_J,axiom,
% 9.04/5.13      ! [N: num] :
% 9.04/5.13        ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 9.04/5.13        = ( some_num @ one ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10122_and__not__num_Osimps_I4_J,axiom,
% 9.04/5.13      ! [M: num] :
% 9.04/5.13        ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 9.04/5.13        = ( some_num @ ( bit0 @ M ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10123_and__not__num_Osimps_I3_J,axiom,
% 9.04/5.13      ! [N: num] :
% 9.04/5.13        ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 9.04/5.13        = none_num ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10124_and__not__num_Osimps_I1_J,axiom,
% 9.04/5.13      ( ( bit_and_not_num @ one @ one )
% 9.04/5.13      = none_num ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10125_and__not__num_Osimps_I7_J,axiom,
% 9.04/5.13      ! [M: num] :
% 9.04/5.13        ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 9.04/5.13        = ( some_num @ ( bit0 @ M ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10126_and__not__num__eq__Some__iff,axiom,
% 9.04/5.13      ! [M: num,N: num,Q3: num] :
% 9.04/5.13        ( ( ( bit_and_not_num @ M @ N )
% 9.04/5.13          = ( some_num @ Q3 ) )
% 9.04/5.13        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13          = ( numeral_numeral_int @ Q3 ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10127_and__not__num_Osimps_I8_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.04/5.13        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.04/5.13          @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 9.04/5.13          @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10128_and__not__num__eq__None__iff,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( ( bit_and_not_num @ M @ N )
% 9.04/5.13          = none_num )
% 9.04/5.13        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13          = zero_zero_int ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10129_int__numeral__and__not__num,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.04/5.13        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10130_int__numeral__not__and__num,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.04/5.13        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10131_Rats__eq__int__div__nat,axiom,
% 9.04/5.13      ( field_5140801741446780682s_real
% 9.04/5.13      = ( collect_real
% 9.04/5.13        @ ^ [Uu3: real] :
% 9.04/5.13          ? [I3: int,N3: nat] :
% 9.04/5.13            ( ( Uu3
% 9.04/5.13              = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 9.04/5.13            & ( N3 != zero_zero_nat ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10132_Rats__abs__iff,axiom,
% 9.04/5.13      ! [X: real] :
% 9.04/5.13        ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
% 9.04/5.13        = ( member_real @ X @ field_5140801741446780682s_real ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10133_Rats__no__top__le,axiom,
% 9.04/5.13      ! [X: real] :
% 9.04/5.13      ? [X3: real] :
% 9.04/5.13        ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 9.04/5.13        & ( ord_less_eq_real @ X @ X3 ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10134_Rats__dense__in__real,axiom,
% 9.04/5.13      ! [X: real,Y: real] :
% 9.04/5.13        ( ( ord_less_real @ X @ Y )
% 9.04/5.13       => ? [X3: real] :
% 9.04/5.13            ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 9.04/5.13            & ( ord_less_real @ X @ X3 )
% 9.04/5.13            & ( ord_less_real @ X3 @ Y ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10135_Rats__no__bot__less,axiom,
% 9.04/5.13      ! [X: real] :
% 9.04/5.13      ? [X3: real] :
% 9.04/5.13        ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 9.04/5.13        & ( ord_less_real @ X3 @ X ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10136_Rats__eq__int__div__int,axiom,
% 9.04/5.13      ( field_5140801741446780682s_real
% 9.04/5.13      = ( collect_real
% 9.04/5.13        @ ^ [Uu3: real] :
% 9.04/5.13          ? [I3: int,J2: int] :
% 9.04/5.13            ( ( Uu3
% 9.04/5.13              = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( ring_1_of_int_real @ J2 ) ) )
% 9.04/5.13            & ( J2 != zero_zero_int ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10137_and__not__num_Oelims,axiom,
% 9.04/5.13      ! [X: num,Xa: num,Y: option_num] :
% 9.04/5.13        ( ( ( bit_and_not_num @ X @ Xa )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( ( X = one )
% 9.04/5.13           => ( ( Xa = one )
% 9.04/5.13             => ( Y != none_num ) ) )
% 9.04/5.13         => ( ( ( X = one )
% 9.04/5.13             => ( ? [N2: num] :
% 9.04/5.13                    ( Xa
% 9.04/5.13                    = ( bit0 @ N2 ) )
% 9.04/5.13               => ( Y
% 9.04/5.13                 != ( some_num @ one ) ) ) )
% 9.04/5.13           => ( ( ( X = one )
% 9.04/5.13               => ( ? [N2: num] :
% 9.04/5.13                      ( Xa
% 9.04/5.13                      = ( bit1 @ N2 ) )
% 9.04/5.13                 => ( Y != none_num ) ) )
% 9.04/5.13             => ( ! [M5: num] :
% 9.04/5.13                    ( ( X
% 9.04/5.13                      = ( bit0 @ M5 ) )
% 9.04/5.13                   => ( ( Xa = one )
% 9.04/5.13                     => ( Y
% 9.04/5.13                       != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 9.04/5.13               => ( ! [M5: num] :
% 9.04/5.13                      ( ( X
% 9.04/5.13                        = ( bit0 @ M5 ) )
% 9.04/5.13                     => ! [N2: num] :
% 9.04/5.13                          ( ( Xa
% 9.04/5.13                            = ( bit0 @ N2 ) )
% 9.04/5.13                         => ( Y
% 9.04/5.13                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) ) ) )
% 9.04/5.13                 => ( ! [M5: num] :
% 9.04/5.13                        ( ( X
% 9.04/5.13                          = ( bit0 @ M5 ) )
% 9.04/5.13                       => ! [N2: num] :
% 9.04/5.13                            ( ( Xa
% 9.04/5.13                              = ( bit1 @ N2 ) )
% 9.04/5.13                           => ( Y
% 9.04/5.13                             != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) ) ) )
% 9.04/5.13                   => ( ! [M5: num] :
% 9.04/5.13                          ( ( X
% 9.04/5.13                            = ( bit1 @ M5 ) )
% 9.04/5.13                         => ( ( Xa = one )
% 9.04/5.13                           => ( Y
% 9.04/5.13                             != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 9.04/5.13                     => ( ! [M5: num] :
% 9.04/5.13                            ( ( X
% 9.04/5.13                              = ( bit1 @ M5 ) )
% 9.04/5.13                           => ! [N2: num] :
% 9.04/5.13                                ( ( Xa
% 9.04/5.13                                  = ( bit0 @ N2 ) )
% 9.04/5.13                               => ( Y
% 9.04/5.13                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.04/5.13                                    @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 9.04/5.13                                    @ ( bit_and_not_num @ M5 @ N2 ) ) ) ) )
% 9.04/5.13                       => ~ ! [M5: num] :
% 9.04/5.13                              ( ( X
% 9.04/5.13                                = ( bit1 @ M5 ) )
% 9.04/5.13                             => ! [N2: num] :
% 9.04/5.13                                  ( ( Xa
% 9.04/5.13                                    = ( bit1 @ N2 ) )
% 9.04/5.13                                 => ( Y
% 9.04/5.13                                   != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10138_and__not__num_Osimps_I5_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.04/5.13        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10139_and__not__num_Osimps_I6_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.13        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10140_and__not__num_Osimps_I9_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.13        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10141_and__not__num_Opelims,axiom,
% 9.04/5.13      ! [X: num,Xa: num,Y: option_num] :
% 9.04/5.13        ( ( ( bit_and_not_num @ X @ Xa )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 9.04/5.13         => ( ( ( X = one )
% 9.04/5.13             => ( ( Xa = one )
% 9.04/5.13               => ( ( Y = none_num )
% 9.04/5.13                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.04/5.13           => ( ( ( X = one )
% 9.04/5.13               => ! [N2: num] :
% 9.04/5.13                    ( ( Xa
% 9.04/5.13                      = ( bit0 @ N2 ) )
% 9.04/5.13                   => ( ( Y
% 9.04/5.13                        = ( some_num @ one ) )
% 9.04/5.13                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 9.04/5.13             => ( ( ( X = one )
% 9.04/5.13                 => ! [N2: num] :
% 9.04/5.13                      ( ( Xa
% 9.04/5.13                        = ( bit1 @ N2 ) )
% 9.04/5.13                     => ( ( Y = none_num )
% 9.04/5.13                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 9.04/5.13               => ( ! [M5: num] :
% 9.04/5.13                      ( ( X
% 9.04/5.13                        = ( bit0 @ M5 ) )
% 9.04/5.13                     => ( ( Xa = one )
% 9.04/5.13                       => ( ( Y
% 9.04/5.13                            = ( some_num @ ( bit0 @ M5 ) ) )
% 9.04/5.13                         => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 9.04/5.13                 => ( ! [M5: num] :
% 9.04/5.13                        ( ( X
% 9.04/5.13                          = ( bit0 @ M5 ) )
% 9.04/5.13                       => ! [N2: num] :
% 9.04/5.13                            ( ( Xa
% 9.04/5.13                              = ( bit0 @ N2 ) )
% 9.04/5.13                           => ( ( Y
% 9.04/5.13                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) )
% 9.04/5.13                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 9.04/5.13                   => ( ! [M5: num] :
% 9.04/5.13                          ( ( X
% 9.04/5.13                            = ( bit0 @ M5 ) )
% 9.04/5.13                         => ! [N2: num] :
% 9.04/5.13                              ( ( Xa
% 9.04/5.13                                = ( bit1 @ N2 ) )
% 9.04/5.13                             => ( ( Y
% 9.04/5.13                                  = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) )
% 9.04/5.13                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 9.04/5.13                     => ( ! [M5: num] :
% 9.04/5.13                            ( ( X
% 9.04/5.13                              = ( bit1 @ M5 ) )
% 9.04/5.13                           => ( ( Xa = one )
% 9.04/5.13                             => ( ( Y
% 9.04/5.13                                  = ( some_num @ ( bit0 @ M5 ) ) )
% 9.04/5.13                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 9.04/5.13                       => ( ! [M5: num] :
% 9.04/5.13                              ( ( X
% 9.04/5.13                                = ( bit1 @ M5 ) )
% 9.04/5.13                             => ! [N2: num] :
% 9.04/5.13                                  ( ( Xa
% 9.04/5.13                                    = ( bit0 @ N2 ) )
% 9.04/5.13                                 => ( ( Y
% 9.04/5.13                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.04/5.13                                        @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 9.04/5.13                                        @ ( bit_and_not_num @ M5 @ N2 ) ) )
% 9.04/5.13                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 9.04/5.13                         => ~ ! [M5: num] :
% 9.04/5.13                                ( ( X
% 9.04/5.13                                  = ( bit1 @ M5 ) )
% 9.04/5.13                               => ! [N2: num] :
% 9.04/5.13                                    ( ( Xa
% 9.04/5.13                                      = ( bit1 @ N2 ) )
% 9.04/5.13                                   => ( ( Y
% 9.04/5.13                                        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N2 ) ) )
% 9.04/5.13                                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10142_and__num_Oelims,axiom,
% 9.04/5.13      ! [X: num,Xa: num,Y: option_num] :
% 9.04/5.13        ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( ( X = one )
% 9.04/5.13           => ( ( Xa = one )
% 9.04/5.13             => ( Y
% 9.04/5.13               != ( some_num @ one ) ) ) )
% 9.04/5.13         => ( ( ( X = one )
% 9.04/5.13             => ( ? [N2: num] :
% 9.04/5.13                    ( Xa
% 9.04/5.13                    = ( bit0 @ N2 ) )
% 9.04/5.13               => ( Y != none_num ) ) )
% 9.04/5.13           => ( ( ( X = one )
% 9.04/5.13               => ( ? [N2: num] :
% 9.04/5.13                      ( Xa
% 9.04/5.13                      = ( bit1 @ N2 ) )
% 9.04/5.13                 => ( Y
% 9.04/5.13                   != ( some_num @ one ) ) ) )
% 9.04/5.13             => ( ( ? [M5: num] :
% 9.04/5.13                      ( X
% 9.04/5.13                      = ( bit0 @ M5 ) )
% 9.04/5.13                 => ( ( Xa = one )
% 9.04/5.13                   => ( Y != none_num ) ) )
% 9.04/5.13               => ( ! [M5: num] :
% 9.04/5.13                      ( ( X
% 9.04/5.13                        = ( bit0 @ M5 ) )
% 9.04/5.13                     => ! [N2: num] :
% 9.04/5.13                          ( ( Xa
% 9.04/5.13                            = ( bit0 @ N2 ) )
% 9.04/5.13                         => ( Y
% 9.04/5.13                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) ) ) )
% 9.04/5.13                 => ( ! [M5: num] :
% 9.04/5.13                        ( ( X
% 9.04/5.13                          = ( bit0 @ M5 ) )
% 9.04/5.13                       => ! [N2: num] :
% 9.04/5.13                            ( ( Xa
% 9.04/5.13                              = ( bit1 @ N2 ) )
% 9.04/5.13                           => ( Y
% 9.04/5.13                             != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) ) ) )
% 9.04/5.13                   => ( ( ? [M5: num] :
% 9.04/5.13                            ( X
% 9.04/5.13                            = ( bit1 @ M5 ) )
% 9.04/5.13                       => ( ( Xa = one )
% 9.04/5.13                         => ( Y
% 9.04/5.13                           != ( some_num @ one ) ) ) )
% 9.04/5.13                     => ( ! [M5: num] :
% 9.04/5.13                            ( ( X
% 9.04/5.13                              = ( bit1 @ M5 ) )
% 9.04/5.13                           => ! [N2: num] :
% 9.04/5.13                                ( ( Xa
% 9.04/5.13                                  = ( bit0 @ N2 ) )
% 9.04/5.13                               => ( Y
% 9.04/5.13                                 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) ) ) )
% 9.04/5.13                       => ~ ! [M5: num] :
% 9.04/5.13                              ( ( X
% 9.04/5.13                                = ( bit1 @ M5 ) )
% 9.04/5.13                             => ! [N2: num] :
% 9.04/5.13                                  ( ( Xa
% 9.04/5.13                                    = ( bit1 @ N2 ) )
% 9.04/5.13                                 => ( Y
% 9.04/5.13                                   != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.04/5.13                                      @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 9.04/5.13                                      @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10143_xor__num_Oelims,axiom,
% 9.04/5.13      ! [X: num,Xa: num,Y: option_num] :
% 9.04/5.13        ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( ( X = one )
% 9.04/5.13           => ( ( Xa = one )
% 9.04/5.13             => ( Y != none_num ) ) )
% 9.04/5.13         => ( ( ( X = one )
% 9.04/5.13             => ! [N2: num] :
% 9.04/5.13                  ( ( Xa
% 9.04/5.13                    = ( bit0 @ N2 ) )
% 9.04/5.13                 => ( Y
% 9.04/5.13                   != ( some_num @ ( bit1 @ N2 ) ) ) ) )
% 9.04/5.13           => ( ( ( X = one )
% 9.04/5.13               => ! [N2: num] :
% 9.04/5.13                    ( ( Xa
% 9.04/5.13                      = ( bit1 @ N2 ) )
% 9.04/5.13                   => ( Y
% 9.04/5.13                     != ( some_num @ ( bit0 @ N2 ) ) ) ) )
% 9.04/5.13             => ( ! [M5: num] :
% 9.04/5.13                    ( ( X
% 9.04/5.13                      = ( bit0 @ M5 ) )
% 9.04/5.13                   => ( ( Xa = one )
% 9.04/5.13                     => ( Y
% 9.04/5.13                       != ( some_num @ ( bit1 @ M5 ) ) ) ) )
% 9.04/5.13               => ( ! [M5: num] :
% 9.04/5.13                      ( ( X
% 9.04/5.13                        = ( bit0 @ M5 ) )
% 9.04/5.13                     => ! [N2: num] :
% 9.04/5.13                          ( ( Xa
% 9.04/5.13                            = ( bit0 @ N2 ) )
% 9.04/5.13                         => ( Y
% 9.04/5.13                           != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) ) )
% 9.04/5.13                 => ( ! [M5: num] :
% 9.04/5.13                        ( ( X
% 9.04/5.13                          = ( bit0 @ M5 ) )
% 9.04/5.13                       => ! [N2: num] :
% 9.04/5.13                            ( ( Xa
% 9.04/5.13                              = ( bit1 @ N2 ) )
% 9.04/5.13                           => ( Y
% 9.04/5.13                             != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) ) ) )
% 9.04/5.13                   => ( ! [M5: num] :
% 9.04/5.13                          ( ( X
% 9.04/5.13                            = ( bit1 @ M5 ) )
% 9.04/5.13                         => ( ( Xa = one )
% 9.04/5.13                           => ( Y
% 9.04/5.13                             != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 9.04/5.13                     => ( ! [M5: num] :
% 9.04/5.13                            ( ( X
% 9.04/5.13                              = ( bit1 @ M5 ) )
% 9.04/5.13                           => ! [N2: num] :
% 9.04/5.13                                ( ( Xa
% 9.04/5.13                                  = ( bit0 @ N2 ) )
% 9.04/5.13                               => ( Y
% 9.04/5.13                                 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) ) ) )
% 9.04/5.13                       => ~ ! [M5: num] :
% 9.04/5.13                              ( ( X
% 9.04/5.13                                = ( bit1 @ M5 ) )
% 9.04/5.13                             => ! [N2: num] :
% 9.04/5.13                                  ( ( Xa
% 9.04/5.13                                    = ( bit1 @ N2 ) )
% 9.04/5.13                                 => ( Y
% 9.04/5.13                                   != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10144_xor__num_Osimps_I5_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.04/5.13        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10145_and__num_Osimps_I5_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.04/5.13        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10146_xor__num_Osimps_I1_J,axiom,
% 9.04/5.13      ( ( bit_un2480387367778600638or_num @ one @ one )
% 9.04/5.13      = none_num ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10147_and__num_Osimps_I1_J,axiom,
% 9.04/5.13      ( ( bit_un7362597486090784418nd_num @ one @ one )
% 9.04/5.13      = ( some_num @ one ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10148_and__num_Osimps_I7_J,axiom,
% 9.04/5.13      ! [M: num] :
% 9.04/5.13        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 9.04/5.13        = ( some_num @ one ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10149_and__num_Osimps_I3_J,axiom,
% 9.04/5.13      ! [N: num] :
% 9.04/5.13        ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 9.04/5.13        = ( some_num @ one ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10150_and__num_Osimps_I2_J,axiom,
% 9.04/5.13      ! [N: num] :
% 9.04/5.13        ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 9.04/5.13        = none_num ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10151_and__num_Osimps_I4_J,axiom,
% 9.04/5.13      ! [M: num] :
% 9.04/5.13        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 9.04/5.13        = none_num ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10152_and__num_Osimps_I8_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.04/5.13        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10153_and__num_Osimps_I6_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.13        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10154_xor__num_Osimps_I9_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.13        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10155_xor__num_Osimps_I7_J,axiom,
% 9.04/5.13      ! [M: num] :
% 9.04/5.13        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 9.04/5.13        = ( some_num @ ( bit0 @ M ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10156_xor__num_Osimps_I4_J,axiom,
% 9.04/5.13      ! [M: num] :
% 9.04/5.13        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 9.04/5.13        = ( some_num @ ( bit1 @ M ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10157_xor__num_Osimps_I3_J,axiom,
% 9.04/5.13      ! [N: num] :
% 9.04/5.13        ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 9.04/5.13        = ( some_num @ ( bit0 @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10158_xor__num_Osimps_I2_J,axiom,
% 9.04/5.13      ! [N: num] :
% 9.04/5.13        ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 9.04/5.13        = ( some_num @ ( bit1 @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10159_and__num_Osimps_I9_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.04/5.13        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.04/5.13          @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 9.04/5.13          @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10160_xor__num_Osimps_I8_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.04/5.13        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10161_xor__num_Osimps_I6_J,axiom,
% 9.04/5.13      ! [M: num,N: num] :
% 9.04/5.13        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.04/5.13        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10162_and__num_Opelims,axiom,
% 9.04/5.13      ! [X: num,Xa: num,Y: option_num] :
% 9.04/5.13        ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 9.04/5.13         => ( ( ( X = one )
% 9.04/5.13             => ( ( Xa = one )
% 9.04/5.13               => ( ( Y
% 9.04/5.13                    = ( some_num @ one ) )
% 9.04/5.13                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.04/5.13           => ( ( ( X = one )
% 9.04/5.13               => ! [N2: num] :
% 9.04/5.13                    ( ( Xa
% 9.04/5.13                      = ( bit0 @ N2 ) )
% 9.04/5.13                   => ( ( Y = none_num )
% 9.04/5.13                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 9.04/5.13             => ( ( ( X = one )
% 9.04/5.13                 => ! [N2: num] :
% 9.04/5.13                      ( ( Xa
% 9.04/5.13                        = ( bit1 @ N2 ) )
% 9.04/5.13                     => ( ( Y
% 9.04/5.13                          = ( some_num @ one ) )
% 9.04/5.13                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 9.04/5.13               => ( ! [M5: num] :
% 9.04/5.13                      ( ( X
% 9.04/5.13                        = ( bit0 @ M5 ) )
% 9.04/5.13                     => ( ( Xa = one )
% 9.04/5.13                       => ( ( Y = none_num )
% 9.04/5.13                         => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 9.04/5.13                 => ( ! [M5: num] :
% 9.04/5.13                        ( ( X
% 9.04/5.13                          = ( bit0 @ M5 ) )
% 9.04/5.13                       => ! [N2: num] :
% 9.04/5.13                            ( ( Xa
% 9.04/5.13                              = ( bit0 @ N2 ) )
% 9.04/5.13                           => ( ( Y
% 9.04/5.13                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) )
% 9.04/5.13                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 9.04/5.13                   => ( ! [M5: num] :
% 9.04/5.13                          ( ( X
% 9.04/5.13                            = ( bit0 @ M5 ) )
% 9.04/5.13                         => ! [N2: num] :
% 9.04/5.13                              ( ( Xa
% 9.04/5.13                                = ( bit1 @ N2 ) )
% 9.04/5.13                             => ( ( Y
% 9.04/5.13                                  = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) )
% 9.04/5.13                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 9.04/5.13                     => ( ! [M5: num] :
% 9.04/5.13                            ( ( X
% 9.04/5.13                              = ( bit1 @ M5 ) )
% 9.04/5.13                           => ( ( Xa = one )
% 9.04/5.13                             => ( ( Y
% 9.04/5.13                                  = ( some_num @ one ) )
% 9.04/5.13                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 9.04/5.13                       => ( ! [M5: num] :
% 9.04/5.13                              ( ( X
% 9.04/5.13                                = ( bit1 @ M5 ) )
% 9.04/5.13                             => ! [N2: num] :
% 9.04/5.13                                  ( ( Xa
% 9.04/5.13                                    = ( bit0 @ N2 ) )
% 9.04/5.13                                 => ( ( Y
% 9.04/5.13                                      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) )
% 9.04/5.13                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 9.04/5.13                         => ~ ! [M5: num] :
% 9.04/5.13                                ( ( X
% 9.04/5.13                                  = ( bit1 @ M5 ) )
% 9.04/5.13                               => ! [N2: num] :
% 9.04/5.13                                    ( ( Xa
% 9.04/5.13                                      = ( bit1 @ N2 ) )
% 9.04/5.13                                   => ( ( Y
% 9.04/5.13                                        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.04/5.13                                          @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 9.04/5.13                                          @ ( bit_un7362597486090784418nd_num @ M5 @ N2 ) ) )
% 9.04/5.13                                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10163_xor__num_Opelims,axiom,
% 9.04/5.13      ! [X: num,Xa: num,Y: option_num] :
% 9.04/5.13        ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 9.04/5.13          = Y )
% 9.04/5.13       => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 9.04/5.13         => ( ( ( X = one )
% 9.04/5.13             => ( ( Xa = one )
% 9.04/5.13               => ( ( Y = none_num )
% 9.04/5.13                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.04/5.13           => ( ( ( X = one )
% 9.04/5.13               => ! [N2: num] :
% 9.04/5.13                    ( ( Xa
% 9.04/5.13                      = ( bit0 @ N2 ) )
% 9.04/5.13                   => ( ( Y
% 9.04/5.13                        = ( some_num @ ( bit1 @ N2 ) ) )
% 9.04/5.13                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 9.04/5.13             => ( ( ( X = one )
% 9.04/5.13                 => ! [N2: num] :
% 9.04/5.13                      ( ( Xa
% 9.04/5.13                        = ( bit1 @ N2 ) )
% 9.04/5.13                     => ( ( Y
% 9.04/5.13                          = ( some_num @ ( bit0 @ N2 ) ) )
% 9.04/5.13                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 9.04/5.13               => ( ! [M5: num] :
% 9.04/5.13                      ( ( X
% 9.04/5.13                        = ( bit0 @ M5 ) )
% 9.04/5.13                     => ( ( Xa = one )
% 9.04/5.13                       => ( ( Y
% 9.04/5.13                            = ( some_num @ ( bit1 @ M5 ) ) )
% 9.04/5.13                         => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 9.04/5.13                 => ( ! [M5: num] :
% 9.04/5.13                        ( ( X
% 9.04/5.13                          = ( bit0 @ M5 ) )
% 9.04/5.13                       => ! [N2: num] :
% 9.04/5.13                            ( ( Xa
% 9.04/5.13                              = ( bit0 @ N2 ) )
% 9.04/5.13                           => ( ( Y
% 9.04/5.13                                = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) )
% 9.04/5.13                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 9.04/5.13                   => ( ! [M5: num] :
% 9.04/5.13                          ( ( X
% 9.04/5.13                            = ( bit0 @ M5 ) )
% 9.04/5.13                         => ! [N2: num] :
% 9.04/5.13                              ( ( Xa
% 9.04/5.13                                = ( bit1 @ N2 ) )
% 9.04/5.13                             => ( ( Y
% 9.04/5.13                                  = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) )
% 9.04/5.13                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 9.04/5.13                     => ( ! [M5: num] :
% 9.04/5.13                            ( ( X
% 9.04/5.13                              = ( bit1 @ M5 ) )
% 9.04/5.13                           => ( ( Xa = one )
% 9.04/5.13                             => ( ( Y
% 9.04/5.13                                  = ( some_num @ ( bit0 @ M5 ) ) )
% 9.04/5.13                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 9.04/5.13                       => ( ! [M5: num] :
% 9.04/5.13                              ( ( X
% 9.04/5.13                                = ( bit1 @ M5 ) )
% 9.04/5.13                             => ! [N2: num] :
% 9.04/5.13                                  ( ( Xa
% 9.04/5.13                                    = ( bit0 @ N2 ) )
% 9.04/5.13                                 => ( ( Y
% 9.04/5.13                                      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) ) )
% 9.04/5.13                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 9.04/5.13                         => ~ ! [M5: num] :
% 9.04/5.13                                ( ( X
% 9.04/5.13                                  = ( bit1 @ M5 ) )
% 9.04/5.13                               => ! [N2: num] :
% 9.04/5.13                                    ( ( Xa
% 9.04/5.13                                      = ( bit1 @ N2 ) )
% 9.04/5.13                                   => ( ( Y
% 9.04/5.13                                        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N2 ) ) )
% 9.04/5.13                                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10164_Bit__Operations_Otake__bit__num__code,axiom,
% 9.04/5.13      ( bit_take_bit_num
% 9.04/5.13      = ( ^ [N3: nat,M6: num] :
% 9.04/5.13            ( produc478579273971653890on_num
% 9.04/5.13            @ ^ [A3: nat,X4: num] :
% 9.04/5.13                ( case_nat_option_num @ none_num
% 9.04/5.13                @ ^ [O: nat] :
% 9.04/5.13                    ( case_num_option_num @ ( some_num @ one )
% 9.04/5.13                    @ ^ [P4: num] :
% 9.04/5.13                        ( case_o6005452278849405969um_num @ none_num
% 9.04/5.13                        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 9.04/5.13                        @ ( bit_take_bit_num @ O @ P4 ) )
% 9.04/5.13                    @ ^ [P4: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P4 ) ) )
% 9.04/5.13                    @ X4 )
% 9.04/5.13                @ A3 )
% 9.04/5.13            @ ( product_Pair_nat_num @ N3 @ M6 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10165_less__eq,axiom,
% 9.04/5.13      ! [M: nat,N: nat] :
% 9.04/5.13        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 9.04/5.13        = ( ord_less_nat @ M @ N ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10166_pow_Osimps_I3_J,axiom,
% 9.04/5.13      ! [X: num,Y: num] :
% 9.04/5.13        ( ( pow @ X @ ( bit1 @ Y ) )
% 9.04/5.13        = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10167_sqr__conv__mult,axiom,
% 9.04/5.13      ( sqr
% 9.04/5.13      = ( ^ [X4: num] : ( times_times_num @ X4 @ X4 ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10168_sqr_Osimps_I2_J,axiom,
% 9.04/5.13      ! [N: num] :
% 9.04/5.13        ( ( sqr @ ( bit0 @ N ) )
% 9.04/5.13        = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10169_sqr_Osimps_I1_J,axiom,
% 9.04/5.13      ( ( sqr @ one )
% 9.04/5.13      = one ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10170_pred__nat__trancl__eq__le,axiom,
% 9.04/5.13      ! [M: nat,N: nat] :
% 9.04/5.13        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 9.04/5.13        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10171_pow_Osimps_I2_J,axiom,
% 9.04/5.13      ! [X: num,Y: num] :
% 9.04/5.13        ( ( pow @ X @ ( bit0 @ Y ) )
% 9.04/5.13        = ( sqr @ ( pow @ X @ Y ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10172_sqr_Osimps_I3_J,axiom,
% 9.04/5.13      ! [N: num] :
% 9.04/5.13        ( ( sqr @ ( bit1 @ N ) )
% 9.04/5.13        = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10173_rat__floor__lemma,axiom,
% 9.04/5.13      ! [A: int,B: int] :
% 9.04/5.13        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 9.04/5.13        & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10174_divide__rat,axiom,
% 9.04/5.13      ! [A: int,B: int,C: int,D: int] :
% 9.04/5.13        ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 9.04/5.13        = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10175_floor__Fract,axiom,
% 9.04/5.13      ! [A: int,B: int] :
% 9.04/5.13        ( ( archim3151403230148437115or_rat @ ( fract @ A @ B ) )
% 9.04/5.13        = ( divide_divide_int @ A @ B ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10176_diff__rat,axiom,
% 9.04/5.13      ! [B: int,D: int,A: int,C: int] :
% 9.04/5.13        ( ( B != zero_zero_int )
% 9.04/5.13       => ( ( D != zero_zero_int )
% 9.04/5.13         => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 9.04/5.13            = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10177_less__rat,axiom,
% 9.04/5.13      ! [B: int,D: int,A: int,C: int] :
% 9.04/5.13        ( ( B != zero_zero_int )
% 9.04/5.13       => ( ( D != zero_zero_int )
% 9.04/5.13         => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 9.04/5.13            = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10178_add__rat,axiom,
% 9.04/5.13      ! [B: int,D: int,A: int,C: int] :
% 9.04/5.13        ( ( B != zero_zero_int )
% 9.04/5.13       => ( ( D != zero_zero_int )
% 9.04/5.13         => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 9.04/5.13            = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10179_le__rat,axiom,
% 9.04/5.13      ! [B: int,D: int,A: int,C: int] :
% 9.04/5.13        ( ( B != zero_zero_int )
% 9.04/5.13       => ( ( D != zero_zero_int )
% 9.04/5.13         => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 9.04/5.13            = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10180_sgn__rat,axiom,
% 9.04/5.13      ! [A: int,B: int] :
% 9.04/5.13        ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
% 9.04/5.13        = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10181_rat__number__collapse_I3_J,axiom,
% 9.04/5.13      ! [W: num] :
% 9.04/5.13        ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 9.04/5.13        = ( numeral_numeral_rat @ W ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10182_rat__number__expand_I3_J,axiom,
% 9.04/5.13      ( numeral_numeral_rat
% 9.04/5.13      = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10183_Fract__of__int__eq,axiom,
% 9.04/5.13      ! [K: int] :
% 9.04/5.13        ( ( fract @ K @ one_one_int )
% 9.04/5.13        = ( ring_1_of_int_rat @ K ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10184_Rat__induct__pos,axiom,
% 9.04/5.13      ! [P: rat > $o,Q3: rat] :
% 9.04/5.13        ( ! [A5: int,B6: int] :
% 9.04/5.13            ( ( ord_less_int @ zero_zero_int @ B6 )
% 9.04/5.13           => ( P @ ( fract @ A5 @ B6 ) ) )
% 9.04/5.13       => ( P @ Q3 ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10185_Fract__coprime,axiom,
% 9.04/5.13      ! [A: int,B: int] :
% 9.04/5.13        ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( divide_divide_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
% 9.04/5.13        = ( fract @ A @ B ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10186_rat__number__collapse_I6_J,axiom,
% 9.04/5.13      ! [K: int] :
% 9.04/5.13        ( ( fract @ K @ zero_zero_int )
% 9.04/5.13        = zero_zero_rat ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10187_rat__number__collapse_I1_J,axiom,
% 9.04/5.13      ! [K: int] :
% 9.04/5.13        ( ( fract @ zero_zero_int @ K )
% 9.04/5.13        = zero_zero_rat ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10188_eq__rat_I3_J,axiom,
% 9.04/5.13      ! [A: int,C: int] :
% 9.04/5.13        ( ( fract @ zero_zero_int @ A )
% 9.04/5.13        = ( fract @ zero_zero_int @ C ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10189_mult__rat__cancel,axiom,
% 9.04/5.13      ! [C: int,A: int,B: int] :
% 9.04/5.13        ( ( C != zero_zero_int )
% 9.04/5.13       => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.04/5.13          = ( fract @ A @ B ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10190_eq__rat_I1_J,axiom,
% 9.04/5.13      ! [B: int,D: int,A: int,C: int] :
% 9.04/5.13        ( ( B != zero_zero_int )
% 9.04/5.13       => ( ( D != zero_zero_int )
% 9.04/5.13         => ( ( ( fract @ A @ B )
% 9.04/5.13              = ( fract @ C @ D ) )
% 9.04/5.13            = ( ( times_times_int @ A @ D )
% 9.04/5.13              = ( times_times_int @ C @ B ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10191_eq__rat_I2_J,axiom,
% 9.04/5.13      ! [A: int] :
% 9.04/5.13        ( ( fract @ A @ zero_zero_int )
% 9.04/5.13        = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10192_One__rat__def,axiom,
% 9.04/5.13      ( one_one_rat
% 9.04/5.13      = ( fract @ one_one_int @ one_one_int ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10193_Fract__of__int__quotient,axiom,
% 9.04/5.13      ( fract
% 9.04/5.13      = ( ^ [K2: int,L3: int] : ( divide_divide_rat @ ( ring_1_of_int_rat @ K2 ) @ ( ring_1_of_int_rat @ L3 ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10194_Fract__of__nat__eq,axiom,
% 9.04/5.13      ! [K: nat] :
% 9.04/5.13        ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
% 9.04/5.13        = ( semiri681578069525770553at_rat @ K ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10195_Zero__rat__def,axiom,
% 9.04/5.13      ( zero_zero_rat
% 9.04/5.13      = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10196_Fract__less__zero__iff,axiom,
% 9.04/5.13      ! [B: int,A: int] :
% 9.04/5.13        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.13       => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 9.04/5.13          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10197_zero__less__Fract__iff,axiom,
% 9.04/5.13      ! [B: int,A: int] :
% 9.04/5.13        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.13       => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 9.04/5.13          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10198_one__less__Fract__iff,axiom,
% 9.04/5.13      ! [B: int,A: int] :
% 9.04/5.13        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.13       => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 9.04/5.13          = ( ord_less_int @ B @ A ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10199_Fract__less__one__iff,axiom,
% 9.04/5.13      ! [B: int,A: int] :
% 9.04/5.13        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.13       => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 9.04/5.13          = ( ord_less_int @ A @ B ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10200_rat__number__collapse_I5_J,axiom,
% 9.04/5.13      ( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 9.04/5.13      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10201_Fract__add__one,axiom,
% 9.04/5.13      ! [N: int,M: int] :
% 9.04/5.13        ( ( N != zero_zero_int )
% 9.04/5.13       => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 9.04/5.13          = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10202_Fract__le__zero__iff,axiom,
% 9.04/5.13      ! [B: int,A: int] :
% 9.04/5.13        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.13       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 9.04/5.13          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10203_zero__le__Fract__iff,axiom,
% 9.04/5.13      ! [B: int,A: int] :
% 9.04/5.13        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.13       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 9.04/5.13          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10204_one__le__Fract__iff,axiom,
% 9.04/5.13      ! [B: int,A: int] :
% 9.04/5.13        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.13       => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 9.04/5.13          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10205_Fract__le__one__iff,axiom,
% 9.04/5.13      ! [B: int,A: int] :
% 9.04/5.13        ( ( ord_less_int @ zero_zero_int @ B )
% 9.04/5.13       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 9.04/5.13          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10206_rat__number__expand_I5_J,axiom,
% 9.04/5.13      ! [K: num] :
% 9.04/5.13        ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 9.04/5.13        = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10207_rat__number__collapse_I4_J,axiom,
% 9.04/5.13      ! [W: num] :
% 9.04/5.13        ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 9.04/5.13        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10208_of__nat__eq__id,axiom,
% 9.04/5.13      semiri1316708129612266289at_nat = id_nat ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10209_less__eq__int__def,axiom,
% 9.04/5.13      ( ord_less_eq_int
% 9.04/5.13      = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 9.04/5.13        @ ( produc8739625826339149834_nat_o
% 9.04/5.13          @ ^ [X4: nat,Y6: nat] :
% 9.04/5.13              ( produc6081775807080527818_nat_o
% 9.04/5.13              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y6 ) ) ) ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(fact_10210_nat__def,axiom,
% 9.04/5.13      ( nat2
% 9.04/5.13      = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 9.04/5.13      ! [X: int,Y: int] :
% 9.04/5.13        ( ( if_int @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 9.04/5.13      ! [X: int,Y: int] :
% 9.04/5.13        ( ( if_int @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 9.04/5.13      ! [X: nat,Y: nat] :
% 9.04/5.13        ( ( if_nat @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 9.04/5.13      ! [X: nat,Y: nat] :
% 9.04/5.13        ( ( if_nat @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 9.04/5.13      ! [X: rat,Y: rat] :
% 9.04/5.13        ( ( if_rat @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 9.04/5.13      ! [X: rat,Y: rat] :
% 9.04/5.13        ( ( if_rat @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 9.04/5.13      ! [X: real,Y: real] :
% 9.04/5.13        ( ( if_real @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 9.04/5.13      ! [X: real,Y: real] :
% 9.04/5.13        ( ( if_real @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 9.04/5.13      ! [P: real > $o] :
% 9.04/5.13        ( ( P @ ( fChoice_real @ P ) )
% 9.04/5.13        = ( ? [X6: real] : ( P @ X6 ) ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 9.04/5.13      ! [X: complex,Y: complex] :
% 9.04/5.13        ( ( if_complex @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 9.04/5.13      ! [X: complex,Y: complex] :
% 9.04/5.13        ( ( if_complex @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 9.04/5.13      ! [X: code_integer,Y: code_integer] :
% 9.04/5.13        ( ( if_Code_integer @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 9.04/5.13      ! [X: code_integer,Y: code_integer] :
% 9.04/5.13        ( ( if_Code_integer @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 9.04/5.13      ! [X: set_int,Y: set_int] :
% 9.04/5.13        ( ( if_set_int @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 9.04/5.13      ! [X: set_int,Y: set_int] :
% 9.04/5.13        ( ( if_set_int @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 9.04/5.13      ! [X: list_int,Y: list_int] :
% 9.04/5.13        ( ( if_list_int @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 9.04/5.13      ! [X: list_int,Y: list_int] :
% 9.04/5.13        ( ( if_list_int @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.04/5.13      ! [X: int > int,Y: int > int] :
% 9.04/5.13        ( ( if_int_int @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.04/5.13      ! [X: int > int,Y: int > int] :
% 9.04/5.13        ( ( if_int_int @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 9.04/5.13      ! [X: option_num,Y: option_num] :
% 9.04/5.13        ( ( if_option_num @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 9.04/5.13      ! [X: option_num,Y: option_num] :
% 9.04/5.13        ( ( if_option_num @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.04/5.13      ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 9.04/5.13        ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.04/5.13      ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 9.04/5.13        ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 9.04/5.13      ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 9.04/5.13        ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 9.04/5.13      ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 9.04/5.13        ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 9.04/5.13      ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 9.04/5.13        ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 9.04/5.13      ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 9.04/5.13        ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.04/5.13      ! [P: $o] :
% 9.04/5.13        ( ( P = $true )
% 9.04/5.13        | ( P = $false ) ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.04/5.13      ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 9.04/5.13        ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
% 9.04/5.13        = Y ) ).
% 9.04/5.13  
% 9.04/5.13  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.04/5.13      ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 9.04/5.13        ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
% 9.04/5.13        = X ) ).
% 9.04/5.13  
% 9.04/5.13  thf(conj_0,conjecture,
% 9.04/5.13      ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ ( vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary ) ) )
% 9.04/5.13      = ( plus_plus_int @ one_one_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ m ) ) ) ) ) ).
% 9.04/5.13  ------------------------
% 9.06/5.59  % (32447)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_32338 for (2995ds/4Mi)
% 9.06/5.60  % (32447)Instruction limit reached!
% 9.06/5.60  % (32447)------------------------------
% 9.06/5.60  % (32447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.60  % (32447)Termination reason: Unknown
% 9.06/5.60  % (32447)Termination phase: shuffling
% 9.06/5.60  
% 9.06/5.60  % (32447)Memory used [KB]: 10874
% 9.06/5.60  % (32447)Time elapsed: 0.005 s
% 9.06/5.60  % (32447)Instructions burned: 4 (million)
% 9.06/5.60  % (32447)------------------------------
% 9.06/5.60  % (32447)------------------------------
% 9.06/5.60  % (32446)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32338 for (2995ds/183Mi)
% 9.06/5.60  % (32451)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_32338 for (2995ds/2Mi)
% 9.06/5.60  % (32449)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_32338 for (2995ds/27Mi)
% 9.06/5.60  % (32451)Instruction limit reached!
% 9.06/5.60  % (32451)------------------------------
% 9.06/5.60  % (32451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.60  % (32451)Termination reason: Unknown
% 9.06/5.60  % (32451)Termination phase: shuffling
% 9.06/5.60  
% 9.06/5.60  % (32451)Memory used [KB]: 10618
% 9.06/5.60  % (32451)Time elapsed: 0.003 s
% 9.06/5.60  % (32451)Instructions burned: 2 (million)
% 9.06/5.60  % (32451)------------------------------
% 9.06/5.60  % (32451)------------------------------
% 9.06/5.60  % (32450)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32338 for (2995ds/2Mi)
% 9.06/5.60  % (32452)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_32338 for (2995ds/275Mi)
% 9.06/5.61  % (32450)Instruction limit reached!
% 9.06/5.61  % (32450)------------------------------
% 9.06/5.61  % (32450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.61  % (32450)Termination reason: Unknown
% 9.06/5.61  % (32450)Termination phase: shuffling
% 9.06/5.61  
% 9.06/5.61  % (32450)Memory used [KB]: 10746
% 9.06/5.61  % (32450)Time elapsed: 0.003 s
% 9.06/5.61  % (32450)Instructions burned: 2 (million)
% 9.06/5.61  % (32450)------------------------------
% 9.06/5.61  % (32450)------------------------------
% 9.06/5.61  % (32454)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32338 for (2995ds/3Mi)
% 9.06/5.61  % (32454)Instruction limit reached!
% 9.06/5.61  % (32454)------------------------------
% 9.06/5.61  % (32454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.61  % (32454)Termination reason: Unknown
% 9.06/5.61  % (32454)Termination phase: shuffling
% 9.06/5.61  
% 9.06/5.61  % (32454)Memory used [KB]: 10746
% 9.06/5.61  % (32454)Time elapsed: 0.004 s
% 9.06/5.61  % (32454)Instructions burned: 4 (million)
% 9.06/5.61  % (32454)------------------------------
% 9.06/5.61  % (32454)------------------------------
% 9.06/5.62  % (32453)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32338 for (2995ds/18Mi)
% 9.06/5.62  % (32455)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_32338 for (2995ds/37Mi)
% 9.06/5.62  % (32457)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_32338 for (2995ds/3Mi)
% 9.06/5.62  % (32456)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_32338 for (2995ds/15Mi)
% 9.06/5.63  % (32457)Instruction limit reached!
% 9.06/5.63  % (32457)------------------------------
% 9.06/5.63  % (32457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.63  % (32457)Termination reason: Unknown
% 9.06/5.63  % (32457)Termination phase: shuffling
% 9.06/5.63  
% 9.06/5.63  % (32457)Memory used [KB]: 10746
% 9.06/5.63  % (32457)Time elapsed: 0.004 s
% 9.06/5.63  % (32457)Instructions burned: 3 (million)
% 9.06/5.63  % (32457)------------------------------
% 9.06/5.63  % (32457)------------------------------
% 9.06/5.63  % (32449)Instruction limit reached!
% 9.06/5.63  % (32449)------------------------------
% 9.06/5.63  % (32449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.63  % (32449)Termination reason: Unknown
% 9.06/5.63  % (32449)Termination phase: shuffling
% 9.06/5.63  
% 9.06/5.63  % (32449)Memory used [KB]: 12537
% 9.06/5.63  % (32449)Time elapsed: 0.028 s
% 9.06/5.63  % (32449)Instructions burned: 27 (million)
% 9.06/5.63  % (32449)------------------------------
% 9.06/5.63  % (32449)------------------------------
% 9.06/5.64  % (32458)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_32338 for (2995ds/1041Mi)
% 9.06/5.64  % (32453)Instruction limit reached!
% 9.06/5.64  % (32453)------------------------------
% 9.06/5.64  % (32453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.64  % (32453)Termination reason: Unknown
% 9.06/5.64  % (32453)Termination phase: shuffling
% 9.06/5.64  
% 9.06/5.64  % (32453)Memory used [KB]: 11513
% 9.06/5.64  % (32453)Time elapsed: 0.024 s
% 9.06/5.64  % (32453)Instructions burned: 19 (million)
% 9.06/5.64  % (32453)------------------------------
% 9.06/5.64  % (32453)------------------------------
% 9.06/5.64  % (32456)Instruction limit reached!
% 9.06/5.64  % (32456)------------------------------
% 9.06/5.64  % (32456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.64  % (32456)Termination reason: Unknown
% 9.06/5.64  % (32456)Termination phase: shuffling
% 9.06/5.64  
% 9.06/5.64  % (32456)Memory used [KB]: 11513
% 9.06/5.64  % (32456)Time elapsed: 0.020 s
% 9.06/5.64  % (32456)Instructions burned: 15 (million)
% 9.06/5.64  % (32456)------------------------------
% 9.06/5.64  % (32456)------------------------------
% 9.06/5.65  % (32455)Instruction limit reached!
% 9.06/5.65  % (32455)------------------------------
% 9.06/5.65  % (32455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.65  % (32455)Termination reason: Unknown
% 9.06/5.65  % (32455)Termination phase: shuffling
% 9.06/5.65  
% 9.06/5.65  % (32455)Memory used [KB]: 12792
% 9.06/5.65  % (32455)Time elapsed: 0.028 s
% 9.06/5.65  % (32455)Instructions burned: 38 (million)
% 9.06/5.65  % (32455)------------------------------
% 9.06/5.65  % (32455)------------------------------
% 9.06/5.65  % (32459)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32338 for (2995ds/7Mi)
% 9.06/5.66  % (32462)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_32338 for (2994ds/3Mi)
% 9.06/5.66  % (32459)Instruction limit reached!
% 9.06/5.66  % (32459)------------------------------
% 9.06/5.66  % (32459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.66  % (32459)Termination reason: Unknown
% 9.06/5.66  % (32459)Termination phase: shuffling
% 9.06/5.66  
% 9.06/5.66  % (32459)Memory used [KB]: 11129
% 9.06/5.66  % (32459)Time elapsed: 0.012 s
% 9.06/5.66  % (32459)Instructions burned: 7 (million)
% 9.06/5.66  % (32459)------------------------------
% 9.06/5.66  % (32459)------------------------------
% 9.06/5.66  % (32460)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32338 for (2995ds/16Mi)
% 9.06/5.66  % (32462)Instruction limit reached!
% 9.06/5.66  % (32462)------------------------------
% 9.06/5.66  % (32462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.66  % (32462)Termination reason: Unknown
% 9.06/5.66  % (32462)Termination phase: shuffling
% 9.06/5.66  
% 9.06/5.66  % (32462)Memory used [KB]: 10746
% 9.06/5.66  % (32462)Time elapsed: 0.003 s
% 9.06/5.66  % (32462)Instructions burned: 3 (million)
% 9.06/5.66  % (32462)------------------------------
% 9.06/5.66  % (32462)------------------------------
% 9.06/5.66  % (32461)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_32338 for (2995ds/3Mi)
% 9.06/5.67  % (32461)Instruction limit reached!
% 9.06/5.67  % (32461)------------------------------
% 9.06/5.67  % (32461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.67  % (32461)Termination reason: Unknown
% 9.06/5.67  % (32461)Termination phase: shuffling
% 9.06/5.67  
% 9.06/5.67  % (32461)Memory used [KB]: 10746
% 9.06/5.67  % (32461)Time elapsed: 0.007 s
% 9.06/5.67  % (32461)Instructions burned: 3 (million)
% 9.06/5.67  % (32461)------------------------------
% 9.06/5.67  % (32461)------------------------------
% 9.06/5.67  % (32464)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_32338 for (2994ds/3Mi)
% 9.06/5.67  % (32464)Instruction limit reached!
% 9.06/5.67  % (32464)------------------------------
% 9.06/5.67  % (32464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.67  % (32464)Termination reason: Unknown
% 9.06/5.67  % (32464)Termination phase: shuffling
% 9.06/5.67  
% 9.06/5.67  % (32464)Memory used [KB]: 10746
% 9.06/5.67  % (32464)Time elapsed: 0.003 s
% 9.06/5.67  % (32464)Instructions burned: 3 (million)
% 9.06/5.67  % (32464)------------------------------
% 9.06/5.67  % (32464)------------------------------
% 9.06/5.68  % (32463)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_32338 for (2994ds/7Mi)
% 9.06/5.68  % (32460)Instruction limit reached!
% 9.06/5.68  % (32460)------------------------------
% 9.06/5.68  % (32460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.68  % (32460)Termination reason: Unknown
% 9.06/5.68  % (32460)Termination phase: shuffling
% 9.06/5.68  
% 9.06/5.68  % (32460)Memory used [KB]: 11513
% 9.06/5.68  % (32460)Time elapsed: 0.022 s
% 9.06/5.68  % (32460)Instructions burned: 17 (million)
% 9.06/5.68  % (32460)------------------------------
% 9.06/5.68  % (32460)------------------------------
% 9.06/5.68  % (32466)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_32338 for (2994ds/18Mi)
% 9.06/5.68  % (32465)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_32338 for (2994ds/4Mi)
% 9.06/5.69  % (32465)Instruction limit reached!
% 9.06/5.69  % (32465)------------------------------
% 9.06/5.69  % (32465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.69  % (32465)Termination reason: Unknown
% 9.06/5.69  % (32465)Termination phase: shuffling
% 9.06/5.69  
% 9.06/5.69  % (32465)Memory used [KB]: 10874
% 9.06/5.69  % (32465)Time elapsed: 0.005 s
% 9.06/5.69  % (32463)Instruction limit reached!
% 9.06/5.69  % (32463)------------------------------
% 9.06/5.69  % (32463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.69  % (32465)Instructions burned: 4 (million)
% 9.06/5.69  % (32465)------------------------------
% 9.06/5.69  % (32465)------------------------------
% 9.06/5.69  % (32463)Termination reason: Unknown
% 9.06/5.69  % (32463)Termination phase: shuffling
% 9.06/5.69  
% 9.06/5.69  % (32463)Memory used [KB]: 11001
% 9.06/5.69  % (32463)Time elapsed: 0.012 s
% 9.06/5.69  % (32463)Instructions burned: 7 (million)
% 9.06/5.69  % (32463)------------------------------
% 9.06/5.69  % (32463)------------------------------
% 9.06/5.69  % (32466)Instruction limit reached!
% 9.06/5.69  % (32466)------------------------------
% 9.06/5.69  % (32466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.69  % (32466)Termination reason: Unknown
% 9.06/5.69  % (32466)Termination phase: shuffling
% 9.06/5.69  
% 9.06/5.69  % (32466)Memory used [KB]: 11513
% 9.06/5.69  % (32466)Time elapsed: 0.014 s
% 9.06/5.69  % (32466)Instructions burned: 21 (million)
% 9.06/5.69  % (32466)------------------------------
% 9.06/5.69  % (32466)------------------------------
% 9.06/5.70  % (32467)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_32338 for (2994ds/710Mi)
% 9.06/5.70  % (32468)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_32338 for (2994ds/6Mi)
% 9.06/5.70  % (32468)Instruction limit reached!
% 9.06/5.70  % (32468)------------------------------
% 9.06/5.70  % (32468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.70  % (32468)Termination reason: Unknown
% 9.06/5.70  % (32468)Termination phase: shuffling
% 9.06/5.70  
% 9.06/5.70  % (32468)Memory used [KB]: 11001
% 9.06/5.70  % (32468)Time elapsed: 0.006 s
% 9.06/5.70  % (32468)Instructions burned: 6 (million)
% 9.06/5.70  % (32468)------------------------------
% 9.06/5.70  % (32468)------------------------------
% 9.06/5.71  % (32470)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_32338 for (2994ds/21Mi)
% 9.06/5.71  % (32469)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_32338 for (2994ds/902Mi)
% 9.06/5.71  % (32471)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_32338 for (2994ds/5Mi)
% 9.06/5.72  % (32446)Instruction limit reached!
% 9.06/5.72  % (32446)------------------------------
% 9.06/5.72  % (32446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.72  % (32446)Termination reason: Unknown
% 9.06/5.72  % (32446)Termination phase: shuffling
% 9.06/5.72  
% 9.06/5.72  % (32446)Memory used [KB]: 14839
% 9.06/5.72  % (32446)Time elapsed: 0.117 s
% 9.06/5.72  % (32446)Instructions burned: 184 (million)
% 9.06/5.72  % (32446)------------------------------
% 9.06/5.72  % (32446)------------------------------
% 9.06/5.72  % (32471)Instruction limit reached!
% 9.06/5.72  % (32471)------------------------------
% 9.06/5.72  % (32471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.72  % (32471)Termination reason: Unknown
% 9.06/5.72  % (32471)Termination phase: shuffling
% 9.06/5.72  
% 9.06/5.72  % (32471)Memory used [KB]: 11001
% 9.06/5.72  % (32471)Time elapsed: 0.006 s
% 9.06/5.72  % (32471)Instructions burned: 5 (million)
% 9.06/5.72  % (32471)------------------------------
% 9.06/5.72  % (32471)------------------------------
% 9.06/5.72  % (32470)Instruction limit reached!
% 9.06/5.72  % (32470)------------------------------
% 9.06/5.72  % (32470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.72  % (32470)Termination reason: Unknown
% 9.06/5.72  % (32470)Termination phase: shuffling
% 9.06/5.72  
% 9.06/5.72  % (32470)Memory used [KB]: 11513
% 9.06/5.72  % (32470)Time elapsed: 0.015 s
% 9.06/5.72  % (32470)Instructions burned: 22 (million)
% 9.06/5.72  % (32470)------------------------------
% 9.06/5.72  % (32470)------------------------------
% 9.06/5.73  % (32472)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_32338 for (2994ds/6Mi)
% 9.06/5.73  % (32473)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_32338 for (2994ds/377Mi)
% 9.06/5.73  % (32474)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_32338 for (2994ds/779Mi)
% 9.06/5.73  % (32472)Instruction limit reached!
% 9.06/5.73  % (32472)------------------------------
% 9.06/5.73  % (32472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.73  % (32472)Termination reason: Unknown
% 9.06/5.73  % (32472)Termination phase: shuffling
% 9.06/5.73  
% 9.06/5.73  % (32472)Memory used [KB]: 11001
% 9.06/5.73  % (32472)Time elapsed: 0.007 s
% 9.06/5.73  % (32472)Instructions burned: 6 (million)
% 9.06/5.73  % (32472)------------------------------
% 9.06/5.73  % (32472)------------------------------
% 9.06/5.74  % (32475)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_32338 for (2994ds/19Mi)
% 9.06/5.76  % (32475)Instruction limit reached!
% 9.06/5.76  % (32475)------------------------------
% 9.06/5.76  % (32475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.76  % (32475)Termination reason: Unknown
% 9.06/5.76  % (32475)Termination phase: shuffling
% 9.06/5.76  
% 9.06/5.76  % (32475)Memory used [KB]: 11513
% 9.06/5.76  % (32475)Time elapsed: 0.014 s
% 9.06/5.76  % (32475)Instructions burned: 19 (million)
% 9.06/5.76  % (32475)------------------------------
% 9.06/5.76  % (32475)------------------------------
% 9.06/5.77  % (32476)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_32338 for (2993ds/879Mi)
% 9.06/5.77  % (32452)Instruction limit reached!
% 9.06/5.77  % (32452)------------------------------
% 9.06/5.77  % (32452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.77  % (32452)Termination reason: Unknown
% 9.06/5.77  % (32452)Termination phase: shuffling
% 9.06/5.77  
% 9.06/5.77  % (32452)Memory used [KB]: 16247
% 9.06/5.77  % (32452)Time elapsed: 0.170 s
% 9.06/5.77  % (32452)Instructions burned: 276 (million)
% 9.06/5.77  % (32452)------------------------------
% 9.06/5.77  % (32452)------------------------------
% 9.06/5.79  % (32477)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_32338 for (2993ds/17Mi)
% 9.06/5.81  % (32477)Instruction limit reached!
% 9.06/5.81  % (32477)------------------------------
% 9.06/5.81  % (32477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.81  % (32477)Termination reason: Unknown
% 9.06/5.81  % (32477)Termination phase: shuffling
% 9.06/5.81  
% 9.06/5.81  % (32477)Memory used [KB]: 11513
% 9.06/5.81  % (32477)Time elapsed: 0.023 s
% 9.06/5.81  % (32477)Instructions burned: 17 (million)
% 9.06/5.81  % (32477)------------------------------
% 9.06/5.81  % (32477)------------------------------
% 9.06/5.84  % (32478)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_32338 for (2993ds/3Mi)
% 9.06/5.84  % (32478)Instruction limit reached!
% 9.06/5.84  % (32478)------------------------------
% 9.06/5.84  % (32478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.84  % (32478)Termination reason: Unknown
% 9.06/5.84  % (32478)Termination phase: shuffling
% 9.06/5.84  
% 9.06/5.84  % (32478)Memory used [KB]: 10746
% 9.06/5.84  % (32478)Time elapsed: 0.005 s
% 9.06/5.84  % (32478)Instructions burned: 3 (million)
% 9.06/5.84  % (32478)------------------------------
% 9.06/5.84  % (32478)------------------------------
% 9.06/5.86  % (32479)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_32338 for (2993ds/30Mi)
% 9.06/5.87  % (32473)Instruction limit reached!
% 9.06/5.87  % (32473)------------------------------
% 9.06/5.87  % (32473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.87  % (32473)Termination reason: Unknown
% 9.06/5.87  % (32473)Termination phase: shuffling
% 9.06/5.87  
% 9.06/5.87  % (32473)Memory used [KB]: 17526
% 9.06/5.87  % (32473)Time elapsed: 0.139 s
% 9.06/5.87  % (32473)Instructions burned: 378 (million)
% 9.06/5.87  % (32473)------------------------------
% 9.06/5.87  % (32473)------------------------------
% 9.06/5.88  % (32480)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_32338 for (2992ds/127Mi)
% 9.06/5.88  % (32479)Instruction limit reached!
% 9.06/5.88  % (32479)------------------------------
% 9.06/5.88  % (32479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.88  % (32479)Termination reason: Unknown
% 9.06/5.88  % (32479)Termination phase: shuffling
% 9.06/5.88  
% 9.06/5.88  % (32479)Memory used [KB]: 12665
% 9.06/5.88  % (32479)Time elapsed: 0.023 s
% 9.06/5.88  % (32479)Instructions burned: 30 (million)
% 9.06/5.88  % (32479)------------------------------
% 9.06/5.88  % (32479)------------------------------
% 9.06/5.89  % (32481)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_32338 for (2992ds/100Mi)
% 9.06/5.93  % (32480)Instruction limit reached!
% 9.06/5.93  % (32480)------------------------------
% 9.06/5.93  % (32480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.93  % (32480)Termination reason: Unknown
% 9.06/5.93  % (32480)Termination phase: shuffling
% 9.06/5.93  
% 9.06/5.93  % (32480)Memory used [KB]: 14072
% 9.06/5.93  % (32480)Time elapsed: 0.050 s
% 9.06/5.93  % (32480)Instructions burned: 127 (million)
% 9.06/5.93  % (32480)------------------------------
% 9.06/5.93  % (32480)------------------------------
% 9.06/5.93  % (32481)Instruction limit reached!
% 9.06/5.93  % (32481)------------------------------
% 9.06/5.93  % (32481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.93  % (32481)Termination reason: Unknown
% 9.06/5.93  % (32481)Termination phase: shuffling
% 9.06/5.93  
% 9.06/5.93  % (32481)Memory used [KB]: 13688
% 9.06/5.93  % (32481)Time elapsed: 0.043 s
% 9.06/5.93  % (32481)Instructions burned: 101 (million)
% 9.06/5.93  % (32481)------------------------------
% 9.06/5.93  % (32481)------------------------------
% 9.06/5.94  % (32482)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_32338 for (2992ds/3Mi)
% 9.06/5.94  % (32482)Instruction limit reached!
% 9.06/5.94  % (32482)------------------------------
% 9.06/5.94  % (32482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.94  % (32482)Termination reason: Unknown
% 9.06/5.94  % (32482)Termination phase: shuffling
% 9.06/5.94  
% 9.06/5.94  % (32482)Memory used [KB]: 10746
% 9.06/5.94  % (32482)Time elapsed: 0.004 s
% 9.06/5.94  % (32482)Instructions burned: 3 (million)
% 9.06/5.94  % (32482)------------------------------
% 9.06/5.94  % (32482)------------------------------
% 9.06/5.94  % (32483)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_32338 for (2992ds/20Mi)
% 9.06/5.95  % (32484)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_32338 for (2992ds/86Mi)
% 9.06/5.95  % (32458)Instruction limit reached!
% 9.06/5.95  % (32458)------------------------------
% 9.06/5.95  % (32458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.95  % (32458)Termination reason: Unknown
% 9.06/5.95  % (32458)Termination phase: shuffling
% 9.06/5.95  
% 9.06/5.95  % (32458)Memory used [KB]: 32877
% 9.06/5.95  % (32458)Time elapsed: 0.319 s
% 9.06/5.95  % (32458)Instructions burned: 1042 (million)
% 9.06/5.95  % (32458)------------------------------
% 9.06/5.95  % (32458)------------------------------
% 9.06/5.96  % (32474)Instruction limit reached!
% 9.06/5.96  % (32474)------------------------------
% 9.06/5.96  % (32474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.96  % (32474)Termination reason: Unknown
% 9.06/5.96  % (32474)Termination phase: shuffling
% 9.06/5.96  
% 9.06/5.96  % (32474)Memory used [KB]: 25330
% 9.06/5.96  % (32474)Time elapsed: 0.227 s
% 9.06/5.96  % (32474)Instructions burned: 781 (million)
% 9.06/5.96  % (32474)------------------------------
% 9.06/5.96  % (32474)------------------------------
% 9.06/5.96  % (32483)Instruction limit reached!
% 9.06/5.96  % (32483)------------------------------
% 9.06/5.96  % (32483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.96  % (32483)Termination reason: Unknown
% 9.06/5.96  % (32483)Termination phase: shuffling
% 9.06/5.96  
% 9.06/5.96  % (32483)Memory used [KB]: 11513
% 9.06/5.96  % (32483)Time elapsed: 0.015 s
% 9.06/5.96  % (32483)Instructions burned: 22 (million)
% 9.06/5.96  % (32483)------------------------------
% 9.06/5.96  % (32483)------------------------------
% 9.06/5.97  % (32486)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_32338 for (2991ds/39Mi)
% 9.06/5.97  % (32487)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_32338 for (2991ds/448Mi)
% 9.06/5.97  % (32485)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_32338 for (2991ds/107Mi)
% 9.06/5.98  % (32467)Instruction limit reached!
% 9.06/5.98  % (32467)------------------------------
% 9.06/5.98  % (32467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.98  % (32467)Termination reason: Unknown
% 9.06/5.98  % (32467)Termination phase: Property scanning
% 9.06/5.98  
% 9.06/5.98  % (32467)Memory used [KB]: 21108
% 9.06/5.98  % (32467)Time elapsed: 0.287 s
% 9.06/5.98  % (32467)Instructions burned: 712 (million)
% 9.06/5.98  % (32467)------------------------------
% 9.06/5.98  % (32467)------------------------------
% 9.06/5.99  % (32484)Instruction limit reached!
% 9.06/5.99  % (32484)------------------------------
% 9.06/5.99  % (32484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.99  % (32484)Termination reason: Unknown
% 9.06/5.99  % (32484)Termination phase: shuffling
% 9.06/5.99  
% 9.06/5.99  % (32484)Memory used [KB]: 13560
% 9.06/5.99  % (32484)Time elapsed: 0.037 s
% 9.06/5.99  % (32484)Instructions burned: 88 (million)
% 9.06/5.99  % (32484)------------------------------
% 9.06/5.99  % (32484)------------------------------
% 9.06/5.99  % (32486)Instruction limit reached!
% 9.06/5.99  % (32486)------------------------------
% 9.06/5.99  % (32486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/5.99  % (32486)Termination reason: Unknown
% 9.06/5.99  % (32486)Termination phase: shuffling
% 9.06/5.99  
% 9.06/5.99  % (32486)Memory used [KB]: 12792
% 9.06/5.99  % (32486)Time elapsed: 0.020 s
% 9.06/5.99  % (32486)Instructions burned: 40 (million)
% 9.06/5.99  % (32486)------------------------------
% 9.06/5.99  % (32486)------------------------------
% 9.06/6.00  % (32488)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_32338 for (2991ds/46Mi)
% 9.06/6.00  % (32489)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_32338 for (2991ds/98Mi)
% 9.06/6.00  % (32490)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_32338 for (2991ds/507Mi)
% 9.06/6.02  % (32485)Instruction limit reached!
% 9.06/6.02  % (32485)------------------------------
% 9.06/6.02  % (32485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/6.02  % (32485)Termination reason: Unknown
% 9.06/6.02  % (32485)Termination phase: shuffling
% 9.06/6.02  
% 9.06/6.02  % (32485)Memory used [KB]: 13816
% 9.06/6.02  % (32485)Time elapsed: 0.049 s
% 9.06/6.02  % (32485)Instructions burned: 109 (million)
% 9.06/6.02  % (32485)------------------------------
% 9.06/6.02  % (32485)------------------------------
% 9.06/6.02  % (32488)Instruction limit reached!
% 9.06/6.02  % (32488)------------------------------
% 9.06/6.02  % (32488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/6.02  % (32488)Termination reason: Unknown
% 9.06/6.02  % (32488)Termination phase: shuffling
% 9.06/6.02  
% 9.06/6.02  % (32488)Memory used [KB]: 12920
% 9.06/6.02  % (32488)Time elapsed: 0.026 s
% 9.06/6.02  % (32488)Instructions burned: 47 (million)
% 9.06/6.02  % (32488)------------------------------
% 9.06/6.02  % (32488)------------------------------
% 12.49/6.03  % (32491)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_32338 for (2991ds/149Mi)
% 12.49/6.03  % (32489)Instruction limit reached!
% 12.49/6.03  % (32489)------------------------------
% 12.49/6.03  % (32489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.49/6.03  % (32489)Termination reason: Unknown
% 12.49/6.03  % (32489)Termination phase: shuffling
% 12.49/6.03  
% 12.49/6.03  % (32489)Memory used [KB]: 13688
% 12.49/6.03  % (32489)Time elapsed: 0.036 s
% 12.49/6.03  % (32489)Instructions burned: 99 (million)
% 12.49/6.03  % (32489)------------------------------
% 12.49/6.03  % (32489)------------------------------
% 12.49/6.03  % (32492)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_32338 for (2991ds/39Mi)
% 12.49/6.04  % (32493)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_32338 for (2990ds/6Mi)
% 12.49/6.05  % (32493)Instruction limit reached!
% 12.49/6.05  % (32493)------------------------------
% 12.49/6.05  % (32493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.49/6.05  % (32493)Termination reason: Unknown
% 12.49/6.05  % (32493)Termination phase: shuffling
% 12.49/6.05  
% 12.49/6.05  % (32493)Memory used [KB]: 11001
% 12.49/6.05  % (32493)Time elapsed: 0.007 s
% 12.49/6.05  % (32493)Instructions burned: 6 (million)
% 12.49/6.05  % (32493)------------------------------
% 12.49/6.05  % (32493)------------------------------
% 12.49/6.05  % (32476)Instruction limit reached!
% 12.49/6.05  % (32476)------------------------------
% 12.49/6.05  % (32476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.49/6.05  % (32469)Instruction limit reached!
% 12.49/6.05  % (32469)------------------------------
% 12.49/6.05  % (32469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.49/6.05  % (32476)Termination reason: Unknown
% 12.49/6.05  % (32476)Termination phase: Property scanning
% 12.49/6.05  
% 12.49/6.05  % (32476)Memory used [KB]: 20980
% 12.49/6.05  % (32476)Time elapsed: 0.286 s
% 12.49/6.05  % (32476)Instructions burned: 880 (million)
% 12.49/6.05  % (32476)------------------------------
% 12.49/6.05  % (32476)------------------------------
% 12.49/6.05  % (32469)Termination reason: Unknown
% 12.49/6.05  % (32469)Termination phase: Property scanning
% 12.49/6.05  
% 12.49/6.05  % (32469)Memory used [KB]: 21108
% 12.49/6.05  % (32469)Time elapsed: 0.332 s
% 12.49/6.05  % (32469)Instructions burned: 902 (million)
% 12.49/6.05  % (32469)------------------------------
% 12.49/6.05  % (32469)------------------------------
% 12.49/6.06  % (32492)Instruction limit reached!
% 12.49/6.06  % (32492)------------------------------
% 12.49/6.06  % (32492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.49/6.06  % (32492)Termination reason: Unknown
% 12.49/6.06  % (32492)Termination phase: shuffling
% 12.49/6.06  
% 12.49/6.06  % (32492)Memory used [KB]: 12792
% 12.49/6.06  % (32492)Time elapsed: 0.027 s
% 12.49/6.06  % (32492)Instructions burned: 41 (million)
% 12.49/6.06  % (32492)------------------------------
% 12.49/6.06  % (32492)------------------------------
% 12.49/6.06  % (32494)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_32338 for (2990ds/1087Mi)
% 12.49/6.06  % (32496)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_32338 for (2990ds/44Mi)
% 12.49/6.06  % (32495)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_32338 for (2990ds/47Mi)
% 12.49/6.07  % (32497)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_32338 for (2990ds/432Mi)
% 12.49/6.08  % (32496)Instruction limit reached!
% 12.49/6.08  % (32496)------------------------------
% 12.49/6.08  % (32496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.49/6.08  % (32496)Termination reason: Unknown
% 12.49/6.08  % (32496)Termination phase: shuffling
% 12.49/6.08  
% 12.49/6.08  % (32496)Memory used [KB]: 12920
% 12.49/6.08  % (32496)Time elapsed: 0.022 s
% 12.49/6.08  % (32496)Instructions burned: 46 (million)
% 12.49/6.08  % (32496)------------------------------
% 12.49/6.08  % (32496)------------------------------
% 12.49/6.09  % (32495)Instruction limit reached!
% 12.49/6.09  % (32495)------------------------------
% 12.49/6.09  % (32495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.49/6.09  % (32495)Termination reason: Unknown
% 12.49/6.09  % (32495)Termination phase: shuffling
% 12.49/6.09  
% 12.49/6.09  % (32495)Memory used [KB]: 12920
% 12.49/6.09  % (32495)Time elapsed: 0.027 s
% 12.49/6.09  % (32495)Instructions burned: 49 (million)
% 12.49/6.09  % (32495)------------------------------
% 12.49/6.09  % (32495)------------------------------
% 12.49/6.10  % (32498)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_32338 for (2990ds/62Mi)
% 12.49/6.10  % (32499)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_32338 for (2990ds/31Mi)
% 12.49/6.10  % (32491)Instruction limit reached!
% 12.49/6.10  % (32491)------------------------------
% 12.49/6.10  % (32491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.49/6.10  % (32491)Termination reason: Unknown
% 12.49/6.10  % (32491)Termination phase: shuffling
% 12.49/6.10  
% 12.49/6.10  % (32491)Memory used [KB]: 14328
% 12.49/6.10  % (32491)Time elapsed: 0.075 s
% 12.49/6.10  % (32491)Instructions burned: 149 (million)
% 12.49/6.10  % (32491)------------------------------
% 12.49/6.10  % (32491)------------------------------
% 12.49/6.11  % (32500)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_32338 for (2990ds/96Mi)
% 12.49/6.12  % (32499)Instruction limit reached!
% 12.49/6.12  % (32499)------------------------------
% 12.49/6.12  % (32499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.49/6.12  % (32499)Termination reason: Unknown
% 12.49/6.12  % (32499)Termination phase: shuffling
% 12.49/6.12  
% 12.49/6.12  % (32499)Memory used [KB]: 12665
% 12.49/6.12  % (32499)Time elapsed: 0.019 s
% 12.49/6.12  % (32499)Instructions burned: 33 (million)
% 12.49/6.12  % (32499)------------------------------
% 12.49/6.12  % (32499)------------------------------
% 12.49/6.12  % (32498)Instruction limit reached!
% 12.49/6.12  % (32498)------------------------------
% 12.49/6.12  % (32498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.49/6.12  % (32498)Termination reason: Unknown
% 12.49/6.12  % (32498)Termination phase: shuffling
% 12.49/6.12  
% 12.49/6.12  % (32498)Memory used [KB]: 13176
% 12.49/6.12  % (32498)Time elapsed: 0.026 s
% 12.49/6.12  % (32498)Instructions burned: 64 (million)
% 12.49/6.12  % (32498)------------------------------
% 12.49/6.12  % (32498)------------------------------
% 13.04/6.13  % (32501)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_32338 for (2990ds/453Mi)
% 13.04/6.13  % (32502)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_32338 for (2990ds/71Mi)
% 13.04/6.15  % (32487)Instruction limit reached!
% 13.04/6.15  % (32487)------------------------------
% 13.04/6.15  % (32487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.04/6.15  % (32487)Termination reason: Unknown
% 13.04/6.15  % (32487)Termination phase: shuffling
% 13.04/6.15  
% 13.04/6.15  % (32487)Memory used [KB]: 20724
% 13.04/6.15  % (32487)Time elapsed: 0.177 s
% 13.04/6.15  % (32487)Instructions burned: 448 (million)
% 13.04/6.15  % (32487)------------------------------
% 13.04/6.15  % (32487)------------------------------
% 13.04/6.15  % (32500)Instruction limit reached!
% 13.04/6.15  % (32500)------------------------------
% 13.04/6.15  % (32500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.04/6.15  % (32500)Termination reason: Unknown
% 13.04/6.15  % (32500)Termination phase: shuffling
% 13.04/6.15  
% 13.04/6.15  % (32500)Memory used [KB]: 13688
% 13.04/6.15  % (32500)Time elapsed: 0.041 s
% 13.04/6.15  % (32500)Instructions burned: 97 (million)
% 13.04/6.15  % (32500)------------------------------
% 13.04/6.15  % (32500)------------------------------
% 13.04/6.16  % (32503)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_32338 for (2989ds/83Mi)
% 13.04/6.16  % (32502)Instruction limit reached!
% 13.04/6.16  % (32502)------------------------------
% 13.04/6.16  % (32502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.04/6.16  % (32502)Termination reason: Unknown
% 13.04/6.16  % (32502)Termination phase: shuffling
% 13.04/6.16  
% 13.04/6.16  % (32502)Memory used [KB]: 13304
% 13.04/6.16  % (32502)Time elapsed: 0.031 s
% 13.04/6.16  % (32502)Instructions burned: 72 (million)
% 13.04/6.16  % (32502)------------------------------
% 13.04/6.16  % (32502)------------------------------
% 13.04/6.17  % (32504)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_32338 for (2989ds/11Mi)
% 13.04/6.17  % (32505)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_32338 for (2989ds/41Mi)
% 13.04/6.18  % (32504)Instruction limit reached!
% 13.04/6.18  % (32504)------------------------------
% 13.04/6.18  % (32504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.04/6.18  % (32504)Termination reason: Unknown
% 13.04/6.18  % (32504)Termination phase: shuffling
% 13.04/6.18  
% 13.04/6.18  % (32504)Memory used [KB]: 11513
% 13.04/6.18  % (32504)Time elapsed: 0.012 s
% 13.04/6.18  % (32504)Instructions burned: 13 (million)
% 13.04/6.18  % (32504)------------------------------
% 13.04/6.18  % (32504)------------------------------
% 13.04/6.19  % (32490)Instruction limit reached!
% 13.04/6.19  % (32490)------------------------------
% 13.04/6.19  % (32490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.04/6.19  % (32490)Termination reason: Unknown
% 13.04/6.19  % (32490)Termination phase: shuffling
% 13.04/6.19  
% 13.04/6.19  % (32490)Memory used [KB]: 18933
% 13.04/6.19  % (32490)Time elapsed: 0.193 s
% 13.04/6.19  % (32490)Instructions burned: 507 (million)
% 13.04/6.19  % (32490)------------------------------
% 13.04/6.19  % (32490)------------------------------
% 13.04/6.19  % (32505)Instruction limit reached!
% 13.04/6.19  % (32505)------------------------------
% 13.04/6.19  % (32505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.04/6.19  % (32505)Termination reason: Unknown
% 13.04/6.19  % (32505)Termination phase: shuffling
% 13.04/6.19  
% 13.04/6.19  % (32505)Memory used [KB]: 12920
% 13.04/6.19  % (32505)Time elapsed: 0.021 s
% 13.04/6.19  % (32505)Instructions burned: 44 (million)
% 13.04/6.19  % (32505)------------------------------
% 13.04/6.19  % (32505)------------------------------
% 13.04/6.20  % (32506)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_32338 for (2989ds/6Mi)
% 13.31/6.20  % (32503)Instruction limit reached!
% 13.31/6.20  % (32503)------------------------------
% 13.31/6.20  % (32503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.31/6.20  % (32503)Termination reason: Unknown
% 13.31/6.20  % (32503)Termination phase: shuffling
% 13.31/6.20  
% 13.31/6.20  % (32503)Memory used [KB]: 13432
% 13.31/6.20  % (32503)Time elapsed: 0.040 s
% 13.31/6.20  % (32503)Instructions burned: 83 (million)
% 13.31/6.20  % (32503)------------------------------
% 13.31/6.20  % (32503)------------------------------
% 13.31/6.20  % (32506)Instruction limit reached!
% 13.31/6.20  % (32506)------------------------------
% 13.31/6.20  % (32506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.31/6.20  % (32506)Termination reason: Unknown
% 13.31/6.20  % (32506)Termination phase: shuffling
% 13.31/6.20  
% 13.31/6.20  % (32506)Memory used [KB]: 11001
% 13.31/6.20  % (32506)Time elapsed: 0.006 s
% 13.31/6.20  % (32506)Instructions burned: 6 (million)
% 13.31/6.20  % (32506)------------------------------
% 13.31/6.20  % (32506)------------------------------
% 13.31/6.20  % (32507)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_32338 for (2989ds/150Mi)
% 13.31/6.20  % (32508)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_32338 for (2989ds/343Mi)
% 13.31/6.20  % (32497)Instruction limit reached!
% 13.31/6.20  % (32497)------------------------------
% 13.31/6.20  % (32497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.31/6.20  % (32497)Termination reason: Unknown
% 13.31/6.20  % (32497)Termination phase: shuffling
% 13.31/6.20  
% 13.31/6.20  % (32497)Memory used [KB]: 20468
% 13.31/6.20  % (32497)Time elapsed: 0.137 s
% 13.31/6.20  % (32497)Instructions burned: 433 (million)
% 13.31/6.20  % (32497)------------------------------
% 13.31/6.20  % (32497)------------------------------
% 13.31/6.21  % (32510)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_32338 for (2989ds/253Mi)
% 13.31/6.21  % (32509)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_32338 for (2989ds/109Mi)
% 13.31/6.22  % (32511)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_32338 for (2989ds/140Mi)
% 13.31/6.25  % (32509)Instruction limit reached!
% 13.31/6.25  % (32509)------------------------------
% 13.31/6.25  % (32509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.31/6.25  % (32509)Termination reason: Unknown
% 13.31/6.25  % (32509)Termination phase: shuffling
% 13.31/6.25  
% 13.31/6.25  % (32509)Memory used [KB]: 13816
% 13.31/6.25  % (32509)Time elapsed: 0.044 s
% 13.31/6.25  % (32509)Instructions burned: 110 (million)
% 13.31/6.25  % (32509)------------------------------
% 13.31/6.25  % (32509)------------------------------
% 13.31/6.26  % (32507)Instruction limit reached!
% 13.31/6.26  % (32507)------------------------------
% 13.31/6.26  % (32507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.31/6.26  % (32507)Termination reason: Unknown
% 13.31/6.26  % (32507)Termination phase: shuffling
% 13.31/6.26  
% 13.31/6.26  % (32507)Memory used [KB]: 14328
% 13.31/6.26  % (32507)Time elapsed: 0.059 s
% 13.31/6.26  % (32507)Instructions burned: 151 (million)
% 13.31/6.26  % (32507)------------------------------
% 13.31/6.26  % (32507)------------------------------
% 13.31/6.27  % (32513)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_32338 for (2988ds/202Mi)
% 13.31/6.27  % (32512)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_32338 for (2988ds/614Mi)
% 13.31/6.27  % (32511)Instruction limit reached!
% 13.31/6.27  % (32511)------------------------------
% 13.31/6.27  % (32511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.31/6.27  % (32511)Termination reason: Unknown
% 13.31/6.27  % (32511)Termination phase: shuffling
% 13.31/6.27  
% 13.31/6.27  % (32511)Memory used [KB]: 14200
% 13.31/6.27  % (32511)Time elapsed: 0.059 s
% 13.31/6.27  % (32511)Instructions burned: 141 (million)
% 13.31/6.27  % (32511)------------------------------
% 13.31/6.27  % (32511)------------------------------
% 13.31/6.29  % (32514)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_32338 for (2988ds/339Mi)
% 13.31/6.29  % (32501)Instruction limit reached!
% 13.31/6.29  % (32501)------------------------------
% 13.31/6.29  % (32501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.31/6.29  % (32501)Termination reason: Unknown
% 13.31/6.29  % (32501)Termination phase: shuffling
% 13.31/6.29  
% 13.31/6.29  % (32501)Memory used [KB]: 20852
% 13.31/6.29  % (32501)Time elapsed: 0.156 s
% 13.31/6.29  % (32501)Instructions burned: 453 (million)
% 13.31/6.29  % (32501)------------------------------
% 13.31/6.29  % (32501)------------------------------
% 13.89/6.30  % (32510)Instruction limit reached!
% 13.89/6.30  % (32510)------------------------------
% 13.89/6.30  % (32510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.89/6.30  % (32510)Termination reason: Unknown
% 13.89/6.30  % (32510)Termination phase: shuffling
% 13.89/6.30  
% 13.89/6.30  % (32510)Memory used [KB]: 15863
% 13.89/6.30  % (32510)Time elapsed: 0.094 s
% 13.89/6.30  % (32510)Instructions burned: 253 (million)
% 13.89/6.30  % (32510)------------------------------
% 13.89/6.30  % (32510)------------------------------
% 13.89/6.31  % (32515)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_32338 for (2988ds/283Mi)
% 13.89/6.31  % (32516)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_32338 for (2988ds/134Mi)
% 13.89/6.33  % (32508)Instruction limit reached!
% 13.89/6.33  % (32508)------------------------------
% 13.89/6.33  % (32508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.89/6.33  % (32508)Termination reason: Unknown
% 13.89/6.33  % (32508)Termination phase: shuffling
% 13.89/6.33  
% 13.89/6.33  % (32508)Memory used [KB]: 17142
% 13.89/6.33  % (32508)Time elapsed: 0.123 s
% 13.89/6.33  % (32508)Instructions burned: 345 (million)
% 13.89/6.33  % (32508)------------------------------
% 13.89/6.33  % (32508)------------------------------
% 13.89/6.34  % (32517)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_32338 for (2988ds/298Mi)
% 13.89/6.35  % (32513)Instruction limit reached!
% 13.89/6.35  % (32513)------------------------------
% 13.89/6.35  % (32513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.89/6.35  % (32513)Termination reason: Unknown
% 13.89/6.35  % (32513)Termination phase: shuffling
% 13.89/6.35  
% 13.89/6.35  % (32513)Memory used [KB]: 15223
% 13.89/6.35  % (32513)Time elapsed: 0.077 s
% 13.89/6.35  % (32513)Instructions burned: 203 (million)
% 13.89/6.35  % (32513)------------------------------
% 13.89/6.35  % (32513)------------------------------
% 13.89/6.36  % (32518)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_32338 for (2987ds/192Mi)
% 13.89/6.37  % (32516)Instruction limit reached!
% 13.89/6.37  % (32516)------------------------------
% 13.89/6.37  % (32516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.89/6.37  % (32516)Termination reason: Unknown
% 13.89/6.37  % (32516)Termination phase: shuffling
% 13.89/6.37  
% 13.89/6.37  % (32516)Memory used [KB]: 14200
% 13.89/6.37  % (32516)Time elapsed: 0.053 s
% 13.89/6.37  % (32516)Instructions burned: 136 (million)
% 13.89/6.37  % (32516)------------------------------
% 13.89/6.37  % (32516)------------------------------
% 13.89/6.38  % (32519)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_32338 for (2987ds/258Mi)
% 14.28/6.41  % (32515)Instruction limit reached!
% 14.28/6.41  % (32515)------------------------------
% 14.28/6.41  % (32515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/6.41  % (32515)Termination reason: Unknown
% 14.28/6.41  % (32515)Termination phase: shuffling
% 14.28/6.41  
% 14.28/6.41  % (32515)Memory used [KB]: 16375
% 14.28/6.41  % (32515)Time elapsed: 0.107 s
% 14.28/6.41  % (32515)Instructions burned: 283 (million)
% 14.28/6.41  % (32515)------------------------------
% 14.28/6.41  % (32515)------------------------------
% 14.28/6.41  % (32514)Instruction limit reached!
% 14.28/6.41  % (32514)------------------------------
% 14.28/6.41  % (32514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/6.41  % (32514)Termination reason: Unknown
% 14.28/6.41  % (32514)Termination phase: shuffling
% 14.28/6.41  
% 14.28/6.41  % (32514)Memory used [KB]: 17014
% 14.28/6.41  % (32514)Time elapsed: 0.128 s
% 14.28/6.41  % (32514)Instructions burned: 341 (million)
% 14.28/6.41  % (32514)------------------------------
% 14.28/6.41  % (32514)------------------------------
% 14.28/6.41  % (32494)Instruction limit reached!
% 14.28/6.41  % (32494)------------------------------
% 14.28/6.41  % (32494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/6.41  % (32494)Termination reason: Unknown
% 14.28/6.41  % (32494)Termination phase: Property scanning
% 14.28/6.41  
% 14.28/6.41  % (32494)Memory used [KB]: 21108
% 14.28/6.41  % (32494)Time elapsed: 0.355 s
% 14.28/6.41  % (32494)Instructions burned: 1087 (million)
% 14.28/6.41  % (32494)------------------------------
% 14.28/6.41  % (32494)------------------------------
% 14.28/6.42  % (32520)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_32338 for (2987ds/90Mi)
% 14.28/6.42  % (32521)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_32338 for (2987ds/231Mi)
% 14.28/6.43  % (32522)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_32338 for (2987ds/458Mi)
% 14.28/6.43  % (32517)Instruction limit reached!
% 14.28/6.43  % (32517)------------------------------
% 14.28/6.43  % (32517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/6.43  % (32517)Termination reason: Unknown
% 14.28/6.43  % (32517)Termination phase: shuffling
% 14.28/6.43  
% 14.28/6.43  % (32517)Memory used [KB]: 18549
% 14.28/6.43  % (32517)Time elapsed: 0.093 s
% 14.28/6.43  % (32517)Instructions burned: 299 (million)
% 14.28/6.43  % (32517)------------------------------
% 14.28/6.43  % (32517)------------------------------
% 14.28/6.43  % (32518)Instruction limit reached!
% 14.28/6.43  % (32518)------------------------------
% 14.28/6.43  % (32518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/6.43  % (32518)Termination reason: Unknown
% 14.28/6.43  % (32518)Termination phase: shuffling
% 14.28/6.43  
% 14.28/6.43  % (32518)Memory used [KB]: 14967
% 14.28/6.43  % (32518)Time elapsed: 0.072 s
% 14.28/6.43  % (32518)Instructions burned: 193 (million)
% 14.28/6.43  % (32518)------------------------------
% 14.28/6.43  % (32518)------------------------------
% 14.28/6.44  % (32524)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_32338 for (2987ds/297Mi)
% 14.28/6.44  % (32523)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_32338 for (2987ds/31Mi)
% 14.28/6.46  % (32523)Instruction limit reached!
% 14.28/6.46  % (32523)------------------------------
% 14.28/6.46  % (32523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/6.46  % (32523)Termination reason: Unknown
% 14.28/6.46  % (32523)Termination phase: shuffling
% 14.28/6.46  
% 14.28/6.46  % (32523)Memory used [KB]: 12665
% 14.28/6.46  % (32523)Time elapsed: 0.019 s
% 14.28/6.46  % (32523)Instructions burned: 31 (million)
% 14.28/6.46  % (32523)------------------------------
% 14.28/6.46  % (32523)------------------------------
% 14.28/6.46  % (32520)Instruction limit reached!
% 14.28/6.46  % (32520)------------------------------
% 14.28/6.46  % (32520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/6.46  % (32520)Termination reason: Unknown
% 14.28/6.46  % (32520)Termination phase: shuffling
% 14.28/6.46  
% 14.28/6.46  % (32520)Memory used [KB]: 13560
% 14.28/6.46  % (32520)Time elapsed: 0.039 s
% 14.28/6.46  % (32520)Instructions burned: 92 (million)
% 14.28/6.46  % (32520)------------------------------
% 14.28/6.46  % (32520)------------------------------
% 14.28/6.47  % (32525)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_32338 for (2986ds/987Mi)
% 14.28/6.47  % (32519)Instruction limit reached!
% 14.28/6.47  % (32519)------------------------------
% 14.28/6.47  % (32519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/6.47  % (32519)Termination reason: Unknown
% 14.28/6.47  % (32519)Termination phase: shuffling
% 14.28/6.47  
% 14.28/6.47  % (32519)Memory used [KB]: 15863
% 14.28/6.47  % (32519)Time elapsed: 0.095 s
% 14.28/6.47  % (32519)Instructions burned: 258 (million)
% 14.28/6.47  % (32519)------------------------------
% 14.28/6.47  % (32519)------------------------------
% 14.28/6.47  % (32526)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_32338 for (2986ds/182Mi)
% 14.28/6.48  % (32527)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_32338 for (2986ds/182Mi)
% 14.28/6.49  % (32512)Instruction limit reached!
% 14.28/6.49  % (32512)------------------------------
% 14.28/6.49  % (32512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/6.49  % (32512)Termination reason: Unknown
% 14.28/6.49  % (32512)Termination phase: shuffling
% 14.28/6.49  
% 14.28/6.49  % (32512)Memory used [KB]: 20596
% 14.28/6.49  % (32512)Time elapsed: 0.220 s
% 14.28/6.49  % (32512)Instructions burned: 614 (million)
% 14.28/6.49  % (32512)------------------------------
% 14.28/6.49  % (32512)------------------------------
% 15.51/6.50  % (32521)Instruction limit reached!
% 15.51/6.50  % (32521)------------------------------
% 15.51/6.50  % (32521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.51/6.50  % (32521)Termination reason: Unknown
% 15.51/6.50  % (32521)Termination phase: shuffling
% 15.51/6.50  
% 15.51/6.50  % (32521)Memory used [KB]: 15607
% 15.51/6.50  % (32521)Time elapsed: 0.080 s
% 15.51/6.50  % (32521)Instructions burned: 232 (million)
% 15.51/6.50  % (32521)------------------------------
% 15.51/6.50  % (32521)------------------------------
% 15.51/6.50  % (32528)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_32338 for (2986ds/154Mi)
% 15.51/6.51  % (32529)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_32338 for (2986ds/79Mi)
% 15.51/6.53  % (32524)Instruction limit reached!
% 15.51/6.53  % (32524)------------------------------
% 15.51/6.53  % (32524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.51/6.53  % (32524)Termination reason: Unknown
% 15.51/6.53  % (32524)Termination phase: shuffling
% 15.51/6.53  
% 15.51/6.53  % (32524)Memory used [KB]: 18549
% 15.51/6.53  % (32524)Time elapsed: 0.096 s
% 15.51/6.53  % (32524)Instructions burned: 297 (million)
% 15.51/6.53  % (32524)------------------------------
% 15.51/6.53  % (32524)------------------------------
% 15.51/6.55  % (32529)Instruction limit reached!
% 15.51/6.55  % (32529)------------------------------
% 15.51/6.55  % (32529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.51/6.55  % (32529)Termination reason: Unknown
% 15.51/6.55  % (32529)Termination phase: shuffling
% 15.51/6.55  
% 15.51/6.55  % (32529)Memory used [KB]: 13432
% 15.51/6.55  % (32529)Time elapsed: 0.036 s
% 15.51/6.55  % (32529)Instructions burned: 81 (million)
% 15.51/6.55  % (32529)------------------------------
% 15.51/6.55  % (32529)------------------------------
% 15.51/6.55  % (32530)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_32338 for (2985ds/216Mi)
% 15.51/6.55  % (32527)Instruction limit reached!
% 15.51/6.55  % (32527)------------------------------
% 15.51/6.55  % (32527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.51/6.55  % (32527)Termination reason: Unknown
% 15.51/6.55  % (32527)Termination phase: shuffling
% 15.51/6.55  
% 15.51/6.55  % (32527)Memory used [KB]: 14839
% 15.51/6.55  % (32527)Time elapsed: 0.071 s
% 15.51/6.55  % (32527)Instructions burned: 183 (million)
% 15.51/6.55  % (32527)------------------------------
% 15.51/6.55  % (32527)------------------------------
% 15.51/6.55  % (32528)Instruction limit reached!
% 15.51/6.55  % (32528)------------------------------
% 15.51/6.55  % (32528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.51/6.55  % (32528)Termination reason: Unknown
% 15.51/6.55  % (32528)Termination phase: shuffling
% 15.51/6.55  
% 15.51/6.55  % (32528)Memory used [KB]: 14583
% 15.51/6.55  % (32528)Time elapsed: 0.053 s
% 15.51/6.55  % (32528)Instructions burned: 156 (million)
% 15.51/6.55  % (32528)------------------------------
% 15.51/6.55  % (32528)------------------------------
% 15.51/6.56  % (32531)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_32338 for (2985ds/114Mi)
% 15.51/6.56  % (32526)Instruction limit reached!
% 15.51/6.56  % (32526)------------------------------
% 15.51/6.56  % (32526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.51/6.56  % (32526)Termination reason: Unknown
% 15.51/6.56  % (32526)Termination phase: shuffling
% 15.51/6.56  
% 15.51/6.56  % (32526)Memory used [KB]: 14839
% 15.51/6.56  % (32526)Time elapsed: 0.089 s
% 15.51/6.56  % (32526)Instructions burned: 182 (million)
% 15.51/6.56  % (32526)------------------------------
% 15.51/6.56  % (32526)------------------------------
% 15.51/6.56  % (32532)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_32338 for (2985ds/922Mi)
% 15.51/6.56  % (32533)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_32338 for (2985ds/182Mi)
% 15.51/6.58  % (32534)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_32338 for (2985ds/164Mi)
% 15.51/6.59  % (32522)Instruction limit reached!
% 15.51/6.59  % (32522)------------------------------
% 15.51/6.59  % (32522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.51/6.59  % (32522)Termination reason: Unknown
% 15.51/6.59  % (32522)Termination phase: shuffling
% 15.51/6.59  
% 15.51/6.59  % (32522)Memory used [KB]: 18549
% 15.51/6.59  % (32522)Time elapsed: 0.167 s
% 15.51/6.59  % (32522)Instructions burned: 459 (million)
% 15.51/6.59  % (32522)------------------------------
% 15.51/6.59  % (32522)------------------------------
% 16.18/6.60  % (32535)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_32338 for (2985ds/50Mi)
% 16.18/6.60  % (32531)Instruction limit reached!
% 16.18/6.60  % (32531)------------------------------
% 16.18/6.60  % (32531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.18/6.60  % (32531)Termination reason: Unknown
% 16.18/6.60  % (32531)Termination phase: shuffling
% 16.18/6.60  
% 16.18/6.60  % (32531)Memory used [KB]: 13944
% 16.18/6.60  % (32531)Time elapsed: 0.047 s
% 16.18/6.60  % (32531)Instructions burned: 115 (million)
% 16.18/6.60  % (32531)------------------------------
% 16.18/6.60  % (32531)------------------------------
% 16.18/6.62  % (32536)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_32338 for (2985ds/368Mi)
% 16.18/6.62  % (32530)Instruction limit reached!
% 16.18/6.62  % (32530)------------------------------
% 16.18/6.62  % (32530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.18/6.62  % (32530)Termination reason: Unknown
% 16.18/6.62  % (32530)Termination phase: shuffling
% 16.18/6.62  
% 16.18/6.62  % (32530)Memory used [KB]: 15479
% 16.18/6.62  % (32530)Time elapsed: 0.070 s
% 16.18/6.62  % (32530)Instructions burned: 218 (million)
% 16.18/6.62  % (32530)------------------------------
% 16.18/6.62  % (32530)------------------------------
% 16.18/6.63  % (32535)Instruction limit reached!
% 16.18/6.63  % (32535)------------------------------
% 16.18/6.63  % (32535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.18/6.63  % (32535)Termination reason: Unknown
% 16.18/6.63  % (32535)Termination phase: shuffling
% 16.18/6.63  
% 16.18/6.63  % (32535)Memory used [KB]: 13048
% 16.18/6.63  % (32535)Time elapsed: 0.024 s
% 16.18/6.63  % (32535)Instructions burned: 51 (million)
% 16.18/6.63  % (32535)------------------------------
% 16.18/6.63  % (32535)------------------------------
% 16.18/6.63  % (32533)Instruction limit reached!
% 16.18/6.63  % (32533)------------------------------
% 16.18/6.63  % (32533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.18/6.63  % (32533)Termination reason: Unknown
% 16.18/6.63  % (32533)Termination phase: shuffling
% 16.18/6.63  
% 16.18/6.63  % (32533)Memory used [KB]: 14967
% 16.18/6.63  % (32533)Time elapsed: 0.066 s
% 16.18/6.63  % (32533)Instructions burned: 185 (million)
% 16.18/6.63  % (32533)------------------------------
% 16.18/6.63  % (32533)------------------------------
% 16.18/6.63  % (32534)Instruction limit reached!
% 16.18/6.63  % (32534)------------------------------
% 16.18/6.63  % (32534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.18/6.63  % (32534)Termination reason: Unknown
% 16.18/6.63  % (32534)Termination phase: shuffling
% 16.18/6.63  
% 16.18/6.63  % (32534)Memory used [KB]: 14711
% 16.18/6.63  % (32534)Time elapsed: 0.059 s
% 16.18/6.63  % (32534)Instructions burned: 166 (million)
% 16.18/6.63  % (32534)------------------------------
% 16.18/6.63  % (32534)------------------------------
% 16.18/6.63  % (32537)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_32338 for (2985ds/1021Mi)
% 16.18/6.64  % (32538)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_32338 for (2985ds/267Mi)
% 16.18/6.64  % (32539)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_32338 for (2985ds/204Mi)
% 16.18/6.64  % (32540)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_32338 for (2985ds/26Mi)
% 16.18/6.66  % (32540)Instruction limit reached!
% 16.18/6.66  % (32540)------------------------------
% 16.18/6.66  % (32540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.18/6.66  % (32540)Termination reason: Unknown
% 16.18/6.66  % (32540)Termination phase: shuffling
% 16.18/6.66  
% 16.18/6.66  % (32540)Memory used [KB]: 12665
% 16.18/6.66  % (32540)Time elapsed: 0.017 s
% 16.18/6.66  % (32540)Instructions burned: 28 (million)
% 16.18/6.66  % (32540)------------------------------
% 16.18/6.66  % (32540)------------------------------
% 16.18/6.67  % (32541)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_32338 for (2984ds/53Mi)
% 16.18/6.69  % (32541)Instruction limit reached!
% 16.18/6.69  % (32541)------------------------------
% 16.18/6.69  % (32541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.18/6.69  % (32541)Termination reason: Unknown
% 16.18/6.69  % (32541)Termination phase: shuffling
% 16.18/6.69  
% 16.18/6.69  % (32541)Memory used [KB]: 13048
% 16.18/6.69  % (32541)Time elapsed: 0.024 s
% 16.18/6.69  % (32541)Instructions burned: 53 (million)
% 16.18/6.69  % (32541)------------------------------
% 16.18/6.69  % (32541)------------------------------
% 16.56/6.70  % (32542)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_32338 for (2984ds/4436Mi)
% 16.56/6.72  % (32539)Instruction limit reached!
% 16.56/6.72  % (32539)------------------------------
% 16.56/6.72  % (32539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.56/6.72  % (32539)Termination reason: Unknown
% 16.56/6.72  % (32539)Termination phase: shuffling
% 16.56/6.72  
% 16.56/6.72  % (32539)Memory used [KB]: 15095
% 16.56/6.72  % (32539)Time elapsed: 0.081 s
% 16.56/6.72  % (32539)Instructions burned: 204 (million)
% 16.56/6.72  % (32539)------------------------------
% 16.56/6.72  % (32539)------------------------------
% 16.56/6.73  % (32538)Instruction limit reached!
% 16.56/6.73  % (32538)------------------------------
% 16.56/6.73  % (32538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.56/6.73  % (32538)Termination reason: Unknown
% 16.56/6.73  % (32538)Termination phase: shuffling
% 16.56/6.73  
% 16.56/6.73  % (32538)Memory used [KB]: 16119
% 16.56/6.73  % (32538)Time elapsed: 0.096 s
% 16.56/6.73  % (32538)Instructions burned: 269 (million)
% 16.56/6.73  % (32538)------------------------------
% 16.56/6.73  % (32538)------------------------------
% 16.56/6.73  % (32543)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_32338 for (2984ds/1539Mi)
% 16.56/6.74  % (32544)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_32338 for (2984ds/408Mi)
% 16.56/6.75  % (32536)Instruction limit reached!
% 16.56/6.75  % (32536)------------------------------
% 16.56/6.75  % (32536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.56/6.75  % (32536)Termination reason: Unknown
% 16.56/6.75  % (32536)Termination phase: shuffling
% 16.56/6.75  
% 16.56/6.75  % (32536)Memory used [KB]: 17270
% 16.56/6.75  % (32536)Time elapsed: 0.134 s
% 16.56/6.75  % (32536)Instructions burned: 369 (million)
% 16.56/6.75  % (32536)------------------------------
% 16.56/6.75  % (32536)------------------------------
% 16.56/6.76  % (32545)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_32338 for (2983ds/5205Mi)
% 16.56/6.78  % (32525)Instruction limit reached!
% 16.56/6.78  % (32525)------------------------------
% 16.56/6.78  % (32525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.56/6.78  % (32525)Termination reason: Unknown
% 16.56/6.78  % (32525)Termination phase: shuffling
% 16.56/6.78  
% 16.56/6.78  % (32525)Memory used [KB]: 32110
% 16.56/6.78  % (32525)Time elapsed: 0.316 s
% 16.56/6.78  % (32525)Instructions burned: 989 (million)
% 16.56/6.78  % (32525)------------------------------
% 16.56/6.78  % (32525)------------------------------
% 17.22/6.80  % (32546)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_32338 for (2983ds/1913Mi)
% 17.22/6.92  % (32532)Instruction limit reached!
% 17.22/6.92  % (32532)------------------------------
% 17.22/6.92  % (32532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.22/6.92  % (32532)Termination reason: Unknown
% 17.22/6.92  % (32532)Termination phase: Property scanning
% 17.22/6.92  
% 17.22/6.92  % (32532)Memory used [KB]: 21108
% 17.22/6.92  % (32532)Time elapsed: 0.355 s
% 17.22/6.92  % (32532)Instructions burned: 923 (million)
% 17.22/6.92  % (32532)------------------------------
% 17.22/6.92  % (32532)------------------------------
% 17.22/6.93  % (32547)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_32338 for (2982ds/4350Mi)
% 17.22/6.94  % (32544)Instruction limit reached!
% 17.22/6.94  % (32544)------------------------------
% 17.22/6.94  % (32544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.22/6.94  % (32544)Termination reason: Unknown
% 17.22/6.94  % (32544)Termination phase: shuffling
% 17.22/6.94  
% 17.22/6.94  % (32544)Memory used [KB]: 18038
% 17.22/6.94  % (32544)Time elapsed: 0.197 s
% 17.22/6.94  % (32544)Instructions burned: 409 (million)
% 17.22/6.94  % (32544)------------------------------
% 17.22/6.94  % (32544)------------------------------
% 17.22/6.96  % (32548)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_32338 for (2982ds/68Mi)
% 17.22/6.97  % (32537)Instruction limit reached!
% 17.22/6.97  % (32537)------------------------------
% 17.22/6.97  % (32537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.22/6.97  % (32537)Termination reason: Unknown
% 17.22/6.97  % (32537)Termination phase: Property scanning
% 17.22/6.97  
% 17.22/6.97  % (32537)Memory used [KB]: 21108
% 17.22/6.97  % (32537)Time elapsed: 0.339 s
% 17.22/6.97  % (32537)Instructions burned: 1022 (million)
% 17.22/6.97  % (32537)------------------------------
% 17.22/6.97  % (32537)------------------------------
% 17.22/6.98  % (32549)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_32338 for (2981ds/358Mi)
% 17.22/7.01  % (32548)Instruction limit reached!
% 17.22/7.01  % (32548)------------------------------
% 17.22/7.01  % (32548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.22/7.01  % (32548)Termination reason: Unknown
% 17.22/7.01  % (32548)Termination phase: shuffling
% 17.22/7.01  
% 17.22/7.01  % (32548)Memory used [KB]: 13176
% 17.22/7.01  % (32548)Time elapsed: 0.055 s
% 17.22/7.01  % (32548)Instructions burned: 69 (million)
% 17.22/7.01  % (32548)------------------------------
% 17.22/7.01  % (32548)------------------------------
% 19.31/7.03  % (32550)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_32338 for (2981ds/31Mi)
% 19.31/7.05  % (32550)Instruction limit reached!
% 19.31/7.05  % (32550)------------------------------
% 19.31/7.05  % (32550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.31/7.05  % (32550)Termination reason: Unknown
% 19.31/7.05  % (32550)Termination phase: shuffling
% 19.31/7.05  
% 19.31/7.05  % (32550)Memory used [KB]: 12665
% 19.31/7.05  % (32550)Time elapsed: 0.024 s
% 19.31/7.05  % (32550)Instructions burned: 33 (million)
% 19.31/7.05  % (32550)------------------------------
% 19.31/7.05  % (32550)------------------------------
% 19.31/7.06  % (32551)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_32338 for (2980ds/279Mi)
% 20.04/7.15  % (32551)Instruction limit reached!
% 20.04/7.15  % (32551)------------------------------
% 20.04/7.15  % (32551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.04/7.15  % (32551)Termination reason: Unknown
% 20.04/7.15  % (32551)Termination phase: shuffling
% 20.04/7.15  
% 20.04/7.15  % (32551)Memory used [KB]: 16375
% 20.04/7.15  % (32551)Time elapsed: 0.087 s
% 20.04/7.15  % (32551)Instructions burned: 281 (million)
% 20.04/7.15  % (32551)------------------------------
% 20.04/7.15  % (32551)------------------------------
% 20.04/7.16  % (32549)Instruction limit reached!
% 20.04/7.16  % (32549)------------------------------
% 20.04/7.16  % (32549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.04/7.16  % (32549)Termination reason: Unknown
% 20.04/7.16  % (32549)Termination phase: shuffling
% 20.04/7.16  
% 20.04/7.16  % (32549)Memory used [KB]: 17270
% 20.04/7.16  % (32549)Time elapsed: 0.180 s
% 20.04/7.16  % (32549)Instructions burned: 358 (million)
% 20.04/7.16  % (32549)------------------------------
% 20.04/7.16  % (32549)------------------------------
% 20.04/7.16  % (32552)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_32338 for (2980ds/258Mi)
% 20.04/7.17  % (32553)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_32338 for (2979ds/26Mi)
% 20.04/7.20  % (32553)Instruction limit reached!
% 20.04/7.20  % (32553)------------------------------
% 20.04/7.20  % (32553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.04/7.20  % (32553)Termination reason: Unknown
% 20.04/7.20  % (32553)Termination phase: shuffling
% 20.04/7.20  
% 20.04/7.20  % (32553)Memory used [KB]: 12665
% 20.04/7.20  % (32553)Time elapsed: 0.023 s
% 20.04/7.20  % (32553)Instructions burned: 28 (million)
% 20.04/7.20  % (32553)------------------------------
% 20.04/7.20  % (32553)------------------------------
% 20.61/7.21  % (32554)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_32338 for (2979ds/995Mi)
% 20.61/7.23  % (32543)Instruction limit reached!
% 20.61/7.23  % (32543)------------------------------
% 20.61/7.23  % (32543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.61/7.23  % (32543)Termination reason: Unknown
% 20.61/7.23  % (32543)Termination phase: Naming
% 20.61/7.23  
% 20.61/7.23  % (32543)Memory used [KB]: 27760
% 20.61/7.23  % (32543)Time elapsed: 0.502 s
% 20.61/7.23  % (32543)Instructions burned: 1540 (million)
% 20.61/7.23  % (32543)------------------------------
% 20.61/7.23  % (32543)------------------------------
% 20.61/7.25  % (32555)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_32338 for (2978ds/373Mi)
% 20.61/7.25  % (32552)Instruction limit reached!
% 20.61/7.25  % (32552)------------------------------
% 20.61/7.25  % (32552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.61/7.25  % (32552)Termination reason: Unknown
% 20.61/7.25  % (32552)Termination phase: shuffling
% 20.61/7.25  
% 20.61/7.25  % (32552)Memory used [KB]: 15863
% 20.61/7.25  % (32552)Time elapsed: 0.095 s
% 20.61/7.25  % (32552)Instructions burned: 258 (million)
% 20.61/7.25  % (32552)------------------------------
% 20.61/7.25  % (32552)------------------------------
% 20.61/7.27  % (32556)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_32338 for (2978ds/253Mi)
% 21.20/7.35  % (32556)Instruction limit reached!
% 21.20/7.35  % (32556)------------------------------
% 21.20/7.35  % (32556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.20/7.35  % (32556)Termination reason: Unknown
% 21.20/7.35  % (32556)Termination phase: shuffling
% 21.20/7.35  
% 21.20/7.35  % (32556)Memory used [KB]: 15991
% 21.20/7.35  % (32556)Time elapsed: 0.081 s
% 21.20/7.35  % (32556)Instructions burned: 254 (million)
% 21.20/7.35  % (32556)------------------------------
% 21.20/7.35  % (32556)------------------------------
% 21.20/7.36  % (32557)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_32338 for (2977ds/832Mi)
% 21.20/7.38  % (32555)Instruction limit reached!
% 21.20/7.38  % (32555)------------------------------
% 21.20/7.38  % (32555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.20/7.38  % (32555)Termination reason: Unknown
% 21.20/7.38  % (32555)Termination phase: shuffling
% 21.20/7.38  
% 21.20/7.38  % (32555)Memory used [KB]: 17398
% 21.20/7.38  % (32555)Time elapsed: 0.134 s
% 21.20/7.38  % (32555)Instructions burned: 375 (million)
% 21.20/7.38  % (32555)------------------------------
% 21.20/7.38  % (32555)------------------------------
% 21.20/7.39  % (32558)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_32338 for (2977ds/668Mi)
% 21.84/7.42  % (32546)Instruction limit reached!
% 21.84/7.42  % (32546)------------------------------
% 21.84/7.42  % (32546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.84/7.42  % (32546)Termination reason: Unknown
% 21.84/7.42  % (32546)Termination phase: Preprocessing 3
% 21.84/7.42  
% 21.84/7.42  % (32546)Memory used [KB]: 26993
% 21.84/7.42  % (32546)Time elapsed: 0.628 s
% 21.84/7.42  % (32546)Instructions burned: 1913 (million)
% 21.84/7.42  % (32546)------------------------------
% 21.84/7.42  % (32546)------------------------------
% 21.84/7.44  % (32559)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_32338 for (2977ds/40Mi)
% 21.84/7.46  % (32559)Instruction limit reached!
% 21.84/7.46  % (32559)------------------------------
% 21.84/7.46  % (32559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.84/7.46  % (32559)Termination reason: Unknown
% 21.84/7.46  % (32559)Termination phase: shuffling
% 21.84/7.46  
% 21.84/7.46  % (32559)Memory used [KB]: 12792
% 21.84/7.46  % (32559)Time elapsed: 0.021 s
% 21.84/7.46  % (32559)Instructions burned: 42 (million)
% 21.84/7.46  % (32559)------------------------------
% 21.84/7.46  % (32559)------------------------------
% 21.84/7.47  % (32560)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_32338 for (2976ds/294Mi)
% 22.21/7.54  % (32554)Instruction limit reached!
% 22.21/7.54  % (32554)------------------------------
% 22.21/7.54  % (32554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.21/7.54  % (32554)Termination reason: Unknown
% 22.21/7.54  % (32554)Termination phase: shuffling
% 22.21/7.54  
% 22.21/7.54  % (32554)Memory used [KB]: 32366
% 22.21/7.54  % (32554)Time elapsed: 0.335 s
% 22.21/7.54  % (32554)Instructions burned: 995 (million)
% 22.21/7.54  % (32554)------------------------------
% 22.21/7.54  % (32554)------------------------------
% 22.21/7.56  % (32581)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_32338 for (2975ds/187Mi)
% 22.21/7.56  % (32560)Instruction limit reached!
% 22.21/7.56  % (32560)------------------------------
% 22.21/7.56  % (32560)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.21/7.56  % (32560)Termination reason: Unknown
% 22.21/7.56  % (32560)Termination phase: shuffling
% 22.21/7.56  
% 22.21/7.56  % (32560)Memory used [KB]: 18549
% 22.21/7.56  % (32560)Time elapsed: 0.094 s
% 22.21/7.56  % (32560)Instructions burned: 295 (million)
% 22.21/7.56  % (32560)------------------------------
% 22.21/7.56  % (32560)------------------------------
% 22.21/7.57  % (32590)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_32338 for (2975ds/371Mi)
% 22.21/7.59  % (32558)Instruction limit reached!
% 22.21/7.59  % (32558)------------------------------
% 22.21/7.59  % (32558)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.21/7.59  % (32558)Termination reason: Unknown
% 22.21/7.59  % (32558)Termination phase: shuffling
% 22.21/7.59  
% 22.21/7.59  % (32558)Memory used [KB]: 23794
% 22.21/7.59  % (32558)Time elapsed: 0.204 s
% 22.21/7.59  % (32558)Instructions burned: 670 (million)
% 22.21/7.59  % (32558)------------------------------
% 22.21/7.59  % (32558)------------------------------
% 23.18/7.61  % (32605)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_32338 for (2975ds/1827Mi)
% 23.18/7.63  % (32581)Instruction limit reached!
% 23.18/7.63  % (32581)------------------------------
% 23.18/7.63  % (32581)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.18/7.63  % (32581)Termination reason: Unknown
% 23.18/7.63  % (32581)Termination phase: shuffling
% 23.18/7.63  
% 23.18/7.63  % (32581)Memory used [KB]: 14967
% 23.18/7.63  % (32581)Time elapsed: 0.072 s
% 23.18/7.63  % (32581)Instructions burned: 189 (million)
% 23.18/7.63  % (32581)------------------------------
% 23.18/7.63  % (32581)------------------------------
% 23.18/7.64  % (32557)Instruction limit reached!
% 23.18/7.64  % (32557)------------------------------
% 23.18/7.64  % (32557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.18/7.64  % (32557)Termination reason: Unknown
% 23.18/7.64  % (32557)Termination phase: Property scanning
% 23.18/7.64  
% 23.18/7.64  % (32557)Memory used [KB]: 21108
% 23.18/7.64  % (32557)Time elapsed: 0.281 s
% 23.18/7.64  % (32557)Instructions burned: 834 (million)
% 23.18/7.64  % (32557)------------------------------
% 23.18/7.64  % (32557)------------------------------
% 23.18/7.64  % (32620)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_32338 for (2975ds/70Mi)
% 23.18/7.65  % (32622)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_32338 for (2974ds/316Mi)
% 23.18/7.67  % (32620)Instruction limit reached!
% 23.18/7.67  % (32620)------------------------------
% 23.18/7.67  % (32620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.18/7.67  % (32620)Termination reason: Unknown
% 23.18/7.67  % (32620)Termination phase: shuffling
% 23.18/7.67  
% 23.18/7.67  % (32620)Memory used [KB]: 13304
% 23.18/7.67  % (32620)Time elapsed: 0.029 s
% 23.18/7.67  % (32620)Instructions burned: 72 (million)
% 23.18/7.67  % (32620)------------------------------
% 23.18/7.67  % (32620)------------------------------
% 23.18/7.68  % (32632)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_32338 for (2974ds/932Mi)
% 23.90/7.71  % (32590)Instruction limit reached!
% 23.90/7.71  % (32590)------------------------------
% 23.90/7.71  % (32590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.90/7.71  % (32590)Termination reason: Unknown
% 23.90/7.71  % (32590)Termination phase: shuffling
% 23.90/7.71  
% 23.90/7.71  % (32590)Memory used [KB]: 17398
% 23.90/7.71  % (32590)Time elapsed: 0.133 s
% 23.90/7.71  % (32590)Instructions burned: 371 (million)
% 23.90/7.71  % (32590)------------------------------
% 23.90/7.71  % (32590)------------------------------
% 23.90/7.72  % (32653)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_32338 for (2974ds/4714Mi)
% 23.90/7.75  % (32622)Instruction limit reached!
% 23.90/7.75  % (32622)------------------------------
% 23.90/7.75  % (32622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.90/7.75  % (32622)Termination reason: Unknown
% 23.90/7.75  % (32622)Termination phase: shuffling
% 23.90/7.75  
% 23.90/7.75  % (32622)Memory used [KB]: 18805
% 23.90/7.75  % (32622)Time elapsed: 0.099 s
% 23.90/7.75  % (32622)Instructions burned: 317 (million)
% 23.90/7.75  % (32622)------------------------------
% 23.90/7.75  % (32622)------------------------------
% 23.90/7.76  % (32678)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_32338 for (2973ds/8250Mi)
% 24.75/7.95  % (32632)Instruction limit reached!
% 24.75/7.95  % (32632)------------------------------
% 24.75/7.95  % (32632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.75/7.95  % (32632)Termination reason: Unknown
% 24.75/7.95  % (32632)Termination phase: shuffling
% 24.75/7.95  
% 24.75/7.95  % (32632)Memory used [KB]: 31470
% 24.75/7.95  % (32632)Time elapsed: 0.277 s
% 24.75/7.95  % (32632)Instructions burned: 934 (million)
% 24.75/7.95  % (32632)------------------------------
% 24.75/7.95  % (32632)------------------------------
% 24.75/7.97  % (32756)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_32338 for (2971ds/770Mi)
% 27.08/8.18  % (32605)Instruction limit reached!
% 27.08/8.18  % (32605)------------------------------
% 27.08/8.18  % (32605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.08/8.18  % (32605)Termination reason: Unknown
% 27.08/8.18  % (32605)Termination phase: Property scanning
% 27.08/8.18  
% 27.08/8.18  % (32605)Memory used [KB]: 31470
% 27.08/8.18  % (32605)Time elapsed: 0.571 s
% 27.08/8.18  % (32605)Instructions burned: 1827 (million)
% 27.08/8.18  % (32605)------------------------------
% 27.08/8.18  % (32605)------------------------------
% 27.08/8.19  % (331)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_32338 for (2969ds/794Mi)
% 27.67/8.20  % (32542)Instruction limit reached!
% 27.67/8.20  % (32542)------------------------------
% 27.67/8.20  % (32542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.67/8.20  % (32542)Termination reason: Unknown
% 27.67/8.20  % (32542)Termination phase: Saturation
% 27.67/8.20  
% 27.67/8.20  % (32542)Memory used [KB]: 33133
% 27.67/8.20  % (32542)Time elapsed: 1.499 s
% 27.67/8.20  % (32542)Instructions burned: 4437 (million)
% 27.67/8.20  % (32542)------------------------------
% 27.67/8.20  % (32542)------------------------------
% 27.67/8.21  % (32545)Instruction limit reached!
% 27.67/8.21  % (32545)------------------------------
% 27.67/8.21  % (32545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.67/8.21  % (32545)Termination reason: Unknown
% 27.67/8.21  % (32545)Termination phase: Property scanning
% 27.67/8.21  
% 27.67/8.21  % (32545)Memory used [KB]: 54881
% 27.67/8.21  % (32545)Time elapsed: 1.457 s
% 27.67/8.21  % (32545)Instructions burned: 5208 (million)
% 27.67/8.21  % (32545)------------------------------
% 27.67/8.21  % (32545)------------------------------
% 27.67/8.22  % (332)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_32338 for (2969ds/464Mi)
% 27.67/8.23  % (333)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_32338 for (2969ds/1376Mi)
% 27.67/8.28  % (32547)Instruction limit reached!
% 27.67/8.28  % (32547)------------------------------
% 27.67/8.28  % (32547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.67/8.28  % (32547)Termination reason: Unknown
% 27.67/8.28  % (32547)Termination phase: Saturation
% 27.67/8.28  
% 27.67/8.28  % (32547)Memory used [KB]: 31086
% 27.67/8.28  % (32547)Time elapsed: 1.374 s
% 27.67/8.28  % (32547)Instructions burned: 4353 (million)
% 27.67/8.28  % (32547)------------------------------
% 27.67/8.28  % (32547)------------------------------
% 27.67/8.29  % (334)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_32338 for (2968ds/906Mi)
% 27.67/8.29  % (32756)Instruction limit reached!
% 27.67/8.29  % (32756)------------------------------
% 27.67/8.29  % (32756)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.67/8.29  % (32756)Termination reason: Unknown
% 27.67/8.29  % (32756)Termination phase: shuffling
% 27.67/8.29  
% 27.67/8.29  % (32756)Memory used [KB]: 24306
% 27.67/8.29  % (32756)Time elapsed: 0.327 s
% 27.67/8.29  % (32756)Instructions burned: 770 (million)
% 27.67/8.29  % (32756)------------------------------
% 27.67/8.29  % (32756)------------------------------
% 27.79/8.31  % (335)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_32338 for (2968ds/434Mi)
% 27.79/8.38  % (332)Instruction limit reached!
% 27.79/8.38  % (332)------------------------------
% 27.79/8.38  % (332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.79/8.38  % (332)Termination reason: Unknown
% 27.79/8.38  % (332)Termination phase: shuffling
% 27.79/8.38  
% 27.79/8.38  % (332)Memory used [KB]: 18677
% 27.79/8.38  % (332)Time elapsed: 0.164 s
% 27.79/8.38  % (332)Instructions burned: 466 (million)
% 27.79/8.38  % (332)------------------------------
% 27.79/8.38  % (332)------------------------------
% 27.79/8.39  % (336)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_32338 for (2967ds/865Mi)
% 28.49/8.45  % (331)Instruction limit reached!
% 28.49/8.45  % (331)------------------------------
% 28.49/8.45  % (331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.49/8.45  % (331)Termination reason: Unknown
% 28.49/8.45  % (331)Termination phase: Property scanning
% 28.49/8.45  
% 28.49/8.45  % (331)Memory used [KB]: 20980
% 28.49/8.45  % (331)Time elapsed: 0.263 s
% 28.49/8.45  % (331)Instructions burned: 795 (million)
% 28.49/8.45  % (331)------------------------------
% 28.49/8.45  % (331)------------------------------
% 28.49/8.46  % (335)Instruction limit reached!
% 28.49/8.46  % (335)------------------------------
% 28.49/8.46  % (335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.49/8.46  % (335)Termination reason: Unknown
% 28.49/8.46  % (335)Termination phase: shuffling
% 28.49/8.46  
% 28.49/8.46  % (335)Memory used [KB]: 18421
% 28.49/8.46  % (335)Time elapsed: 0.155 s
% 28.49/8.46  % (335)Instructions burned: 434 (million)
% 28.49/8.46  % (335)------------------------------
% 28.49/8.46  % (335)------------------------------
% 28.49/8.46  % (337)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_32338 for (2966ds/593Mi)
% 28.49/8.47  % (338)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_32338 for (2966ds/443Mi)
% 29.14/8.59  % (334)Instruction limit reached!
% 29.14/8.59  % (334)------------------------------
% 29.14/8.59  % (334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.14/8.59  % (334)Termination reason: Unknown
% 29.14/8.59  % (334)Termination phase: Property scanning
% 29.14/8.59  
% 29.14/8.59  % (334)Memory used [KB]: 20980
% 29.14/8.59  % (334)Time elapsed: 0.296 s
% 29.14/8.59  % (334)Instructions burned: 906 (million)
% 29.14/8.59  % (334)------------------------------
% 29.14/8.59  % (334)------------------------------
% 29.66/8.60  % (339)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_32338 for (2965ds/594Mi)
% 29.66/8.63  % (338)Instruction limit reached!
% 29.66/8.63  % (338)------------------------------
% 29.66/8.63  % (338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.66/8.63  % (338)Termination reason: Unknown
% 29.66/8.63  % (338)Termination phase: shuffling
% 29.66/8.63  
% 29.66/8.63  % (338)Memory used [KB]: 18293
% 29.66/8.63  % (338)Time elapsed: 0.161 s
% 29.66/8.63  % (338)Instructions burned: 443 (million)
% 29.66/8.63  % (338)------------------------------
% 29.66/8.63  % (338)------------------------------
% 29.66/8.63  % (333)Instruction limit reached!
% 29.66/8.63  % (333)------------------------------
% 29.66/8.63  % (333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.66/8.63  % (333)Termination reason: Unknown
% 29.66/8.63  % (333)Termination phase: shuffling
% 29.66/8.63  
% 29.66/8.63  % (333)Memory used [KB]: 37483
% 29.66/8.63  % (333)Time elapsed: 0.405 s
% 29.66/8.63  % (333)Instructions burned: 1379 (million)
% 29.66/8.63  % (333)------------------------------
% 29.66/8.63  % (333)------------------------------
% 29.66/8.64  % (337)Instruction limit reached!
% 29.66/8.64  % (337)------------------------------
% 29.66/8.64  % (337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.66/8.64  % (337)Termination reason: Unknown
% 29.66/8.64  % (337)Termination phase: shuffling
% 29.66/8.64  
% 29.66/8.64  % (337)Memory used [KB]: 22771
% 29.66/8.64  % (337)Time elapsed: 0.181 s
% 29.66/8.64  % (337)Instructions burned: 596 (million)
% 29.66/8.64  % (337)------------------------------
% 29.66/8.64  % (337)------------------------------
% 29.66/8.64  % (340)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_32338 for (2965ds/327Mi)
% 29.66/8.64  % (336)Instruction limit reached!
% 29.66/8.64  % (336)------------------------------
% 29.66/8.64  % (336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.66/8.64  % (336)Termination reason: Unknown
% 29.66/8.64  % (336)Termination phase: shuffling
% 29.66/8.64  
% 29.66/8.64  % (336)Memory used [KB]: 26481
% 29.66/8.64  % (336)Time elapsed: 0.256 s
% 29.66/8.64  % (336)Instructions burned: 865 (million)
% 29.66/8.64  % (336)------------------------------
% 29.66/8.64  % (336)------------------------------
% 29.66/8.65  % (341)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_32338 for (2965ds/5411Mi)
% 29.66/8.65  % (342)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_32338 for (2964ds/1762Mi)
% 29.66/8.66  % (343)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_32338 for (2964ds/96Mi)
% 29.66/8.69  % (343)Instruction limit reached!
% 29.66/8.69  % (343)------------------------------
% 29.66/8.69  % (343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.66/8.69  % (343)Termination reason: Unknown
% 29.66/8.69  % (343)Termination phase: shuffling
% 29.66/8.69  
% 29.66/8.69  % (343)Memory used [KB]: 13688
% 29.66/8.69  % (343)Time elapsed: 0.035 s
% 29.66/8.69  % (343)Instructions burned: 97 (million)
% 29.66/8.69  % (343)------------------------------
% 29.66/8.69  % (343)------------------------------
% 30.96/8.70  % (344)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_32338 for (2964ds/503Mi)
% 30.96/8.76  % (340)Instruction limit reached!
% 30.96/8.76  % (340)------------------------------
% 30.96/8.76  % (340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.96/8.76  % (340)Termination reason: Unknown
% 30.96/8.76  % (340)Termination phase: shuffling
% 30.96/8.76  
% 30.96/8.76  % (340)Memory used [KB]: 16886
% 30.96/8.76  % (340)Time elapsed: 0.119 s
% 30.96/8.76  % (340)Instructions burned: 328 (million)
% 30.96/8.76  % (340)------------------------------
% 30.96/8.76  % (340)------------------------------
% 30.96/8.77  % (345)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_32338 for (2963ds/90Mi)
% 31.29/8.81  % (339)Instruction limit reached!
% 31.29/8.81  % (339)------------------------------
% 31.29/8.81  % (339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.29/8.81  % (339)Termination reason: Unknown
% 31.29/8.81  % (339)Termination phase: shuffling
% 31.29/8.81  
% 31.29/8.81  % (339)Memory used [KB]: 20340
% 31.29/8.81  % (339)Time elapsed: 0.208 s
% 31.29/8.81  % (339)Instructions burned: 594 (million)
% 31.29/8.81  % (339)------------------------------
% 31.29/8.81  % (339)------------------------------
% 31.29/8.81  % (345)Instruction limit reached!
% 31.29/8.81  % (345)------------------------------
% 31.29/8.81  % (345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.29/8.81  % (345)Termination reason: Unknown
% 31.29/8.81  % (345)Termination phase: shuffling
% 31.29/8.81  
% 31.29/8.81  % (345)Memory used [KB]: 13560
% 31.29/8.81  % (345)Time elapsed: 0.038 s
% 31.29/8.81  % (345)Instructions burned: 90 (million)
% 31.29/8.81  % (345)------------------------------
% 31.29/8.81  % (345)------------------------------
% 31.29/8.82  % (346)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_32338 for (2963ds/593Mi)
% 31.29/8.82  % (347)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_32338 for (2963ds/1512Mi)
% 31.29/8.88  % (344)Instruction limit reached!
% 31.29/8.88  % (344)------------------------------
% 31.29/8.88  % (344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.29/8.88  % (344)Termination reason: Unknown
% 31.29/8.88  % (344)Termination phase: shuffling
% 31.29/8.88  
% 31.29/8.88  % (344)Memory used [KB]: 19189
% 31.29/8.88  % (344)Time elapsed: 0.175 s
% 31.29/8.88  % (344)Instructions burned: 505 (million)
% 31.29/8.88  % (344)------------------------------
% 31.29/8.88  % (344)------------------------------
% 31.29/8.89  % (348)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_32338 for (2962ds/569Mi)
% 31.94/8.96  % (32653)Instruction limit reached!
% 31.94/8.96  % (32653)------------------------------
% 31.94/8.96  % (32653)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.94/8.96  % (32653)Termination reason: Unknown
% 31.94/8.96  % (32653)Termination phase: Function definition elimination
% 31.94/8.96  
% 31.94/8.96  % (32653)Memory used [KB]: 45542
% 31.94/8.96  % (32653)Time elapsed: 1.244 s
% 31.94/8.96  % (32653)Instructions burned: 4716 (million)
% 31.94/8.96  % (32653)------------------------------
% 31.94/8.96  % (32653)------------------------------
% 31.94/8.98  % (349)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_32338 for (2961ds/501Mi)
% 33.39/9.03  % (346)Instruction limit reached!
% 33.39/9.03  % (346)------------------------------
% 33.39/9.03  % (346)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.39/9.03  % (346)Termination reason: Unknown
% 33.39/9.03  % (346)Termination phase: shuffling
% 33.39/9.03  
% 33.39/9.03  % (346)Memory used [KB]: 20340
% 33.39/9.03  % (346)Time elapsed: 0.210 s
% 33.39/9.03  % (346)Instructions burned: 593 (million)
% 33.39/9.03  % (346)------------------------------
% 33.39/9.03  % (346)------------------------------
% 33.39/9.04  % (350)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_32338 for (2961ds/562Mi)
% 33.39/9.06  % (348)Instruction limit reached!
% 33.39/9.06  % (348)------------------------------
% 33.39/9.06  % (348)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.39/9.06  % (348)Termination reason: Unknown
% 33.39/9.06  % (348)Termination phase: shuffling
% 33.39/9.06  
% 33.39/9.06  % (348)Memory used [KB]: 22387
% 33.39/9.06  % (348)Time elapsed: 0.172 s
% 33.39/9.06  % (348)Instructions burned: 571 (million)
% 33.39/9.06  % (348)------------------------------
% 33.39/9.06  % (348)------------------------------
% 33.39/9.07  % (351)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_32338 for (2960ds/406Mi)
% 34.07/9.15  % (349)Instruction limit reached!
% 34.07/9.15  % (349)------------------------------
% 34.07/9.15  % (349)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.07/9.15  % (349)Termination reason: Unknown
% 34.07/9.15  % (349)Termination phase: shuffling
% 34.07/9.15  
% 34.07/9.15  % (349)Memory used [KB]: 18933
% 34.07/9.15  % (349)Time elapsed: 0.177 s
% 34.07/9.15  % (349)Instructions burned: 502 (million)
% 34.07/9.15  % (349)------------------------------
% 34.07/9.15  % (349)------------------------------
% 34.07/9.16  % (342)Instruction limit reached!
% 34.07/9.16  % (342)------------------------------
% 34.07/9.16  % (342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.07/9.16  % (342)Termination reason: Unknown
% 34.07/9.16  % (342)Termination phase: Property scanning
% 34.07/9.16  
% 34.07/9.16  % (342)Memory used [KB]: 37739
% 34.07/9.16  % (342)Time elapsed: 0.503 s
% 34.07/9.16  % (342)Instructions burned: 1762 (million)
% 34.07/9.16  % (342)------------------------------
% 34.07/9.16  % (342)------------------------------
% 34.07/9.16  % (352)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_32338 for (2959ds/582Mi)
% 34.07/9.17  % (353)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_32338 for (2959ds/456Mi)
% 34.37/9.21  % (350)Instruction limit reached!
% 34.37/9.21  % (350)------------------------------
% 34.37/9.21  % (350)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.37/9.21  % (350)Termination reason: Unknown
% 34.37/9.21  % (350)Termination phase: shuffling
% 34.37/9.21  
% 34.37/9.21  % (350)Memory used [KB]: 22259
% 34.37/9.21  % (350)Time elapsed: 0.169 s
% 34.37/9.21  % (350)Instructions burned: 562 (million)
% 34.37/9.21  % (350)------------------------------
% 34.37/9.21  % (350)------------------------------
% 34.37/9.21  % (351)Instruction limit reached!
% 34.37/9.21  % (351)------------------------------
% 34.37/9.21  % (351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.37/9.21  % (351)Termination reason: Unknown
% 34.37/9.21  % (351)Termination phase: shuffling
% 34.37/9.21  
% 34.37/9.21  % (351)Memory used [KB]: 17654
% 34.37/9.21  % (351)Time elapsed: 0.142 s
% 34.37/9.21  % (351)Instructions burned: 406 (million)
% 34.37/9.21  % (351)------------------------------
% 34.37/9.21  % (351)------------------------------
% 34.37/9.22  % (354)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_32338 for (2959ds/393Mi)
% 34.37/9.22  % (355)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_32338 for (2959ds/1335Mi)
% 35.00/9.30  % (347)Instruction limit reached!
% 35.00/9.30  % (347)------------------------------
% 35.00/9.30  % (347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.00/9.30  % (347)Termination reason: Unknown
% 35.00/9.30  % (347)Termination phase: SInE selection
% 35.00/9.30  
% 35.00/9.30  % (347)Memory used [KB]: 26097
% 35.00/9.30  % (347)Time elapsed: 0.483 s
% 35.00/9.30  % (347)Instructions burned: 1514 (million)
% 35.00/9.30  % (347)------------------------------
% 35.00/9.30  % (347)------------------------------
% 35.00/9.32  % (356)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_32338 for (2958ds/3373Mi)
% 35.00/9.33  % (353)Instruction limit reached!
% 35.00/9.33  % (353)------------------------------
% 35.00/9.33  % (353)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.00/9.33  % (353)Termination reason: Unknown
% 35.00/9.33  % (353)Termination phase: shuffling
% 35.00/9.33  
% 35.00/9.33  % (353)Memory used [KB]: 18549
% 35.00/9.33  % (353)Time elapsed: 0.162 s
% 35.00/9.33  % (353)Instructions burned: 456 (million)
% 35.00/9.33  % (353)------------------------------
% 35.00/9.33  % (353)------------------------------
% 35.00/9.34  % (354)Instruction limit reached!
% 35.00/9.34  % (354)------------------------------
% 35.00/9.34  % (354)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.00/9.34  % (354)Termination reason: Unknown
% 35.00/9.34  % (354)Termination phase: shuffling
% 35.00/9.34  
% 35.00/9.34  % (354)Memory used [KB]: 19957
% 35.00/9.34  % (354)Time elapsed: 0.122 s
% 35.00/9.34  % (354)Instructions burned: 395 (million)
% 35.00/9.34  % (354)------------------------------
% 35.00/9.34  % (354)------------------------------
% 35.00/9.34  % (357)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_32338 for (2958ds/474Mi)
% 35.00/9.35  % (358)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_32338 for (2957ds/157Mi)
% 35.00/9.37  % (352)Instruction limit reached!
% 35.00/9.37  % (352)------------------------------
% 35.00/9.37  % (352)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.00/9.37  % (352)Termination reason: Unknown
% 35.00/9.37  % (352)Termination phase: shuffling
% 35.00/9.37  
% 35.00/9.37  % (352)Memory used [KB]: 20468
% 35.00/9.37  % (352)Time elapsed: 0.210 s
% 35.00/9.37  % (352)Instructions burned: 583 (million)
% 35.00/9.37  % (352)------------------------------
% 35.00/9.37  % (352)------------------------------
% 35.00/9.38  % (359)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_32338 for (2957ds/590Mi)
% 35.75/9.40  % (358)Instruction limit reached!
% 35.75/9.40  % (358)------------------------------
% 35.75/9.40  % (358)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.75/9.40  % (358)Termination reason: Unknown
% 35.75/9.40  % (358)Termination phase: shuffling
% 35.75/9.40  
% 35.75/9.40  % (358)Memory used [KB]: 14583
% 35.75/9.40  % (358)Time elapsed: 0.053 s
% 35.75/9.40  % (358)Instructions burned: 159 (million)
% 35.75/9.40  % (358)------------------------------
% 35.75/9.40  % (358)------------------------------
% 35.75/9.42  % (360)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_32338 for (2957ds/104Mi)
% 35.75/9.46  % (360)Instruction limit reached!
% 35.75/9.46  % (360)------------------------------
% 35.75/9.46  % (360)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.75/9.46  % (360)Termination reason: Unknown
% 35.75/9.46  % (360)Termination phase: shuffling
% 35.75/9.46  
% 35.75/9.46  % (360)Memory used [KB]: 13816
% 35.75/9.46  % (360)Time elapsed: 0.043 s
% 35.75/9.46  % (360)Instructions burned: 107 (million)
% 35.75/9.46  % (360)------------------------------
% 35.75/9.46  % (360)------------------------------
% 35.75/9.47  % (361)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_32338 for (2956ds/414Mi)
% 35.75/9.48  % (357)Instruction limit reached!
% 35.75/9.48  % (357)------------------------------
% 35.75/9.48  % (357)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.75/9.48  % (357)Termination reason: Unknown
% 35.75/9.48  % (357)Termination phase: shuffling
% 35.75/9.48  
% 35.75/9.48  % (357)Memory used [KB]: 21108
% 35.75/9.48  % (357)Time elapsed: 0.142 s
% 35.75/9.48  % (357)Instructions burned: 477 (million)
% 35.75/9.48  % (357)------------------------------
% 35.75/9.48  % (357)------------------------------
% 35.75/9.50  % (362)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_32338 for (2956ds/715Mi)
% 36.22/9.59  % (359)Instruction limit reached!
% 36.22/9.59  % (359)------------------------------
% 36.22/9.59  % (359)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.22/9.59  % (359)Termination reason: Unknown
% 36.22/9.59  % (359)Termination phase: shuffling
% 36.22/9.59  
% 36.22/9.59  % (359)Memory used [KB]: 20340
% 36.22/9.59  % (359)Time elapsed: 0.205 s
% 36.22/9.59  % (359)Instructions burned: 591 (million)
% 36.22/9.59  % (359)------------------------------
% 36.22/9.59  % (359)------------------------------
% 36.22/9.59  % (361)Instruction limit reached!
% 36.22/9.59  % (361)------------------------------
% 36.22/9.59  % (361)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.22/9.59  % (361)Termination reason: Unknown
% 36.22/9.59  % (361)Termination phase: shuffling
% 36.22/9.59  
% 36.22/9.59  % (361)Memory used [KB]: 20340
% 36.22/9.59  % (361)Time elapsed: 0.125 s
% 36.22/9.59  % (361)Instructions burned: 415 (million)
% 36.22/9.59  % (361)------------------------------
% 36.22/9.59  % (361)------------------------------
% 36.77/9.60  % (363)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_32338 for (2955ds/961Mi)
% 36.77/9.60  % (364)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_32338 for (2955ds/1213Mi)
% 36.77/9.61  % (355)Instruction limit reached!
% 36.77/9.61  % (355)------------------------------
% 36.77/9.61  % (355)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.77/9.61  % (355)Termination reason: Unknown
% 36.77/9.61  % (355)Termination phase: shuffling
% 36.77/9.61  
% 36.77/9.61  % (355)Memory used [KB]: 36843
% 36.77/9.61  % (355)Time elapsed: 0.391 s
% 36.77/9.61  % (355)Instructions burned: 1337 (million)
% 36.77/9.61  % (355)------------------------------
% 36.77/9.61  % (355)------------------------------
% 36.77/9.63  % (365)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_32338 for (2955ds/1979Mi)
% 37.37/9.70  % (341)First to succeed.
% 37.37/9.71  % (341)Refutation found. Thanks to Tanya!
% 37.37/9.71  % SZS status Theorem for DTF2THF_32338
% 37.37/9.71  % SZS output start Proof for DTF2THF_32338
% 37.37/9.71  thf(type_def_5, type, rat: $tType).
% 37.37/9.71  thf(type_def_6, type, int: $tType).
% 37.37/9.71  thf(type_def_8, type, real: $tType).
% 37.37/9.71  thf(type_def_9, type, nat: $tType).
% 37.37/9.71  thf(type_def_10, type, complex: $tType).
% 37.37/9.71  thf(type_def_11, type, product_prod_int_int: $tType).
% 37.37/9.71  thf(type_def_12, type, num: $tType).
% 37.37/9.71  thf(type_def_13, type, option_num: $tType).
% 37.37/9.71  thf(type_def_14, type, product_prod_num_num: $tType).
% 37.37/9.71  thf(type_def_15, type, code_integer: $tType).
% 37.37/9.71  thf(type_def_16, type, produc6271795597528267376eger_o: $tType).
% 37.37/9.71  thf(type_def_17, type, produc8923325533196201883nteger: $tType).
% 37.37/9.71  thf(type_def_18, type, set_fi4554929511873752355omplex: $tType).
% 37.37/9.71  thf(type_def_19, type, filter6041513312241820739omplex: $tType).
% 37.37/9.71  thf(type_def_20, type, set_fi7789364187291644575l_real: $tType).
% 37.37/9.71  thf(type_def_21, type, filter2146258269922977983l_real: $tType).
% 37.37/9.71  thf(type_def_22, type, set_real: $tType).
% 37.37/9.71  thf(type_def_23, type, set_set_nat: $tType).
% 37.37/9.71  thf(type_def_24, type, set_nat: $tType).
% 37.37/9.71  thf(type_def_25, type, set_int: $tType).
% 37.37/9.71  thf(type_def_26, type, filter_real: $tType).
% 37.37/9.71  thf(type_def_27, type, product_prod_nat_nat: $tType).
% 37.37/9.71  thf(type_def_28, type, filter_int: $tType).
% 37.37/9.71  thf(type_def_29, type, filter_nat: $tType).
% 37.37/9.71  thf(type_def_30, type, set_Pr5085853215250843933omplex: $tType).
% 37.37/9.71  thf(type_def_31, type, set_Pr6218003697084177305l_real: $tType).
% 37.37/9.71  thf(type_def_32, type, set_o: $tType).
% 37.37/9.71  thf(type_def_33, type, set_complex: $tType).
% 37.37/9.71  thf(type_def_34, type, set_Product_unit: $tType).
% 37.37/9.71  thf(type_def_35, type, set_char: $tType).
% 37.37/9.71  thf(type_def_36, type, char: $tType).
% 37.37/9.71  thf(type_def_37, type, extended_enat: $tType).
% 37.37/9.71  thf(type_def_38, type, set_rat: $tType).
% 37.37/9.71  thf(type_def_39, type, set_VEBT_VEBT: $tType).
% 37.37/9.71  thf(type_def_40, type, list_o: $tType).
% 37.37/9.71  thf(type_def_41, type, list_int: $tType).
% 37.37/9.71  thf(type_def_42, type, set_Pr958786334691620121nt_int: $tType).
% 37.37/9.71  thf(type_def_43, type, set_Code_integer: $tType).
% 37.37/9.71  thf(type_def_44, type, list_nat: $tType).
% 37.37/9.71  thf(type_def_45, type, list_complex: $tType).
% 37.37/9.71  thf(type_def_46, type, list_real: $tType).
% 37.37/9.71  thf(type_def_47, type, list_set_nat: $tType).
% 37.37/9.71  thf(type_def_48, type, list_VEBT_VEBT: $tType).
% 37.37/9.71  thf(type_def_49, type, vEBT_VEBT: $tType).
% 37.37/9.71  thf(type_def_50, type, list_Code_integer: $tType).
% 37.37/9.71  thf(type_def_51, type, list_num: $tType).
% 37.37/9.71  thf(type_def_52, type, list_P8526636022914148096eger_o: $tType).
% 37.37/9.71  thf(type_def_53, type, list_P7333126701944960589_nat_o: $tType).
% 37.37/9.71  thf(type_def_54, type, product_prod_nat_o: $tType).
% 37.37/9.71  thf(type_def_55, type, list_P3521021558325789923at_int: $tType).
% 37.37/9.71  thf(type_def_56, type, product_prod_nat_int: $tType).
% 37.37/9.71  thf(type_def_57, type, list_P6011104703257516679at_nat: $tType).
% 37.37/9.71  thf(type_def_58, type, list_P1726324292696863441at_num: $tType).
% 37.37/9.71  thf(type_def_59, type, product_prod_nat_num: $tType).
% 37.37/9.71  thf(type_def_60, type, list_P5647936690300460905T_VEBT: $tType).
% 37.37/9.71  thf(type_def_61, type, produc8025551001238799321T_VEBT: $tType).
% 37.37/9.71  thf(type_def_62, type, list_P3744719386663036955um_num: $tType).
% 37.37/9.71  thf(type_def_63, type, list_P3126845725202233233VEBT_o: $tType).
% 37.37/9.71  thf(type_def_64, type, produc334124729049499915VEBT_o: $tType).
% 37.37/9.71  thf(type_def_65, type, list_P7037539587688870467BT_nat: $tType).
% 37.37/9.71  thf(type_def_66, type, produc9072475918466114483BT_nat: $tType).
% 37.37/9.71  thf(type_def_67, type, list_P7413028617227757229T_VEBT: $tType).
% 37.37/9.71  thf(type_def_68, type, produc8243902056947475879T_VEBT: $tType).
% 37.37/9.71  thf(type_def_69, type, list_P4002435161011370285od_o_o: $tType).
% 37.37/9.71  thf(type_def_70, type, list_P3795440434834930179_o_int: $tType).
% 37.37/9.71  thf(type_def_71, type, list_P7495141550334521929T_VEBT: $tType).
% 37.37/9.71  thf(type_def_72, type, list_P5087981734274514673_int_o: $tType).
% 37.37/9.71  thf(type_def_73, type, list_P5707943133018811711nt_int: $tType).
% 37.37/9.71  thf(type_def_74, type, list_P7524865323317820941T_VEBT: $tType).
% 37.37/9.71  thf(type_def_75, type, list_P4547456442757143711BT_int: $tType).
% 37.37/9.71  thf(type_def_76, type, option4927543243414619207at_nat: $tType).
% 37.37/9.71  thf(type_def_77, type, produc4411394909380815293omplex: $tType).
% 37.37/9.71  thf(type_def_78, type, produc2422161461964618553l_real: $tType).
% 37.37/9.71  thf(type_def_79, type, set_Pr1261947904930325089at_nat: $tType).
% 37.37/9.71  thf(type_def_80, type, set_num: $tType).
% 37.37/9.71  thf(type_def_81, type, set_option_num: $tType).
% 37.37/9.71  thf(type_def_82, type, set_op4508134149509766951at_nat: $tType).
% 37.37/9.71  thf(type_def_83, type, set_set_o: $tType).
% 37.37/9.71  thf(type_def_84, type, set_set_complex: $tType).
% 37.37/9.71  thf(type_def_85, type, set_set_int: $tType).
% 37.37/9.71  thf(type_def_86, type, set_set_real: $tType).
% 37.37/9.71  thf(type_def_87, type, set_set_set_nat: $tType).
% 37.37/9.71  thf(type_def_88, type, set_set_VEBT_VEBT: $tType).
% 37.37/9.71  thf(func_def_0, type, set_fi4554929511873752355omplex: $tType).
% 37.37/9.71  thf(func_def_1, type, list_P7413028617227757229T_VEBT: $tType).
% 37.37/9.71  thf(func_def_2, type, set_fi7789364187291644575l_real: $tType).
% 37.37/9.71  thf(func_def_3, type, set_op4508134149509766951at_nat: $tType).
% 37.37/9.71  thf(func_def_4, type, filter6041513312241820739omplex: $tType).
% 37.37/9.71  thf(func_def_5, type, list_P7037539587688870467BT_nat: $tType).
% 37.37/9.71  thf(func_def_6, type, list_P4547456442757143711BT_int: $tType).
% 37.37/9.71  thf(func_def_7, type, list_P5647936690300460905T_VEBT: $tType).
% 37.37/9.71  thf(func_def_8, type, list_P7524865323317820941T_VEBT: $tType).
% 37.37/9.71  thf(func_def_9, type, produc8243902056947475879T_VEBT: $tType).
% 37.37/9.71  thf(func_def_10, type, set_Pr5085853215250843933omplex: $tType).
% 37.37/9.71  thf(func_def_11, type, produc8923325533196201883nteger: $tType).
% 37.37/9.71  thf(func_def_12, type, list_P3126845725202233233VEBT_o: $tType).
% 37.37/9.71  thf(func_def_13, type, list_P7495141550334521929T_VEBT: $tType).
% 37.37/9.71  thf(func_def_14, type, filter2146258269922977983l_real: $tType).
% 37.37/9.71  thf(func_def_15, type, list_P8526636022914148096eger_o: $tType).
% 37.37/9.71  thf(func_def_16, type, option4927543243414619207at_nat: $tType).
% 37.37/9.71  thf(func_def_17, type, set_Pr6218003697084177305l_real: $tType).
% 37.37/9.71  thf(func_def_18, type, list_P3744719386663036955um_num: $tType).
% 37.37/9.71  thf(func_def_19, type, list_P1726324292696863441at_num: $tType).
% 37.37/9.71  thf(func_def_20, type, list_P6011104703257516679at_nat: $tType).
% 37.37/9.71  thf(func_def_21, type, list_P3521021558325789923at_int: $tType).
% 37.37/9.71  thf(func_def_22, type, list_P5707943133018811711nt_int: $tType).
% 37.37/9.71  thf(func_def_23, type, produc9072475918466114483BT_nat: $tType).
% 37.37/9.71  thf(func_def_24, type, produc8025551001238799321T_VEBT: $tType).
% 37.37/9.71  thf(func_def_25, type, set_Pr1261947904930325089at_nat: $tType).
% 37.37/9.71  thf(func_def_26, type, set_Pr958786334691620121nt_int: $tType).
% 37.37/9.71  thf(func_def_27, type, produc4411394909380815293omplex: $tType).
% 37.37/9.71  thf(func_def_28, type, list_P7333126701944960589_nat_o: $tType).
% 37.37/9.71  thf(func_def_29, type, list_P5087981734274514673_int_o: $tType).
% 37.37/9.71  thf(func_def_30, type, list_P3795440434834930179_o_int: $tType).
% 37.37/9.71  thf(func_def_31, type, produc334124729049499915VEBT_o: $tType).
% 37.37/9.71  thf(func_def_32, type, produc6271795597528267376eger_o: $tType).
% 37.37/9.71  thf(func_def_33, type, set_set_VEBT_VEBT: $tType).
% 37.37/9.71  thf(func_def_34, type, set_set_set_nat: $tType).
% 37.37/9.71  thf(func_def_35, type, produc2422161461964618553l_real: $tType).
% 37.37/9.71  thf(func_def_36, type, product_prod_num_num: $tType).
% 37.37/9.71  thf(func_def_37, type, product_prod_nat_num: $tType).
% 37.37/9.71  thf(func_def_38, type, product_prod_nat_nat: $tType).
% 37.37/9.71  thf(func_def_39, type, product_prod_nat_int: $tType).
% 37.37/9.71  thf(func_def_40, type, product_prod_int_int: $tType).
% 37.37/9.71  thf(func_def_41, type, list_P4002435161011370285od_o_o: $tType).
% 37.37/9.71  thf(func_def_42, type, set_set_complex: $tType).
% 37.37/9.71  thf(func_def_43, type, set_option_num: $tType).
% 37.37/9.71  thf(func_def_44, type, list_VEBT_VEBT: $tType).
% 37.37/9.71  thf(func_def_45, type, set_set_real: $tType).
% 37.37/9.71  thf(func_def_46, type, product_prod_nat_o: $tType).
% 37.37/9.71  thf(func_def_47, type, list_set_nat: $tType).
% 37.37/9.71  thf(func_def_48, type, list_Code_integer: $tType).
% 37.37/9.71  thf(func_def_49, type, set_VEBT_VEBT: $tType).
% 37.37/9.71  thf(func_def_50, type, set_set_nat: $tType).
% 37.37/9.71  thf(func_def_51, type, set_set_int: $tType).
% 37.37/9.71  thf(func_def_52, type, set_Code_integer: $tType).
% 37.37/9.71  thf(func_def_53, type, set_Product_unit: $tType).
% 37.37/9.71  thf(func_def_54, type, list_complex: $tType).
% 37.37/9.71  thf(func_def_55, type, set_complex: $tType).
% 37.37/9.71  thf(func_def_56, type, filter_real: $tType).
% 37.37/9.71  thf(func_def_57, type, set_set_o: $tType).
% 37.37/9.71  thf(func_def_58, type, option_num: $tType).
% 37.37/9.71  thf(func_def_59, type, filter_nat: $tType).
% 37.37/9.71  thf(func_def_60, type, filter_int: $tType).
% 37.37/9.71  thf(func_def_61, type, set_char: $tType).
% 37.37/9.71  thf(func_def_62, type, list_real: $tType).
% 37.37/9.71  thf(func_def_63, type, set_real: $tType).
% 37.37/9.71  thf(func_def_64, type, list_num: $tType).
% 37.37/9.71  thf(func_def_65, type, list_nat: $tType).
% 37.37/9.71  thf(func_def_66, type, list_int: $tType).
% 37.37/9.71  thf(func_def_67, type, vEBT_VEBT: $tType).
% 37.37/9.71  thf(func_def_68, type, set_rat: $tType).
% 37.37/9.71  thf(func_def_69, type, set_num: $tType).
% 37.37/9.71  thf(func_def_70, type, set_nat: $tType).
% 37.37/9.71  thf(func_def_71, type, set_int: $tType).
% 37.37/9.71  thf(func_def_72, type, code_integer: $tType).
% 37.37/9.71  thf(func_def_73, type, extended_enat: $tType).
% 37.37/9.71  thf(func_def_74, type, list_o: $tType).
% 37.37/9.71  thf(func_def_75, type, complex: $tType).
% 37.37/9.71  thf(func_def_76, type, set_o: $tType).
% 37.37/9.71  thf(func_def_77, type, char: $tType).
% 37.37/9.71  thf(func_def_78, type, real: $tType).
% 37.37/9.71  thf(func_def_79, type, rat: $tType).
% 37.37/9.71  thf(func_def_80, type, num: $tType).
% 37.37/9.71  thf(func_def_81, type, nat: $tType).
% 37.37/9.71  thf(func_def_82, type, int: $tType).
% 37.37/9.71  thf(func_def_83, type, archim2889992004027027881ng_rat: rat > int).
% 37.37/9.71  thf(func_def_84, type, archim7802044766580827645g_real: real > int).
% 37.37/9.71  thf(func_def_85, type, archim3151403230148437115or_rat: rat > int).
% 37.37/9.71  thf(func_def_86, type, archim6058952711729229775r_real: real > int).
% 37.37/9.71  thf(func_def_87, type, archimedean_frac_rat: rat > rat).
% 37.37/9.71  thf(func_def_88, type, archim2898591450579166408c_real: real > real).
% 37.37/9.71  thf(func_def_89, type, archim7778729529865785530nd_rat: rat > int).
% 37.37/9.71  thf(func_def_90, type, archim8280529875227126926d_real: real > int).
% 37.37/9.71  thf(func_def_91, type, binomial: nat > nat > nat).
% 37.37/9.71  thf(func_def_92, type, gbinomial_complex: complex > nat > complex).
% 37.37/9.71  thf(func_def_93, type, gbinomial_int: int > nat > int).
% 37.37/9.71  thf(func_def_94, type, gbinomial_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_95, type, gbinomial_rat: rat > nat > rat).
% 37.37/9.71  thf(func_def_96, type, gbinomial_real: real > nat > real).
% 37.37/9.71  thf(func_def_97, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 37.37/9.71  thf(func_def_98, type, bit_and_not_num: num > num > option_num).
% 37.37/9.71  thf(func_def_99, type, bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o).
% 37.37/9.71  thf(func_def_100, type, bit_concat_bit: nat > int > int > int).
% 37.37/9.71  thf(func_def_101, type, bit_or_not_num_neg: num > num > num).
% 37.37/9.71  thf(func_def_102, type, bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o).
% 37.37/9.71  thf(func_def_103, type, bit_ri7632146776885996613nteger: code_integer > code_integer).
% 37.37/9.71  thf(func_def_104, type, bit_ri7919022796975470100ot_int: int > int).
% 37.37/9.71  thf(func_def_105, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 37.37/9.71  thf(func_def_106, type, bit_ri631733984087533419it_int: nat > int > int).
% 37.37/9.71  thf(func_def_107, type, bit_se3949692690581998587nteger: code_integer > code_integer > code_integer).
% 37.37/9.71  thf(func_def_108, type, bit_se725231765392027082nd_int: int > int > int).
% 37.37/9.71  thf(func_def_109, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_110, type, bit_se8568078237143864401it_int: nat > int > int).
% 37.37/9.71  thf(func_def_111, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_112, type, bit_se1345352211410354436nteger: nat > code_integer > code_integer).
% 37.37/9.71  thf(func_def_113, type, bit_se2159334234014336723it_int: nat > int > int).
% 37.37/9.71  thf(func_def_114, type, bit_se2161824704523386999it_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_115, type, bit_se2119862282449309892nteger: nat > code_integer).
% 37.37/9.71  thf(func_def_116, type, bit_se2000444600071755411sk_int: nat > int).
% 37.37/9.71  thf(func_def_117, type, bit_se2002935070580805687sk_nat: nat > nat).
% 37.37/9.71  thf(func_def_118, type, bit_se1409905431419307370or_int: int > int > int).
% 37.37/9.71  thf(func_def_119, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_120, type, bit_se7788150548672797655nteger: nat > code_integer > code_integer).
% 37.37/9.71  thf(func_def_121, type, bit_se545348938243370406it_int: nat > int > int).
% 37.37/9.71  thf(func_def_122, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_123, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 37.37/9.71  thf(func_def_124, type, bit_se7879613467334960850it_int: nat > int > int).
% 37.37/9.71  thf(func_def_125, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_126, type, bit_se1745604003318907178nteger: nat > code_integer > code_integer).
% 37.37/9.71  thf(func_def_127, type, bit_se2923211474154528505it_int: nat > int > int).
% 37.37/9.71  thf(func_def_128, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_129, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 37.37/9.71  thf(func_def_130, type, bit_se4203085406695923979it_int: nat > int > int).
% 37.37/9.71  thf(func_def_131, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_132, type, bit_se3222712562003087583nteger: code_integer > code_integer > code_integer).
% 37.37/9.71  thf(func_def_133, type, bit_se6526347334894502574or_int: int > int > int).
% 37.37/9.71  thf(func_def_134, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_135, type, bit_se9216721137139052372nteger: code_integer > nat > $o).
% 37.37/9.71  thf(func_def_136, type, bit_se1146084159140164899it_int: int > nat > $o).
% 37.37/9.71  thf(func_def_137, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 37.37/9.71  thf(func_def_138, type, bit_take_bit_num: nat > num > option_num).
% 37.37/9.71  thf(func_def_139, type, bit_un7362597486090784418nd_num: num > num > option_num).
% 37.37/9.71  thf(func_def_140, type, bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o).
% 37.37/9.71  thf(func_def_141, type, bit_un2480387367778600638or_num: num > num > option_num).
% 37.37/9.71  thf(func_def_142, type, bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o).
% 37.37/9.71  thf(func_def_143, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 37.37/9.71  thf(func_def_144, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 37.37/9.71  thf(func_def_145, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 37.37/9.71  thf(func_def_146, type, code_int_of_integer: code_integer > int).
% 37.37/9.71  thf(func_def_147, type, code_integer_of_int: int > code_integer).
% 37.37/9.71  thf(func_def_148, type, code_integer_of_num: num > code_integer).
% 37.37/9.71  thf(func_def_149, type, code_nat_of_integer: code_integer > nat).
% 37.37/9.71  thf(func_def_150, type, code_negative: num > code_integer).
% 37.37/9.71  thf(func_def_151, type, code_positive: num > code_integer).
% 37.37/9.71  thf(func_def_152, type, code_Target_negative: num > int).
% 37.37/9.71  thf(func_def_153, type, code_Target_positive: num > int).
% 37.37/9.71  thf(func_def_154, type, code_T6385005292777649522of_nat: nat > int).
% 37.37/9.71  thf(func_def_155, type, comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex).
% 37.37/9.71  thf(func_def_156, type, comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real).
% 37.37/9.71  thf(func_def_157, type, comple4887499456419720421f_real: set_real > real).
% 37.37/9.71  thf(func_def_158, type, comple7806235888213564991et_nat: set_set_nat > set_nat).
% 37.37/9.71  thf(func_def_159, type, complete_Sup_Sup_int: set_int > int).
% 37.37/9.71  thf(func_def_160, type, complete_Sup_Sup_nat: set_nat > nat).
% 37.37/9.71  thf(func_def_161, type, comple1385675409528146559p_real: set_real > real).
% 37.37/9.71  thf(func_def_162, type, arg: complex > real).
% 37.37/9.71  thf(func_def_163, type, cis: real > complex).
% 37.37/9.71  thf(func_def_164, type, cnj: complex > complex).
% 37.37/9.71  thf(func_def_165, type, complex2: real > real > complex).
% 37.37/9.71  thf(func_def_166, type, im: complex > real).
% 37.37/9.71  thf(func_def_167, type, re: complex > real).
% 37.37/9.71  thf(func_def_168, type, csqrt: complex > complex).
% 37.37/9.71  thf(func_def_169, type, imaginary_unit: complex).
% 37.37/9.71  thf(func_def_170, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 37.37/9.71  thf(func_def_171, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 37.37/9.71  thf(func_def_172, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 37.37/9.71  thf(func_def_173, type, adjust_div: product_prod_int_int > int).
% 37.37/9.71  thf(func_def_174, type, adjust_mod: int > int > int).
% 37.37/9.71  thf(func_def_175, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 37.37/9.71  thf(func_def_176, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 37.37/9.71  thf(func_def_177, type, unique5706413561485394159nteger: produc8923325533196201883nteger > $o).
% 37.37/9.71  thf(func_def_178, type, unique6319869463603278526ux_int: product_prod_int_int > $o).
% 37.37/9.71  thf(func_def_179, type, unique6322359934112328802ux_nat: product_prod_nat_nat > $o).
% 37.37/9.71  thf(func_def_180, type, unique3479559517661332726nteger: num > num > produc8923325533196201883nteger).
% 37.37/9.71  thf(func_def_181, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 37.37/9.71  thf(func_def_182, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 37.37/9.71  thf(func_def_183, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 37.37/9.71  thf(func_def_184, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 37.37/9.71  thf(func_def_185, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 37.37/9.71  thf(func_def_186, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 37.37/9.71  thf(func_def_187, type, comm_s2602460028002588243omplex: complex > nat > complex).
% 37.37/9.71  thf(func_def_188, type, comm_s4660882817536571857er_int: int > nat > int).
% 37.37/9.71  thf(func_def_189, type, comm_s4663373288045622133er_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_190, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 37.37/9.71  thf(func_def_191, type, comm_s7457072308508201937r_real: real > nat > real).
% 37.37/9.71  thf(func_def_192, type, semiri3624122377584611663nteger: nat > code_integer).
% 37.37/9.71  thf(func_def_193, type, semiri5044797733671781792omplex: nat > complex).
% 37.37/9.71  thf(func_def_194, type, semiri1406184849735516958ct_int: nat > int).
% 37.37/9.71  thf(func_def_195, type, semiri1408675320244567234ct_nat: nat > nat).
% 37.37/9.71  thf(func_def_196, type, semiri773545260158071498ct_rat: nat > rat).
% 37.37/9.71  thf(func_def_197, type, semiri2265585572941072030t_real: nat > real).
% 37.37/9.71  thf(func_def_198, type, invers8013647133539491842omplex: complex > complex).
% 37.37/9.71  thf(func_def_199, type, inverse_inverse_rat: rat > rat).
% 37.37/9.71  thf(func_def_200, type, inverse_inverse_real: real > real).
% 37.37/9.71  thf(func_def_201, type, at_bot_real: filter_real).
% 37.37/9.71  thf(func_def_202, type, at_top_int: filter_int).
% 37.37/9.71  thf(func_def_203, type, at_top_nat: filter_nat).
% 37.37/9.71  thf(func_def_204, type, at_top_real: filter_real).
% 37.37/9.71  thf(func_def_205, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 37.37/9.71  thf(func_def_206, type, eventually_real: (real > $o) > filter_real > $o).
% 37.37/9.71  thf(func_def_207, type, filterlim_int_real: (int > real) > filter_real > filter_int > $o).
% 37.37/9.71  thf(func_def_208, type, filterlim_nat_int: (nat > int) > filter_int > filter_nat > $o).
% 37.37/9.71  thf(func_def_209, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 37.37/9.71  thf(func_def_210, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 37.37/9.71  thf(func_def_211, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 37.37/9.71  thf(func_def_212, type, princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex).
% 37.37/9.71  thf(func_def_213, type, princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real).
% 37.37/9.71  thf(func_def_214, type, finite_card_o: set_o > nat).
% 37.37/9.71  thf(func_def_215, type, finite_card_complex: set_complex > nat).
% 37.37/9.71  thf(func_def_216, type, finite_card_int: set_int > nat).
% 37.37/9.71  thf(func_def_217, type, finite_card_nat: set_nat > nat).
% 37.37/9.71  thf(func_def_218, type, finite410649719033368117t_unit: set_Product_unit > nat).
% 37.37/9.71  thf(func_def_219, type, finite_card_set_nat: set_set_nat > nat).
% 37.37/9.71  thf(func_def_220, type, finite_card_char: set_char > nat).
% 37.37/9.71  thf(func_def_221, type, finite3207457112153483333omplex: set_complex > $o).
% 37.37/9.71  thf(func_def_222, type, finite_finite_int: set_int > $o).
% 37.37/9.71  thf(func_def_223, type, finite_finite_nat: set_nat > $o).
% 37.37/9.71  thf(func_def_224, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 37.37/9.71  thf(func_def_225, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 37.37/9.71  thf(func_def_226, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 37.37/9.71  thf(func_def_227, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 37.37/9.71  thf(func_def_228, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 37.37/9.71  thf(func_def_229, type, comp_C3531382070062128313er_num: (code_integer > code_integer) > (num > code_integer) > num > code_integer).
% 37.37/9.71  thf(func_def_230, type, comp_int_int_num: (int > int) > (num > int) > num > int).
% 37.37/9.71  thf(func_def_231, type, comp_int_nat_int: (int > nat) > (int > int) > int > nat).
% 37.37/9.71  thf(func_def_232, type, comp_int_real_real: (int > real) > (real > int) > real > real).
% 37.37/9.71  thf(func_def_233, type, comp_nat_nat_nat: (nat > nat) > (nat > nat) > nat > nat).
% 37.37/9.71  thf(func_def_234, type, comp_nat_real_nat: (nat > real) > (nat > nat) > nat > real).
% 37.37/9.71  thf(func_def_235, type, id_o: $o > $o).
% 37.37/9.71  thf(func_def_236, type, id_nat: nat > nat).
% 37.37/9.71  thf(func_def_237, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 37.37/9.71  thf(func_def_238, type, inj_on_nat_char: (nat > char) > set_nat > $o).
% 37.37/9.71  thf(func_def_239, type, inj_on_real_real: (real > real) > set_real > $o).
% 37.37/9.71  thf(func_def_240, 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 > $o).
% 37.37/9.71  thf(func_def_241, type, map_fu4960017516451851995nt_int: (int > product_prod_nat_nat) > ((product_prod_nat_nat > product_prod_nat_nat) > int > int) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > int > int > int).
% 37.37/9.71  thf(func_def_242, type, map_fu4826362097070443709at_o_o: (int > product_prod_nat_nat) > ($o > $o) > (product_prod_nat_nat > $o) > int > $o).
% 37.37/9.71  thf(func_def_243, type, map_fu2345160673673942751at_nat: (int > product_prod_nat_nat) > (nat > nat) > (product_prod_nat_nat > nat) > int > nat).
% 37.37/9.71  thf(func_def_244, type, map_fu3667384564859982768at_int: (int > product_prod_nat_nat) > (product_prod_nat_nat > int) > (product_prod_nat_nat > product_prod_nat_nat) > int > int).
% 37.37/9.71  thf(func_def_245, type, strict1292158309912662752at_nat: (nat > nat) > set_nat > $o).
% 37.37/9.71  thf(func_def_246, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 37.37/9.71  thf(func_def_247, type, fun_is_measure_int: (int > nat) > $o).
% 37.37/9.71  thf(func_def_248, type, gcd_Gcd_int: set_int > int).
% 37.37/9.71  thf(func_def_249, type, gcd_Gcd_nat: set_nat > nat).
% 37.37/9.71  thf(func_def_250, type, bezw: nat > nat > product_prod_int_int).
% 37.37/9.71  thf(func_def_251, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 37.37/9.71  thf(func_def_252, type, gcd_gcd_Code_integer: code_integer > code_integer > code_integer).
% 37.37/9.71  thf(func_def_253, type, gcd_gcd_int: int > int > int).
% 37.37/9.71  thf(func_def_254, type, gcd_gcd_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_255, type, gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 37.37/9.71  thf(func_def_256, type, abs_abs_Code_integer: code_integer > code_integer).
% 37.37/9.71  thf(func_def_257, type, abs_abs_complex: complex > complex).
% 37.37/9.71  thf(func_def_258, type, abs_abs_int: int > int).
% 37.37/9.71  thf(func_def_259, type, abs_abs_rat: rat > rat).
% 37.37/9.71  thf(func_def_260, type, abs_abs_real: real > real).
% 37.37/9.71  thf(func_def_261, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 37.37/9.71  thf(func_def_262, type, minus_minus_complex: complex > complex > complex).
% 37.37/9.71  thf(func_def_263, type, minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat).
% 37.37/9.71  thf(func_def_264, type, minus_minus_int: int > int > int).
% 37.37/9.71  thf(func_def_265, type, minus_minus_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_266, type, minus_minus_rat: rat > rat > rat).
% 37.37/9.71  thf(func_def_267, type, minus_minus_real: real > real > real).
% 37.37/9.71  thf(func_def_268, type, minus_minus_set_o: set_o > set_o > set_o).
% 37.37/9.71  thf(func_def_269, type, minus_811609699411566653omplex: set_complex > set_complex > set_complex).
% 37.37/9.71  thf(func_def_270, type, minus_minus_set_int: set_int > set_int > set_int).
% 37.37/9.71  thf(func_def_271, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 37.37/9.71  thf(func_def_272, type, minus_minus_set_rat: set_rat > set_rat > set_rat).
% 37.37/9.71  thf(func_def_273, type, minus_minus_set_real: set_real > set_real > set_real).
% 37.37/9.71  thf(func_def_274, type, minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat).
% 37.37/9.71  thf(func_def_275, type, minus_5127226145743854075T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_276, type, one_one_Code_integer: code_integer).
% 37.37/9.71  thf(func_def_277, type, one_one_complex: complex).
% 37.37/9.71  thf(func_def_278, type, one_on7984719198319812577d_enat: extended_enat).
% 37.37/9.71  thf(func_def_279, type, one_one_int: int).
% 37.37/9.71  thf(func_def_280, type, one_one_nat: nat).
% 37.37/9.71  thf(func_def_281, type, one_one_rat: rat).
% 37.37/9.71  thf(func_def_282, type, one_one_real: real).
% 37.37/9.71  thf(func_def_283, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 37.37/9.71  thf(func_def_284, type, plus_plus_complex: complex > complex > complex).
% 37.37/9.71  thf(func_def_285, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 37.37/9.71  thf(func_def_286, type, plus_plus_int: int > int > int).
% 37.37/9.71  thf(func_def_287, type, plus_plus_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_288, type, plus_plus_num: num > num > num).
% 37.37/9.71  thf(func_def_289, type, plus_plus_rat: rat > rat > rat).
% 37.37/9.71  thf(func_def_290, type, plus_plus_real: real > real > real).
% 37.37/9.71  thf(func_def_291, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 37.37/9.71  thf(func_def_292, type, sgn_sgn_complex: complex > complex).
% 37.37/9.71  thf(func_def_293, type, sgn_sgn_int: int > int).
% 37.37/9.71  thf(func_def_294, type, sgn_sgn_rat: rat > rat).
% 37.37/9.71  thf(func_def_295, type, sgn_sgn_real: real > real).
% 37.37/9.71  thf(func_def_296, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 37.37/9.71  thf(func_def_297, type, times_times_complex: complex > complex > complex).
% 37.37/9.71  thf(func_def_298, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 37.37/9.71  thf(func_def_299, type, times_times_int: int > int > int).
% 37.37/9.71  thf(func_def_300, type, times_times_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_301, type, times_times_num: num > num > num).
% 37.37/9.71  thf(func_def_302, type, times_times_rat: rat > rat > rat).
% 37.37/9.71  thf(func_def_303, type, times_times_real: real > real > real).
% 37.37/9.71  thf(func_def_304, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 37.37/9.71  thf(func_def_305, type, uminus1482373934393186551omplex: complex > complex).
% 37.37/9.71  thf(func_def_306, type, uminus_uminus_int: int > int).
% 37.37/9.71  thf(func_def_307, type, uminus_uminus_rat: rat > rat).
% 37.37/9.71  thf(func_def_308, type, uminus_uminus_real: real > real).
% 37.37/9.71  thf(func_def_309, type, uminus_uminus_set_o: set_o > set_o).
% 37.37/9.71  thf(func_def_310, type, uminus8566677241136511917omplex: set_complex > set_complex).
% 37.37/9.71  thf(func_def_311, type, uminus1532241313380277803et_int: set_int > set_int).
% 37.37/9.71  thf(func_def_312, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 37.37/9.71  thf(func_def_313, type, uminus2201863774496077783et_rat: set_rat > set_rat).
% 37.37/9.71  thf(func_def_314, type, uminus612125837232591019t_real: set_real > set_real).
% 37.37/9.71  thf(func_def_315, type, uminus613421341184616069et_nat: set_set_nat > set_set_nat).
% 37.37/9.71  thf(func_def_316, type, uminus8041839845116263051T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_317, type, zero_z3403309356797280102nteger: code_integer).
% 37.37/9.71  thf(func_def_318, type, zero_zero_complex: complex).
% 37.37/9.71  thf(func_def_319, type, zero_z5237406670263579293d_enat: extended_enat).
% 37.37/9.71  thf(func_def_320, type, zero_zero_int: int).
% 37.37/9.71  thf(func_def_321, type, zero_zero_nat: nat).
% 37.37/9.71  thf(func_def_322, type, zero_zero_rat: rat).
% 37.37/9.71  thf(func_def_323, type, zero_zero_real: real).
% 37.37/9.71  thf(func_def_324, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 37.37/9.71  thf(func_def_325, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 37.37/9.71  thf(func_def_326, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 37.37/9.71  thf(func_def_327, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 37.37/9.71  thf(func_def_328, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 37.37/9.71  thf(func_def_329, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 37.37/9.71  thf(func_def_330, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 37.37/9.71  thf(func_def_331, type, groups3417619833198082522nteger: ($o > code_integer) > code_integer > list_o > code_integer).
% 37.37/9.71  thf(func_def_332, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 37.37/9.71  thf(func_def_333, type, groups9119017779487936845_o_nat: ($o > nat) > nat > list_o > nat).
% 37.37/9.71  thf(func_def_334, type, the_int: (int > $o) > int).
% 37.37/9.71  thf(func_def_335, type, the_real: (real > $o) > real).
% 37.37/9.71  thf(func_def_336, type, if_int_int: $o > (int > int) > (int > int) > int > int).
% 37.37/9.71  thf(func_def_337, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 37.37/9.71  thf(func_def_338, type, if_complex: $o > complex > complex > complex).
% 37.37/9.71  thf(func_def_339, type, if_int: $o > int > int > int).
% 37.37/9.71  thf(func_def_340, type, if_list_int: $o > list_int > list_int > list_int).
% 37.37/9.71  thf(func_def_341, type, if_nat: $o > nat > nat > nat).
% 37.37/9.71  thf(func_def_342, type, if_option_num: $o > option_num > option_num > option_num).
% 37.37/9.71  thf(func_def_343, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 37.37/9.71  thf(func_def_344, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 37.37/9.71  thf(func_def_345, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 37.37/9.71  thf(func_def_346, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 37.37/9.71  thf(func_def_347, type, if_rat: $o > rat > rat > rat).
% 37.37/9.71  thf(func_def_348, type, if_real: $o > real > real > real).
% 37.37/9.71  thf(func_def_349, type, if_set_int: $o > set_int > set_int > set_int).
% 37.37/9.71  thf(func_def_350, type, abs_Integ: product_prod_nat_nat > int).
% 37.37/9.71  thf(func_def_351, type, rep_Integ: int > product_prod_nat_nat).
% 37.37/9.71  thf(func_def_352, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 37.37/9.71  thf(func_def_353, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 37.37/9.71  thf(func_def_354, type, nat2: int > nat).
% 37.37/9.71  thf(func_def_355, type, power_int_real: real > int > real).
% 37.37/9.71  thf(func_def_356, type, ring_11222124179247155820nteger: set_Code_integer).
% 37.37/9.71  thf(func_def_357, type, ring_1_Ints_complex: set_complex).
% 37.37/9.71  thf(func_def_358, type, ring_1_Ints_int: set_int).
% 37.37/9.71  thf(func_def_359, type, ring_1_Ints_rat: set_rat).
% 37.37/9.71  thf(func_def_360, type, ring_1_Ints_real: set_real).
% 37.37/9.71  thf(func_def_361, type, ring_18347121197199848620nteger: int > code_integer).
% 37.37/9.71  thf(func_def_362, type, ring_17405671764205052669omplex: int > complex).
% 37.37/9.71  thf(func_def_363, type, ring_1_of_int_int: int > int).
% 37.37/9.71  thf(func_def_364, type, ring_1_of_int_rat: int > rat).
% 37.37/9.71  thf(func_def_365, type, ring_1_of_int_real: int > real).
% 37.37/9.71  thf(func_def_366, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 37.37/9.71  thf(func_def_367, type, sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat).
% 37.37/9.71  thf(func_def_368, type, sup_sup_int: int > int > int).
% 37.37/9.71  thf(func_def_369, type, sup_sup_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_370, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 37.37/9.71  thf(func_def_371, type, lattic1921953407002678535_Max_o: set_o > $o).
% 37.37/9.71  thf(func_def_372, type, lattic8263393255366662781ax_int: set_int > int).
% 37.37/9.71  thf(func_def_373, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 37.37/9.71  thf(func_def_374, type, lattic4275903605611617917x_real: set_real > real).
% 37.37/9.71  thf(func_def_375, type, bfun_nat_real: (nat > real) > filter_nat > $o).
% 37.37/9.71  thf(func_def_376, type, at_infinity_real: filter_real).
% 37.37/9.71  thf(func_def_377, type, append_int: list_int > list_int > list_int).
% 37.37/9.71  thf(func_def_378, type, append_nat: list_nat > list_nat > list_nat).
% 37.37/9.71  thf(func_def_379, type, distinct_int: list_int > $o).
% 37.37/9.71  thf(func_def_380, type, linord2614967742042102400et_nat: set_nat > list_nat).
% 37.37/9.71  thf(func_def_381, type, cons_int: int > list_int > list_int).
% 37.37/9.71  thf(func_def_382, type, cons_nat: nat > list_nat > list_nat).
% 37.37/9.71  thf(func_def_383, type, nil_int: list_int).
% 37.37/9.71  thf(func_def_384, type, nil_nat: list_nat).
% 37.37/9.71  thf(func_def_385, type, set_o2: list_o > set_o).
% 37.37/9.71  thf(func_def_386, type, set_complex2: list_complex > set_complex).
% 37.37/9.71  thf(func_def_387, type, set_int2: list_int > set_int).
% 37.37/9.71  thf(func_def_388, type, set_nat2: list_nat > set_nat).
% 37.37/9.71  thf(func_def_389, type, set_real2: list_real > set_real).
% 37.37/9.71  thf(func_def_390, type, set_set_nat2: list_set_nat > set_set_nat).
% 37.37/9.71  thf(func_def_391, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_392, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_393, type, nth_o: list_o > nat > $o).
% 37.37/9.71  thf(func_def_394, type, nth_Code_integer: list_Code_integer > nat > code_integer).
% 37.37/9.71  thf(func_def_395, type, nth_complex: list_complex > nat > complex).
% 37.37/9.71  thf(func_def_396, type, nth_int: list_int > nat > int).
% 37.37/9.71  thf(func_def_397, type, nth_nat: list_nat > nat > nat).
% 37.37/9.71  thf(func_def_398, type, nth_num: list_num > nat > num).
% 37.37/9.71  thf(func_def_399, type, nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o).
% 37.37/9.71  thf(func_def_400, type, nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o).
% 37.37/9.71  thf(func_def_401, type, nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int).
% 37.37/9.71  thf(func_def_402, type, nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat).
% 37.37/9.71  thf(func_def_403, type, nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num).
% 37.37/9.71  thf(func_def_404, type, nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT).
% 37.37/9.71  thf(func_def_405, type, nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num).
% 37.37/9.71  thf(func_def_406, type, nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o).
% 37.37/9.71  thf(func_def_407, type, nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat).
% 37.37/9.71  thf(func_def_408, type, nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT).
% 37.37/9.71  thf(func_def_409, type, nth_real: list_real > nat > real).
% 37.37/9.71  thf(func_def_410, type, nth_set_nat: list_set_nat > nat > set_nat).
% 37.37/9.71  thf(func_def_411, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 37.37/9.71  thf(func_def_412, type, product_o_o: list_o > list_o > list_P4002435161011370285od_o_o).
% 37.37/9.71  thf(func_def_413, type, product_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 37.37/9.71  thf(func_def_414, type, product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 37.37/9.71  thf(func_def_415, type, produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o).
% 37.37/9.71  thf(func_def_416, type, product_int_o: list_int > list_o > list_P5087981734274514673_int_o).
% 37.37/9.71  thf(func_def_417, type, product_int_int: list_int > list_int > list_P5707943133018811711nt_int).
% 37.37/9.71  thf(func_def_418, type, produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT).
% 37.37/9.71  thf(func_def_419, type, product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o).
% 37.37/9.71  thf(func_def_420, type, product_nat_int: list_nat > list_int > list_P3521021558325789923at_int).
% 37.37/9.71  thf(func_def_421, type, product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat).
% 37.37/9.71  thf(func_def_422, type, product_nat_num: list_nat > list_num > list_P1726324292696863441at_num).
% 37.37/9.71  thf(func_def_423, type, produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 37.37/9.71  thf(func_def_424, type, product_num_num: list_num > list_num > list_P3744719386663036955um_num).
% 37.37/9.71  thf(func_def_425, type, product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 37.37/9.71  thf(func_def_426, type, produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 37.37/9.71  thf(func_def_427, type, produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 37.37/9.71  thf(func_def_428, type, produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 37.37/9.71  thf(func_def_429, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_430, type, upto: int > int > list_int).
% 37.37/9.71  thf(func_def_431, type, upto_aux: int > int > list_int > list_int).
% 37.37/9.71  thf(func_def_432, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 37.37/9.71  thf(func_def_433, type, suc: nat > nat).
% 37.37/9.71  thf(func_def_434, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 37.37/9.71  thf(func_def_435, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 37.37/9.71  thf(func_def_436, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 37.37/9.71  thf(func_def_437, type, case_nat_option_num: option_num > (nat > option_num) > nat > option_num).
% 37.37/9.71  thf(func_def_438, type, pred: nat > nat).
% 37.37/9.71  thf(func_def_439, type, semiri3842193898606819883omplex: set_complex).
% 37.37/9.71  thf(func_def_440, type, semiring_1_Nats_int: set_int).
% 37.37/9.71  thf(func_def_441, type, semiri4939895301339042750nteger: nat > code_integer).
% 37.37/9.71  thf(func_def_442, type, semiri8010041392384452111omplex: nat > complex).
% 37.37/9.71  thf(func_def_443, type, semiri1314217659103216013at_int: nat > int).
% 37.37/9.71  thf(func_def_444, type, semiri1316708129612266289at_nat: nat > nat).
% 37.37/9.71  thf(func_def_445, type, semiri681578069525770553at_rat: nat > rat).
% 37.37/9.71  thf(func_def_446, type, semiri5074537144036343181t_real: nat > real).
% 37.37/9.71  thf(func_def_447, type, size_size_list_o: list_o > nat).
% 37.37/9.71  thf(func_def_448, type, size_s3445333598471063425nteger: list_Code_integer > nat).
% 37.37/9.71  thf(func_def_449, type, size_s3451745648224563538omplex: list_complex > nat).
% 37.37/9.71  thf(func_def_450, type, size_size_list_int: list_int > nat).
% 37.37/9.71  thf(func_def_451, type, size_size_list_nat: list_nat > nat).
% 37.37/9.71  thf(func_def_452, type, size_size_list_num: list_num > nat).
% 37.37/9.71  thf(func_def_453, type, size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat).
% 37.37/9.71  thf(func_def_454, type, size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat).
% 37.37/9.71  thf(func_def_455, type, size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat).
% 37.37/9.71  thf(func_def_456, type, size_s4246224855604898693_int_o: list_P5087981734274514673_int_o > nat).
% 37.37/9.71  thf(func_def_457, type, size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat).
% 37.37/9.71  thf(func_def_458, type, size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > nat).
% 37.37/9.71  thf(func_def_459, type, size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat).
% 37.37/9.71  thf(func_def_460, type, size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat).
% 37.37/9.71  thf(func_def_461, type, size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat).
% 37.37/9.71  thf(func_def_462, type, size_size_list_real: list_real > nat).
% 37.37/9.71  thf(func_def_463, type, size_s3254054031482475050et_nat: list_set_nat > nat).
% 37.37/9.71  thf(func_def_464, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_465, type, size_size_num: num > nat).
% 37.37/9.71  thf(func_def_466, type, size_size_option_num: option_num > nat).
% 37.37/9.71  thf(func_def_467, type, size_s170228958280169651at_nat: option4927543243414619207at_nat > nat).
% 37.37/9.71  thf(func_def_468, type, size_size_char: char > nat).
% 37.37/9.71  thf(func_def_469, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_470, type, nat_list_encode: list_nat > nat).
% 37.37/9.71  thf(func_def_471, type, nat_list_encode_rel: list_nat > list_nat > $o).
% 37.37/9.71  thf(func_def_472, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 37.37/9.71  thf(func_def_473, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 37.37/9.71  thf(func_def_474, type, nat_prod_encode: product_prod_nat_nat > nat).
% 37.37/9.71  thf(func_def_475, type, nat_set_decode: nat > set_nat).
% 37.37/9.71  thf(func_def_476, type, nat_set_encode: set_nat > nat).
% 37.37/9.71  thf(func_def_477, type, nat_triangle: nat > nat).
% 37.37/9.71  thf(func_def_478, type, root: nat > real > real).
% 37.37/9.71  thf(func_def_479, type, sqrt: real > real).
% 37.37/9.71  thf(func_def_480, type, bitM: num > num).
% 37.37/9.71  thf(func_def_481, type, inc: num > num).
% 37.37/9.71  thf(func_def_482, type, neg_nu8804712462038260780nteger: code_integer > code_integer).
% 37.37/9.71  thf(func_def_483, type, neg_nu7009210354673126013omplex: complex > complex).
% 37.37/9.71  thf(func_def_484, type, neg_numeral_dbl_int: int > int).
% 37.37/9.71  thf(func_def_485, type, neg_numeral_dbl_rat: rat > rat).
% 37.37/9.71  thf(func_def_486, type, neg_numeral_dbl_real: real > real).
% 37.37/9.71  thf(func_def_487, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 37.37/9.71  thf(func_def_488, type, neg_nu6511756317524482435omplex: complex > complex).
% 37.37/9.71  thf(func_def_489, type, neg_nu3811975205180677377ec_int: int > int).
% 37.37/9.71  thf(func_def_490, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 37.37/9.71  thf(func_def_491, type, neg_nu6075765906172075777c_real: real > real).
% 37.37/9.71  thf(func_def_492, type, neg_nu5831290666863070958nteger: code_integer > code_integer).
% 37.37/9.71  thf(func_def_493, type, neg_nu8557863876264182079omplex: complex > complex).
% 37.37/9.71  thf(func_def_494, type, neg_nu5851722552734809277nc_int: int > int).
% 37.37/9.71  thf(func_def_495, type, neg_nu5219082963157363817nc_rat: rat > rat).
% 37.37/9.71  thf(func_def_496, type, neg_nu8295874005876285629c_real: real > real).
% 37.37/9.71  thf(func_def_497, type, neg_numeral_sub_int: num > num > int).
% 37.37/9.71  thf(func_def_498, type, bit0: num > num).
% 37.37/9.71  thf(func_def_499, type, bit1: num > num).
% 37.37/9.71  thf(func_def_500, type, one: num).
% 37.37/9.71  thf(func_def_501, type, case_num_option_num: option_num > (num > option_num) > (num > option_num) > num > option_num).
% 37.37/9.71  thf(func_def_502, type, size_num: num > nat).
% 37.37/9.71  thf(func_def_503, type, num_of_nat: nat > num).
% 37.37/9.71  thf(func_def_504, type, numera6620942414471956472nteger: num > code_integer).
% 37.37/9.71  thf(func_def_505, type, numera6690914467698888265omplex: num > complex).
% 37.37/9.71  thf(func_def_506, type, numera1916890842035813515d_enat: num > extended_enat).
% 37.37/9.71  thf(func_def_507, type, numeral_numeral_int: num > int).
% 37.37/9.71  thf(func_def_508, type, numeral_numeral_nat: num > nat).
% 37.37/9.71  thf(func_def_509, type, numeral_numeral_rat: num > rat).
% 37.37/9.71  thf(func_def_510, type, numeral_numeral_real: num > real).
% 37.37/9.71  thf(func_def_511, type, pow: num > num > num).
% 37.37/9.71  thf(func_def_512, type, pred_numeral: num > nat).
% 37.37/9.71  thf(func_def_513, type, sqr: num > num).
% 37.37/9.71  thf(func_def_514, type, none_num: option_num).
% 37.37/9.71  thf(func_def_515, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 37.37/9.71  thf(func_def_516, type, some_num: num > option_num).
% 37.37/9.71  thf(func_def_517, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 37.37/9.71  thf(func_def_518, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 37.37/9.71  thf(func_def_519, type, case_option_int_num: int > (num > int) > option_num > int).
% 37.37/9.71  thf(func_def_520, type, case_option_num_num: num > (num > num) > option_num > num).
% 37.37/9.71  thf(func_def_521, type, case_o6005452278849405969um_num: option_num > (num > option_num) > option_num > option_num).
% 37.37/9.71  thf(func_def_522, type, map_option_num_num: (num > num) > option_num > option_num).
% 37.37/9.71  thf(func_def_523, type, size_option_num: (num > nat) > option_num > nat).
% 37.37/9.71  thf(func_def_524, type, size_o8335143837870341156at_nat: (product_prod_nat_nat > nat) > option4927543243414619207at_nat > nat).
% 37.37/9.71  thf(func_def_525, type, bot_bot_o_o: $o > $o).
% 37.37/9.71  thf(func_def_526, type, bot_bot_complex_o: complex > $o).
% 37.37/9.71  thf(func_def_527, type, bot_bot_int_o: int > $o).
% 37.37/9.71  thf(func_def_528, type, bot_bot_nat_o: nat > $o).
% 37.37/9.71  thf(func_def_529, type, bot_bot_real_o: real > $o).
% 37.37/9.71  thf(func_def_530, type, bot_bot_set_nat_o: set_nat > $o).
% 37.37/9.71  thf(func_def_531, type, bot_bo4199563552545308370d_enat: extended_enat).
% 37.37/9.71  thf(func_def_532, type, bot_bot_nat: nat).
% 37.37/9.71  thf(func_def_533, type, bot_bot_set_o: set_o).
% 37.37/9.71  thf(func_def_534, type, bot_bot_set_complex: set_complex).
% 37.37/9.71  thf(func_def_535, type, bot_bot_set_int: set_int).
% 37.37/9.71  thf(func_def_536, type, bot_bot_set_nat: set_nat).
% 37.37/9.71  thf(func_def_537, type, bot_bot_set_real: set_real).
% 37.37/9.71  thf(func_def_538, type, bot_bot_set_set_nat: set_set_nat).
% 37.37/9.71  thf(func_def_539, type, bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_540, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 37.37/9.71  thf(func_def_541, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 37.37/9.71  thf(func_def_542, type, ord_less_int: int > int > $o).
% 37.37/9.71  thf(func_def_543, type, ord_less_nat: nat > nat > $o).
% 37.37/9.71  thf(func_def_544, type, ord_less_num: num > num > $o).
% 37.37/9.71  thf(func_def_545, type, ord_less_rat: rat > rat > $o).
% 37.37/9.71  thf(func_def_546, type, ord_less_real: real > real > $o).
% 37.37/9.71  thf(func_def_547, type, ord_less_set_o: set_o > set_o > $o).
% 37.37/9.71  thf(func_def_548, type, ord_less_set_complex: set_complex > set_complex > $o).
% 37.37/9.71  thf(func_def_549, type, ord_less_set_int: set_int > set_int > $o).
% 37.37/9.71  thf(func_def_550, type, ord_less_set_nat: set_nat > set_nat > $o).
% 37.37/9.71  thf(func_def_551, type, ord_less_set_real: set_real > set_real > $o).
% 37.37/9.71  thf(func_def_552, type, ord_less_set_set_nat: set_set_nat > set_set_nat > $o).
% 37.37/9.71  thf(func_def_553, type, ord_le3480810397992357184T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_554, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 37.37/9.71  thf(func_def_555, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 37.37/9.71  thf(func_def_556, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 37.37/9.71  thf(func_def_557, type, ord_le4104064031414453916r_real: filter_real > filter_real > $o).
% 37.37/9.71  thf(func_def_558, type, ord_less_eq_int: int > int > $o).
% 37.37/9.71  thf(func_def_559, type, ord_less_eq_nat: nat > nat > $o).
% 37.37/9.71  thf(func_def_560, type, ord_less_eq_num: num > num > $o).
% 37.37/9.71  thf(func_def_561, type, ord_less_eq_rat: rat > rat > $o).
% 37.37/9.71  thf(func_def_562, type, ord_less_eq_real: real > real > $o).
% 37.37/9.71  thf(func_def_563, type, ord_less_eq_set_o: set_o > set_o > $o).
% 37.37/9.71  thf(func_def_564, type, ord_le211207098394363844omplex: set_complex > set_complex > $o).
% 37.37/9.71  thf(func_def_565, type, ord_less_eq_set_int: set_int > set_int > $o).
% 37.37/9.71  thf(func_def_566, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 37.37/9.71  thf(func_def_567, type, ord_less_eq_set_real: set_real > set_real > $o).
% 37.37/9.71  thf(func_def_568, type, ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o).
% 37.37/9.71  thf(func_def_569, type, ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_570, type, ord_max_Code_integer: code_integer > code_integer > code_integer).
% 37.37/9.71  thf(func_def_571, type, ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat).
% 37.37/9.71  thf(func_def_572, type, ord_max_int: int > int > int).
% 37.37/9.71  thf(func_def_573, type, ord_max_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_574, type, order_Greatest_nat: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_575, type, order_9091379641038594480t_real: (nat > real) > $o).
% 37.37/9.71  thf(func_def_576, type, order_mono_nat_nat: (nat > nat) > $o).
% 37.37/9.71  thf(func_def_577, type, order_mono_nat_real: (nat > real) > $o).
% 37.37/9.71  thf(func_def_578, type, order_mono_real_real: (real > real) > $o).
% 37.37/9.71  thf(func_def_579, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 37.37/9.71  thf(func_def_580, type, order_7092887310737990675l_real: (real > real) > $o).
% 37.37/9.71  thf(func_def_581, type, top_top_set_o: set_o).
% 37.37/9.71  thf(func_def_582, type, top_top_set_int: set_int).
% 37.37/9.71  thf(func_def_583, type, top_top_set_nat: set_nat).
% 37.37/9.71  thf(func_def_584, type, top_to1996260823553986621t_unit: set_Product_unit).
% 37.37/9.71  thf(func_def_585, type, top_top_set_real: set_real).
% 37.37/9.71  thf(func_def_586, type, top_top_set_char: set_char).
% 37.37/9.71  thf(func_def_587, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 37.37/9.71  thf(func_def_588, type, power_power_complex: complex > nat > complex).
% 37.37/9.71  thf(func_def_589, type, power_power_int: int > nat > int).
% 37.37/9.71  thf(func_def_590, type, power_power_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_591, type, power_power_rat: rat > nat > rat).
% 37.37/9.71  thf(func_def_592, type, power_power_real: real > nat > real).
% 37.37/9.71  thf(func_def_593, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 37.37/9.71  thf(func_def_594, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 37.37/9.71  thf(func_def_595, type, product_Pair_int_int: int > int > product_prod_int_int).
% 37.37/9.71  thf(func_def_596, type, product_Pair_nat_o: nat > $o > product_prod_nat_o).
% 37.37/9.71  thf(func_def_597, type, product_Pair_nat_int: nat > int > product_prod_nat_int).
% 37.37/9.71  thf(func_def_598, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 37.37/9.71  thf(func_def_599, type, product_Pair_nat_num: nat > num > product_prod_nat_num).
% 37.37/9.71  thf(func_def_600, type, produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT).
% 37.37/9.71  thf(func_def_601, type, product_Pair_num_num: num > num > product_prod_num_num).
% 37.37/9.71  thf(func_def_602, type, produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o).
% 37.37/9.71  thf(func_def_603, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 37.37/9.71  thf(func_def_604, type, produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT).
% 37.37/9.71  thf(func_def_605, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 37.37/9.71  thf(func_def_606, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 37.37/9.71  thf(func_def_607, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 37.37/9.71  thf(func_def_608, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 37.37/9.71  thf(func_def_609, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 37.37/9.71  thf(func_def_610, type, produc6771430404735790350plex_o: (complex > complex > $o) > produc4411394909380815293omplex > $o).
% 37.37/9.71  thf(func_def_611, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 37.37/9.71  thf(func_def_612, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 37.37/9.71  thf(func_def_613, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 37.37/9.71  thf(func_def_614, 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).
% 37.37/9.71  thf(func_def_615, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 37.37/9.71  thf(func_def_616, type, produc6842872674320459806at_nat: (nat > nat > nat) > product_prod_nat_nat > nat).
% 37.37/9.71  thf(func_def_617, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 37.37/9.71  thf(func_def_618, type, produc478579273971653890on_num: (nat > num > option_num) > product_prod_nat_num > option_num).
% 37.37/9.71  thf(func_def_619, type, produc5414030515140494994real_o: (real > real > $o) > produc2422161461964618553l_real > $o).
% 37.37/9.71  thf(func_def_620, type, produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer).
% 37.37/9.71  thf(func_def_621, type, product_fst_int_int: product_prod_int_int > int).
% 37.37/9.71  thf(func_def_622, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 37.37/9.71  thf(func_def_623, type, product_snd_int_int: product_prod_int_int > int).
% 37.37/9.71  thf(func_def_624, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 37.37/9.71  thf(func_def_625, type, fract: int > int > rat).
% 37.37/9.71  thf(func_def_626, type, frct: product_prod_int_int > rat).
% 37.37/9.71  thf(func_def_627, type, field_5140801741446780682s_real: set_real).
% 37.37/9.71  thf(func_def_628, type, normalize: product_prod_int_int > product_prod_int_int).
% 37.37/9.71  thf(func_def_629, type, of_int: int > rat).
% 37.37/9.71  thf(func_def_630, type, quotient_of: rat > product_prod_int_int).
% 37.37/9.71  thf(func_def_631, type, real_V2521375963428798218omplex: set_complex).
% 37.37/9.71  thf(func_def_632, type, real_V3694042436643373181omplex: complex > complex > real).
% 37.37/9.71  thf(func_def_633, type, real_V975177566351809787t_real: real > real > real).
% 37.37/9.71  thf(func_def_634, type, real_V1022390504157884413omplex: complex > real).
% 37.37/9.71  thf(func_def_635, type, real_V7735802525324610683m_real: real > real).
% 37.37/9.71  thf(func_def_636, type, real_V4546457046886955230omplex: real > complex).
% 37.37/9.71  thf(func_def_637, type, real_V1803761363581548252l_real: real > real).
% 37.37/9.71  thf(func_def_638, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 37.37/9.71  thf(func_def_639, type, divide1717551699836669952omplex: complex > complex > complex).
% 37.37/9.71  thf(func_def_640, type, divide_divide_int: int > int > int).
% 37.37/9.71  thf(func_def_641, type, divide_divide_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_642, type, divide_divide_rat: rat > rat > rat).
% 37.37/9.71  thf(func_def_643, type, divide_divide_real: real > real > real).
% 37.37/9.71  thf(func_def_644, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 37.37/9.71  thf(func_def_645, type, dvd_dvd_complex: complex > complex > $o).
% 37.37/9.71  thf(func_def_646, type, dvd_dvd_int: int > int > $o).
% 37.37/9.71  thf(func_def_647, type, dvd_dvd_nat: nat > nat > $o).
% 37.37/9.71  thf(func_def_648, type, dvd_dvd_rat: rat > rat > $o).
% 37.37/9.71  thf(func_def_649, type, dvd_dvd_real: real > real > $o).
% 37.37/9.71  thf(func_def_650, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 37.37/9.71  thf(func_def_651, type, modulo_modulo_int: int > int > int).
% 37.37/9.71  thf(func_def_652, type, modulo_modulo_nat: nat > nat > nat).
% 37.37/9.71  thf(func_def_653, type, zero_n356916108424825756nteger: $o > code_integer).
% 37.37/9.71  thf(func_def_654, type, zero_n1201886186963655149omplex: $o > complex).
% 37.37/9.71  thf(func_def_655, type, zero_n2684676970156552555ol_int: $o > int).
% 37.37/9.71  thf(func_def_656, type, zero_n2687167440665602831ol_nat: $o > nat).
% 37.37/9.71  thf(func_def_657, type, zero_n2052037380579107095ol_rat: $o > rat).
% 37.37/9.71  thf(func_def_658, type, zero_n3304061248610475627l_real: $o > real).
% 37.37/9.71  thf(func_def_659, type, suminf_real: (nat > real) > real).
% 37.37/9.71  thf(func_def_660, type, summable_real: (nat > real) > $o).
% 37.37/9.71  thf(func_def_661, type, sums_real: (nat > real) > real > $o).
% 37.37/9.71  thf(func_def_662, type, collect_o: ($o > $o) > set_o).
% 37.37/9.71  thf(func_def_663, type, collect_complex: (complex > $o) > set_complex).
% 37.37/9.71  thf(func_def_664, type, collect_int: (int > $o) > set_int).
% 37.37/9.71  thf(func_def_665, type, collect_nat: (nat > $o) > set_nat).
% 37.37/9.71  thf(func_def_666, type, collec8663557070575231912omplex: (produc4411394909380815293omplex > $o) > set_Pr5085853215250843933omplex).
% 37.37/9.71  thf(func_def_667, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 37.37/9.71  thf(func_def_668, type, collec3392354462482085612at_nat: (product_prod_nat_nat > $o) > set_Pr1261947904930325089at_nat).
% 37.37/9.71  thf(func_def_669, type, collec3799799289383736868l_real: (produc2422161461964618553l_real > $o) > set_Pr6218003697084177305l_real).
% 37.37/9.71  thf(func_def_670, type, collect_real: (real > $o) > set_real).
% 37.37/9.71  thf(func_def_671, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 37.37/9.71  thf(func_def_672, type, pow_nat: set_nat > set_set_nat).
% 37.37/9.71  thf(func_def_673, type, image_o_o: ($o > $o) > set_o > set_o).
% 37.37/9.71  thf(func_def_674, type, image_o_int: ($o > int) > set_o > set_int).
% 37.37/9.71  thf(func_def_675, type, image_o_nat: ($o > nat) > set_o > set_nat).
% 37.37/9.71  thf(func_def_676, type, image_o_real: ($o > real) > set_o > set_real).
% 37.37/9.71  thf(func_def_677, type, image_complex_o: (complex > $o) > set_complex > set_o).
% 37.37/9.71  thf(func_def_678, type, image_1468599708987790691omplex: (complex > complex) > set_complex > set_complex).
% 37.37/9.71  thf(func_def_679, type, image_complex_int: (complex > int) > set_complex > set_int).
% 37.37/9.71  thf(func_def_680, type, image_complex_nat: (complex > nat) > set_complex > set_nat).
% 37.37/9.71  thf(func_def_681, type, image_complex_real: (complex > real) > set_complex > set_real).
% 37.37/9.71  thf(func_def_682, type, image_932796090930683071T_VEBT: (complex > vEBT_VEBT) > set_complex > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_683, type, image_int_o: (int > $o) > set_int > set_o).
% 37.37/9.71  thf(func_def_684, type, image_int_complex: (int > complex) > set_int > set_complex).
% 37.37/9.71  thf(func_def_685, type, image_int_int: (int > int) > set_int > set_int).
% 37.37/9.71  thf(func_def_686, type, image_int_nat: (int > nat) > set_int > set_nat).
% 37.37/9.71  thf(func_def_687, type, image_int_real: (int > real) > set_int > set_real).
% 37.37/9.71  thf(func_def_688, type, image_nat_o: (nat > $o) > set_nat > set_o).
% 37.37/9.71  thf(func_def_689, type, image_nat_complex: (nat > complex) > set_nat > set_complex).
% 37.37/9.71  thf(func_def_690, type, image_nat_int: (nat > int) > set_nat > set_int).
% 37.37/9.71  thf(func_def_691, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 37.37/9.71  thf(func_def_692, type, image_nat_real: (nat > real) > set_nat > set_real).
% 37.37/9.71  thf(func_def_693, type, image_nat_set_nat: (nat > set_nat) > set_nat > set_set_nat).
% 37.37/9.71  thf(func_def_694, type, image_nat_char: (nat > char) > set_nat > set_char).
% 37.37/9.71  thf(func_def_695, type, image_nat_VEBT_VEBT: (nat > vEBT_VEBT) > set_nat > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_696, type, image_num_option_num: (num > option_num) > set_num > set_option_num).
% 37.37/9.71  thf(func_def_697, type, image_4198897800814241419at_nat: (product_prod_nat_nat > option4927543243414619207at_nat) > set_Pr1261947904930325089at_nat > set_op4508134149509766951at_nat).
% 37.37/9.71  thf(func_def_698, type, image_rat_rat: (rat > rat) > set_rat > set_rat).
% 37.37/9.71  thf(func_def_699, type, image_real_o: (real > $o) > set_real > set_o).
% 37.37/9.71  thf(func_def_700, type, image_real_complex: (real > complex) > set_real > set_complex).
% 37.37/9.71  thf(func_def_701, type, image_5971271580939081552omplex: (real > filter6041513312241820739omplex) > set_real > set_fi4554929511873752355omplex).
% 37.37/9.71  thf(func_def_702, type, image_2178119161166701260l_real: (real > filter2146258269922977983l_real) > set_real > set_fi7789364187291644575l_real).
% 37.37/9.71  thf(func_def_703, type, image_real_int: (real > int) > set_real > set_int).
% 37.37/9.71  thf(func_def_704, type, image_real_nat: (real > nat) > set_real > set_nat).
% 37.37/9.71  thf(func_def_705, type, image_real_real: (real > real) > set_real > set_real).
% 37.37/9.71  thf(func_def_706, type, image_real_VEBT_VEBT: (real > vEBT_VEBT) > set_real > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_707, type, image_set_o_set_o: (set_o > set_o) > set_set_o > set_set_o).
% 37.37/9.71  thf(func_def_708, type, image_7998606247489673935omplex: (set_complex > set_complex) > set_set_complex > set_set_complex).
% 37.37/9.71  thf(func_def_709, type, image_524474410958335435et_int: (set_int > set_int) > set_set_int > set_set_int).
% 37.37/9.71  thf(func_def_710, type, image_7916887816326733075et_nat: (set_nat > set_nat) > set_set_nat > set_set_nat).
% 37.37/9.71  thf(func_def_711, type, image_2436557299294012491t_real: (set_real > set_real) > set_set_real > set_set_real).
% 37.37/9.71  thf(func_def_712, type, image_7884819252390400639et_nat: (set_set_nat > set_set_nat) > set_set_set_nat > set_set_set_nat).
% 37.37/9.71  thf(func_def_713, type, image_1661326939266726661T_VEBT: (set_VEBT_VEBT > set_VEBT_VEBT) > set_set_VEBT_VEBT > set_set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_714, type, image_char_nat: (char > nat) > set_char > set_nat).
% 37.37/9.71  thf(func_def_715, type, image_VEBT_VEBT_o: (vEBT_VEBT > $o) > set_VEBT_VEBT > set_o).
% 37.37/9.71  thf(func_def_716, type, image_VEBT_VEBT_int: (vEBT_VEBT > int) > set_VEBT_VEBT > set_int).
% 37.37/9.71  thf(func_def_717, type, image_VEBT_VEBT_nat: (vEBT_VEBT > nat) > set_VEBT_VEBT > set_nat).
% 37.37/9.71  thf(func_def_718, type, image_VEBT_VEBT_real: (vEBT_VEBT > real) > set_VEBT_VEBT > set_real).
% 37.37/9.71  thf(func_def_719, type, image_3375948659692109573T_VEBT: (vEBT_VEBT > vEBT_VEBT) > set_VEBT_VEBT > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_720, type, insert_o: $o > set_o > set_o).
% 37.37/9.71  thf(func_def_721, type, insert_complex: complex > set_complex > set_complex).
% 37.37/9.71  thf(func_def_722, type, insert_int: int > set_int > set_int).
% 37.37/9.71  thf(func_def_723, type, insert_nat: nat > set_nat > set_nat).
% 37.37/9.71  thf(func_def_724, type, insert_real: real > set_real > set_real).
% 37.37/9.71  thf(func_def_725, type, insert_set_nat: set_nat > set_set_nat > set_set_nat).
% 37.37/9.71  thf(func_def_726, type, insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_727, type, is_singleton_o: set_o > $o).
% 37.37/9.71  thf(func_def_728, type, is_singleton_int: set_int > $o).
% 37.37/9.71  thf(func_def_729, type, is_singleton_nat: set_nat > $o).
% 37.37/9.71  thf(func_def_730, type, is_singleton_real: set_real > $o).
% 37.37/9.71  thf(func_def_731, type, is_sin24926331636114728T_VEBT: set_VEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_732, type, the_elem_o: set_o > $o).
% 37.37/9.71  thf(func_def_733, type, the_elem_int: set_int > int).
% 37.37/9.71  thf(func_def_734, type, the_elem_nat: set_nat > nat).
% 37.37/9.71  thf(func_def_735, type, the_elem_real: set_real > real).
% 37.37/9.71  thf(func_def_736, type, the_elem_set_nat: set_set_nat > set_nat).
% 37.37/9.71  thf(func_def_737, type, the_elem_VEBT_VEBT: set_VEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_738, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 37.37/9.71  thf(func_def_739, type, set_or1266510415728281911st_int: int > int > set_int).
% 37.37/9.71  thf(func_def_740, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 37.37/9.71  thf(func_def_741, type, set_or1222579329274155063t_real: real > real > set_real).
% 37.37/9.71  thf(func_def_742, type, set_or4662586982721622107an_int: int > int > set_int).
% 37.37/9.71  thf(func_def_743, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 37.37/9.71  thf(func_def_744, type, set_ord_atLeast_nat: nat > set_nat).
% 37.37/9.71  thf(func_def_745, type, set_ord_atLeast_real: real > set_real).
% 37.37/9.71  thf(func_def_746, type, set_ord_atMost_int: int > set_int).
% 37.37/9.71  thf(func_def_747, type, set_ord_atMost_nat: nat > set_nat).
% 37.37/9.71  thf(func_def_748, type, set_or6656581121297822940st_int: int > int > set_int).
% 37.37/9.71  thf(func_def_749, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 37.37/9.71  thf(func_def_750, type, set_or5832277885323065728an_int: int > int > set_int).
% 37.37/9.71  thf(func_def_751, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 37.37/9.71  thf(func_def_752, type, set_or1633881224788618240n_real: real > real > set_real).
% 37.37/9.71  thf(func_def_753, type, set_or1210151606488870762an_nat: nat > set_nat).
% 37.37/9.71  thf(func_def_754, type, set_or5849166863359141190n_real: real > set_real).
% 37.37/9.71  thf(func_def_755, type, set_ord_lessThan_int: int > set_int).
% 37.37/9.71  thf(func_def_756, type, set_ord_lessThan_nat: nat > set_nat).
% 37.37/9.71  thf(func_def_757, type, set_or5984915006950818249n_real: real > set_real).
% 37.37/9.71  thf(func_def_758, type, ascii_of: char > char).
% 37.37/9.71  thf(func_def_759, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 37.37/9.71  thf(func_def_760, type, size_char: char > nat).
% 37.37/9.71  thf(func_def_761, type, comm_s629917340098488124ar_nat: char > nat).
% 37.37/9.71  thf(func_def_762, type, integer_of_char: char > code_integer).
% 37.37/9.71  thf(func_def_763, type, unique3096191561947761185of_nat: nat > char).
% 37.37/9.71  thf(func_def_764, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 37.37/9.71  thf(func_def_765, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 37.37/9.71  thf(func_def_766, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 37.37/9.71  thf(func_def_767, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 37.37/9.71  thf(func_def_768, type, topolo2815343760600316023s_real: real > filter_real).
% 37.37/9.71  thf(func_def_769, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 37.37/9.71  thf(func_def_770, type, topolo896644834953643431omplex: filter6041513312241820739omplex).
% 37.37/9.71  thf(func_def_771, type, topolo1511823702728130853y_real: filter2146258269922977983l_real).
% 37.37/9.71  thf(func_def_772, type, arccos: real > real).
% 37.37/9.71  thf(func_def_773, type, arcosh_real: real > real).
% 37.37/9.71  thf(func_def_774, type, arcsin: real > real).
% 37.37/9.71  thf(func_def_775, type, arctan: real > real).
% 37.37/9.71  thf(func_def_776, type, arsinh_real: real > real).
% 37.37/9.71  thf(func_def_777, type, artanh_real: real > real).
% 37.37/9.71  thf(func_def_778, type, cos_complex: complex > complex).
% 37.37/9.71  thf(func_def_779, type, cos_real: real > real).
% 37.37/9.71  thf(func_def_780, type, cos_coeff: nat > real).
% 37.37/9.71  thf(func_def_781, type, cosh_complex: complex > complex).
% 37.37/9.71  thf(func_def_782, type, cosh_real: real > real).
% 37.37/9.71  thf(func_def_783, type, cot_complex: complex > complex).
% 37.37/9.71  thf(func_def_784, type, cot_real: real > real).
% 37.37/9.71  thf(func_def_785, type, diffs_real: (nat > real) > nat > real).
% 37.37/9.71  thf(func_def_786, type, exp_complex: complex > complex).
% 37.37/9.71  thf(func_def_787, type, exp_real: real > real).
% 37.37/9.71  thf(func_def_788, type, ln_ln_real: real > real).
% 37.37/9.71  thf(func_def_789, type, log: real > real > real).
% 37.37/9.71  thf(func_def_790, type, pi: real).
% 37.37/9.71  thf(func_def_791, type, powr_real: real > real > real).
% 37.37/9.71  thf(func_def_792, type, sin_complex: complex > complex).
% 37.37/9.71  thf(func_def_793, type, sin_real: real > real).
% 37.37/9.71  thf(func_def_794, type, sin_coeff: nat > real).
% 37.37/9.71  thf(func_def_795, type, sinh_complex: complex > complex).
% 37.37/9.71  thf(func_def_796, type, sinh_real: real > real).
% 37.37/9.71  thf(func_def_797, type, tan_complex: complex > complex).
% 37.37/9.71  thf(func_def_798, type, tan_real: real > real).
% 37.37/9.71  thf(func_def_799, type, tanh_complex: complex > complex).
% 37.37/9.71  thf(func_def_800, type, tanh_real: real > real).
% 37.37/9.71  thf(func_def_801, type, transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 37.37/9.71  thf(func_def_802, type, transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 37.37/9.71  thf(func_def_803, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 37.37/9.71  thf(func_def_804, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_805, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_806, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_807, type, vEBT_VEBT_high: nat > nat > nat).
% 37.37/9.71  thf(func_def_808, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_809, type, vEBT_VEBT_low: nat > nat > nat).
% 37.37/9.71  thf(func_def_810, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_811, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 37.37/9.71  thf(func_def_812, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_813, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 37.37/9.71  thf(func_def_814, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_815, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 37.37/9.71  thf(func_def_816, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_817, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 37.37/9.71  thf(func_def_818, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 37.37/9.71  thf(func_def_819, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 37.37/9.71  thf(func_def_820, type, vEBT_VEBT_height: vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_821, type, vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_822, type, accp_list_nat: (list_nat > list_nat > $o) > list_nat > $o).
% 37.37/9.71  thf(func_def_823, type, accp_nat: (nat > nat > $o) > nat > $o).
% 37.37/9.71  thf(func_def_824, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 37.37/9.71  thf(func_def_825, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 37.37/9.71  thf(func_def_826, type, accp_P3113834385874906142um_num: (product_prod_num_num > product_prod_num_num > $o) > product_prod_num_num > $o).
% 37.37/9.71  thf(func_def_827, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 37.37/9.71  thf(func_def_828, type, accp_VEBT_VEBT: (vEBT_VEBT > vEBT_VEBT > $o) > vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_829, type, pred_nat: set_Pr1261947904930325089at_nat).
% 37.37/9.71  thf(func_def_830, type, fChoice_real: (real > $o) > real).
% 37.37/9.71  thf(func_def_831, type, member_o: $o > set_o > $o).
% 37.37/9.71  thf(func_def_832, type, member_Code_integer: code_integer > set_Code_integer > $o).
% 37.37/9.71  thf(func_def_833, type, member_complex: complex > set_complex > $o).
% 37.37/9.71  thf(func_def_834, type, member_int: int > set_int > $o).
% 37.37/9.71  thf(func_def_835, type, member_nat: nat > set_nat > $o).
% 37.37/9.71  thf(func_def_836, type, member_option_num: option_num > set_option_num > $o).
% 37.37/9.71  thf(func_def_837, type, member3954567711264315760at_nat: option4927543243414619207at_nat > set_op4508134149509766951at_nat > $o).
% 37.37/9.71  thf(func_def_838, type, member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o).
% 37.37/9.71  thf(func_def_839, type, member_rat: rat > set_rat > $o).
% 37.37/9.71  thf(func_def_840, type, member_real: real > set_real > $o).
% 37.37/9.71  thf(func_def_841, type, member_set_o: set_o > set_set_o > $o).
% 37.37/9.71  thf(func_def_842, type, member_set_complex: set_complex > set_set_complex > $o).
% 37.37/9.71  thf(func_def_843, type, member_set_int: set_int > set_set_int > $o).
% 37.37/9.71  thf(func_def_844, type, member_set_nat: set_nat > set_set_nat > $o).
% 37.37/9.71  thf(func_def_845, type, member_set_real: set_real > set_set_real > $o).
% 37.37/9.71  thf(func_def_846, type, member_set_set_nat: set_set_nat > set_set_set_nat > $o).
% 37.37/9.71  thf(func_def_847, type, member_set_VEBT_VEBT: set_VEBT_VEBT > set_set_VEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_848, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_849, type, deg: nat).
% 37.37/9.71  thf(func_def_850, type, m: nat).
% 37.37/9.71  thf(func_def_851, type, na: nat).
% 37.37/9.71  thf(func_def_852, type, summary: vEBT_VEBT).
% 37.37/9.71  thf(func_def_853, type, treeList: list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_872, type, sP0: real > real > real > $o).
% 37.37/9.71  thf(func_def_873, type, sP1: num > real > real > $o).
% 37.37/9.71  thf(func_def_874, type, sP2: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 37.37/9.71  thf(func_def_875, type, sP3: real > real > real > $o).
% 37.37/9.71  thf(func_def_876, type, sP4: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 37.37/9.71  thf(func_def_877, type, sP5: real > real > real > $o).
% 37.37/9.71  thf(func_def_878, type, sP6: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 37.37/9.71  thf(func_def_879, type, sP7: real > num > real > $o).
% 37.37/9.71  thf(func_def_880, type, sP8: (int > $o) > int > $o).
% 37.37/9.71  thf(func_def_881, type, sP9: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 37.37/9.71  thf(func_def_882, type, sP10: list_VEBT_VEBT > nat > nat > $o).
% 37.37/9.71  thf(func_def_883, type, sP11: num > real > real > $o).
% 37.37/9.71  thf(func_def_884, type, sP12: set_VEBT_VEBT > set_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_885, type, sP13: set_VEBT_VEBT > set_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_886, type, sP14: (int > $o) > int > int > $o).
% 37.37/9.71  thf(func_def_887, type, sP15: (int > $o) > int > int > $o).
% 37.37/9.71  thf(func_def_888, type, sP16: nat > nat > nat > $o).
% 37.37/9.71  thf(func_def_889, type, sP17: nat > nat > nat > $o).
% 37.37/9.71  thf(func_def_890, type, sP18: nat > nat > nat > $o).
% 37.37/9.71  thf(func_def_891, type, sP19: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 37.37/9.71  thf(func_def_892, type, sP20: int > int > int > $o).
% 37.37/9.71  thf(func_def_893, type, sP21: int > int > int > $o).
% 37.37/9.71  thf(func_def_894, type, sP22: int > int > int > $o).
% 37.37/9.71  thf(func_def_895, type, sP23: real > num > real > $o).
% 37.37/9.71  thf(func_def_896, type, sP24: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 37.37/9.71  thf(func_def_897, type, sP25: (int > $o) > $o).
% 37.37/9.71  thf(func_def_898, type, sP26: real > real > real > $o).
% 37.37/9.71  thf(func_def_899, type, sP27: real > real > real > $o).
% 37.37/9.71  thf(func_def_900, type, sP28: real > num > real > $o).
% 37.37/9.71  thf(func_def_901, type, sP29: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 37.37/9.71  thf(func_def_902, type, sP30: (real > $o) > (real > $o) > (real > $o) > (real > $o) > $o).
% 37.37/9.71  thf(func_def_903, type, sP31: real > real > real > $o).
% 37.37/9.71  thf(func_def_904, type, sP32: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 37.37/9.71  thf(func_def_905, type, sP33: list_VEBT_VEBT > vEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_906, type, sP34: nat > nat > nat > list_VEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_907, type, sP35: list_VEBT_VEBT > vEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_908, type, sP36: nat > vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_909, type, sP37: nat > vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_910, type, sP38: vEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_911, type, sP39: vEBT_VEBT > nat > $o).
% 37.37/9.71  thf(func_def_912, type, sP40: real > num > real > $o).
% 37.37/9.71  thf(func_def_913, type, sP41: real > real > (real > real > $o) > $o).
% 37.37/9.71  thf(func_def_914, type, sP42: real > real > real > $o).
% 37.37/9.71  thf(func_def_915, type, sP43: (real > $o) > (real > $o) > (real > $o) > (real > $o) > $o).
% 37.37/9.71  thf(func_def_916, type, sP44: nat > (nat > $o) > nat > $o).
% 37.37/9.71  thf(func_def_917, type, sP45: real > num > real > $o).
% 37.37/9.71  thf(func_def_918, type, sP46: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 37.37/9.71  thf(func_def_919, type, sP47: (real > $o) > (real > $o) > (real > $o) > (real > $o) > $o).
% 37.37/9.71  thf(func_def_920, type, sP48: (real > $o) > (real > $o) > (real > $o) > (real > $o) > $o).
% 37.37/9.71  thf(func_def_921, type, sP49: real > real > real > $o).
% 37.37/9.71  thf(func_def_922, type, sP50: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 37.37/9.71  thf(func_def_923, type, sP51: nat > vEBT_VEBT > list_VEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_924, type, sP52: num > real > real > $o).
% 37.37/9.71  thf(func_def_925, type, sP53: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 37.37/9.71  thf(func_def_926, type, sK54: (real > real) > real).
% 37.37/9.71  thf(func_def_927, type, sK55: (real > real) > real).
% 37.37/9.71  thf(func_def_928, type, sK56: nat > nat > nat).
% 37.37/9.71  thf(func_def_929, type, sK57: real > (int > $o) > int).
% 37.37/9.71  thf(func_def_930, type, sK58: nat > nat > nat).
% 37.37/9.71  thf(func_def_931, type, sK59: nat > nat > nat).
% 37.37/9.71  thf(func_def_932, type, sK60: nat > real > real).
% 37.37/9.71  thf(func_def_933, type, sK61: real > real > real).
% 37.37/9.71  thf(func_def_934, type, sK62: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_935, type, sK63: int > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_936, type, sK64: int > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_937, type, sK65: real > int).
% 37.37/9.71  thf(func_def_938, type, sK66: nat > list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_939, type, sK67: real > nat).
% 37.37/9.71  thf(func_def_940, type, sK68: nat > nat > nat).
% 37.37/9.71  thf(func_def_941, type, sK69: vEBT_VEBT > list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_942, type, sK70: nat > list_VEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_943, type, sK71: real > real).
% 37.37/9.71  thf(func_def_944, type, sK72: int > int > int).
% 37.37/9.71  thf(func_def_945, type, sK73: nat > nat).
% 37.37/9.71  thf(func_def_946, type, sK74: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_947, type, sK75: nat > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_948, type, sK76: nat > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_949, type, sK77: real > real > nat).
% 37.37/9.71  thf(func_def_950, type, sK78: (real > real) > real).
% 37.37/9.71  thf(func_def_951, type, sK79: (real > real) > real).
% 37.37/9.71  thf(func_def_952, type, sK80: list_VEBT_VEBT > (vEBT_VEBT > $o) > nat).
% 37.37/9.71  thf(func_def_953, type, sK81: list_VEBT_VEBT > (vEBT_VEBT > $o) > vEBT_VEBT).
% 37.37/9.71  thf(func_def_954, type, sK82: int > nat).
% 37.37/9.71  thf(func_def_955, type, sK83: int > nat).
% 37.37/9.71  thf(func_def_956, type, sK84: nat > nat > nat).
% 37.37/9.71  thf(func_def_957, type, sK85: int > int).
% 37.37/9.71  thf(func_def_958, type, sK86: int > (int > $o) > int).
% 37.37/9.71  thf(func_def_959, type, sK87: int > (int > $o) > int).
% 37.37/9.71  thf(func_def_960, type, sK88: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_961, type, sK89: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_962, type, sK90: real > real > real).
% 37.37/9.71  thf(func_def_963, type, sK91: nat > nat > nat).
% 37.37/9.71  thf(func_def_964, type, sK92: (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_965, type, sK93: (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_966, type, sK94: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_967, type, sK95: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_968, type, sK96: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_969, type, sK97: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_970, type, sK98: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_971, type, sK99: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_972, type, sK100: (int > $o) > int > int).
% 37.37/9.71  thf(func_def_973, type, sK101: nat > nat).
% 37.37/9.71  thf(func_def_974, type, sK102: int > nat).
% 37.37/9.71  thf(func_def_975, type, sK103: (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_976, type, sK104: (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_977, type, sK105: int > nat).
% 37.37/9.71  thf(func_def_978, type, sK106: real > (int > $o) > int).
% 37.37/9.71  thf(func_def_979, type, sK107: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_980, type, sK108: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_981, type, sK109: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_982, type, sK110: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_983, type, sK111: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_984, type, sK112: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_985, type, sK113: int > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_986, type, sK114: int > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_987, type, sK115: real > real).
% 37.37/9.71  thf(func_def_988, type, sK116: real > nat).
% 37.37/9.71  thf(func_def_989, type, sK117: nat > (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_990, type, sK118: (nat > nat) > nat).
% 37.37/9.71  thf(func_def_991, type, sK119: int > nat).
% 37.37/9.71  thf(func_def_992, type, sK120: (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_993, type, sK121: nat > nat).
% 37.37/9.71  thf(func_def_994, type, sK122: int > (int > $o) > int).
% 37.37/9.71  thf(func_def_995, type, sK123: (real > real) > real).
% 37.37/9.71  thf(func_def_996, type, sK124: (real > real) > real).
% 37.37/9.71  thf(func_def_997, type, sK125: nat > nat).
% 37.37/9.71  thf(func_def_998, type, sK126: int > int > int).
% 37.37/9.71  thf(func_def_999, type, sK127: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1000, type, sK128: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1001, type, sK129: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1002, type, sK130: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1003, type, sK131: int > int).
% 37.37/9.71  thf(func_def_1004, type, sK132: nat > nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1005, type, sK133: int > int).
% 37.37/9.71  thf(func_def_1006, type, sK134: (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1007, type, sK135: (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1008, type, sK136: (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1009, type, sK137: (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1010, type, sK138: (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1011, type, sK139: (int > $o) > int > int).
% 37.37/9.71  thf(func_def_1012, type, sK140: real > nat).
% 37.37/9.71  thf(func_def_1013, type, sK141: real > real).
% 37.37/9.71  thf(func_def_1014, type, sK142: nat > nat > nat).
% 37.37/9.71  thf(func_def_1015, type, sK143: nat > nat > nat).
% 37.37/9.71  thf(func_def_1016, type, sK144: nat > nat > nat).
% 37.37/9.71  thf(func_def_1017, type, sK145: int > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_1018, type, sK146: int > (int > $o) > int).
% 37.37/9.71  thf(func_def_1019, type, sK147: int > (int > $o) > int).
% 37.37/9.71  thf(func_def_1020, type, sK148: (int > $o) > int).
% 37.37/9.71  thf(func_def_1021, type, sK149: (int > $o) > int > int).
% 37.37/9.71  thf(func_def_1022, type, sK150: int > (int > $o) > int).
% 37.37/9.71  thf(func_def_1023, type, sK151: nat > nat).
% 37.37/9.71  thf(func_def_1024, type, sK152: nat > nat > nat).
% 37.37/9.71  thf(func_def_1025, type, sK153: int > int).
% 37.37/9.71  thf(func_def_1026, type, sK154: int > int).
% 37.37/9.71  thf(func_def_1027, type, sK155: (vEBT_VEBT > $o) > list_VEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1028, type, sK156: list_VEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1029, type, sK157: list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1030, type, sK158: nat > list_VEBT_VEBT > nat > nat > nat > nat).
% 37.37/9.71  thf(func_def_1031, type, sK159: nat > list_VEBT_VEBT > nat > nat > nat > nat).
% 37.37/9.71  thf(func_def_1032, type, sK160: nat > list_VEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1033, type, sK161: vEBT_VEBT > list_VEBT_VEBT > nat > nat).
% 37.37/9.71  thf(func_def_1034, type, sK162: vEBT_VEBT > list_VEBT_VEBT > nat > nat).
% 37.37/9.71  thf(func_def_1035, type, sK163: nat > nat).
% 37.37/9.71  thf(func_def_1036, type, sK164: nat > nat > nat).
% 37.37/9.71  thf(func_def_1037, type, sK165: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1038, type, sK166: int > nat).
% 37.37/9.71  thf(func_def_1039, type, sK167: (list_VEBT_VEBT > $o) > list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1040, type, sK168: set_VEBT_VEBT > set_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1041, type, sK169: real > real).
% 37.37/9.71  thf(func_def_1042, type, sK170: nat > nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1043, type, sK171: (int > $o) > int > int > int).
% 37.37/9.71  thf(func_def_1044, type, sK172: (int > $o) > int > int > int).
% 37.37/9.71  thf(func_def_1045, type, sK173: (int > $o) > int > int > int).
% 37.37/9.71  thf(func_def_1046, type, sK174: (int > $o) > int > int > int).
% 37.37/9.71  thf(func_def_1047, type, sK175: (nat > real) > nat).
% 37.37/9.71  thf(func_def_1048, type, sK176: (nat > real) > nat).
% 37.37/9.71  thf(func_def_1049, type, sK177: nat > nat > nat).
% 37.37/9.71  thf(func_def_1050, type, sK178: nat > nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1051, type, sK179: nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1052, type, sK180: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1053, type, sK181: (int > int > $o) > int).
% 37.37/9.71  thf(func_def_1054, type, sK182: (int > int > $o) > int).
% 37.37/9.71  thf(func_def_1055, type, sK183: (int > int > $o) > int).
% 37.37/9.71  thf(func_def_1056, type, sK184: (int > int > $o) > int).
% 37.37/9.71  thf(func_def_1057, type, sK185: nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1058, type, sK186: nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1059, type, sK187: (nat > real) > nat).
% 37.37/9.71  thf(func_def_1060, type, sK188: (nat > real) > nat).
% 37.37/9.71  thf(func_def_1061, type, sK189: nat > nat).
% 37.37/9.71  thf(func_def_1062, type, sK190: (nat > real) > nat).
% 37.37/9.71  thf(func_def_1063, type, sK191: (nat > real) > nat).
% 37.37/9.71  thf(func_def_1064, type, sK192: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_1065, type, sK193: int > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_1066, type, sK194: int > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_1067, type, sK195: int > int).
% 37.37/9.71  thf(func_def_1068, type, sK196: (nat > nat) > nat).
% 37.37/9.71  thf(func_def_1069, type, sK197: (nat > nat) > nat).
% 37.37/9.71  thf(func_def_1070, type, sK198: real > int).
% 37.37/9.71  thf(func_def_1071, type, sK199: real > nat > real).
% 37.37/9.71  thf(func_def_1072, type, sK200: nat > nat > nat).
% 37.37/9.71  thf(func_def_1073, type, sK201: nat > nat > nat).
% 37.37/9.71  thf(func_def_1074, type, sK202: nat > nat > nat).
% 37.37/9.71  thf(func_def_1075, type, sK203: nat > nat).
% 37.37/9.71  thf(func_def_1076, type, sK204: (nat > nat) > nat).
% 37.37/9.71  thf(func_def_1077, type, sK205: (nat > nat) > nat).
% 37.37/9.71  thf(func_def_1078, type, sK206: set_VEBT_VEBT > vEBT_VEBT > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1079, type, sK207: (int > real) > int).
% 37.37/9.71  thf(func_def_1080, type, sK208: (int > real) > int).
% 37.37/9.71  thf(func_def_1081, type, sK209: real > real).
% 37.37/9.71  thf(func_def_1082, type, sK210: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1083, type, sK211: real > real > real > real).
% 37.37/9.71  thf(func_def_1084, type, sK212: (nat > nat) > nat).
% 37.37/9.71  thf(func_def_1085, type, sK213: (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_1086, type, sK214: (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_1087, type, sK215: nat > nat > nat).
% 37.37/9.71  thf(func_def_1088, type, sK216: nat > nat).
% 37.37/9.71  thf(func_def_1089, type, sK217: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1090, type, sK218: nat > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1091, type, sK219: nat > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1092, type, sK220: int > (int > $o) > int).
% 37.37/9.71  thf(func_def_1093, type, sK221: (int > $o) > int).
% 37.37/9.71  thf(func_def_1094, type, sK222: (int > $o) > int).
% 37.37/9.71  thf(func_def_1095, type, sK223: real > real).
% 37.37/9.71  thf(func_def_1096, type, sK224: nat > nat).
% 37.37/9.71  thf(func_def_1097, type, sK225: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1098, type, sK226: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1099, type, sK227: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1100, type, sK228: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1101, type, sK229: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1102, type, sK230: nat > nat > nat).
% 37.37/9.71  thf(func_def_1103, type, sK231: (nat > int) > nat).
% 37.37/9.71  thf(func_def_1104, type, sK232: real > real).
% 37.37/9.71  thf(func_def_1105, type, sK233: int > int).
% 37.37/9.71  thf(func_def_1106, type, sK234: int > int > int).
% 37.37/9.71  thf(func_def_1107, type, sK235: nat > nat).
% 37.37/9.71  thf(func_def_1108, type, sK236: nat > nat).
% 37.37/9.71  thf(func_def_1109, type, sK237: int > int).
% 37.37/9.71  thf(func_def_1110, type, sK238: nat > nat > nat > nat).
% 37.37/9.71  thf(func_def_1111, type, sK239: nat > nat > nat > nat).
% 37.37/9.71  thf(func_def_1112, type, sK240: nat > nat).
% 37.37/9.71  thf(func_def_1113, type, sK241: (real > $o) > real).
% 37.37/9.71  thf(func_def_1114, type, sK242: (real > $o) > real).
% 37.37/9.71  thf(func_def_1115, type, sK243: (real > $o) > nat).
% 37.37/9.71  thf(func_def_1116, type, sK244: (nat > real) > nat).
% 37.37/9.71  thf(func_def_1117, type, sK245: real > real > nat).
% 37.37/9.71  thf(func_def_1118, type, sK246: real > real).
% 37.37/9.71  thf(func_def_1119, type, sK247: nat > nat > nat).
% 37.37/9.71  thf(func_def_1120, type, sK248: (real > $o) > nat).
% 37.37/9.71  thf(func_def_1121, type, sK249: (real > $o) > real).
% 37.37/9.71  thf(func_def_1122, type, sK250: (real > $o) > real).
% 37.37/9.71  thf(func_def_1123, type, sK251: real > int).
% 37.37/9.71  thf(func_def_1124, type, sK252: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1125, type, sK253: nat > nat).
% 37.37/9.71  thf(func_def_1126, type, sK254: (nat > $o) > int > nat).
% 37.37/9.71  thf(func_def_1127, type, sK255: real > real > real > real).
% 37.37/9.71  thf(func_def_1128, type, sK256: nat > nat).
% 37.37/9.71  thf(func_def_1129, type, sK257: nat > nat > nat).
% 37.37/9.71  thf(func_def_1130, type, sK258: nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1131, type, sK259: nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1132, type, sK260: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_1133, type, sK261: int > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_1134, type, sK262: int > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_1135, type, sK263: int > int).
% 37.37/9.71  thf(func_def_1136, type, sK264: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1137, type, sK265: real > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1138, type, sK266: real > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1139, type, sK267: int > int > int).
% 37.37/9.71  thf(func_def_1140, type, sK268: nat > nat).
% 37.37/9.71  thf(func_def_1141, type, sK269: real > real).
% 37.37/9.71  thf(func_def_1142, type, sK270: list_VEBT_VEBT > nat > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1143, type, sK271: vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1144, type, sK272: list_VEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1145, type, sK273: list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1146, type, sK274: int > (int > $o) > int).
% 37.37/9.71  thf(func_def_1147, type, sK275: real > nat).
% 37.37/9.71  thf(func_def_1148, type, sK276: (int > real) > int).
% 37.37/9.71  thf(func_def_1149, type, sK277: (int > real) > int).
% 37.37/9.71  thf(func_def_1150, type, sK278: real > real).
% 37.37/9.71  thf(func_def_1151, type, sK279: int > int).
% 37.37/9.71  thf(func_def_1152, type, sK280: vEBT_VEBT > option4927543243414619207at_nat).
% 37.37/9.71  thf(func_def_1153, type, sK281: vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1154, type, sK282: vEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1155, type, sK283: vEBT_VEBT > list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1156, type, sK284: vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_1157, type, sK285: vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_1158, type, sK286: nat > nat > nat).
% 37.37/9.71  thf(func_def_1159, type, sK287: nat > nat).
% 37.37/9.71  thf(func_def_1160, type, sK288: nat > nat).
% 37.37/9.71  thf(func_def_1161, type, sK289: int > int).
% 37.37/9.71  thf(func_def_1162, type, sK290: int > (int > $o) > int).
% 37.37/9.71  thf(func_def_1163, type, sK291: (real > int) > real).
% 37.37/9.71  thf(func_def_1164, type, sK292: (real > int) > real).
% 37.37/9.71  thf(func_def_1165, type, sK293: int > int > int).
% 37.37/9.71  thf(func_def_1166, type, sK294: nat > nat).
% 37.37/9.71  thf(func_def_1167, type, sK295: list_VEBT_VEBT > (vEBT_VEBT > $o) > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1168, type, sK296: real > real > real).
% 37.37/9.71  thf(func_def_1169, type, sK297: int > nat).
% 37.37/9.71  thf(func_def_1170, type, sK298: int > nat).
% 37.37/9.71  thf(func_def_1171, type, sK299: real > real).
% 37.37/9.71  thf(func_def_1172, type, sK300: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1173, type, sK301: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1174, type, sK302: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1175, type, sK303: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1176, type, sK304: real > nat > real).
% 37.37/9.71  thf(func_def_1177, type, sK305: nat > nat).
% 37.37/9.71  thf(func_def_1178, type, sK306: (real > real) > real).
% 37.37/9.71  thf(func_def_1179, type, sK307: (real > real) > real).
% 37.37/9.71  thf(func_def_1180, type, sK308: real > real > nat).
% 37.37/9.71  thf(func_def_1181, type, sK309: real > nat).
% 37.37/9.71  thf(func_def_1182, type, sK310: real > real > real).
% 37.37/9.71  thf(func_def_1183, type, sK311: real > nat).
% 37.37/9.71  thf(func_def_1184, type, sK312: vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_1185, type, sK313: vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_1186, type, sK314: nat > vEBT_VEBT > option4927543243414619207at_nat).
% 37.37/9.71  thf(func_def_1187, type, sK315: nat > vEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1188, type, sK316: nat > vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1189, type, sK317: nat > vEBT_VEBT > list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1190, type, sK318: real > real).
% 37.37/9.71  thf(func_def_1191, type, sK319: nat > nat > nat).
% 37.37/9.71  thf(func_def_1192, type, sK320: vEBT_VEBT > nat > nat).
% 37.37/9.71  thf(func_def_1193, type, sK321: vEBT_VEBT > nat > nat).
% 37.37/9.71  thf(func_def_1194, type, sK322: vEBT_VEBT > nat > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1195, type, sK323: vEBT_VEBT > nat > list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1196, type, sK324: vEBT_VEBT > nat > nat).
% 37.37/9.71  thf(func_def_1197, type, sK325: vEBT_VEBT > nat > nat).
% 37.37/9.71  thf(func_def_1198, type, sK326: vEBT_VEBT > nat > nat).
% 37.37/9.71  thf(func_def_1199, type, sK327: vEBT_VEBT > nat > nat).
% 37.37/9.71  thf(func_def_1200, type, sK328: vEBT_VEBT > nat > nat).
% 37.37/9.71  thf(func_def_1201, type, sK329: vEBT_VEBT > nat > nat).
% 37.37/9.71  thf(func_def_1202, type, sK330: vEBT_VEBT > nat > list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1203, type, sK331: vEBT_VEBT > nat > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1204, type, sK332: nat > vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1205, type, sK333: nat > vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1206, type, sK334: nat > vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1207, type, sK335: nat > vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1208, type, sK336: nat > vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1209, type, sK337: nat > vEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1210, type, sK338: nat > vEBT_VEBT > list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1211, type, sK339: nat > vEBT_VEBT > list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1212, type, sK340: nat > vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1213, type, sK341: nat > vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1214, type, sK342: nat > vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1215, type, sK343: nat > vEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1216, type, sK344: nat > list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1217, type, sK345: nat > list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1218, type, sK346: vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_1219, type, sK347: vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_1220, type, sK348: nat > nat > nat).
% 37.37/9.71  thf(func_def_1221, type, sK349: (int > $o) > nat).
% 37.37/9.71  thf(func_def_1222, type, sK350: (int > $o) > nat).
% 37.37/9.71  thf(func_def_1223, type, sK351: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1224, type, sK352: real > int).
% 37.37/9.71  thf(func_def_1225, type, sK353: int > int).
% 37.37/9.71  thf(func_def_1226, type, sK354: real > real > real).
% 37.37/9.71  thf(func_def_1227, type, sK355: (nat > int) > nat).
% 37.37/9.71  thf(func_def_1228, type, sK356: (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_1229, type, sK357: (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_1230, type, sK358: (real > int) > real).
% 37.37/9.71  thf(func_def_1231, type, sK359: (real > int) > real).
% 37.37/9.71  thf(func_def_1232, type, sK360: (real > real) > real).
% 37.37/9.71  thf(func_def_1233, type, sK361: (real > real) > real).
% 37.37/9.71  thf(func_def_1234, type, sK362: (nat > $o) > nat > nat > nat).
% 37.37/9.71  thf(func_def_1235, type, sK363: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1236, type, sK364: nat > nat > nat).
% 37.37/9.71  thf(func_def_1237, type, sK365: int > int).
% 37.37/9.71  thf(func_def_1238, type, sK366: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1239, type, sK367: list_VEBT_VEBT > nat > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1240, type, sK368: list_VEBT_VEBT > nat > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1241, type, sK369: (real > nat) > real).
% 37.37/9.71  thf(func_def_1242, type, sK370: (real > nat) > real).
% 37.37/9.71  thf(func_def_1243, type, sK371: (nat > nat) > nat).
% 37.37/9.71  thf(func_def_1244, type, sK372: list_VEBT_VEBT > nat > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1245, type, sK373: list_VEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1246, type, sK374: list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1247, type, sK375: vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1248, type, sK376: (nat > int) > nat).
% 37.37/9.71  thf(func_def_1249, type, sK377: real > int).
% 37.37/9.71  thf(func_def_1250, type, sK378: nat > nat).
% 37.37/9.71  thf(func_def_1251, type, sK379: nat > nat).
% 37.37/9.71  thf(func_def_1252, type, sK380: int > nat).
% 37.37/9.71  thf(func_def_1253, type, sK381: int > nat).
% 37.37/9.71  thf(func_def_1254, type, sK382: real > real > (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1255, type, sK383: real > real > real > (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1256, type, sK384: real > real > real > (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1257, type, sK385: (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1258, type, sK386: (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1259, type, sK387: (real > real > $o) > real).
% 37.37/9.71  thf(func_def_1260, type, sK388: real > real).
% 37.37/9.71  thf(func_def_1261, type, sK389: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1262, type, sK390: real > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1263, type, sK391: real > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1264, type, sK392: int > nat).
% 37.37/9.71  thf(func_def_1265, type, sK393: nat > nat).
% 37.37/9.71  thf(func_def_1266, type, sK394: nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1267, type, sK395: (nat > real) > nat).
% 37.37/9.71  thf(func_def_1268, type, sK396: int > int).
% 37.37/9.71  thf(func_def_1269, type, sK397: real > real > nat).
% 37.37/9.71  thf(func_def_1270, type, sK398: real > real).
% 37.37/9.71  thf(func_def_1271, type, sK399: nat > (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_1272, type, sK400: nat > (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_1273, type, sK401: real > real > real).
% 37.37/9.71  thf(func_def_1274, type, sK402: (real > nat) > real).
% 37.37/9.71  thf(func_def_1275, type, sK403: (real > nat) > real).
% 37.37/9.71  thf(func_def_1276, type, sK404: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1277, type, sK405: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1278, type, sK406: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1279, type, sK407: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1280, type, sK408: (nat > nat > $o) > nat).
% 37.37/9.71  thf(func_def_1281, type, sK409: vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_1282, type, sK410: vEBT_VEBT > $o).
% 37.37/9.71  thf(func_def_1283, type, sK411: vEBT_VEBT > option4927543243414619207at_nat).
% 37.37/9.71  thf(func_def_1284, type, sK412: vEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1285, type, sK413: vEBT_VEBT > list_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1286, type, sK414: vEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1287, type, sK415: int > int > int).
% 37.37/9.71  thf(func_def_1288, type, sK416: real > int).
% 37.37/9.71  thf(func_def_1289, type, sK417: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1290, type, sK418: int > nat).
% 37.37/9.71  thf(func_def_1291, type, sK419: (int > int > $o) > int).
% 37.37/9.71  thf(func_def_1292, type, sK420: (int > int > $o) > int).
% 37.37/9.71  thf(func_def_1293, type, sK421: (int > int > $o) > int).
% 37.37/9.71  thf(func_def_1294, type, sK422: (int > int > $o) > int).
% 37.37/9.71  thf(func_def_1295, type, sK423: (int > int > $o) > int).
% 37.37/9.71  thf(func_def_1296, type, sK424: (real > nat) > real).
% 37.37/9.71  thf(func_def_1297, type, sK425: (real > nat) > real).
% 37.37/9.71  thf(func_def_1298, type, sK426: (real > int) > real).
% 37.37/9.71  thf(func_def_1299, type, sK427: (real > int) > real).
% 37.37/9.71  thf(func_def_1300, type, sK428: real > int).
% 37.37/9.71  thf(func_def_1301, type, sK429: int > nat).
% 37.37/9.71  thf(func_def_1302, type, sK430: real > real).
% 37.37/9.71  thf(func_def_1303, type, sK431: nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1304, type, sK432: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1305, type, sK433: nat > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1306, type, sK434: nat > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1307, type, sK435: list_VEBT_VEBT > (vEBT_VEBT > $o) > nat).
% 37.37/9.71  thf(func_def_1308, type, sK436: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1309, type, sK437: real > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1310, type, sK438: real > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1311, type, sK439: (nat > int) > nat).
% 37.37/9.71  thf(func_def_1312, type, sK440: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1313, type, sK441: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1314, type, sK442: real > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1315, type, sK443: real > (real > $o) > (real > $o) > real).
% 37.37/9.71  thf(func_def_1316, type, sK444: (nat > nat) > nat).
% 37.37/9.71  thf(func_def_1317, type, sK445: nat > nat).
% 37.37/9.71  thf(func_def_1318, type, sK446: (int > $o) > int > int).
% 37.37/9.71  thf(func_def_1319, type, sK447: (int > $o) > int > int).
% 37.37/9.71  thf(func_def_1320, type, sK448: int > (int > $o) > (int > $o) > int).
% 37.37/9.71  thf(func_def_1321, type, sK449: (int > $o) > int).
% 37.37/9.71  thf(func_def_1322, type, sK450: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1323, type, sK451: int > int).
% 37.37/9.71  thf(func_def_1324, type, sK452: real > real > nat).
% 37.37/9.71  thf(func_def_1325, type, sK453: real > real).
% 37.37/9.71  thf(func_def_1326, type, sK454: (nat > $o) > nat > nat).
% 37.37/9.71  thf(func_def_1327, type, sK455: (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1328, type, sK456: nat > nat > nat).
% 37.37/9.71  thf(func_def_1329, type, sK457: int > nat).
% 37.37/9.71  thf(func_def_1330, type, sK458: int > nat).
% 37.37/9.71  thf(func_def_1331, type, sK459: (real > real) > real).
% 37.37/9.71  thf(func_def_1332, type, sK460: (real > real) > real).
% 37.37/9.71  thf(func_def_1333, type, sK461: (int > real) > int).
% 37.37/9.71  thf(func_def_1334, type, sK462: (int > real) > int).
% 37.37/9.71  thf(func_def_1335, type, sK463: nat > vEBT_VEBT > list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1336, type, sK464: nat > vEBT_VEBT > list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1337, type, sK465: nat > list_VEBT_VEBT > nat > nat > nat > nat).
% 37.37/9.71  thf(func_def_1338, type, sK466: nat > list_VEBT_VEBT > nat > nat > nat > nat).
% 37.37/9.71  thf(func_def_1339, type, sK467: list_VEBT_VEBT > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1340, type, sK468: list_VEBT_VEBT > nat).
% 37.37/9.71  thf(func_def_1341, type, sK469: list_VEBT_VEBT > nat > vEBT_VEBT).
% 37.37/9.71  thf(func_def_1342, type, sK470: int > nat).
% 37.37/9.71  thf(func_def_1343, type, sK471: int > nat).
% 37.37/9.71  thf(func_def_1344, type, sK472: nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1345, type, sK473: nat > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1346, type, sK474: int > int).
% 37.37/9.71  thf(func_def_1347, type, sK475: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT).
% 37.37/9.71  thf(func_def_1348, type, sK476: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1349, type, sK477: nat > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1350, type, sK478: nat > (nat > $o) > (nat > $o) > nat).
% 37.37/9.71  thf(func_def_1352, type, ph480: !>[X0: $tType]:(X0)).
% 37.37/9.71  thf(f40037,plain,(
% 37.37/9.71    $false),
% 37.37/9.71    inference(subsumption_resolution,[status(thm)],[f40036,f39951])).
% 37.37/9.71  thf(f39951,plain,(
% 37.37/9.71    ( ! [X0 : nat,X1 : nat] : (((plus_plus_int @ (semiri1314217659103216013at_int @ X0) @ (semiri1314217659103216013at_int @ X1)) = (semiri1314217659103216013at_int @ (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ X0) @ (semiri1314217659103216013at_int @ X1)))))) )),
% 37.37/9.71    inference(beta_eta_normalization,[status(thm)],[f33249])).
% 37.37/9.71  thf(f33249,plain,(
% 37.37/9.71    ( ! [X0 : nat,X1 : nat] : (((plus_plus_int @ (semiri1314217659103216013at_int @ X0) @ (semiri1314217659103216013at_int @ X1)) = (semiri1314217659103216013at_int @ ((^[Y0 : nat]: ((^[Y1 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ Y1)))))) @ X0 @ X1)))) )),
% 37.37/9.71    inference(definition_unfolding,[status(thm)],[f28634,f32230])).
% 37.37/9.71  thf(f32230,plain,(
% 37.37/9.71    (plus_plus_nat = (^[Y0 : nat]: ((^[Y1 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ Y1)))))))),
% 37.37/9.71    inference(cnf_transformation,[status(thm)],[f11142])).
% 37.37/9.71  thf(f11142,plain,(
% 37.37/9.71    (plus_plus_nat = (^[Y0 : nat]: ((^[Y1 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ Y1)))))))),
% 37.37/9.71    inference(fool_elimination,[status(thm)],[f11141])).
% 37.37/9.71  thf(f11141,plain,(
% 37.37/9.71    ((^[X0 : nat, X1 : nat] : (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ X0) @ (semiri1314217659103216013at_int @ X1)))) = plus_plus_nat)),
% 37.37/9.71    inference(rectify,[status(thm)],[f9157])).
% 37.37/9.71  thf(f9157,axiom,(
% 37.37/9.71    ((^[X39 : nat, X40 : nat] : (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ X39) @ (semiri1314217659103216013at_int @ X40)))) = plus_plus_nat)),
% 37.37/9.71    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_9156_nat__plus__as__int)).
% 37.37/9.71  thf(f28634,plain,(
% 37.37/9.71    ( ! [X0 : nat,X1 : nat] : (((plus_plus_int @ (semiri1314217659103216013at_int @ X0) @ (semiri1314217659103216013at_int @ X1)) = (semiri1314217659103216013at_int @ (plus_plus_nat @ X0 @ X1)))) )),
% 37.37/9.71    inference(cnf_transformation,[status(thm)],[f22808])).
% 37.37/9.71  thf(f22808,plain,(
% 37.37/9.71    ! [X0 : nat,X1 : nat] : ((plus_plus_int @ (semiri1314217659103216013at_int @ X0) @ (semiri1314217659103216013at_int @ X1)) = (semiri1314217659103216013at_int @ (plus_plus_nat @ X0 @ X1)))),
% 37.37/9.71    inference(rectify,[status(thm)],[f155])).
% 37.37/9.71  thf(f155,axiom,(
% 37.37/9.71    ! [X12 : nat,X13 : nat] : ((semiri1314217659103216013at_int @ (plus_plus_nat @ X12 @ X13)) = (plus_plus_int @ (semiri1314217659103216013at_int @ X12) @ (semiri1314217659103216013at_int @ X13)))),
% 37.37/9.71    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_154_int__ops_I5_J)).
% 37.37/9.71  thf(f40036,plain,(
% 37.37/9.71    ((semiri1314217659103216013at_int @ (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ one_one_nat) @ (semiri1314217659103216013at_int @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ summary @ (set_VEBT_VEBT2 @ treeList)))))))) != (plus_plus_int @ (semiri1314217659103216013at_int @ one_one_nat) @ (semiri1314217659103216013at_int @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ summary @ (set_VEBT_VEBT2 @ treeList)))))))),
% 37.37/9.71    inference(forward_demodulation,[status(thm)],[f40035,f39818])).
% 37.37/9.71  thf(f39818,plain,(
% 37.37/9.71    ((semiri1314217659103216013at_int @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ summary @ (set_VEBT_VEBT2 @ treeList))))) = (uminus_uminus_int @ (archim6058952711729229775r_real @ (uminus_uminus_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ na) @ (semiri1314217659103216013at_int @ one_one_nat)))))))))),
% 37.37/9.71    inference(beta_eta_normalization,[status(thm)],[f36599])).
% 37.37/9.71  thf(f36599,plain,(
% 37.37/9.71    ((semiri1314217659103216013at_int @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ summary @ (set_VEBT_VEBT2 @ treeList))))) = ((^[Y0 : real]: (uminus_uminus_int @ (archim6058952711729229775r_real @ (uminus_uminus_real @ Y0)))) @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ ((^[Y0 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ one_one_nat)))) @ na)))))),
% 37.37/9.71    inference(definition_unfolding,[status(thm)],[f32612,f32511,f32629])).
% 37.37/9.71  thf(f32629,plain,(
% 37.37/9.71    (m = ((^[Y0 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ one_one_nat)))) @ na))),
% 37.37/9.71    inference(definition_unfolding,[status(thm)],[f30171,f32625])).
% 37.37/9.71  thf(f32625,plain,(
% 37.37/9.71    (suc = (^[Y0 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ one_one_nat)))))),
% 37.37/9.71    inference(definition_unfolding,[status(thm)],[f29899,f28641])).
% 37.37/9.71  thf(f28641,plain,(
% 37.37/9.71    (one_one_int = (semiri1314217659103216013at_int @ one_one_nat))),
% 37.37/9.71    inference(cnf_transformation,[status(thm)],[f137])).
% 37.37/9.71  thf(f137,axiom,(
% 37.37/9.71    (one_one_int = (semiri1314217659103216013at_int @ one_one_nat))),
% 37.37/9.71    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_136_int__ops_I2_J)).
% 37.37/9.71  thf(f29899,plain,(
% 37.37/9.71    (suc = (^[Y0 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ one_one_int))))),
% 37.37/9.71    inference(cnf_transformation,[status(thm)],[f19014])).
% 37.37/9.71  thf(f19014,plain,(
% 37.37/9.71    (suc = (^[Y0 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ one_one_int))))),
% 37.37/9.71    inference(fool_elimination,[status(thm)],[f19013])).
% 37.37/9.71  thf(f19013,plain,(
% 37.37/9.71    ((^[X0 : nat] : (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ X0) @ one_one_int))) = suc)),
% 37.37/9.71    inference(rectify,[status(thm)],[f7503])).
% 37.37/9.71  thf(f7503,axiom,(
% 37.37/9.71    ((^[X39 : nat] : (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ X39) @ one_one_int))) = suc)),
% 37.37/9.71    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_7502_Suc__as__int)).
% 37.37/9.71  thf(f30171,plain,(
% 37.37/9.71    (m = (suc @ na))),
% 37.37/9.71    inference(cnf_transformation,[status(thm)],[f19921])).
% 37.37/9.71  thf(f19921,plain,(
% 37.37/9.71    (m = (suc @ na)) & ((ord_less_eq_nat @ one_one_nat @ na) = $true)),
% 37.37/9.71    inference(fool_elimination,[status(thm)],[f19920])).
% 37.37/9.71  thf(f19920,plain,(
% 37.37/9.71    (ord_less_eq_nat @ one_one_nat @ na) & (m = (suc @ na))),
% 37.37/9.71    inference(rectify,[status(thm)],[f320])).
% 37.37/9.71  thf(f320,axiom,(
% 37.37/9.71    (ord_less_eq_nat @ one_one_nat @ na) & (m = (suc @ na))),
% 37.37/9.71    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_319__C00_C)).
% 37.37/9.71  thf(f32511,plain,(
% 37.37/9.71    (archim7802044766580827645g_real = (^[Y0 : real]: (uminus_uminus_int @ (archim6058952711729229775r_real @ (uminus_uminus_real @ Y0)))))),
% 37.37/9.71    inference(cnf_transformation,[status(thm)],[f19665])).
% 37.37/9.71  thf(f19665,plain,(
% 37.37/9.71    (archim7802044766580827645g_real = (^[Y0 : real]: (uminus_uminus_int @ (archim6058952711729229775r_real @ (uminus_uminus_real @ Y0)))))),
% 37.37/9.71    inference(fool_elimination,[status(thm)],[f19664])).
% 37.37/9.71  thf(f19664,plain,(
% 37.37/9.71    (archim7802044766580827645g_real = (^[X0 : real] : (uminus_uminus_int @ (archim6058952711729229775r_real @ (uminus_uminus_real @ X0)))))),
% 37.37/9.71    inference(rectify,[status(thm)],[f5036])).
% 37.37/9.71  thf(f5036,axiom,(
% 37.37/9.71    (archim7802044766580827645g_real = (^[X22 : real] : (uminus_uminus_int @ (archim6058952711729229775r_real @ (uminus_uminus_real @ X22)))))),
% 37.37/9.71    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_5035_ceiling__def)).
% 37.37/9.71  thf(f32612,plain,(
% 37.37/9.71    ((archim7802044766580827645g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ m))) = (semiri1314217659103216013at_int @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ summary @ (set_VEBT_VEBT2 @ treeList))))))),
% 37.37/9.71    inference(cnf_transformation,[status(thm)],[f3])).
% 37.37/9.71  thf(f3,axiom,(
% 37.37/9.71    ((archim7802044766580827645g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ m))) = (semiri1314217659103216013at_int @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ summary @ (set_VEBT_VEBT2 @ treeList))))))),
% 37.37/9.71    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_2__C3_C)).
% 37.37/9.71  thf(f40035,plain,(
% 37.37/9.71    ((semiri1314217659103216013at_int @ (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ one_one_nat) @ (semiri1314217659103216013at_int @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ summary @ (set_VEBT_VEBT2 @ treeList)))))))) != (plus_plus_int @ (semiri1314217659103216013at_int @ one_one_nat) @ (uminus_uminus_int @ (archim6058952711729229775r_real @ (uminus_uminus_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ na) @ (semiri1314217659103216013at_int @ one_one_nat))))))))))),
% 37.37/9.71    inference(forward_demodulation,[status(thm)],[f40034,f38868])).
% 37.37/9.71  thf(f38868,plain,(
% 37.37/9.71    ( ! [X2 : nat,X3 : list_VEBT_VEBT,X0 : vEBT_VEBT,X1 : option4927543243414619207at_nat] : (((vEBT_VEBT_height @ (vEBT_Node @ X1 @ X2 @ X3 @ X0)) = (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ one_one_nat) @ (semiri1314217659103216013at_int @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ X3))))))))) )),
% 37.37/9.71    inference(beta_eta_normalization,[status(thm)],[f36003])).
% 37.37/9.71  thf(f36003,plain,(
% 37.37/9.71    ( ! [X2 : nat,X3 : list_VEBT_VEBT,X0 : vEBT_VEBT,X1 : option4927543243414619207at_nat] : ((((^[Y0 : nat]: ((^[Y1 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ Y1)))))) @ one_one_nat @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ X3))))) = (vEBT_VEBT_height @ (vEBT_Node @ X1 @ X2 @ X3 @ X0)))) )),
% 37.37/9.71    inference(definition_unfolding,[status(thm)],[f31903,f32230])).
% 37.37/9.71  thf(f31903,plain,(
% 37.37/9.71    ( ! [X2 : nat,X3 : list_VEBT_VEBT,X0 : vEBT_VEBT,X1 : option4927543243414619207at_nat] : (((plus_plus_nat @ one_one_nat @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ X3))))) = (vEBT_VEBT_height @ (vEBT_Node @ X1 @ X2 @ X3 @ X0)))) )),
% 37.37/9.71    inference(cnf_transformation,[status(thm)],[f23897])).
% 37.37/9.71  thf(f23897,plain,(
% 37.37/9.71    ! [X0 : vEBT_VEBT,X1 : option4927543243414619207at_nat,X2 : nat,X3 : list_VEBT_VEBT] : ((plus_plus_nat @ one_one_nat @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ X3))))) = (vEBT_VEBT_height @ (vEBT_Node @ X1 @ X2 @ X3 @ X0)))),
% 37.37/9.71    inference(rectify,[status(thm)],[f91])).
% 37.37/9.71  thf(f91,axiom,(
% 37.37/9.71    ! [X11 : vEBT_VEBT,X8 : option4927543243414619207at_nat,X9 : nat,X10 : list_VEBT_VEBT] : ((vEBT_VEBT_height @ (vEBT_Node @ X8 @ X9 @ X10 @ X11)) = (plus_plus_nat @ one_one_nat @ (lattic8265883725875713057ax_nat @ (image_VEBT_VEBT_nat @ vEBT_VEBT_height @ (insert_VEBT_VEBT @ X11 @ (set_VEBT_VEBT2 @ X10))))))),
% 37.37/9.71    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_90_VEBT__internal_Oheight_Osimps_I2_J)).
% 37.37/9.71  thf(f40034,plain,(
% 37.37/9.71    ((plus_plus_int @ (semiri1314217659103216013at_int @ one_one_nat) @ (uminus_uminus_int @ (archim6058952711729229775r_real @ (uminus_uminus_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ na) @ (semiri1314217659103216013at_int @ one_one_nat))))))))) != (semiri1314217659103216013at_int @ (vEBT_VEBT_height @ (vEBT_Node @ none_P5556105721700978146at_nat @ (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ na) @ (semiri1314217659103216013at_int @ (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ na) @ (semiri1314217659103216013at_int @ one_one_nat)))))) @ treeList @ summary))))),
% 37.37/9.71    inference(beta_eta_normalization,[status(thm)],[f36014])).
% 37.37/9.71  thf(f36014,plain,(
% 37.37/9.71    ((semiri1314217659103216013at_int @ (vEBT_VEBT_height @ (vEBT_Node @ none_P5556105721700978146at_nat @ ((^[Y0 : nat]: ((^[Y1 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ Y1)))))) @ na @ ((^[Y0 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ one_one_nat)))) @ na)) @ treeList @ summary))) != (plus_plus_int @ (semiri1314217659103216013at_int @ one_one_nat) @ ((^[Y0 : real]: (uminus_uminus_int @ (archim6058952711729229775r_real @ (uminus_uminus_real @ Y0)))) @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ ((^[Y0 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ one_one_nat)))) @ na))))))),
% 37.37/9.71    inference(definition_unfolding,[status(thm)],[f31918,f32630,f28641,f32511,f32629])).
% 37.37/9.71  thf(f32630,plain,(
% 37.37/9.71    (deg = ((^[Y0 : nat]: ((^[Y1 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ Y1)))))) @ na @ ((^[Y0 : nat]: (nat2 @ (plus_plus_int @ (semiri1314217659103216013at_int @ Y0) @ (semiri1314217659103216013at_int @ one_one_nat)))) @ na)))),
% 37.37/9.71    inference(definition_unfolding,[status(thm)],[f31916,f32230,f32629])).
% 37.37/9.71  thf(f31916,plain,(
% 37.37/9.71    (deg = (plus_plus_nat @ na @ m))),
% 37.37/9.71    inference(cnf_transformation,[status(thm)],[f86])).
% 37.37/9.71  thf(f86,axiom,(
% 37.37/9.71    (deg = (plus_plus_nat @ na @ m))),
% 37.37/9.71    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',fact_85__C3_Ohyps_C_I4_J)).
% 37.37/9.71  thf(f31918,plain,(
% 37.37/9.71    ((semiri1314217659103216013at_int @ (vEBT_VEBT_height @ (vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary))) != (plus_plus_int @ one_one_int @ (archim7802044766580827645g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ m)))))),
% 37.37/9.71    inference(cnf_transformation,[status(thm)],[f22710])).
% 37.37/9.71  thf(f22710,plain,(
% 37.37/9.71    ((semiri1314217659103216013at_int @ (vEBT_VEBT_height @ (vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary))) != (plus_plus_int @ one_one_int @ (archim7802044766580827645g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ m)))))),
% 37.37/9.71    inference(flattening,[status(thm)],[f10243])).
% 37.37/9.71  thf(f10243,negated_conjecture,(
% 37.37/9.71    ~((semiri1314217659103216013at_int @ (vEBT_VEBT_height @ (vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary))) = (plus_plus_int @ one_one_int @ (archim7802044766580827645g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ m)))))),
% 37.37/9.71    inference(negated_conjecture,[status(cth)],[f10242])).
% 37.37/9.71  thf(f10242,conjecture,(
% 37.37/9.71    ((semiri1314217659103216013at_int @ (vEBT_VEBT_height @ (vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary))) = (plus_plus_int @ one_one_int @ (archim7802044766580827645g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri5074537144036343181t_real @ m)))))),
% 37.37/9.71    file('/export/starexec/sandbox2/tmp/tmp.6exu7wMTxS/DTF2THF_32338.p',conj_0)).
% 37.37/9.71  % SZS output end Proof for DTF2THF_32338
% 37.37/9.71  % (341)------------------------------
% 37.37/9.71  % (341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.37/9.71  % (341)Termination reason: Refutation
% 37.37/9.71  
% 37.37/9.71  % (341)Memory used [KB]: 41065
% 37.37/9.71  % (341)Time elapsed: 1.055 s
% 37.37/9.71  % (341)Instructions burned: 3417 (million)
% 37.37/9.71  % (341)------------------------------
% 37.37/9.71  % (341)------------------------------
% 37.37/9.71  % (32445)Success in time 4.569 s
%------------------------------------------------------------------------------